Volume 0

1Pre-book to Tractatus Principio | On Difference · Volume I – Book of Mathematics

1. Prelude: From Stroke to Reading

1This book does not begin with a finished object, but with a trace. A stroke is visible; what it means has not yet been decided by that fact. It may be a notch in bone, a mark on clay, a boundary in a drawing, or a sign on a page. Only in a reading does the visible trace become a determinate distinction.

2The question that will guide us through the following chapters is therefore not at first: What is number? What is the atom? What is a property? We ask first: What must happen for something to become readable as something determinate? Between the trace and what it designates lies an enactment. Someone sets a mark, orders it with other marks, compares it, counts it, or uses it within a rule. The result may later appear so self-evident that this enactment is no longer read along with it.

3Such a shortening is not in itself an error. Signs and rules make memory, comparison, and transmission possible. Whoever notes down a number does not have to perform the counting process anew each time; whoever writes down a measurement can continue calculating with the result. But the possibility of using a result without repeating the process by which it arose raises a question: Does the result remain relationally the same in its later use, or do the new contexts determine it differently?

4The historical part follows this question through concrete practices. It begins with marked traces and numeral signs, moves to Greek concepts and constructions, follows the transfer of philosophical words into Latin and their scholastic transmission, and finally leads to mathematical physics, measuring apparatuses, and particle theories. This is not the linear history of a single thought. Between the stations lie differences, breaks, reinterpretations, and gaps in transmission. Where a direct transfer is not attested, it is not asserted.

5For the reading, four levels must be kept apart: what a source actually shows; how its historical transmission can be reconstructed; what relational consequence follows from it; and what stronger thesis still remains unproven. The Tractatus Principio serves here as an internal reading anchor for the work, not as proof of the intentions of earlier authors. Its concepts must not be projected back into the historical texts.

6We therefore begin with a modest but consequential difference: A trace is not already the meaning of its trace. Between the two stands a reading.

2. The Notch: A Trace of Enactment

1A notch first shows that a surface has been altered. It is at once a material trace and the result of an action: A tool struck a bone or some other material, was applied, moved, and left behind a visible boundary. What the notch preserves cannot in every case be read from its form alone.

2Prehistoric bones with incised rows are often interpreted as counting or recording devices. Such readings are possible, but they are not automatically secured. For each findspot, form, manufacture, traces of use, find context, and comparative finds must be taken into account. A regular row can support a counting interpretation; it may also have served other purposes. Archaeological research therefore cautions against inferring immediately from the trace to a determinate concept of number or to a complete calculating process.[1]

3The Ishango bone makes this caution especially vivid. It was found in 1950 in the Ishango area on Lake Edward. Its notches lie in grouped rows; some groups exhibit striking numerical relations. The Royal Belgian Institute of Natural Sciences dates the object to roughly twenty thousand years ago and describes the groupings without thereby definitively fixing a single function.[2] In research, counting, calendrical, and mathematical interpretations, among others, have been proposed. That such patterns are readable does not by itself prove which reading the person who made them intended.

4For our argument this boundary is important. We do not need to know whether each notch designated a counted object, a day, or another unit in order to hold fast to this: The object preserves a trace of intentional working whose meaning is not identical with its material shape. The notch is not already the number we hope to recognize in it. It becomes a counting trace only within a determinate practice of comparison and reading.

5With this, a first relational order comes forward. A single notch can be read as a difference between an unworked and a worked surface. Several notches can appear as row, group, repetition, or boundary. Which of these relations is decisive depends on what is compared and what the trace is used for. The determination does not arise from an isolated stroke alone.

6This does not mean that the enactment arbitrarily produces the material trace, or that the trace offers no resistance. Notches differ in depth, direction, spacing, and manner of production. These features limit plausible interpretations. But even a careful description of the features does not replace the historical rule of reading that makes them intelligible as a determinate sequence of signs.

7Thus the bone becomes the first touchstone of this book: It shows how an enactment can remain publicly preserved while the meaning of its trace remains open. The later history of numeral signs will not simply recount the same action on a larger scale. It will show how signs enter into orders in which their position and their use are systematically regulated.

3. Mesopotamia: Signs Order Quantities and Measures

1With the earliest written records from Mesopotamia, the recording of quantities becomes part of institutional practices. On clay tablets, numeral signs appear together with entries concerning goods, labor services, dues, and measures. The evidence belongs within a history of administration, storage, and training; it is not merely the discovery of an abstract number that is then applied to things.

2Early proto-cuneiform systems used different numeral signs and counting methods for different orders of magnitude: for example, for counted objects, areas, capacities, or weight. The value of a sign was therefore not always independent of what was being counted or measured. The same visible form could have one place and function in one metrological system and another in another. The signs were read within a practice of units and kinds of objects.[3]

3Even before writing, there were clay tokens and sealed clay envelopes in Western Asia, so-called bullae, whose contents were in part made recognizable on the outside by impressions or signs. A connection between such tokens and later numeral signs is plausible and is investigated on the basis of individual finds; but it is not in every case secured as a gapless direct descent. The preserved objects allow for comparison, not for a blanket narrative according to which a finished number system emerged step by step from a single precursor.[3]

4Rather, the tablets show various concrete ways of recording and working with quantities. An entry can connect type of goods, system of measure, and numeral sign with one another. In early calculation exercises, work is done with capacities, rations, weights, or areas; later we encounter mathematical texts in which numeral signs enter into regulated sequences of calculation. The result of such a calculation can be written down, transmitted, and used again. The clay tablet then preserves not only a quantity, but a readable order of units and operations.

5It does not follow from this practice that Mesopotamians knew no abstract numbers. By the Old Babylonian period at the latest, texts also contain calculations with place-value numbers that are no longer directly bound to a concrete indication of measure. At the same time, metrological quantities remained important in problems and answers. Practical measures and calculationally available numeral signs stood alongside one another and could be translated into one another.[3]

6This is decisive for our question: Between counted good, unit of measure, numeral sign, and calculating operation there is no single, unchanging relation. A calculation can pass from concrete measures into a more generally usable numerical notation and then translate the result back again into a measure. The mathematical achievement consists precisely in reliably organizing such transitions.

7At the same time, this makes visible what may not yet be asserted. The later availability of place-value numbers does not prove that every number was always already treated as an independent entity. Conversely, the binding of early numeral signs to goods and measures does not prove that an abstract conception of number was absent. The evidence shows different roles that numeral signs assume in calculations and records.

4. When a Result Becomes Transferable

1A numerical notation gains a special reach when its signs not only count objects, but enter into further calculations according to fixed rules. In Mesopotamia, a sexagesimal number system developed over centuries; alongside it, the earliest written records contained several metrological systems. Later place-value notation made longer calculation procedures possible whose results did not each time have to be represented by a collection of individual objects.[3]

2This development was not a simple leap from concrete counting to “pure” mathematics. In Old Babylonian schools, scribes learned, among other things, metrological conversions, multiplication tables, and reciprocal tables. A problem could be formulated with measures, translated into place-value numbers for the calculation, and after the operation led back again into measures. The path of calculation presupposed prior knowledge of conversion rules; these rules could remain unspoken in an individual problem because they counted as learned.[3]

3Here we encounter an early form of transferability. A result is preserved not only as the trace of a one-time enactment of counting or measuring. It can enter into a new calculation. A reciprocal or multiplication table makes it possible to carry out an operation without constructing it from the ground up at every use. This is no devaluation of the original practice. It is a cultural and mathematical achievement: procedures become storable, teachable, and reusable.

4But the role of the result changes with its use. In a first problem, a number may arise from the conversion of determinate measures. In a later problem, the same numerical notation may already serve as the input of an operation. The name or sign may persist; its function within the context is now another one. From this alone no reification yet follows. It first shows why, at every historical stage, we must ask anew by what a number is determined there and which earlier relations continue to be carried along.

5Nor may zero at this point be inserted into prehistory as a timelessly identical sign. A placeholder in a place-value notation, an empty place, a number zero, and a philosophical concept of nothingness are different roles. The history of numerical signs shows that these functions arise at different times and in different systems; their equation requires its own evidence in each case.[4]

6With the Greeks, the state of transmission shifts. We can now read philosophical texts and mathematical textbooks in which unity, number, boundary, figure, and proof are explicitly determined. The Mesopotamian evidence does not thereby disappear from the history; it remains an early, independent strand of regulated numerical and calculational practice. The following section therefore asks not whether the Greeks “invented mathematics,” but how their texts make the conditions of mathematical determination themselves into an object.

5. When Difference Itself Becomes Visible

1With the Greeks, it is not simply the quantity of available knowledge that changes. For the history we are following here, what is decisive is that the conditions of determination now explicitly become the object of mathematics and philosophy.

2Human beings had counted, measured, recorded, and calculated long before the Greeks. Mesopotamian mathematicians possessed elaborated calculation procedures and regulated numeral signs. The Greek practices do not stand outside this history. At this point, however, we are not yet asserting a direct transmission of every single mathematical idea, documented without gaps. What interests us first is a structure that becomes readable again in different practices: a sign receives its determinacy not from its visible body alone. It depends on how it is set, ordered, and read.

3On the tally bone, the notch was not simply a small thing in the bone. It was the visible trace of an enactment: someone had incised something in order to represent or hold something fast. In the Mesopotamian numeral signs, such a trace became part of a regulated order of signs. The reading of the signs followed an order in which their position and their connection counted for what they designated.

4Among the Greeks we can now observe how related questions emerge in transmitted concepts, mathematical procedures, and philosophical arguments. What does it mean for something to be the same? What does it mean for it to differ? When can a multiplicity be read as a determinate number? What counts as a boundary? What makes a point a point? When does a division generate two viable determinations? And what does it mean when something is called indivisible?

5These questions do not join Heraclitus, Democritus, the mathematical tradition, Plato, and Aristotle into a single doctrine. Their texts differ in object, time, transmission, and intention. The connection reconstructed by this section is a movement of reading: we follow the words and the enactments through which differences are set, figures are produced, numbers are determined, and concepts are divided. The relational reading is thereby a premise of our inquiry. It is tested against the texts; it is not smuggled into them as a shared confession.

6The secondary literature helps us examine word-forms, textual transmission, and competing interpretations. But it does not decide in place of the reading which conceptual movement becomes readable in the context of the passages examined here. Where the wording supports a finding, and where we develop a more far-reaching reading from it, will therefore be marked in each case.

7The relational reading must also state what would count against it. For an individual case, it would be weakened if the text treated a relation explicitly only as an external access to an independently determined magnitude, and if definition, role in proof, and admissible operations remained stable across the contexts of use examined. Then the relation would be epistemically important for this text, but not demonstrably constitutive for the meaning of the determination. A drift would likewise not be documented if later texts expressly preserved the older role, grounded its transfer, and kept differences between source concept and target concept recognizable. One may speak of retrojection only if a later reader actually attributes the later role to the earlier text. If the source chains predominantly show such counter-findings, the drift thesis must be restricted or withdrawn for the strand under examination; the mere persistence of a name suffices neither for confirmation nor for refutation.

6. Heraclitus: The Enactment That Lets Things Diverge and Come Together

1No complete work by Heraclitus has been preserved. We possess fragments and quotations transmitted by later authors. Every interpretation must therefore keep distinct the respective wording, its transmission, and the limits of what a fragment can document.

2For the connection with the tally bone, B1 is important first. There the speaker describes his own doing with διαιρέων ἕκαστον κατὰ φύσιν καὶ φράζων ὅκως ἔχει: he separates off each individual thing according to its nature and sets out how it is.[5] The fragment shows distinction not only as a finished result. It also names an enactment of distinguishing and presenting. Here lies the connection to the notch: a difference is not merely found; it is marked and made readable in a determinate way.

3B51 carries this movement further linguistically. In a widespread textual form, the transmitted sentence reads: οὐ ξυνιᾶσιν ὅκως διαφερόμενον ἑωυτῶι ξυμφέρεται· παλίντονος ἁρμονίη ὅκωσπερ τόξου καὶ λύρης. The core of the sentence is that something that goes apart or differs at the same time comes together with itself; bow and lyre serve as images.[6]

4The two movements must not be split into two processes independent of one another: first a difference and then an external joining-together. The drawn bow becomes determinable as this bow through sides acting against one another. The tension separates and holds together. Correspondingly, the lyre does not become a sounding instrument despite the tension of its strings; its determinate possibility of sound depends on these relations of tension.

5Ἁρμονίη should not here be equated without further ado with the modern word “harmony.” The image suggests the fitting-together or attunement of different parts. The determinate unity does not arise through the erasure of difference. For our reading this means: what separates makes the sides determinable; and this determinability is their belonging-together. The poles are not first present as finished entities and then enter into a relation. Their determinacy becomes visible in the relation that distinguishes them.

6This reading is not a total theory of Heraclitus reconstructed from a single fragment. It takes the wording and the image seriously and reads them together with B1: there, distinguishing is named as enactment; here, going-apart appears as coming-together with itself. B53, “Polemos is father and king of all things,” supplements the linguistic movement insofar as, as the fragment proceeds, differences between gods and humans, free persons and slaves emerge.[7] It does not follow from this that Heraclitus advocates a modern ontology of conflict. For our context, rather, the fragment shows that determinations are made visible in the linguistic enactment of distinguishing.

7Thus the Greek line of thought connects with the preceding history of the trace: the bone preserves the marking of a representational enactment. Democritus, according to our perspectival reading, lets us ask where distinguishing reaches a limit. Heraclitus makes visible the tension in which distinction and belonging-together are enacted jointly.

7. Democritus: The Indivisible and the Limit of Distinguishing

1If Heraclitus shows that the distinguishing poles belong together in the enactment of their tension, the gaze now turns to the limit of this dividing. In Democritus we encounter for this the word ἄτομος. Hardly any ancient concept seems more familiar to us today than the atom. We hear ἄτομος and easily imagine a very small material particle. This idea is understandable, but it must not be inserted without examination as the meaning of the Greek word.

2Ἄτομος first means “not cut,” “uncut,” or “indivisible.” In Aristotle’s report on Leucippus and Democritus, the indivisible stands in connection with the full and the void. According to Aristotle, the beings also differ through ῥυσμός, διαθιγή, and τροπή; he explains these expressions with σχῆμα, τάξις, and θέσις, that is, roughly shape, ordering, and position.[8]

3For our inquiry, the meaning of the word is the starting point, but not the whole interpretation. We ask what “indivisible” can mean when we read the concept from a relational perspective. On this reading, something is ἄτομος insofar as it can no longer be distinguished: within the respective enactment, it can no longer be decomposed into distinguishable determinations. The indivisible then does not necessarily designate an independent smallest thing that stands fixed independently of every perspective of reading. It marks the limit of what is still distinguished in a determinate order.

4This is a perspectival reading of the word, not a translation meant to replace the transmitted finding. According to the preserved testimonies, ancient atomism conceives of full, indivisible bodies and the void; this finding remains in force. Our question is whether the word ἄτομος at the same time makes visible a relational conceptual role: what is determined as indivisible depends on which division or distinction can be carried out.

5Aristotle’s example with letters makes this question especially vivid. An Α differs from a Ν through its shape. ΑΝ differs from ΝΑ through the ordering of the same letters. A letter differs from a differently oriented letter through its position.[8] The letters show that differences do not always have to be traced back to additional material. The determination can change through shape, sequence, or orientation.

6Taken by itself, the example does not prove that Democritus understood atoms exclusively relationally. Aristotle expressly reports a doctrine of indivisible bodies. Yet his linguistic example exposes a second level: even where indivisible bodies are presupposed, the diversity of their configurations is made recognizable by means of differences in shape, ordering, and position. For our reading, therefore, ἄτομος is the limiting concept of distinguishing: that which, in a perspective, is not further distinguished, while the configurations in which it occurs become determinable through relations.

7The transmitted Democritean expressions also demand caution. Alongside ῥυσμός, διαθιγή, and τροπή, Aristotle offers his own explanatory words σχῆμα, τάξις, and θέσις. These series must not be treated as congruent without examination. Ῥυσμός in particular is disputed in the research. Gustavo Laet Gomes shows that its meaning and its function in the atomistic explanatory context are not simply disposed of by Aristotle’s translation into “shape.”[9] This does not support a finished process ontology of Democritus. But it does show why our reading should not prematurely translate the movement of the original word-use into a modern notion of static particles.

8Together with Heraclitus, Democritus makes two sides of the same question visible: Heraclitus shows the enactment in which distinguishing poles belong together; Democritus lets us ask after the limit at which something can no longer be distinguished. In number and geometry, this relational determination now becomes visible in regulated mathematical enactments.

8. Number and Geometry: Sign, Enactment, and Relation

1Greek mathematics had a long prehistory and did not consist of a single practice. For the line of thought pursued here, two things are especially illuminating: the determination of number as a multiplicity of units, and the role of construction, language, and diagram in geometry.

2At the beginning of the seventh book, Euclid’s Elements first define the μονάς, the unit: it is that by which each existing thing is called one. The following definition determines ἀριθμός, number, as πλῆθος ἐκ μονάδων συγκείμενον, as a multiplicity composed of units.[10] In this definition, then, number does not first stand there as an isolated, independent something. It is a determinate multiplicity whose connection becomes readable through units and through their gathering-together.

3From here, the number four can be read as a sign for an enactment of counting: units are gathered together, and the number designates the multiplicity thus determined. The modern notation 1 + 1 + 1 + 1 = 4 illustrates this thought; it is not Euclid’s own formulation and must not be presented as ancient notation. Nor, in Euclid, is the unit itself a number in the sense of Definition VII.2; number is a multiplicity of units. Precisely this distinction makes visible that number and unit stand in a relation of determination.

4The definition taken by itself does not prove that Euclid understood a number exclusively as an action. But it permits a relational reading: a number becomes determinable as number through its units and their gathering-together. In Aristotelian language, a related structure appears when Aristotle describes the unit as the measure of a multiplicity and number as measured multiplicity or as a multiplicity of measures.[11] The common line we reconstruct from this lies not in an identical doctrine, but in the linguistic binding of number, unit, multiplicity, and measure.

5In geometry, the connection of form, enactment, and result becomes especially vivid. Euclid’s first postulates require drawing a straight line from any point to any point, extending a bounded straight line continuously, and describing a circle around a center with a given distance.[12] The verbs describe operations. In a construction problem, the sought figure is brought forth through permitted steps. A solution is thereby not merely a name or an abstractly stated result: it also consists in producing the figure by way of the permitted construction and proving the relevant properties.

6This does not mean that every Greek mathematical statement had to be drawn, or that a diagram alone bears the proof. What is decisive is the concrete cooperation of text, named points, and figure. Reviel Netz describes the labelled diagram as a necessary component in the reading of Greek mathematical proofs; mathematical language and diagram work together in the process.[13]

7A drawn line is not simply already “the line” at issue in the proof. It is named, extended, connected with other lines, or bounded by points. The linguistic instructions govern which differences in the diagram are relevant; the figure in turn gives the instructions a visible form. In a construction, therefore, the form of the figure, the enactment of its production, and the result of the proof depend on one another. They are different roles within a coherent mathematical process.

8Euclid’s definitions of point and line fit this context. A point is “that which has no part”; a line is “breadthless length”; the boundaries of a line are points, and the boundaries of a surface are lines.[14] These definitions do not anticipate a modern theory of dimensions. They determine point, line, and surface through demarcations that refer to one another. The point appears at once as an undivided case of determination and as the boundary of a line; the line as length and as the boundary of a surface.

9The mathematical object therefore does not simply stand independently beside the procedure through which it is constructed and determined in the proof. For the relational reconstruction, the form remains inseparable from the enactment of its production; the result becomes readable through the construction and the relations in the diagram. Here the idea that became visible in trace and number is elaborated in a regulated deductive practice.

9. Plato: A Boundary Does Not Yet Create a Viable Kind

1With Socrates, the first thing to note is the state of the sources: he left behind no philosophical writings. When we speak of Socratic questioning, we are referring above all to the literary figure of Socrates in Plato’s dialogues. The historical person and the dialogical figure must not simply be equated.

2In these dialogues a practice becomes visible in which determinations emerge through questions, answers, and tests. The question “What is this?” is not answered by merely pointing to a thing. Answers are tested against examples; concepts are brought together, distinguished, and rejected again. A name does not yet guarantee a viable determination.

3In the Phaedrus, Socrates describes two procedures that belong together: collecting and dividing. What appears scattered is first to be grasped in a common form; then it is to be divided again, but not arbitrarily. The division should proceed “at the natural joints,” κατ’ ἄρθρα ᾗ πέφυκεν, and not tear off parts like a bad butcher.[15] The image binds conceptual division to a question about the constitution of that which is divided.

4The Statesman makes visible why a formally enacted dichotomy does not yet yield a successful determination. The Stranger criticizes the division of human beings into “Greeks” and “barbarians.” The expression “barbarians” gathers together an innumerable multitude whose members, as the text says, are connected neither by common descent nor by a common language. As an analogy, the Stranger names a division in which one separated the number ten thousand from all other numbers and then treated the indeterminate remainder, on the basis of a single name, as if it were a unified kind.[16]

5In both cases the cut has been made; the names are present. Yet one side gathers an open, manifold set together and thereby gives it the appearance of internal unity. It describes more than the alleged kind actually determines. The error is not merely that a word would be political or unfriendly. The division forms a pole whose extension appears closed by the name, while its contents bear no corresponding common determination.

6This allows the Platonic critique to be placed within the reading movement so far. A boundary can bring forth sides, but from the presence of the boundary it does not follow that each side is determined as its own kind. A distinction can succeed formally and still divide its subject matter incorrectly. The outside of a posited boundary must not be treated as an independent thing or a unified genus for that reason alone.

7The Sophist expressly formulates the relational side of the problem. In the context of the great kinds, it says: τὸ δέ γ᾽ ἕτερον ἀεὶ πρὸς ἕτερον — the Other is always other in relation to an Other.[17] The determination as other does not stand here by itself like a color. It requires a relation: something is other than something else. The sentence does not claim that all objects can be reduced to relations. It establishes, for the concept of otherness, that this determination is readable only within a relation.

8Thus the systematicity of distinguishing itself becomes testable. Not every cut generates two determinations that can bear their names; and “other” does not straightforwardly designate a property that exists without its counterpart.

10. Aristotle: Boundary, Mode of Consideration, and the Continuation of the Question

1Aristotle was Plato’s student. This biographical connection alone proves no continuity of doctrines. The connection we are tracing here must show itself in the questions and concepts of the texts. Plato had examined when a division is appropriate and how otherness remains related to an Other. Aristotle, for his part, examines what status mathematical objects, units, points, and boundaries possess. We read this investigation as a continuation and transformation of the problem field, not as a repetition of Plato.

2In Metaphysics III, Aristotle asks what ontological status surfaces, lines, and points have. The text formulates an aporia: bodies seem to be determined by surfaces, lines, and points; at the same time, it threatens to become problematic to treat these mathematical determinations as independent substances. Aristotle notes that surfaces, lines, and points are, in a corresponding way, πέρατα or διαιρέσεις — boundaries or divisions (Metaphysics III, 1002a–b).[18]

3In the argument, surfaces and points appear when bodies touch and are divided: when bodies are joined or separated, a boundary can be determined as one or as two. Aristotle asks whether such boundaries come to be and pass away as independent things. The passage is an aporia, not an isolated theorem according to which “the point is in truth only a boundary.” For our reading, precisely the tension is important: the mathematical object can neither be held fast effortlessly as an independent smallest substance nor understood without reference to division, boundary, and continuity.

4In Metaphysics XIII, Aristotle discusses how mathematical objects exist. His answer cannot be reduced to the alternative that they are either independent bodies or mere inventions. The mathematician investigates objects in a specific respect. Aristotle’s expression ᾗ — “insofar as” or “from the point of view that” — helps describe how an investigation takes certain properties into account and sets others aside. Scholarship discusses whether this is to be understood more precisely as abstraction, as a special mode of consideration, or otherwise.[19]

5Within the frame of our premise, what matters here is that what is mathematically determinable need not be explained solely by an allegedly isolated thing. A geometrical figure can be treated in a proof as a figure of a certain kind, even though the drawn trace consists at the same time of ink, sand, or another material. The mathematical aspect under consideration determines which relations and properties are relevant in the given proof. The procedure therefore does not necessarily isolate a hidden object; it forms a regulated perspective of consideration.

6The unit and number also appear in Aristotle within such an order of determination. In Metaphysics X, the One is described as the measure of a multiplicity; in Metaphysics XIV, number is said to be a measured multiplicity, or a multiplicity of measures.[20] This is not the claim that Aristotle had no doctrine of substance of his own. He distinguishes ways of being and gives οὐσία a central position. Rather, it means that in these passages certain mathematical concepts are determined through measures, multiplicities, and their relations.

7For the connection to Plato it is therefore not enough to say: Aristotle was his student, so he must represent the same doctrine. We are tracing the continuity more precisely. In Plato it becomes visible that a boundary can bring forth a side without already justifying it as a kind; otherness is determined as being-other in relation to another. In Aristotle, the question concerning the conditions of mathematical determinability remains in place: point, line, and surface appear in contexts of division and boundary; number and unit in contexts of multiplicity and measure; and mathematical objects in a particular way of considering. The concepts change their role, but the enactment of determination remains bound up with them.

Interlude: Archimedes — Ratio, Figure, and Physical Determination

8In Archimedes, this mathematical practice is extended to the investigation of floating bodies. In On Floating Bodies, he does not first determine an isolated property that we could straightforwardly equate with the modern concept “density.” Rather, he investigates how the weight and size of a body relate to the weight of an equal volume of fluid.[21]

9In Book I, Proposition 5, a body that is lighter than the fluid will be immersed to such an extent that the weight of the displaced fluid corresponds to the weight of the whole body. The state of floating is thus determined by a ratio: body weight to fluid weight of the same volume. “Density” can be used here as a modern analytical designation for such a relational determination; it is not to be presupposed as an independent thing detached from comparison.

10Shape, too, remains part of the nexus. In Book II, Archimedes examines the equilibrium positions of floating paraboloid bodies; for their stability, both the shape and the ratio of the body’s weight to the weight of the fluid are significant. The figure, its position in the fluid, and the weight ratios interlock. Archimedes therefore does not merely describe a property of the body, but a determination that results in the relation between body, fluid, volume, weight, and position.[22]

11This passage extends the trace of number and geometry. As, in the case of number, multiplicity from units is co-determining, and, in the case of the figure, the enactment of its construction is co-determining, so here “density” is determined through a comparison. The ratio is not a subsequent measurement of two already fully determined things; it is the procedure through which their floating behavior becomes quantitatively determinable.

12For precisely this reason, Archimedes is suitable as a transition to the later question concerning the reification of properties. What we today call “the density of a body” appears in this reading not as an inner thing that the body carries within itself independently of all relations. The floating behavior becomes quantitatively determinable only through body, volume, weight, fluid, and the comparison of the weights of equal volumes; in Book II, shape and position are added. For the observer, the body is given to perception as a physical object; the thesis is not that it is first brought forth by the relation. Rather, what is relationally determined is the quantitative magnitude that we call “density.” If the comparison is linguistically shortened to “the body possesses a certain density,” this relational determination can appear as its property, while the comparative operation recedes from view. Here this is a systematic consequence of the reading, not yet a claim about an already demonstrated historical change in language.

13The chronological distance is significant for the subject matter here: Archimedes belongs to the Hellenistic mathematics of the third century BCE. His investigation is not a continuation of Euclid in the narrow sense, but a case of its own in which ratio and geometrical figure carry a physical determination.

11. The Greek Trace of Relational Reading

1We have not shown that Heraclitus, Democritus, Euclid, Plato, Aristotle, and Archimedes represented a shared philosophical system. Such a claim would flatten their differences. What we have reconstructed instead is a problem field in which linguistic and mathematical determinations become readable relationally.

2In Heraclitus, the enactment of separating off and presenting is expressly named; divergent sides come together in a tensed joining. In Democritus, ἄτομος leads to the boundary of what can be further distinguished within a perspective; Aristotle’s letter examples make differences of shape, arrangement, and position visible. Euclid’s number is defined as a multiplicity from units, and his geometry connects construction, labeled diagram, and proof. Plato shows that a posited cut does not guarantee a viable genus and that otherness needs a reference. Aristotle investigates mathematical objects in connection with division, boundary, measure, and a particular mode of consideration. Archimedes brings this question to an object of the physical world given to perception: the floating behavior of a body becomes quantitatively determinable through weight, volume, fluid, shape, and their comparison.

3The supporting claim of this section is not that all these authors say the same thing. It is this: in these texts, objects cannot always be determined independently of the relations, distinctions, and enactments through which they become readable. The point, the number, the Other, and the geometrical figure each appear within a systematicity that helps bring forth their determination.

4This is the Greek trace that continues from the history of bones and number signs. A notch is visible difference and the remaining trace of an enactment. A number is the sign of an articulated multiplicity. A geometrical figure is form, construction, and object of proof within a regulated nexus. In Archimedes, relational determination is related to bodies in a physical investigation: the body is given to perception, but the quantitative magnitude by which its floating behavior is determined is determinable only in comparison. A later shorthand such as “the body has this magnitude” can condense the same measuring relation; if the magnitude enters a new system of measurement, theory, or instruction, further relations can help determine its role. The same applies to the number four: the enactment of counting, Euclid’s definition of multiplicity, and a set-theoretical representation must not be treated as an unchanged determination merely because the sign 4 persists. The historical question is which relations are preserved in each case and which are newly added.

5Here our reconstruction of the Greek section comes to an initial close. Whether and how this relational systematicity is shifted in later translations and conceptual orders is another historical question. It follows only in the next part.

12. From the Greek Conceptual World to the Latin System of Reading

1What happens to a concept when not only its word changes, but also the language in which its relations are read? This question opens the next historical section. It first requires a distinction: a Latin word can translate a Greek expression without fully covering its field of meaning; it can set a new accent, philosophically extend an existing Latin usage, or bring several Greek distinctions together under one Latin word. Translation is thus an enactment of distinguishing. It establishes which similarity between the words is to count, and makes a particular reading transferable.

2From this observation it does not yet follow that Rome reified Greek thought. The Greek texts themselves already distinguish what is said of an underlying thing \(καθ’ ὑποκειμένου\) and what is in an underlying thing \(ἐν ὑποκειμένῳ\). But to render these formulas too quickly as “subject,” “bearer,” or “property” would mean importing modern German meanings into the Greek text. Conversely, Latin is not a neutral vessel either: a translation calls up, for Latin-reading people, the already existing fields of use of the chosen word. The translator can mediate between languages, but his Latin word choice reorders the Greek problem for Latin reading. What must therefore be examined is which meanings a word could carry at its respective time, what the translation highlights from them, and what later readers make of it.

12.1 Translation as Selection: Cicero and Lucretius

3Cicero describes the work of constructing a Latin philosophical vocabulary with unusual openness. In De finibus, he explains that new words and names must be found for new objects: verba parienda sunt … nova rebus novis nomina. Technical languages do this in any case; philosophy cannot simply take over the words of the forum. At the same time, Cicero rejects a mechanical word-for-word translation. A Greek expression may require a paraphrase; if there is no appropriate Latin equivalent, one should leave the Greek word standing or explain the matter.[23] In this self-description, translation is not neutral labeling. It decides between existing fields of meaning, new formations, and paraphrases.

4This becomes especially clear in Cicero’s Academica. He explains that he has rendered ποιότης with qualitas: qualitates igitur appellavi, quas ποιότητας Graeci vocant. The Greek word, he adds, is not an expression of everyday language even in Greek, but one used by philosophers. Cicero forms qualitas after the already familiar qualis — “of what kind / how constituted?” — and says expressly that through use one can make the new word more familiar. In the same argument he describes how, from an active principle and matter, a common body and, as it were, a qualitas emerges; shortly afterward he calls matter subiectam omnibus, laid beneath things. The text thus connects new formation and stabilization with a concrete explanatory context of active principle and matter. It does not demonstrate that the determination so named is now understood as an independent thing. Rather, it shows how Cicero introduces a Greek technical expression into existing Latin word fields and first seeks to establish its use.[24]

5Lucretius shows another possibility. He opens De rerum natura with a reference to the difficulty of expressing Greek discoveries in Latin verse: the egestas linguae, the poverty or scarcity of the language, meets the rerum novitas, the novelty of the things. For the Epicurean atoms he does not choose a single consistently fixed equivalent. He speaks of primordia, principia, semina, corpora prima, and occasionally corpuscula. These words direct attention respectively to beginning, principle, seed, or body. They do not allow the atoms to shrink into a single German or Latin thing-designation; at the same time, they are no evidence that Lucretius represented a modern relational atomic theory. His verses open up the doctrine through movements, combinations, separations, and processes of generation, while the shifting word choice calls up different aspects of what is being explained.[25]

6οὐσία, too, receives no consistently fixed Latin equivalent in Cicero. In his fragmentarily preserved translation of Plato’s Timaeus, he renders the expression according to context, among other ways, with aeternitas and materia. One choice brings the duration and unchangeability of the intelligible model to the fore; the other makes the material aspect visible in a cosmological context. These renderings are not interchangeable. Rather, they show that Cicero interprets the Greek expression within the particular argument and distributes it across existing Latin fields of meaning. The Roman conceptual world therefore does not begin with a stable list of one-to-one equivalents.[26]

7Nor can a single, steadily progressing line of translation be observed between Cicero and Boethius. Apuleius’ Latin adaptations of philosophical texts and Calcidius’ translation, together with his commentary on Plato’s Timaeus, show how existing Latin words are extended and explained in commentary. In Calcidius, essentia and substantia can overlap; in other places, the Latin designation silva for ὕλη appears, as does the description of a substrate in which qualities, quantities, and further accidents appear. Marius Victorinus, in the fourth century, develops in turn a more explicit system of distinction: in him, substantia can render οὐσία, subsistentia ὑπόστασις, and existentia ὕπαρξις. These stations do not show an already completed terminology, but competing attempts to make Greek word- and concept-relations readable in Latin. Boethius’ later assignment stands within this history; it is not its presuppositionless beginning.[27]

12.2 οὐσία, essentia, substantia, and ὑπόστασις

8οὐσία already has more than one use. In Greek, the word can designate possession or property; in Plato and Aristotle it also becomes the object of the question concerning what is. In Metaphysics Z, Aristotle asks what is to count as οὐσία. In the Categories, he calls individual human beings and horses first οὐσίαι: other things are said of them, while they themselves are neither said of another nor “in an underlying thing” \(ἐν ὑποκειμένῳ\). Species and genera he calls second οὐσίαι. Metaphysics Z separately examines whether οὐσία is rather the underlying thing \(ὑποκείμενον\), form, or the composite.[28] The expressions “bearer” and “subject” are widespread modern aids of interpretation and translation, not neutral words of the Aristotelian text. What is decisive first is the distinction Aristotle makes with his formulas and examples; the question whether one should reconstruct from this a bearer-model already belongs to interpretation.

9For a Latin translation, the Greek word formation itself posed a problem. οὐσία belongs to the word field of εἶναι, “to be”; no common Latin noun of the same formation stood ready. Seneca demonstrates in Letter 58 how difficult the transfer could be. In the widespread printed reading, he asks permission to say essentia and asks how οὐσία is otherwise to be expressed except as the necessary and as the nature that lies beneath everything. For τὸ ὄν, he proposes quod est, but remarks that this renders a Greek noun by a Latin verbal form. In this way he makes not only a decision of translation, but also the grammatical friction between the languages into a topic.[29]

10The famous wording, however, is not text-critically smooth: in the manuscript tradition, at the place where printed editions read essentiam, the sequence quid sentiam stands; essentiam is a conjecture that goes back to Muretus. The following sentence — Seneca asking to be allowed to use this word — makes the conjecture plausible, but does not replace secure manuscript evidence. Seneca also appeals to Cicero as authority for essentia; a corresponding attestation in Cicero’s surviving writings has not so far been demonstrable. Quintilian names essentia and ens among Greek-shaped new formations that appeared harsh or unfamiliar to some. The textual evidence therefore attests a discussion about the possibility and acceptance of the word, not beyond doubt Cicero’s authorship or a straight path to the later technical language.[30]

11Nor may substantia be interpreted from the sound it has in present-day German. The word is indeed connected with substare (“to stand beneath,” “to have standing”); in Latin it could also designate means or possessions. The historical dictionaries place the abstract-philosophical sense — what something consists of, subsistence, or essence — predominantly only after the Augustan period. The Latin-reading person thus already brought along a word with its own field of meaning, but it does not follow from this that every early attestation already expresses an elaborated metaphysics. Likewise, the Greek ὑπόστασις designates, depending on the text, concrete subsistence, reality, or individual being; the literal notion of standing-beneath does not by itself decide the meaning.[31]

12The Christian disputes over the Trinity make the ambiguity historically visible. Tertullian formulates the programmatic phrase una substantia, tres personae: one substance, three persons. This is meant to assert unity without abolishing distinction. Augustine later reports that Greek speakers speak of one οὐσία and three ὑποστάσεις; he renders this formula in Latin as una essentia, tres substantiae. At the same time, he explains that essentia names what in Greek is called οὐσία, which in Latin is more commonly called substantia. The words are thus not distributed one-to-one: Tertullian’s formula connects substantia with unity and personae with distinction; Augustine’s report on translation renders hypostaseis with substantiae, but at the same time treats substantia as the more customary equivalent of οὐσία. Latin conceptual choice here must make intelligible not only Greek philosophy, but also a theological distinction. Translation therefore does not simply stabilize a philosophical counterpart; it shows competing sortings in linguistic usage.[32]

13Boethius makes this work of assignment especially clear — and at the same time shows that it does not turn out the same everywhere. In his translation of the Categories, he renders Aristotle’s οὐσία with substantia. In Contra Eutychen et Nestorium, by contrast, he offers an explicit assignment: οὐσία corresponds to essentia, οὐσίωσις to subsistentia, ὑπόστασις to substantia, and πρόσωπον to persona. For substantia he offers his own explanation: it is so called because it underlies the accidents. In the Categories translation he renders the Greek formulas καθ’ ὑποκειμένου / ἐν ὑποκειμένῳ with de subiecto praedicari / in subiecto esse. The Latin word family of subicere already included meanings such as “to lay beneath,” “to suppose,” “to subordinate,” and “to set before the eyes”; Cicero can speak in the same passage of the Academica of subiecta materia. The substantival technical use of subiectum, which Boethius’ translation systematizes, must be distinguished from this. Here it serves as a member of an order of predication and dependence and is not to be equated straightforwardly with the modern “subject” as knowing I or with a finished “property-bearer.” In his theological treatise, Boethius explains that substantia underlies the accidents, but adds that ecclesiastical usage decides how the word may be used of God and creatures. His interpretation is therefore a concrete historical step, not a translation key valid everywhere.[33]

14Here there is a reliable indication that a Latin interpretation can strengthen a possible reading: Boethius explicitly explains substantia by way of underlying in relation to accidents and elsewhere explains subiectum as that which something needs in order to be able to exist. This is more than a mere word-for-word translation. It is not, however, evidence that the Romans first invented such dependence. Aristotle already distinguishes between what is said of a ὑποκείμενον and what is in a ὑποκείμενον; at the same time he examines whether οὐσία is to be understood as underlying thing, form, or composite. The Latin terminology makes certain lines of this Greek discussion repeatable by linking them with the fields of meaning of substantia and subiectum. Which of these fields a reader actually activates must be shown from the respective text.

12.3 ποιότης and ποσότης: A Shared Latin Morphology?

15Aristotle’s Categories define ποιότης not as a hidden little thing in the object, but as that “according to which we are said to be such-and-such”: the formulation binds quality to the naming of what is ποιόν in a certain respect. Boethius translates accordingly: qualitas est secundum quam quales quidam dicimur. The “according to which” and the “as what sort” remain visible in Latin as well. In Book I of his Academica, Cicero shows how qualitas arises as a Latin philosophical new formation; it joins itself to the comparison with qualis. The transition from a predicate (“of what kind?”) to a noun (“quality”) makes it possible to name the determination in its own right. It does not yet prove that it is understood as an independent thing — neither in Latin nor in the Greek ποιότης.[34]

16For ποσότης the evidentiary situation is different. The Latin quantitas appears formally parallel to qualitas; a simultaneous coinage by Cicero, however, is not securely attested. Vitruvius explains ποσότης with quantitas and describes it as the determination of modules from the individual parts of a building, through which the whole receives an appropriate measure. Here quantitas is bound into a relation between parts, module, and whole. Quintilian uses quantitas and qualitas for the volume and constitution of the voice; in prosody, they concern duration and order. These uses introduce the words into technical languages without already attesting a unified metaphysics of categories. The historical dictionaries place the broad abstract use of quantitas predominantly after the Augustan period. Boethius’ translation later stabilizes ποσότης as a category name and transmits Aristotle’s differentiations between discrete and continuous quantities, parts, and common boundaries. The history of this word therefore runs from context-bound measures and features toward a teachable category, not demonstrably from a relation to an independent thing.[35]

17Nor is the proximity of qualitas and quantitas a neutral mirror of Greek word forms. Cicero marks qualitas as a new formation; quantitas is not attested in the same way as his coinage and appears in Vitruvius first as a measure determined from modules and building parts. The later list of categories places the two words formally side by side and makes them easier to teach together. This stabilization changes the comparability of the concepts; whether it also reinterprets a relational determination of measure or constitution into something that an object “has” is decided neither by the ending -tas nor by the list alone. It must be examined in the definitions, examples, and arguments of each text.

12.4 συμβεβηκός and accidens: That Which Comes Along

18The Greek συμβεβηκός is a participle of συμβαίνειν: something that occurs, goes along with, or happens. In philosophical usage it can designate a circumstance or a non-essential determination. The Latin accidens, from accidere, likewise brings with it the meaning of what happens or befalls. Historical dictionaries list both the sense “chance/circumstance” and the philosophical sense of a non-essential feature. The word choice thus possesses a shared field of occurrence and belonging; by itself it compels no reading of a finished property-thing.[36]

19The well-known definition that an accident can be present or absent without destroying that in which it is stands especially clearly in Porphyry’s Isagoge and in Boethius’ Latin transmission. In the Latin version, the word subiectum appears here. It is tempting to paraphrase the definition immediately as “property that belongs to a bearer”; historically more cautiously, one should first note that the definition describes a relation of dependence between the accident and that in which it is. Aristotle also uses συμβεβηκός in several contexts; at the same time, in the Categories he develops the distinction between “being said of an underlying thing” and “being in an underlying thing.” The logical division of predicables, the doctrine of categories, and a general ontology of the property-bearer belong together, but they are not the same theorem.[37]

20In Boethius, the distinction between substantia and accidens becomes a durably usable Latin schema. His translation of the Categories renders primary substances as that which is neither said of a subject nor in a subject; his commentary explains that accidents are in subjects. Cassiodorus, in the sixth century, transmits the ten categories together with their Latin names and definitions within a teaching context. In this way the Greek categorical order becomes a repeatable aid to reading in the Latin educational sphere. That it becomes transferable in this way is historically attested. That every later use thereby automatically presupposes the same ontology would be too far-reaching a conclusion.[38]

12.5 πρός τι, ad aliquid, and relatio

21For the guiding question, the transfer of πρὸς τι is especially revealing. Aristotle’s Categories VII defines the πρὸς τι as that which is what it is from another or toward another. He tests examples such as the double and the half, master and slave, and knowledge and the knowable. The relation is not merely an additional label: whoever knows a relative term in a determinate way must also know that to which it refers. The text also asks whether an οὐσία can itself be relative; for individual substances Aristotle denies this, while in the case of body parts such as hand or head he treats the question as difficult.[39]

22Boethius’ translation reads: ad aliquid vero talia dicuntur quaecumque hoc ipsum quod sunt aliorum dicuntur, vel quomodolibet aliter ad aliud — “related to something” are those things of which precisely what they are is said from others, or which are otherwise in some way related to another. The heading of his chapter connects de relativis with ad aliquid. In the Latin prose, the relational expression ad aliquid and the substantivized relativum thus remain visible. The abstract relatio must be distinguished from this: in classical Latin it had above all rhetorical, juridical, and political uses such as reference back or reply; as a general philosophical designation it becomes graspable later. Boethius’ commentary emphasizes that relatives cannot be understood by themselves; what is understood in them as relation must be considered together with another. The translation therefore preserves the dependence on the correlate, while the later technical language can additionally systematize the reference as a nameable object.[40]

23At the same time, Aristotle lists πρὸς τι as one of the ten categories. Relation thereby stands in a systematic order beside οὐσία, ποιότης, and ποσότης. From this alone it does not yet follow that every relation “inheres” as an additional feature in an already fully determined substance: this formulation is a later theoretical reconstruction. The Greek text rather holds two observations together: a relative term is determined by reference to its correlate, and relative expressions form their own section of the categorical order. The tension between dependency on relation and categorial classification is thus already present in the Greek text; its interpretation did not first arise through Latin.

24Boethius’ further texts keep this tension open. In De Trinitate, he distinguishes relational assertions such as “Father” and “Son” or “right” and “left” from a quality such as “white.” The relational does not change the substance of that about which one speaks; it holds only in reference to another. Augustine formulates with respect to Father and Son that they are named not according to substance, but according to relation. These sources refute the simple assumption that Boethius or the Latin authors transformed πρὸς τι into a mere inner property of an isolated thing. But they also show that “relation” can be described as a distinct object of systematic inquiry.[41]

12.6 From Translation to School: Boethius and the Middle Ages

25Boethius’ importance does not lie only in transferring individual Greek words into Latin. His translations and commentaries on the Categories, on On Interpretation, and on Porphyry’s Isagoge become central texts of the so-called Logica vetus. In the centuries when the Latin West had only limited access to Greek philosophical commentaries, Boethius’ translation and commentary on the Categories, together with the pseudo-Augustinian Categoriae decem, were decisive channels of mediation. Cassiodorus takes the categorical order into his Institutiones and describes the work as a key to the understanding of further texts.[42]

26In this way a Latin table of concepts is stabilized: substantia, quantitas, qualitas, ad aliquid, facere, pati, quando, ubi, situs, and habere. This stabilization is not identical with an unchangeable semantics. Augustine still holds essentia and substantia alongside one another as competing or explanatory renderings; Boethius uses different assignments for οὐσία and ὑπόστασις depending on the work. But the list makes concepts comparable and teachable. It gives later readers a recognizable grid into which new objects and assertions can be ordered.

27In the twelfth century, πρὸς τι / ad aliquid becomes the object of new debates. Alongside the expression ad aliquid, formed from Aristotle’s phrase, relativum and relatio appear; their use and conceptual delimitation develop gradually. The noun relatio brings its own older meanings — for instance reference back, counter-speech, or report — into philosophical language. Later, authors such as Albertus Magnus explain that ad aliquid is the most precise designation because the preposition ad indicates the orientation toward another and the accusative the target of this reference. The terminology can thus not only mark a relation linguistically, but designate it as something analyzable. Christopher J. Martin describes, for the discussions of the early twelfth century, how individual relations and their ontological role are worked out more precisely.[43]

28Here a historical change can in fact be demonstrated, but it is not a linear transformation of relation into an inner property. In the medieval commentaries, it is asked whether a relation is a type of accident of its own in a substance, whether it can be traced back to non-relational properties, or whether it possesses a special mode of being. Thomas Aquinas, for example, in his discussion of relation, distinguishes the being of a creature in itself from its being-related to another; in creatures, relation can be grasped as an accident in the subject, while divine relations do not simply fall under the same doctrine of accident. The existence of these alternatives shows that the table of concepts enables questions that are not settled with a single answer.[44]

12.7 Interim Finding: A Relation Becomes Readable — and Can Become an Object

29Textual finding: Cicero comments on his new formation qualitas and explains difficulties of translation. Lucretius distributes the atomistic explanation across several Latin words. Seneca’s Letter 58 and Quintilian attest that essentia was discussed as a word formation, though Seneca’s famous reading is text-critically uncertain. Boethius renders οὐσία in the Categories as substantia, but in a theological treatise distinguishes οὐσία as essentia from ὑπόστασις as substantia. For πρὸς τι he preserves ad aliquid and emphasizes the necessary co-consideration of the correlate. Late-antique teaching texts and medieval commentators carry this conceptual systematicity forward.

30Philological-historical reconstruction: Latin terminology stabilizes gradually, but not in a single translation decision. Cicero sets a visibly new form; later authors expand, reinterpret, or reorder the available words. Boethius’ translations and commentaries, Cassiodorus’ teaching presentation, and medieval instruction make certain category names reusable over a long period. The stabilization lies first in the repeatable distinction — in the assignment of words, definitions, and examples — and not in a suddenly completed fixation of meaning.

31Relational consequence: Once a linguistic determination receives a fixed name, it can be listed as category, quality, or relation. That does not yet mean that the same meaning passes unchanged from the Greek nexus into the Latin list. In the new Latin nexus, the expression receives further relations: it stands beside other categories, belongs in a definition, and can serve as the premise of a commentary. Boethius’ explanations of substantia and subiectum give accidents a specific dependency role; his and Augustine’s examples for ad aliquid expressly preserve dependence on the correlate. The question is therefore not only whether a relational member later appears as a property, but whether and how its role in the new system changes the readability of the adopted determination.

32Still unproven hypothesis: The repeatable Latin table of concepts may have stabilized an altered relational role; this does not follow from the translation words alone. What is well grounded is that the terminology is transmitted more systematically and that medieval authors expressly discuss relation as a possible type of accident of an οὐσία/substantia. Whether substantia, relatio, or qualitas thereby performs the same determination as in a Greek source must be examined in the respective arguments. If the Latin list places a determination into new relations with definition, category, and object, a new reading can arise under the same name. Only if later readers project this newly stabilized role back onto the earlier source would the strongest form of closure be demonstrated.

13. The School of Copying: How a Reading Rule Is Passed On

1The Latin table of concepts did not arise once and for all at the moment of translation. It was copied in manuscripts, entered into commentaries, shortened in textbooks, and read anew in schools. The hypothesis to be tested is therefore this: a system of reading reproduced over centuries could have made translation decisions so familiar that later readers no longer had to reenact the history of these decisions. Manuscript reproduction, however, never preserves only an unchanged word. It can correct a text, explain it, rearrange it, shorten it, or connect it with a divergent reading. The sources must therefore show where repetition stabilizes and where it opens new distinctions.

13.1 Latin: Still Used, Increasingly Learned

2The formula that Latin became a “dead language” at some point obscures the historical process that we must understand here. In the Romance-speaking regions, the spoken forms of Latin shifted over centuries into the early Romance languages; learned written Latin was increasingly acquired in some regions through grammar, reading, and school. This change proceeded unevenly by region and social group. At the same time, Latin remained a language of liturgy, the Church, administration, law, scholarship, and instruction. It was also spoken—for example in teaching situations, in monastic and ecclesial communities, in negotiations, or at courts—but not in the same way for all groups and not always as a first language.[45]

3The decrees of Tours from the year 813 are an often-cited, but interpretation-dependent, piece of evidence for this. Canon 17 requires bishops to provide homilies and have them transferred into the rustica Romana lingua or the theodiscam language, so that all can better understand what is said. The text documents a concern for intelligibility and a perceived difference between the language of preaching and the spoken languages of the listeners. It does not prove that Latin suddenly was no longer spoken in 813. Nor is the interpretation that the canon marks a sharp break between Latin and Romance undisputed in research: Roger Wright emphasizes the role of Carolingian reforms in the perception of a separation; other approaches stress older differences of register, multilingualism, and regional communication situations. The more recent discussion of Tours warns us first to determine the target group within the ecclesial context and not to derive a general history of language from a single canon.[46]

4For the history of concepts, what changed with the acquisition of Latin is decisive. A learner could hear a technical word in a spoken conversation; but he could just as well become familiar with it in a grammar, a glossary, a definition, a sermon, or a commentary. The more strongly the learned norm was mediated through school and text, the more weight the transmitted examples and definitions could gain for technical usage. This is a plausible mechanism, not an automatic process: even a learned language remains capable of change, and learned readers could dispute definitions or interpret them anew.[45]

13.2 Copying, Correcting, Glossing

5A manuscript materially carries with it the reading history of a text. In Cassiodorus’ Institutiones, the work of the antiquarii, the scribes and copyists, is expressly honored. Through his work, the scribe disseminates words written down in one place across many provinces; at the same time, Cassiodorus recommends orthographic aids so that scribes do not pass on faulty forms and correctors can recognize errors. Charlemagne’s Admonitio generalis of 789 orders monasteries and dioceses to teach psalms, notation, chants, computus, and grammar, and to correct Catholic books carefully. These texts attest an ideal of controlled copying and correction. They do not show that there were uniform scriptoria everywhere or a centrally enforced textual version.[47]

6“Scriptorium” should therefore not be understood as a workshop organized in the same way everywhere. In monasteries such as Vivarium, Cassiodorus connected library order, instruction, and book work in a programmatic design; other books were copied in episcopal seats, palace schools, cathedral and monastic schools, later also by scribes outside such centers. The norms that Cassiodorus or the Carolingian reform formulate are important witnesses to what learned circles considered desirable. Their actual implementation must be reconstructed from surviving codices, corrections, ownership notes, and layers of glosses.[47]

7Precisely the glosses show why copying had both a stabilizing and a changing effect. A marginal or interlinear gloss can explain a rare word, add a Greek equivalent, correct a reading, or preserve a question from a teaching situation. But it can also have been taken over from an older gloss without the scribe himself having been taught at the relevant passage. John Marenbon therefore warns against treating collections of glosses as complete records of instruction. The surviving gloss manuscripts reflect the interests of particular places of learning; copies of the same group of glosses differ both in selection and in interpretive depth. Copying thus disseminates reading rules, while the manuscript itself keeps local deviations and new explanations visible.[48]

8A similar caution applies to the transmission of Greek texts. In the Byzantine sphere, many ancient works were transferred in the ninth century from older majuscule exemplars into minuscule script. For Aristotle, later textual witnesses partly go back to very few such exemplars. This transcription could make the text materially more manageable and enable its further dissemination; at the same time, it could allow older witnesses, variants, and editorial decisions to recede from view. Here too, no pure preservation without intervention can be described.[49]

13.3 What the Latin School Placed Side by Side

9For a medieval learner, the concepts were not only a series of individual translations. They were encountered in ordered lists, definitions, and example sentences. In Institutiones II.3, Cassiodorus lists the ten Categoriae or Praedicamenta and places Latin designations beside their Greek keywords: substantia — οὐσία, quantitas — ποσότης, ad aliquid — πρὸς τί, qualitas — ποιότης, and the remaining categories. He defines quantitas as discrete or continuous and names number, speech, line, surface, body, place, motion, and time. Ad aliquid he calls that which is said to be what it is from another; qualitas is that according to which some are called quales. At the end he demands that Aristotle’s work be read attentively, for everything that human beings say can be found under the ten praedicamenta. The categories appear here at once as a reading instrument for speech and as an object of instruction.[50]

10Isidore of Seville transmits the same basic list in the Etymologiae II.26, but explains it differently. He explains relatio through reference to something else and gives father and son as well as master and servant as correlative names: according to his explanation, both designations begin at the same time. He calls substantia that which is neither said of a subiectum nor is in one; immediately afterward he says that the remaining nine are accidentia, which lie in substantia, among them quantitas, qualitas, and figure. Subiectum, in subiecto, substantia, and accidentia thereby stand in an explicit explanatory connection. Unlike Cassiodorus, Isidore uses relatio instead of ad aliquid in his list. Thus two influential teaching texts already reproduce a common categorial order, but not the same terminological surface or the same degree of elaboration.[51]

11Alcuin’s Dialectica lets the concepts appear in an exercise form. In answer to the question of how a complete statement can be formed from the ten categories, the dialogue assembles the sentence: “Augustinus magnus orator, filius illius, stans in templo, hodie infolatus, disputando fatigatur.” Then each member is assigned: Augustinus as usia, magnus as quantitas, orator as qualitas, filius illius as ad aliquid, stans as situs, in templo as locus, hodie as tempus, infolatus as habitus, disputando as agere, and fatigatur as pati. The dialogue then expressly asks whether the first usia underlies the nine accidents. Here the list of categories becomes a rule for reading sentences: a sentence is decomposed into a first naming member and subsequent determinations.[52]

12What, in this, was learned by heart? The sources permit a precise but limited answer. They show which category names, formulations, and examples were repeated in teaching texts; Cassiodorus says that the writing should be read attentively, Isidore orders it encyclopedically, Alcuin dramatizes the assignment in a question-and-answer dialogue. It follows from this that a list and its definitions were available and repeatable. It does not follow that all pupils learned the same sentences verbatim by heart. For a claim about concrete memorization practice, additional evidence such as teaching instructions, pupils’ copies, or examination questions would be necessary. Even a didactically structured text was not used in the same way in every place.[50] [51] [52]

13The words examined earlier received different places in this instruction. Essentia is not simply one more member of the list of ten. In the category lists available here, learners regularly encounter substantia; essentia belongs to other translation debates and theological debates and must not be treated without evidence as an equally frequent school category. Subiectum becomes repeatable above all in the formulas de subiecto praedicari and in subiecto esse. Accidens is encountered both in the Porphyrian list of the five predicables and in the explanation of the nine categories that are in a subiectum. Qualitas and quantitas stand as neighboring categories beside substantia. For relatio, ad aliquid remains visible as a formulation closer to the Greek; the two names are not used interchangeably everywhere.[51] [52]

14The table therefore does not bundle all concepts into an unchanged ontology. A hierarchy emerges most strongly where substantia appears as the first category and the remaining nine are called accidentia by Isidore. Qualitas, quantitas, and accidens can thereby be read as determinations that are in a substantia. With relatio and ad aliquid, at the same time, dependence on the counterpart remains linguistically visible; father and son are not explained without reference to one another. A medieval learner could thus take up both the substantia-accident reading and relational dependence from the same teaching tradition.[50] [51] [52]

13.4 More Than One Line of Transmission

15Greek, Syriac, Arabic, and Latin mediation must not be drawn as a straight route from Greece to Rome and from there into the Middle Ages. Aristotelian texts were continuously read, ordered, commented on, and copied in Greek-speaking schools. In Christian-Syriac educational centers, logical writings of Aristotle and Porphyry were translated and read with commentaries; the connection between translation and commentary was part of a school context. In the Abbasid realm, new Arabic translations and commentaries arose in the ninth and tenth centuries, especially in the environment of al-Kindī, Ḥunayn ibn Isḥāq, and the Baghdad scholars. These lines were not mere transport routes: they developed their own terminologies and independent questions for Aristotle.[53]

16One example of this independent work is al-Kindī’s treatment of the categories. In his treatise on the number of the Aristotelian books, he assigns kammiyya (quantity) and kayfiyya (quality) a special rank and understands further categories as combinations of these two with jawhar (substance). This is a systematic rearrangement within the Arabic tradition. Arabic category words such as jawhar, kammiyya, kayfiyya, iḍāfa, and ʿaraḍ possessed their own language histories and could assume new technical meanings through translation. The Latin concepts were not thereby automatically replaced; but new commentaries and conceptual interpretations did enter that transmission which Latin readers could later use.[54]

17Until the eleventh century, Latin knowledge of Aristotelian logic in the West was shaped above all by Boethius’ direct translations from Greek, his commentaries and teaching texts, Porphyry’s Isagoge, and the pseudo-Augustinian Categoriae decem. The core of the logica vetus included especially Isagoge, Categories, and De interpretatione. Aristotle’s remaining logical writings were long not available in the West in the same way. For πρὸς τι this meant: many Latin readers encountered the systematic treatment of relations above all in the categorial tradition mediated through Boethius. Knowledge of Greek certainly existed, but school readings could widely rely on the Latin version and its commentaries.[55]

18In the twelfth and thirteenth centuries, this stock was expanded by several translation movements proceeding side by side. Gerard of Cremona translated works in Toledo on the basis of Arabic exemplars. James of Venice and Henricus Aristippus worked directly from Greek; in the thirteenth century William of Moerbeke translated and revised many Aristotelian works on the basis of Greek manuscripts. The early corpus vetustius connected Boethius’ older versions with new translations, but remained incomplete and in places difficult to understand. The later so-called corpus recentius brought revised and additional Greek-Latin versions into circulation. Alongside these, Arabic philosophical works and commentaries—for example by Avicenna and Averroes—were translated into Latin. For many medieval readers, Aristotle therefore consisted of several textual and commentary traditions.[56]

19Mathematics, too, followed more than one path. The Latin tradition preserved teachings on calculation and music theory, among other things, through Boethius; the complete Elements of Euclid were transferred from Arabic into Latin in several versions in the twelfth century. An influential version attributed to Adelard stands within a more complex history of versions than the name of a single translator suggests. Archimedes’ works later reached Latin readers also through direct translations from Greek, especially through William of Moerbeke. A drawn proof or mathematical theorem thus likewise moved through copying, translation, diagram, and commentary. The multilingual path alone does not yet document any ontological shift.[57]

13.5 University and Book Market: Institutional Repetition

20With the universities, instruction and book production changed again. The Paris statutes of 1255 prescribe how long masters had to lecture on particular texts. They place the logica vetus—including Porphyry’s Isagoge, the Praedicamenta and Perihermenias, as well as Boethius’ logical writings—beside newly available Analytics and works of natural philosophy by Aristotle. The statutes complain that masters might pass too quickly over difficult texts and that students would thereby learn less. This documents that textual reading, prescribed duration, and institutionally shared textual stocks were negotiated as conditions of learning.[58]

21In the thirteenth century, the pecia system supported the reproduction of university books. A stationarius kept an exemplar; scribes could borrow individual gatherings—peciae—and produce their own books from them. In this way, teaching texts and commentaries could be disseminated more quickly and in several copies. The procedure aimed at availability and a controllable exemplar, not at identical copies without any deviation. Professional book production stepped alongside monastic and school copying and increased the reach and speed of transmission. This system too remained open to variants and new glosses.[59]

22The historical movement that can be documented with this is more concrete than the claim of a general reification. A Greek text was translated; the Latin version gained material duration through copying; commentaries and textbooks connected words with definitions and examples; category lists regularly placed them side by side; schools and universities repeated this order. With substantia as the first point in the list, with quantitas and qualitas as further categories, and with accidens as dependent determination, there is clear evidence for an ontological direction of reading—especially when Isidore designates the remaining nine as accidentia that are in substantia. With relatio and ad aliquid, the same teaching texts at the same time secure correlative dependence: father and son, master and servant are not explained without reference to one another.[50] [51] [52]

13.6 Textual Finding, Reconstruction, and Hypothesis

23Textual finding: Cassiodorus, Isidore, and Alcuin present category lists, definitions, and example sentences in the Latin language; Cassiodorus couples Latin category names to Greek lemmata, Isidore explains relatio and the accidentia, Alcuin practices the assignment of sentence members. Cassiodorus describes copyists and correction, Carolingian norm texts recommend schools and emended books, gloss manuscripts preserve different explanations, and the Paris statutes attest a regulated reading plan. The texts therefore document repetition and institutional mediation of particular reading rules.

24Historical reconstruction: For numerous medieval pupils, the Latin doctrine of categories was a textually mediated form of knowledge. In the early phase, they encountered above all Boethius’ Latin versions, commentaries, Isidore, and the Categoriae decem. From the twelfth century onward, new Greek-Latin and Arabic-Latin translations and commentaries were added; in the thirteenth century, curricula and book production connected these different stocks. In many places, Latin thereby became a written and learned language acquired through grammar and reading, but it also remained spoken in ecclesial, school, and learned contexts. Every new teaching or commentary relation could expand the role of a transmitted word; the recurrence of the name therefore proves no unchanged meaning.

25Relational consequence: If a translation decision appears in repeated lists, definitions, and teaching examples, the new teaching structure can co-determine the meaning of the word. The pupil then learns substantia not only without the history of translation; he learns it as first category, as counterpole to nine accidentia, or in particular sentence patterns. This is already a relational role in the teaching system, which can differ from the role of a Greek source expression. The question is therefore which relations instruction preserves, which it shortens, and which new determinacy the examples and operations give to the word.

26Hypothesis still unproven: The sources support the possibility that institutional repetition stabilized a particular reading; they do not document that copying and instruction generally or necessarily produced reification. Layers of glosses vary, commentaries pose new questions, access through Arabic and Greek sources changes the Latin body of texts, and relational definitions remain visible in instruction. More than the loss of a history of translation must therefore be tested: did the relational role of substantia or relatio change in the respective teaching texts, and was this later role then read back onto the ancient source? Only such a comparison of source text, commentary, textbook, and concrete application could document a drift in meaning.

27A trace remains preserved. A reading rule arises. It is repeated, copied, commented on, and taught. With every repetition, what is read can become more familiar, until its provenance no longer has to be read along for understanding. The manuscript makes this movement visible, but not automatically: on the same margin on which a definition is handed down, there can stand variants, doubts, and new distinctions.

14. Between Figure and Sign: How Mathematical Determinations Are Passed On

1The finding so far ends with a practice: Euclid's proof begins with a linguistically guided construction and works on a labelled figure; Archimedes determines the behavior of floating bodies through comparable weights, equal volumes, shape, and position. The question now is what happens to these procedures when mathematical texts are passed on through other languages, forms of teaching, numerals, and rules of calculation. The following history therefore does not trace the rise of an ever more abstract mathematics. It asks what readers could do, in each case, with a transmitted mathematical presentation.

Gassendi as the starting point of the backward search

2The selection of the following history of transmission is first oriented by Gassendi's own mathematical practice. In his Institutio astronomica, in observations such as the transit of Mercury, and in De apparente magnitudine, he connects tables, instrument readings, and visual angles with the comparison of observed appearances and astronomical models.[60] In De motu impresso he treats the composition of horizontal and vertical motion geometrically; De proportione uses and corrects a Galilean rule of ratio without presenting a new experimental proof within the treatise itself.[61] In the Syntagma philosophicum, finally, magnitudo, figura, pondus, and soliditas stand beside an explicit distinction between mathematical divisibility and physical matter.[62]

3These works point to three connected but not identical fields of work: mathematical astronomy, geometrical theory of motion, and natural philosophy. They limit the backward search to procedures that Gassendi actually used. The following history of transmission must show, from the sources in each case, which traditions these works continue or change.

14.1 A figure is transmitted

4The late antique Greek tradition, too, did not preserve Euclid and Archimedes as unchangeable books. Theon of Alexandria edited the text of the Elements in the fourth century; his version became the authoritative Greek form of transmission. Proclus' commentary on Book I explains definitions, postulates, kinds of proof, and the role of mathematics in teaching. In later copies, text and diagrams were passed on together, but not always identically. Additional postulates, divergent formulations of proofs, and changing diagrams show that a geometrical theorem remained recognizable while its textual and pictorial form could be edited, commented on, and corrected.[63]

5The Greek texts remained accessible in the Byzantine Empire. At the same time, mathematical works were transferred into Syriac and Arabic educational contexts. The paths were not uniform: Syriac often served as a mediating language, while other texts were transferred directly from Greek into Arabic or edited in several versions. In the Abbasid empire, translations of Euclid, Archimedes, Ptolemy, and other authors arose in the ninth century. Translators such as al-Ḥajjāj ibn Yūsuf and Isḥāq ibn Ḥunayn, as well as scholars such as Thābit ibn Qurra, did not simply transfer sequences of signs: they explained technical terms, checked proofs, added commentaries, and worked over mathematical problems. Thābit, for example, translated or commented on parts of the Archimedean lemmas and the Conics of Apollonius. In astronomy, an important path ran through Ptolemy's Almagest: an Arabic version was translated into Latin by Gerard of Cremona in the twelfth century; astronomical tables then made the calculated positions usable for calendars, observation, and model comparison. Arabic mathematics thus became a working and developmental tradition of its own, in which Greek procedures stood beside Indian modes of calculation and new algebraic methods.[64] [65]

6In the Latin West, the Elements were initially read in the early Middle Ages only in limited excerpts and compilatory adaptations. In the twelfth century, a Latin translation from an Arabic version arose and is attributed to Adelard of Bath; whether all textual witnesses connected with this name go back to the same act of translation is disputed. Campanus of Novara later compiled an influential version that was widely used in teaching. It contained constructions and proofs with diagrams whose arrangement and labelling had to follow the linguistic instructions. In 1482 Erhard Ratdolt printed a Latin edition of the Elements in Venice. Print disseminated a model of geometrical reading, but it did not create a uniform text from one day to the next.[66] [67]

7Latin knowledge of Archimedes long remained narrower. William of Moerbeke translated several surviving works from Greek in 1269; his texts, including On the Sphere and Cylinder, On the Equilibrium of Planes, and On Floating Bodies, initially circulated in only a few manuscripts. Other Archimedean results reached the West through Arabic adaptations and Latin translations. The state of transmission was uneven: not every work read in Greek or Arabic circles was accessible to Latin mathematicians.[68]

8In the fifteenth century, Archimedes' texts and diagrams were again collected, translated, and checked. Jacopo da San Cassiano made a Latin translation; Regiomontanus revised text and figures and planned an edition. In 1544 the first printed complete edition of the then available works of Archimedes appeared in Basel, with Greek text and Latin translation. This edition made Archimedean proofs more accessible for Renaissance readers. Publication in print, however, eliminated neither variants nor the need to check the arguments against figures and constructions. Moreover, On the Method, Archimedes' investigation of mechanical heuristics, remained unknown until its rediscovery in 1906. The mathematicians of the sixteenth and seventeenth centuries therefore did not simply possess “Archimedes,” but a limited stock mediated by the history of editions.[69]

9Here a shift in transmission becomes visible, but not yet any demonstrated drift of meaning. A theorem can be passed on even if the occasion of its discovery is not fully transmitted with it. What matters is whether definition, steps of proof, lettered figure, and instruction for construction preserve the relations through which the theorem is determined in the transmitted version. A later reader can enact the construction again, even though she does not pass through the same movement of search. The result therefore need not already mean something different; yet the same theorem-name alone also does not prove that its determinacy has remained unchanged. This must be checked in each case in the continued proof and its use.

14.2 Ratio is not yet number

10Euclid's Book V offers a useful test. Ratio, λόγος, is there a relation between magnitudes of the same kind; Definition V.5 compares ratios by setting suitable multiples of the magnitudes over against one another. The definition makes an apparently abstract equality checkable by going back to multiplications and comparisons of magnitudes. In Book VI such ratios are used for similar figures and ratios of areas. A ratio here is not simply a number that one writes down like a length. It orders comparable magnitudes and allows statements about them to be proved.[70]

11This separation of ratio and number remained important in the medieval tradition, but it changed in practice. In geometry, a ratio could be checked through magnitudes of the same kind and repeated multiplication; in astronomy, angles, times, and periods of motion were calculated through tables and sexagesimal number series. The tables make results storable and transferable: for a later calculation the observation need not be made anew. At the same time, their use requires the user to know which magnitudes, units, times, and model assumptions a tabular value connects. A stored numerical value does not by itself cancel its order of reference.

14.3 Digits and operations of calculation

12In the Middle Ages different practices of calculation existed side by side. Roman numerals remained in use for dating, bookkeeping, and public inscriptions. Calculation could also be carried out with a counting board or abacus on which counters were moved. Depending on place and text, the word abacus denoted a calculating table, but also calculation itself. The Italian abaco tradition of the late Middle Ages is therefore not simply to be equated with the ancient counting board.[71]

13Another path led through Indian place-value calculation, which was further developed by Arabic-speaking scholars. In Latin Europe it was taught from the twelfth century onward in algorism texts. These texts describe procedures of calculation with nine digits and the sign for the empty place-value, which was received as cifra or zephirum. The transmitted Latin version of al-Khwārizmī's teaching of calculation is an important witness; the Arabic original is not preserved. The operations are carried out step by step on digits: places are aligned, sums are carried, products are treated according to place-values. The procedure no longer brings about a result by moving individual counters, but by a regulated sequence of written operations.[72] [73]

14Leonardo of Pisa, later called Fibonacci, presented the “method of the Indians” in the Liber abaci in 1202 and revised the book in 1228. He describes digits and place-value calculation by means of buying, exchange rates, profit, mixtures, and other problems, not as the isolated introduction of a finished notation. The transfer was a longer process of contact between Mediterranean mercantile and scholarly milieus. Fibonacci is an important witness for this, but not the sole origin of the later Italian schools of calculation. The abbacus tradition developed its own textbooks and procedures; some mathematical practices may have been mediated through Provence, Catalonia, and Arabic-speaking commercial spaces.[74] [75]

15A place-value system changes what a numeral does in a calculation. The digit 4 stands in the units place for four units, in the tens place for four tens, and so on; its function therefore results from sign-form, position, and operational rule. The zero marks an empty place-value and makes it possible for the position of the remaining digits to remain stably readable. Numbers can thus be more easily notated, transferred, and transformed according to uniform rules. It does not follow from this that medieval calculators therefore understood number as an independent thing. In their account books, digits remain bound to coins, measures, goods, debt relations, and concrete problems of calculation. Yet the enactment becomes repeatable as an algorithm: another person can process the same signs according to the same rules without again experiencing the original act of counting.

14.4 From calculation in words to algebraic transformation

16Algebra, too, did not develop in a single invention. Al-Khwārizmī's ninth-century Kitāb al-jabr wa-l-muqābala treated equations in words and ordered them by types. al-jabr, “completion” or restoration, and al-muqābala, balancing, denote operations by which terms are brought from one side of an equation to the other or like terms are combined. In the Latin translations of the twelfth century, especially Robert of Chester's version of 1145, this teaching of calculation was transferred into a new language and into the teaching stock of Latin schools. The equation remained expressed linguistically, but the operations on its members received systematic rules.[76]

17Italian masters of calculation developed from this a broad practice between the thirteenth and sixteenth centuries. In the abbacus manuscripts, mercantile problems, root extraction, and algebra appear side by side. Between rhetorical calculation, which expresses problems and solutions completely in words, and the later alphabetic symbolism lies a syncopated notation: words such as cosa, censo, and cubo designate unknowns and powers, while abbreviations shorten acts of calculation. Luca Pacioli's Summa de arithmetica of 1494 collected procedures of calculation, commerce, and algebra. Cardano's Ars magna of 1545 presented procedures for solving cubic equations; Bombelli, in his Algebra of 1572, developed rules by which expressions with square roots of negative numbers too could be operated on. These texts make relations between sought and known magnitudes processable before a concrete figure is constructed. Nevertheless, the unknown is often introduced as cosa, res, or “thing,” and the equation is first formulated in words. The algebraic operation remains linked with an applied problem and its magnitudes.[77]

18With François Viète this transformation received a new generality. In the Isagoge of 1591 he uses letters for known and unknown magnitudes—traditionally consonants for given magnitudes and vowels for sought magnitudes—and describes algebra as logistica speciosa, an art of calculation with signs for kinds of magnitudes. The letters are not merely abbreviations for individual numbers. They allow general relations to be written and operations to be carried out on them before concrete numbers are inserted. A geometrical relation can thereby be transferred into an equation and algebraically transformed. This strengthens the manipulability of relations independently of any particular drawn figure. But Viète's signs continue to stand for magnitudes and dimensions: his algebra is not a purely syntactic world of symbols detached from mathematical objects.[78]

19A double possibility arises here. Signs can shorten and generalize the enactment; at the same time, the notated relation can be present as a finished structure whose production from a concrete problem need no longer be retraced at every use. Whether this detachment occurs, however, depends on how the sign is interpreted and which operation is recognized as admissible.

14.5 Change becomes geometrically readable

20In the fourteenth century, the so-called Oxford Calculators—including Thomas Bradwardine, William Heytesbury, and Richard Swineshead—investigated questions concerning the change of speed and other intensive qualities. Their works arose in logical disputations and sophismata: one examined what follows from a statement when a quality increases uniformly or non-uniformly. Bradwardine analyzed the relation of speed to resistance; Heytesbury formulated rules for uniformly accelerated motion. These authors did not simply treat change as a modern physical magnitude. They asked how a degree of motion, its increase, and the distance traversed thereby can be set in ratio.[79]

21Nicole Oresme developed in De configurationibus qualitatum et motuum geometrical presentations of such intensities. A quality, such as heat or speed, receives a “breadth” or intensity that varies along a “length” or extension. Oresme draws the intensities as lines above a baseline; the figure arising in this way can be regarded as a configuration of change. In the case of a uniformly difform quality, the triangular outline yields a geometrical presentation with which the total intensity of a process can be compared. In certain cases Oresme shows that a uniformly variable quality has the same total value as a uniform quality at the mean degree. This is a geometrical argument about change, not yet a coordinate system with measurable x- and y-values, nor calculus in the modern sense.[80]

22For our guiding question, what happens here is decisive: change becomes readable as an area or configuration, and geometrical relations between figures carry statements about a temporal enactment. The diagram is not merely an illustration, but also not yet a neutral graph that depicts a finished natural magnitude. It is a means of argumentation in which the interpretation of the baseline, the intensity, and the whole course must be fixed. The mathematical presentation generates possibilities of comparison by transferring a change into an ordered figure.

14.6 Renaissance, print, and mathematical description of nature

23In the Renaissance these procedures were further disseminated through print, schools, and professional mathematicians. Regiomontanus renewed astronomical tables and trigonometric procedures. Pacioli and the masters of calculation connected algebra with commercial arithmetic. In his Quesiti et inventioni diverse of 1546, Tartaglia treated, among other things, artillery and practical mechanics. Cardano and Bombelli made algebraic transformations more productive. Simon Stevin, with De Thiende (1585), advocated decimal fractions and their application in surveying, commerce, weighing, and astronomy. Johannes Kepler used geometry in celestial mechanics and investigated the volumes of wine casks in the Nova stereometria doliorum of 1615. In these cases, mathematical magnitudes are not only proved, but transferred into practices of measurement, calculation, and administration.[81]

24The printed book stabilized visible signs, diagrams, and tables. A figure could now be present in the same arrangement in many copies, while readers enacted it again in constructions and proofs. Procedures of calculation could be taught in handbooks as sequences of operations. Tables stored results of observations and calculations. None of this means that mathematical presentation was now automatically separated from its applications: Stevin justified decimal calculation precisely by the trouble of converting rods, feet, and inches; Kepler's cask calculation was meant to make a concrete problem of volume manageable. Practical transferability and theoretical generalization strengthened one another.

25Galileo is a nodal point because he applies Archimedean geometry of proportion to questions about bodies and motion. In his early studies of floating, levers, and inclined planes, he initially describes “heaviness” through effective weight and comparison: in floating, what counts is the weight of the body in relation to the weight of the displaced medium; on the lever, the effect of a weight also depends on its distance from the pivot. In Le Meccaniche he uses the term momento for an effective magnitude in a mechanical context. The expression is not to be equated with modern momentum or torque; Galileo uses it flexibly for weight-effect, lever arm, and motion.[82]

26In the Discorsi e dimostrazioni matematiche intorno a due nuove scienze (1638), Galileo presents his investigation of falling motion. Salviati describes how a ball rolls down an inclined groove; the distance is marked, time is compared through an outflowing quantity of water that is weighed after each run. The text recounts that repeated runs and time intervals were compared. Whether every detail is to be read as a literal laboratory report or as a dialogically sharpened description of an experiment remains a separate source-critical question. Mathematically, however, the setup is clear: time is operationalized through a measurable portion of water, path through the length of the groove, and change through the comparison of ratios of time and path.[83]

27Galileo then formulates a rule of naturally accelerated motion: equal increases of speed occur in equal times. From this he derives, with the principle of mean speed, that a body starting from rest traverses in the same time the same distance as a body that moves uniformly with the mean speed. The distances therefore stand as the squares of the times. In the second part, he connects uniform horizontal motion with accelerated vertical motion and derives the parabolic trajectory of a projectile. Time, distance, and speed are not here simply pre-given properties of a body: through definitions, comparisons, procedures of measurement, and geometrical proofs, they become magnitudes with which a motion can be calculated.

28At the same time, magnitude now appears more clearly as something that can be attributed to a body at a particular moment. Galileo speaks of speed as a magnitude of the state of motion; this differs from a description that treats motion only as change of place over a duration. But the attribution does not arise from nothing: it continues to depend on the relation between distance traversed, time interval, rule of acceleration, and procedure of measurement. Galileo's laws of falling receive their mathematical form precisely because these relations are defined and stabilized.[84]

29In his investigation of the resistance of materials, Galileo also uses mathematical limiting cases and infinitely many indivisible parts. His First New Science treats how beams break and how the structure of a body determines its resistance. In the dialogues he argues with mathematical points and continua that are not straightforwardly physical atoms. This solution differs from Gassendi's assertion of finite, corporeal atoms. Galileo therefore offers no simple transition from geometrical figures to atomistic bodies. The mathematical readability of nature can be compatible with different ontologies.[85]

14.7 Sunspots, Tycho, and the changing readability of the world-picture

30Tycho Brahe's world system was not a mere intermediate stage on the path from Ptolemy to Copernicus. In De mundi aetherei recentioribus phaenomenis (1588), Tycho kept the Earth at the center and at rest; Sun and Moon moved around the Earth, while the remaining planets orbited the Sun. This arrangement preserved a resting Earth and at the same time could present many of the observable planetary motions in a geometry that came close to the Copernican one. Agreement among the calculated appearances, however, did not decide which center was to count as physically moved. Tycho's choice also depended on questions of parallax, the size of the fixed stars, and whether a heavy Earth could plausibly be set in motion. The world system was thus more than a calculating aid: it connected geometrical relations with statements about what actually moves.[86]

31In the years 1611 and 1612, this question was negotiated around a new telescopic phenomenon. Christoph Scheiner observed sunspots independently of Galileo and published his first letters to the Augsburg councilor Marcus Welser under the name “Apelles.” Welser sent the letters to Galileo and asked for his response. From the beginning, the observation was therefore not only a look through a telescope, but a sequence of observing, drawing, transmitting, translating, and comparing. The surviving letters also show that Galileo and Scheiner did not simply write in a common scientific language-space: Scheiner wrote in Latin, Galileo in Tuscan, and translations as well as intermediaries shaped what each could read from the other.[87]

32Nor did the image of the Sun arise immediately in the eyepiece. Galileo and his collaborator Benedetto Castelli drew the solar image projected through the telescope onto paper and marked the position of the spots on successive days. The plates produced in this way made changes of form, shifts, and disappearance at the solar limb comparable. Galileo argued that the spots changed their shape and appeared shortened when approaching the limb; this spoke in favor of their lying on or near the surface of the Sun and being carried along by the rotation of the Sun. Scheiner had initially interpreted them as small celestial bodies in regular orbit. In this dispute, “satelles”—literally a companion or attendant—designated them as independent bodies whose orbit around the Sun was supposed to explain their appearance. The relational word-form was preserved, while the interpretation made the spots into orbiting objects.[87]

33This opposition did not remain stable. Scheiner abandoned his first satellite hypothesis in later letters and finally acknowledged that the spots belonged to the Sun or were very near it. This does not mean, however, that he thereby adopted Galileo's Copernicanism. In 1614 the Disquisitiones mathematicae de controversiis et novitatibus astronomicis appeared in Ingolstadt, composed and defended under Scheiner's direction by his student Johannes Georg Locher. The work set different world systems over against one another and argued against the motion of the Earth. The Tychonic configuration offered a viable possibility of newly arranging the observed planetary motions without presupposing the Earth as a moved body. Galileo later answered Locher's and Scheiner's objections in the Dialogo: on the Second Day he mocks the illustrations with which the consequences of an Earth's rotation for falling bodies, birds, and projectiles are visualized. The controversy thus concerns not only what observation shows, but also which context of motion enters into the observation as a presupposition.[88]

34In the Rosa Ursina sive Sol (1626–1630), Scheiner transformed his drawings, gathered over years, into an extensive work of observation and calculation. He compared the paths of the spots at different seasons and determined from their changing inclination the inclination of the solar axis to the ecliptic. In this way he corrected Galileo's earlier assumption that the monthly spot-paths always ran parallel to the ecliptic. The spots were now quantitatively comparable through recurring positions, daily drawings, and seasonal patterns. The evaluation made a stable class of observable appearances out of individual dark markings; but their measurability still presupposed the relation of spot, solar disk, time of observation, and ecliptic.[89]

35The effect ran in both directions, but not as a simple personal exchange between three simultaneous interlocutors. Tycho had been dead since 1601; his world-picture continued to be read in printed writings, teaching traditions, and controversies. Galileo and Scheiner, by contrast, stood in direct, though polemical, connection through Welser, printed letters, and later books. For the Third Day of the Dialogo, Galileo took over a particularly difficult finding of measurement that Scheiner had published in the Rosa Ursina and made it into an argument for the annual motion of the Earth. The observation of the opponent could thus receive a new justification within another world system. At the same time this was no simple confirmation: Galileo's account in the Dialogo of his early discovery of the “deviating” spot-paths is a retrospective presentation; which parts of the argument come from his own earlier observation and which from Scheiner's published data remains disputed in scholarship.[90]

36For the relational question, the finding lies in a change of role, not in a single unchanged result of measurement that would merely be handed on. A drawn trace becomes a repeatable data set; from data sets a solar rotation or axial inclination is determined; in the later argument this determination can appear as a premise about Sun or Earth. The new role changes what the determination does in the respective context: the spot as orbiting satellite, as appearance on the Sun, and as evidence within a world system are not meaning-identical uses, even if trace of observation and vocabulary establish continuity. The axial inclination remains determined by spot-path, time of observation, and ecliptic; its return as a property of the Sun is a new shorthand for these measurement relations. Galileo's recourse to Scheiner's data adds a further relation: a published finding of measurement becomes evidence in another cosmological argumentation. This is a concrete case of relational restabilization; it does not yet prove a retroprojection of the later role onto the earlier observation.

37Textual finding: Scheiner and Galileo stabilized observations through projection, drawing, temporal repetition, and geometrical comparison. Scheiner revised his initial interpretation of the spots, but resisted the Copernican interpretation of the Earth's motion; Galileo used Scheiner's later measurements in his own Copernican argument.

38Historical reconstruction: Tycho did not act as a direct interlocutor, but through a world system that remained available in the astronomical practice of the early seventeenth century and compatible with new observations. The concrete mediation between Scheiner and Galileo ran through Welser, translations, printed letters, Scheiner's student Locher, and the later works of both astronomers.

39Relational consequence: A measurement determination can enter into another theory, yet it does not necessarily enter there with an unchanged role. The comparison that generated it can be shortened, newly weighted, or supplemented by new model relations. The shorthand “inclination of the Sun's axis” preserves a reference to the measurement series, while Galileo's use of Scheiner's data places it in a new proof relation. Only if the later reading were presented as the original meaning of the earlier observation would the retroprojection meant here be present; this stronger step is not proven by this example.

40Hypothesis not yet proven: The case shows how a measurement determination can be restabilized through new proof and model relations. It proves neither that Tychonic astronomy caused a reification nor that Galileo's system eliminated relations. A back-reading in which the later meaning of the spot or of the measured magnitude were presented as the original meaning of the earlier source would have to be separately demonstrated. Gassendi knew these astronomical and mechanical practices; whether and how they changed the relational role of his atomistic concepts must be examined separately.

14.8 What Gassendi actually does with mathematics

41Gassendi had been appointed royal professor of mathematics at the Collège Royal in 1645. His inaugural lecture and the Institutio astronomica published in 1647 rest on an astronomical teaching practice. The Institutio presents the astronomical hypotheses of the ancients, of Copernicus, and of Tycho Brahe comparatively. Gassendi's mathematics appears here in the calculation of celestial positions, in observation protocols, in tables, and in the handling of instruments. His observation of the transit of Mercury of 1631 checked an appearance predicted by Kepler's Rudolphine Tables. For astronomical observations he used, among other things, an alidade and a graduated circle; numbers from tables and instrument readings were compared with the observed celestial phenomenon. In De apparente magnitudine solis humilis et sublimis (1642), he investigates the apparent size of Sun and Moon at the horizon and high in the sky, connects observation with optical explanation, and works with geometrical visual angles. The measurement here first concerns the appearance for a particular observer and a particular geometry of sight, not a directly accessible magnitude of the celestial body.[60]

42His mathematical practice is also visible in short treatises on motion. In De motu impresso a motore translato (1642), he discusses the experiment of letting a body fall from the mast of a moving ship. The question is whether the body carries along the horizontal motion of the ship during its fall. Gassendi presents the process as a connection of horizontal and vertical motion and argues geometrically with the path that arises. The treatise is thus a physical case of comparison, not an algebraic calculation.

43In De proportione qua gravia decidentia accelerantur (1646), Gassendi investigates the proportion according to which falling heavy bodies are accelerated. He defends Galileo's law that, in uniformly accelerated motion, the spaces traversed grow with the squares of the times, and in doing so corrects an error in his earlier presentation. The text is part of a written engagement with Galileo's theory and its critics; it shows that Gassendi applies proportional-geometrical arguments, but not that he himself experimentally measured falling motion in these writings.[61]

44In Gassendi's atomistic natural philosophy, too, mathematical considerations receive a precise but limited role. In the Syntagma philosophicum he lists, among properties of atoms, moles or magnitudo, figura, pondus, and soliditas. For pondus the modern concept “mass” must not be inserted without marking; likewise this list by itself does not attest a modern magnitude of density. In the discussion of divisibility he expressly warns against inferring the infinite divisibility of physical bodies from the mathematical divisibility of a continuum. Mathematical magnitude and physical matter are not simply the same for him. Geometrical considerations can test or refute arguments about matter, but by themselves they do not yet provide an observation of an atom.[62]

45After this historical check, the source map set out in advance can be made more precise: Gassendi's procedures do not stem from a single strand of tradition, but connect mathematical astronomy, geometrical theory of motion, and a natural philosophy that expressly keeps mathematical and physical divisibility apart.

46The sources therefore do not support any sweeping claim that Gassendi simply treated mathematical magnitudes as properties of bodies. His astronomical magnitudes arise through observation, angle measurement, and tables; his magnitudes of motion through ratio and geometrical proof; his atomism does take size, shape, weight, and solidity as marks of corpuscles, but keeps their physical existence apart from mathematical constructions. The relevant question is not whether Gassendi uses “mathematics,” but when and through which procedures a relation is determined as a magnitude and when this magnitude can be attributed to the body.

14.9 Descartes as counter-move: place and motion through neighborhood

47The usage of language in this controversy warns against too quickly making a common conceptual system out of jointly used words. Scheiner wrote his letters on the sunspots in Latin; Galileo answered Welser in Italian, and Scheiner read Galileo's letters only in translation. Even the title of Galileo's print, Istoria e dimostrazioni intorno alle macchie solari e loro accidenti, designates the spots as macchie and their accidenti. The word accidenti is here to be read first from the Italian title and from the course of the observation reports: it means phenomena, occurrences, or modes of appearance. The title alone does not show that Galileo classified the spots as Aristotelian accidents in the technical sense. Conversely, Scheiner designates them with Latin words such as maculae and satellites. That satelles originally designates a companion foregrounds the relational word-form; in the astronomical dispute, however, the companion becomes an independent celestial body that enacts an observed orbit. The same appearance is brought into different orders of objects by word choice and description.[87]

48Descartes' Principia philosophiae appeared in Latin in 1644; the French version approved by him followed in 1647. Here too, the change between Latin and French is not a mere exchange of labels. The Latin words locus, spatium, situs, vicinia, and translatio receive their determinacy in the distinctions that Descartes carries out in the text. In doing so, he expressly begins from the common meanings in order to reorder them.[91]

49In Part II, Article 13, Descartes says that the words locus and spatium designate nothing different from body; they indicate its magnitudo, figura, and situs inter alia corpora—its size, shape, and position among other bodies. In order to establish this position, one must regard other bodies as unmoved. The same person in the stern of a ship can therefore at once remain in one place if one takes the ship's parts as reference, and change his place if one looks to the shore. Thus there is not first an empty place into which a body enters; place is determined by comparison with selected bodies. In Article 15 Descartes calls the boundary between surrounding and surrounded body a modus. And Article 16 excludes a vacuum understood philosophically: where there is extension, there must be corporeal substance. Locus and spatium are therefore not simply interchangeable, but both are bound to extension and positional relations between bodies.[91]

50This move becomes still clearer in the definition of motion. In Article 25 Descartes determines motion in the proper sense as translationem unius partis materiae, sive unius corporis, ex vicinia eorum corporum, quae illud immediate contingunt et tanquam quiescentia spectantur, in viciniam aliorum: the transfer of one part of matter, or of one body, from the neighborhood of those bodies that immediately touch it and are regarded as at rest into the neighborhood of others. Vicinia here does not simply mean a modern coordinate frame. What is decisive is the relation to adjacent bodies that are posited as at rest for the determination. In Article 29 Descartes calls the transfer reciprocal: when a body separates from the neighborhood of another, the same change can also be described from the opposite side. Only the selection of what is to count as at rest allows ordinary usage to attribute motion to one of the two bodies.[91]

51Here lies the reason for reading Descartes in this historical strand as the last relationalist counter-pole. His definitions do not lead place and motion back to a property that a body would carry in itself independently of others. Situs arises between bodies; motus is determined as change of neighborhood; the boundary of a body is to be understood relationally as modus. The enactment of determination is therefore held fast in the definition itself. In this limited sense, Descartes can count as the “last relationalist” of the section followed here—not as the last thinker at all who investigates relations, but as the last counter-pole in the line that leads from mathematically stabilized determinations of bodies to Gassendi's atomism.[91] [92]

52At the same time, this designation must be limited. Descartes rejects the atom and the empty between, but not every concept of a corporeal entity. For him matter is res extensa: an extended substance; shape and motion are modes in which extension is determined. The relational definition of place and motion thus stands in an ontology in which extension is the basic determination of corporeal substance. In addition, in Article 27 Descartes speaks of motion and rest as different modes of a body. His theory of relation is therefore not free of tension: motion is determined through neighborhood, but at the same time treated as a mode of the body. Recent research discusses precisely this tension between reciprocal relation of motion and corporeal state.[91] [92] [93]

53The contrast with Gassendi is instructive. Gassendi's atoms are corporeal units with magnitudo, figura, pondus, and soliditas, which move in the void; Descartes' matter forms a continuum without vacuum, whose divisions and boundaries do not lead to indivisible ultimate bodies. In Gassendi, the relations can appear as relationships between already assumed atoms. In Descartes, at least place and motion are determinable through the relations to other bodies. Yet this opposition too is not completely symmetrical: Gassendi distinguishes the physical existence of atoms from mathematical divisibility, while Descartes grasps corporeal substance itself as extension. Both connect relation and object, but they set the connection at different points.[92] [93]

54Textual finding: Descartes determines place, extension, and motion in the Principia through situs, vicinia, corpora contigua, and the selection of bodies regarded as at rest. At the same time, he calls extension the essence of corporeal substance and treats motion as a mode.

55Historical reconstruction: Scheiner's Latin letters, Galileo's Italian replies, and Descartes' Latin and French Principia show that learned language around 1600/1640 was neither uniform nor a merely neutral vessel. Translation, addressees, and textual genre helped determine how macula, satelles, accidenti, locus, spatium, and motus were read.

56Relational consequence: Descartes provides strong counter-evidence to the straight-line thesis that mathematical stabilization necessarily leads to reification. Precisely where he presupposes a fixed corporeal world, he continues to determine place and motion through relations. At the same time he provides extension as the basic mark of the res extensa, to which further modes can be attached.

57Hypothesis not yet proven: Within the frame of this narrative, Descartes is the last major counter-design that keeps the production of determinations of place and motion visible in relational definitions. Whether and how this rule of reading weakens in later mechanics must be checked in the subsequent historical strand; here the finding is not sufficient to assert a final historical boundary.

58Textual finding: From Euclid and Archimedes to Gassendi and Descartes, construction, figure, ratio, and proof remain effective. Added to them are place-value notation, algebra, printed tables, geometrical presentations of change, procedures of measurement, and explicitly relational definitions of place and motion.

59Historical reconstruction: These procedures run through Greek, Byzantine, Syriac, Arabic, and Latin contexts. They are mediated in different languages and textual forms. Scheiner and Galileo show how drawings and measurements enter new world models through translation and controversy; Gassendi and Descartes take up mathematical and physical procedures, but order body, motion, and extension differently.

60Relational consequence: Measured values, diagrams, and signs can make results storable and transferable. The original enactment of determination therefore need not be repeated at every use. This can move relations out of view. Descartes, however, makes visible that a stably determined natural magnitude can continue to be relationally defined; Gassendi conversely shows how relational magnitudes can be referred to physical units.

61Hypothesis not yet proven: The chapter findings prove neither a continuous movement from relation to property nor a historical endpoint. They show a tension: a result can appear as transferable determination and later be read as presupposition of the object; at the same time, authors such as Descartes can preserve the relation in the definition itself. In the next step it must be examined whether later orders of concepts and calculation redistribute this tension.

62The two strands thus do not come together in a single solution. Gassendi posits physical atoms with determinable marks; Descartes determines place and motion through the neighborhood of other bodies, but anchors these relations in extended substance. Their opposition leaves the guiding question open: When does a result of determination become the presupposition of what is determined?

15. When Determinations Become Presuppositions: Gassendi, Newton, Leibniz, and Euler

Reading Key to the Following Examples

1E5 designates a determination that arises in the earlier context R0 and is reused in the later context R1. Its name can remain while its role changes. Later in the volume, “level 5” designates already stabilized particle determinations as inputs to a new systematics. Working maps bring together the relations under examination; they replace neither historical evidence nor laws of nature.

2The preceding chapters have followed two strands of transmission into the early modern period. On one side stand the Latin names and categories that became readable through translation, commentary, and instruction. On the other side stand mathematical procedures whose constructions, relations, measurements, and signs could be preserved and applied again. Now the question is not only whether a result can serve as a starting point without renewed emergence. What is decisive is whether, in this new use, it still has the same relational role. E5 can be determined in R0 by E1–E4 and enter new relations in R1 with F1, F2, and F3. The name or the sign can persist while what determines E5 and what it accomplishes in the argument changes. It must also be examined whether the later role R1 is subsequently read back into R0.

3This is not a history in which a relational antiquity necessarily issues into a reifying modernity. The authors examined do not simply stand in a closed chain of influence. Gassendi reads ancient sources with philological care and changes his reconstruction; this already provides a direct counterexample to a completely closed retroprojection. Newton mathematically distinguishes absolute magnitudes from the relative measures through which human beings infer them. Leibniz makes relations symbolically manageable and at the same time defends a relational determination of space and time. Euler transfers sine and cosine from geometric determinations into an analytic functional role; under the same name, therefore, one must in each case examine whether the same determination or a newly restabilized relation is present. The guiding question is: Which relations determine an expression at each station, which new ones are added, and does a later system read its own role back into the earlier source?

15.1 Gassendi reads back — and rereads

4Gassendi is suitable as a first test because, as his texts show, his atomism arises through intensive work on ancient testimonies. For Epicurus’ doctrine, Book X of Diogenes Laertius’ Lives and Opinions of Eminent Philosophers is central. Gassendi’s Animadversiones in decimum librum Diogenis Laertii of 1649 contains the Greek text, a Latin translation, and extensive commentary. Lucretius is another explicitly present source; Gassendi has access to Democritus above all through ancient reports and controversies, including Aristotle’s testimonies, not through a surviving work by Democritus. These differences in transmission are decisive for the reading: in Gassendi, a thesis about “atomism” can be assembled from a cited Epicurean sentence, a Latin poem, and an Aristotelian counterstatement, without these testimonies having the same status.[94]

5The comparison of Gassendi’s early manuscript De atomis (1636–1637) with the Animadversiones (1649) and the posthumously printed Syntagma philosophicum (1658) shows that the source is not simply inserted into an already finished systematics. Jo Coture has traced, on the basis of the explanation of atomic motion and the shape of the earth, how Gassendi changed his early reconstruction in order, from his point of view, to bring it closer to Epicurus’ transmitted doctrine. The textual finding is therefore double: Gassendi orders the ancient statements into his own system of natural philosophy; at the same time, a concrete revision of his reading can be followed. This revision is a counterexample to the idea that a stable systematics only reproduces older sources in its own image.[95]

6In his account of Epicurean physics, Gassendi distinguishes features that he attributes to the atoms themselves from sensibly perceptible qualities that arise from combination and arrangement. Among the atomic determinations he names, among others, size (magnitudo), shape (figura), weight or heaviness (pondus), and solidity (soliditas). In his account of Epicurean physics he records that the atomists do not attribute sensuous qualities such as color, smell, taste, and warmth to the atoms; rather, these arise from the atoms and their determinate position and order. The Latin formulation is explicitly genetic: sensuous qualities are supposed to arise “from these” and “from determinate position and order” and to change through removal, addition, or rearrangement.[96]

7Gassendi thereby preserves a relational explanatory trait: perceptible quality is not simply one more small thing in the atom, but depends on arrangement and interaction. At the same time, his explanation presupposes individual atoms as corporeal units in which size, shape, and weight are already determined. The direction is therefore not uniform. At the microlevel, atomic bodies are treated as bearers of determinate features; at the level of sensuous qualities, properties are explained from relations between bodies. In addition, Gassendi distinguishes mathematical divisibility from physical divisibility: that a magnitude can be thought as geometrically further divisible does not prove for him that a physical atomic body can actually be broken down into parts. Indivisibility thus becomes a physical limit, not a claim that the atom has no distinguishable determinations at all.[97]

8The relational reading must therefore neither replace Gassendi’s source with its own premise nor assert an unambiguous reversal. The text shows that sensuous qualities emerge from arrangement and combination, while size, shape, weight, and solidity appear as determinations of atomic bodies. This list of features does not simply assume unchanged the role of the ancient atomos: Gassendi places it in an early modern system of corpus, cause, physical divisibility, and sense quality. The source and Gassendi’s version share words and doctrinal pieces, but their roles must be compared in each case. Gassendi’s revisions also show that the transmitted text can correct his own systematics. What is documented is an active retro-reading with alteration, not a closed loop in which later categories merely confirm the earlier source.

15.2 Newton: mathematically absolute magnitudes and relative accesses

9In the Principia (1687), Newton places a striking distinction at the beginning. In the definitions it says: Quantitas materiae est mensura ejusdem orta ex illius densitate et magnitudine conjunctim — the quantity of matter is a measure of matter that arises jointly from its density and magnitude, or volume. The translation as “mass” is common, but the Latin definition must not be transferred unnoticed into the later, fully standardized concept of mass. The text binds the magnitude to density and extension; immediately afterward Newton determines the quantity of motion as a measure that arises jointly from velocity and quantity of matter. The definitions of force also describe quantified changes of motion and their relation to time.[98]

10In the Scholium on time, space, place, and motion, Newton distinguishes “absolute, true, and mathematical” magnitudes from “relative, apparent, and common” ones. According to his formulation, absolute time flows uniformly and without relation to anything external; common time, by contrast, is a sensibly accessible measure of duration by means of motion, such as hour, day, or year. Absolute space remains, by its nature, uniform and immovable without relation to external things; relative space is a movable measure of this space, which the senses determine through its position relative to bodies. Motion is correspondingly distinguished as absolute or relative.[99]

11Here lies the sharpest contrast to this investigation’s Heraclitus reading. In Heraclitus, the difference becomes readable in the moving-apart and coming-together; it determines the poles instead of merely measuring an already fixed magnitude between them. Newton, by contrast, explicitly posits a space and a time that exist “without relation to anything external.” The observable, body-related measures thereby receive a different position: they are measures of a mathematically presupposed absolute magnitude. Within our relational reading perspective, this is a clear change in the direction of reading. It is not, however, a claim that Newton historically reacts to Heraclitus or adopts him directly. Newton’s argumentative structure also remains more differentiated. Relative measures are not superfluous errors, but the common access from which mathematical natural philosophy first proceeds. The motion of a ship, of a body in the ship, and of the earth must be described relative to the respective reference bodies. Newton explains, however, that relative motions are not simply additively sufficient for true motion, and develops the distinction further by means of dynamic effects: in circular motion, tensions and forces show effects that are not exhausted by mere motion relative to nearby bodies. The rotating water in the bucket is meant to make visible the difference between an observed positional relation and the motion indicated by the forces.[100]

12The absolute magnitude in Newton’s text is therefore neither merely an immediately measured property nor only a metaphysical postulate. It is a mathematical presupposition of the definitions and at the same time part of a physical argumentation that relies on relative observations and effects. Newton grants that absolute time and space are not directly distinguished by the senses; relative measures are used in everyday life and in observation. The change therefore also concerns the relational role: “absolute time” in Newton is not an unchanged continued name for observed duration, but a posited magnitude to which common time measures stand in a derived measuring relation. The text provides no simple transition of “relation disappears, entity remains.” It explicitly places two differently determined magnitudes and their relation of access side by side.[101]

13The definition of the quantity of matter also limits a crude reification thesis. Newton introduces quantitas materiae with a measuring connection: density and magnitude are brought together; then he says that this quantity is also called “body” or “mass,” is made known through the weight of each body, and has been found through pendulum experiments to be proportional to weight. Within this definition, the magnitude is accessible relationally through density, extension, and weight. When quantitas materiae later appears as a dynamic term in a calculation of motion, it must be examined whether the later role is the same as that of the initial measure or whether it is newly determined in the law-connection. The mere recurrence of the word quantitas does not prove this identity of meaning.

15.2.1 Fluxions: from flowing to limiting ratio

14The same tension appears again in Newton’s differential calculus, this time within the mathematical procedure. In De quadratura curvarum, Newton asks: if an equation with arbitrarily many fluentes quantitates is given, how can their fluxiones be determined? The fluentes are the variable quantities treated as increasing or decreasing; the fluxiones designate their rates of change. The procedure therefore translates a relation between variable quantities into a relation between their fluxions. In the Principia, this operation is conducted geometrically through variable figures and ever narrower parallelograms: their number grows, their width decreases without limit, and their sums and ratios approach the magnitudes of the curvilinear figure.[102]

15Newton himself makes the critical step thematic in the Scholium to the lemmas. Against the objection that before disappearance there is no “last” ratio and afterward no ratio at all, he draws the comparison with a body that reaches a last place: the last velocity is the one with which it reaches this place; correspondingly, the last ratio is that “with which” the magnitudes disappear. Immediately thereafter, however, Newton restricts this way of speaking: such ratios are not ratios of “last magnitudes,” but limits to which the ratios of magnitudes decreasing without limit can come arbitrarily close, without exceeding them or reaching them before the unlimited decrease. When he speaks of vanishing or last magnitudes, therefore, one should not understand fixed determinate magnitudes, but always think of magnitudes that continue to decrease without limit.[102]

16The text thus explicitly preserves the process. Newton does not simply assert a last, tiny object that would remain at the end of the flowing. And yet the limit value receives the status of a “certain and determinate” geometric object: it can be determined and used in further proofs. This is where the relational reading of the book begins. A process of change is transferred into a determinate limiting ratio; this limiting ratio can subsequently enter the next operation as a result, without the approximation having to be produced anew for every step of the proof. Newton’s text keeps the genesis visible, while his mathematical method at the same time makes its repeated enactment unnecessary. The small break that this section adds lies precisely in this combination.

17The analogy to the opposition “A / not A” is an interpretation of this investigation, not Newtonian set theory or logic. If the question whether a determinate limit point has been reached is posed binarily, then “not yet reached” gathers together very different states of approximation. Non-reaching is then not a uniform counterworld, but an open multiplicity of possible courses. Newton’s mathematical concept of the limit value cannot be reduced to this binary division; his text defines the limit value precisely through ratio, approximation, and a given difference. The parallel concerns rather the direction of reading: a relational movement can be abbreviated by a fixed decision schema, and the result gained thereby can continue to work as an independent presupposition.

18Thus, in Newton, two differently directed modes of determination stand side by side. In the Scholium on space and time he posits absolute magnitudes that are not determined by external relations. In the fluxional calculus, by contrast, he determines limiting ratios through variable magnitudes and approximation. A later use of the limiting ratio as a fixed mathematical starting point does not simply mean that the original process remains preserved as unchanged meaning. In the limit proof, the limit has a new role: it orders a sequence of ratios and makes possible a proof concerning the magnitude they approach. Newton keeps this determination reconstructible through his definition and the surrounding argumentation. The case documents operative detachment, but no self-evident identity of meaning between the enactment of approximation and the limit value.

15.3 Leibniz: relational spatial order and calculable differences

19Leibniz offers a countertest because his mathematics makes the continued use of stabilized determinations especially clear, while his doctrine of space explicitly defends a relational reading. In the Nova methodus, published in 1684, he introduces differential signs such as dx and dy into general rules for tangents, maxima, and minima. A geometric problem can now be worked on through algebraic expressions; the differences are transformed according to rules of calculation, and a solution can be obtained from the symbols without reconstructing the figure anew for every individual step.[103]

20This operational capacity does not make the relation meaningless, but it changes its role in the calculus. Leibniz derives differentials from changes of magnitudes and their relations; in geometric examples, dy/dx corresponds to the relation between changes of ordinate and abscissa values. At the same time, the signs become reusable elements of a calculus: one can add, multiply, substitute, and transform them according to fixed rules. The differentials in the computational procedure are therefore not without further qualification identical in meaning with the geometric changes from which they are initially obtained or explained. Leibniz later defends infinitesimal magnitudes as “well-founded fictions”: their usefulness depends on a general method and on the justifications of the calculus, not on their being ordinary finite magnitudes or physical things. Here lies a countertest: operative detachment and abstract use can go together with an explicitly preserved relational justification.[104]

21In the correspondence with Samuel Clarke of 1715–1716, Leibniz determines space as the order of simultaneously existing things and time as the order of successive events. Space is not an absolute container alongside things; it is the order through which things receive their position relative to one another. Clarke replies that space is a quantity and not merely a relation of position and order. Here “relation” and “thing” do not simply stand opposed to one another. Leibniz’s concept of order is itself structured and meant objectively; Clarke, in turn, attempts to ground space as an independent magnitude whose parts and quantity cannot be reduced to bodily relations.[105]

22The correspondence is an unusually clear historical piece of evidence that the direction of reading was disputed. Leibniz’s spatium as the order of coexisting things and Clarke’s space as quantity are not two labels for the same determination: they order position and extension into different grounding contexts. At the same time, the dispute forbids a teleological interpretation of Newton: Leibniz’s relational position arises in a concrete confrontation, not as the later discovery of an allegedly lost connection. Newton’s theory cannot be reconstructed solely from Clarke’s formulations; Clarke represents it, but his answers are not simply identical with every aspect of Newton’s own philosophical position.[106]

23Leibniz’s doctrine of monads should be included in this question only cautiously. In the Monadology of 1714, monads are determined as simple substances without parts; each represents the universe from its own point of view. This explains how a manifold world appears in perspectives without the monads acting directly on one another. The texts make unity, perception, change, and perspectival representation central problems. They do not, however, without further qualification document that the doctrine of monads was developed as an answer to a tension between mathematical relation and physical bearer. Such a connection remains a further-reaching interpretation and should not be presented here as a historical cause.[107]

15.4 Euler: sine and cosine receive a new analytic role

24With Euler, what changes is not simply the ontology of mathematics; what becomes visible is rather a new reach of mathematical operations. In the Introductio in analysin infinitorum (1748), Euler makes functions the central object of analysis. His definition is: Functio quantitatis variabilis est expressio analytica quomodocunque composita ex illa quantitate variabili et numeris seu quantitatibus constantibus — a function of a variable magnitude is an analytic expression composed in whatever way from this variable magnitude and numbers or constant magnitudes. In the Institutiones calculi differentialis (1755), he formulates the thought of dependence further: when a variable magnitude depends on another, the magnitudes depending on it are called functions of this variable.[108]

25These formulations must remain in their historical difference. Euler’s function is neither simply the modern set-theoretical mapping nor merely a geometric curve. The decisive test case here is the sine and the cosine. Euler does not invent these magnitudes; they have a long astronomical and geometric history as determinations of the circle: at first chord lengths were calculated, later line lengths and ratios were assigned to a circular arc. Euler’s Chapter VIII, too, bears the title “On transcendental magnitudes that arise from the circle” and initially starts from the circle and a fixed radius. The textual finding therefore contradicts the literal claim that sine and cosine arise in Euler wholly without geometry.[109]

26The shift lies in the next step. Euler treats sine and cosine not only as lines that are read off from a respectively constructed circle figure. Once in the analytic functional frame, they become expressions of a variable arc length, which are further determined by series and relations to exponential magnitudes. Their role as analytic functions is not simply the same as their earlier role as lines or ratios in the circle. Euler transfers the geometric determination into a new context: circle and arc ground the introduction; series, equations, and transformation rules determine the function within analysis. Later these functions can be applied to geometry and used for calculating arc and circle relations. The return path is therefore not a mere application of the same unchanged E5: the analytically restabilized E5 now enters new relations and thereby makes the circle readable in a new way.[110]

27It is precisely in this sense that a possible reification can be examined here: not because sine and cosine would literally be things or because Euler offered no derivation. He shows the geometric starting point and presents the analytic relations. What is decisive is that the later function receives a new relational determinacy in calculation: it is now a transformable expression in an apparatus of functions and series. If this analytic form is subsequently used to determine the circle, the return path is mediated by the new mathematical role. The analytic E5 is not without further qualification identical in meaning with the geometric E5 from which it emerged. “Without foundation” would therefore be too crude; more precisely, the earlier enactment of determination is replaced or supplemented in the new context by other relations.

28Euler’s function is therefore not a relation that disappears from mathematics. Sine and cosine in particular carry complex relations within themselves: between arc, circle, ratio, series, and exponential expression. Yet systematic analysis allows a once-gained relation to be named as a function, developed, transformed, and inserted into further relations. The function makes the relation operationally available. From a function, derivatives, integrals, series, and new functions can emerge: the relation becomes an operation on relations.

29Euler makes this gain explicit, not invisible. He defines variables and functions, distinguishes kinds of expression, and lays open computational procedures. The Mechanica of 1736 also shows a new systematization: motion under forces is transferred into analytically treatable problems, although Euler does not yet simply adopt there the later, popularized short formula F = ma as Newton’s second law. The transformation of mechanical problems into analysis is a historically specific achievement, not a mere translation of a timeless law of nature into more convenient signs.[111]

30At the same time, the expressions stabilized in this way can appear as given objects of further mathematics. A function is differentiated; its derivative in turn becomes a function; the derivative is integrated or developed into a series. In this process, its problem of origin does not have to be retold step by step. This is a clear case of reusability, but not yet proof that Euler understands functions as ontologically independent things or that the physical world is thereby necessarily read as an inventory of property-bearers. His definitions keep present the dependence on variable and constant magnitudes. The systematics closes certain operations; it does not completely exclude the origin.

15.5 Countertest: continuation proves no identity of meaning

31The sources examined yield no uniform chain in which scholastic categories immediately produce Newton’s space, Leibniz’s differentials, or Euler’s functions. There are concrete transmissions: Gassendi receives and comments on ancient authorities; Newton works within a mathematical mechanics whose older procedures and teaching traditions are historically mediable; Euler reads Newton and analytically reshapes mechanical problems. Between these authors, however, there are different textual paths and questions. Chronological succession, demonstrable transmission, and structural similarity must be kept apart.[112]

32The new examination is directed toward the role of an expression in each context. A transmitted word or symbol can remain while its determinacy changes:

33These comparisons show three different cases. In Gassendi, the source remains effective and can correct his reconstruction. In Leibniz, the relational justification of the calculus explicitly remains at issue, although the signs are operatively reused. In Euler, a geometrically introduced magnitude is transferred into a new analytic role. In Newton, relative measurement and absolute theoretical magnitude stand side by side. There is thus relational back-binding, operative detachment, and relational restabilization; these cases must not be pulled together under “forgetting.”

34The central fallacy would be to infer complete original grounding from continued use. If E5 occurs in E6, E7, and E8 as a presupposition, this first shows that the later steps proceed from E5. It does not prove that E5 alone, or in identity of meaning, explains the conditions E1–E4 through which it was originally determined. Likewise, the later recurrence of a concept in a theory does not prove that this theory preserves its earlier meaning. Conversely, no change of meaning follows merely from the absent repetition of a derivation step: what is decisive is whether the constitutive relations are preserved in the new context, reconstructible there, or replaced by new relations.

35A first, limited drift can therefore be described: determinations change their relational environment and can function differently under a persisting name. The stronger claim — that a later role is projected back into the earlier source and presented there as original meaning — is not documented throughout for these four cases. Gassendi’s revision speaks against a closed cycle; Leibniz preserves a relational justification; Newton and Euler also show how a measurement or construction determination can become a new theoretical or analytic role. The later chapters must examine whether Riemann, Kant, Hilbert, Gödel, and school instruction provide more concrete evidence of such retro-readings.

36“Forgetting the foundation” must therefore be made more precise. It is not enough that an origin is not repeated in every calculation. The stronger question is whether, with the omitted relations, part of the determinacy of E5 is lost and whether a new order occupies this position differently. If E5 is determined differently in R1 than in R0, then the later use does not simply presuppose the same result; it produces a relational restabilization. Reification would be the possible case in which this newly determined role appears as the original, independent character of E5.

37Thus there is no gapless chain of transmission and no necessary reification of every concept. Historically documented are individual translations, mathematical transformations, measurements, and new system-roles. The relational interpretation asks whether the new roles alter the older meanings and whether later readers retroproject this alteration. Δ := |(| remains for this an internal reading anchor from the Tractatus, not historical proof: what must be examined is whether the back-reference remains visible as alteration or whether its later determination appears as the original one.

16. When the Rule System Meets Its Limit: Riemann, Hilbert, and Gödel

1The drift so far can now be tested at another site. A mathematical determination can be reused as a number, function, or concept; but it can also enter into a rule system that specifies which propositions follow from which presuppositions. Then the systematic order itself becomes the object: What presuppositions does it set? What consequences do they allow? Can individual rules be omitted? Can the system secure its own freedom from contradiction? The path from Riemann through Hilbert to Gödel does not lead to a single answer. Rather, it shows how making presuppositions explicit first enables new control, and how a formal self-examination at the same time makes a boundary visible.

16.1 Riemann: Presuppositions are expressly interrogated

2Riemann's Göttingen habilitation lecture of 1854 does not begin with a finished assertion that space must be Euclidean. He first distinguishes relations of extension or domain from relations of measure and asks which assumptions are required in each case in order to determine a geometry. For space, one might presuppose, for example, that lines exist independently of their position, that bodies likewise possess this independence, or that length and direction remain unchanged when place changes. Different presuppositions lead to different relations of measure. Riemann expressly asks “how, to what degree, and to what extent” these presuppositions are warranted by experience.[113]

3This is an important counter-finding to the idea that mathematical abstraction simply leaves its foundation behind. Riemann releases a determination from familiar Euclidean intuition, but he makes the hypotheses chosen for this visible and investigates their empirical reach. His geometry generalizes the possible relations between place, direction, length, and curvature. The mathematical form is therefore not without presuppositions; rather, it makes different presuppositions comparable and shows which consequences arise from them. The relational reading can hold here: A metric determination results from a chosen connection between relations of place and measure. Historically documented at the same time is that Riemann expressly problematizes the selection of this connection instead of presenting it as a self-evidently given property of space.

16.2 Hilbert: Relations are determined axiomatically and their consequences tested

4With Hilbert, mathematical determination as an axiom system becomes especially clear. The Foundations of Geometry go back to Göttingen lectures from the winter semester of 1898/99 and appeared in 1899 in the Festschrift for the unveiling of the Gauss-Weber monument. Hilbert begins with three “systems of things,” which are called points, lines, and planes. What these elements mean geometrically is not first fixed by an intuitive definition; their relations are determined by axioms concerning incidence, order, congruence, parallels, and continuity. Hilbert then investigates what follows from the axioms and what role each individual one plays in the proofs.[114]

5This formulation can be read as if the points were already provided as finished things and subsequently equipped with relations. But Hilbert's methodological step is more precise: The axioms lay down which relations hold between the three systems; the names alone do not decide what geometric points or lines are “in themselves.” The system can be interpreted through different models. A geometric statement is thereby released from its individual drawing and bound to the structure of the fulfilled axioms. A diagram remains useful, but a proof must indicate which axioms support it.

6In doing this, Hilbert makes visible precisely what runs along as a presupposition in a proof. The five axiom groups are tested for freedom from contradiction and mutual independence; models show that certain groups do not simply follow from the others. For some propositions it can be indicated which axioms make their proof possible; other geometries show what changes when an axiom is omitted. The result is a systematic order in which presuppositions are reusable and individually testable. A proposition first receives its role within this axiomatic network: Its meaning in the proof depends on which axioms support it and which other statements are connected with it. The relation to a drawn figure can recede in the process, without the same geometric determination thereby already being preserved unchanged. Whoever uses the proposition in another interpretation or application must examine which relations are carried over and which are newly added. Hilbert's models and independence proofs keep this dependence visible; they prevent the names “point” and “line” alone from guaranteeing a meaning identical in all contexts.

7Hilbert extends this movement in 1900, in his second Paris problem, to arithmetic: He demands a proof of its freedom from contradiction. In the later Hilbert program, classical mathematics is to be formalized and its proofs investigated as sequences of signs according to fixed rules. The metamathematical examination itself presupposes elementary, intuitively secure modes of inference. The presupposition therefore does not disappear; it changes level. What counts within formalized mathematics as axiom and rule is examined from outside as a finite sequence of signs and derivation.[115]

16.3 Gödel: Self-reference and the boundary of internal proof

8Gödel does not belong to the Göttingen group. He worked in Vienna and presented his first incompleteness result in September 1930 at a conference in Königsberg. The historical connection to Hilbert is nevertheless concrete: Gödel investigates formal systems such as the Principia Mathematica and expressly includes the systems developed by Hilbert and his collaborators. His work thus arises as the mathematical examination of a program whose aim was the formal securing of mathematics, not as a late chapter of a Göttingen course.[116]

9The first incompleteness theorem does not simply say that “every system” is limited. In Gödel's 1931 version, it concerns formally precise systems with an effective proof procedure that can represent enough arithmetic; for the original undecidability theorem, Gödel uses the stronger presupposition of ω-consistency. In such systems he constructs a proposition that is not provable within the system. The form frequently cited today, which gets by with mere consistency, is due to Rosser (1936) and must not be retroactively presented as Gödel's exact presupposition of 1931. The second theorem additionally concerns the consistency statement formulable in the system: Under the required conditions, the system cannot itself prove its own freedom from contradiction.[117]

10The decisive enactment consists in coding signs and proofs by numbers. Formulae, derivations, and the relation “is a proof of” become arithmetically representable. A system can thereby formulate statements about its own possible proofs. Gödel's proposition is not a merely external critique, but a result that proceeds from the self-application of the arithmetized proof rules. The formal relation of provability itself becomes the object of the system. Yet precisely this operation does not generate a complete self-overview: The external demonstration that the constructed proposition is unprovable in the system takes place on a metamathematical level and in turn presupposes means of inference there.

11Here the drift of this investigation meets a precise boundary case. A rule system uses axioms and rules of inference as given initial determinations. It can code part of its own proof enactment and further process it as an arithmetical relation. But under the stated conditions, the systematic order cannot transform all presuppositions back into an internally secured proof of its own freedom from contradiction. The boundary is not a mysterious deficiency, but a mathematically proven difference between internal derivability and a metamathematical statement about derivability.

12Gödel himself, too, does not formulate the boundary simply as a “forgotten foundation.” In a footnote to his 1931 work he explains incompleteness through the possibility of always forming higher types, while a single formal system grasps only a limited domain of such types; by suitable extension, a previously undecidable proposition could become decidable. This is especially illuminating for our question: The boundary of a system can become the starting point of an expanded system. But this does not prove that the historical origin of its axioms was forgotten. What is shown is that formal closure is always relative to a particular language, set of axioms, and proof relation; an extension changes the frame and thereby also what counts as provable.[118]

13Gödel himself drew further-reaching philosophical consequences from the theorems. In his Gibbs lecture of 1951, published only posthumously, he argued that either human mathematical thought surpasses every finite machine or there are absolutely unsolvable Diophantine problems; he preferred the first alternative. This is Gödel's own philosophical interpretation, but not a consequence that the formal theorem alone compels. Nor does the theorem prove that mathematics as a whole is without foundation, or that its users had psychologically forgotten presuppositions. Such generalizations exceed the mathematical finding.[119]

16.4 What can be shown in the drift

14The three stations therefore do not describe a linear movement from intuitive enactment to forgotten rule. Riemann makes geometric hypotheses explicit and asks after their empirical justification. Hilbert fixes relations as axioms and investigates their consequences, dependencies, and models. Gödel shows that a sufficiently expressive, effectively axiomatized system can indeed internally represent certain statements about its own provability, but cannot internally prove its own consistency under the relevant presuppositions. At every stage something is stabilized; at no stage does it follow from this that all origins become invisible.

15The relational consequence is narrower: When the proof enactment is formalized, the relation “derivable from these axioms according to these rules” can be represented arithmetically. A Gödel number used in this process is not identical in meaning with a number that counts a plurality; its role is determined by coding function, syntax, and proof relation. This stabilization enables new operations and at the same time a precise boundary of what it can accomplish internally. Gödel does not show that a result forgets its presupposition in the historical sense. He shows that a system cannot secure its own derivability in every respect from out of the same level. The boundary remains visible as a relation between system and metasystem.

16The conjecture of the book thereby receives a stricter form. The loss of the foundation is not Gödel's theorem. What can be tested, rather, is whether later readers read the presuppositions of formal rules as absolutely given because of those rules' operative availability. Gödel provides no historical proof for this, but he does provide a mathematical boundary case: Even the formally explicated systematic order cannot completely dissolve its conditions of possibility into itself. Where it is expanded, a new frame arises with new presuppositions. The result once again becomes the starting point—and the question concerning its foundation begins again from the beginning on the next level.

16.5 The four as element: set theory, concept of number, and the omitted enactment

17The drift can be grasped once more, and more sharply, with the number four. In an identitarily ordered number sequence, 4 can appear as a determinate member: It comes after 3, before 5, is unambiguously distinguishable by its place and its relations, and can enter into further calculations as an operand. For this use, no one has to incise four strokes anew or count out four things. The enactment is no longer necessary for the number to operate. Precisely therein lies the strength of the stabilization—and the point at which its origin in use can disappear.

18Historically, however, it would be too crude to have all the mathematicians treated here already working “with set theory.” Euclid's Elements VII.2 determines number as a “plurality composed of units”; the unit is treated separately. With Hilbert, points, lines, and planes are introduced as three systems whose geometric relations are axiomatically fixed. This is an axiomatic structuring, but not already the claim that all mathematics ontologically consists of sets. In 1931 Gödel initially investigates the Principia Mathematica and related typed formal systems. His arithmetization assigns numbers to sign sequences, formulae, and proofs. These three practices—Euclidean plurality, Hilbert's axiom system, and Gödel's numerical coding—are related in their stabilization of determinations, but they are not historically and conceptually identical.[120]

19With Dedekind, the movement between enactment and abstract number can be documented unusually directly. In What Are Numbers and What Should They Be? (1888) he first explains that the concept of number is independent of spatial and temporal intuition; numbers are “free creations of the human mind.” Yet immediately afterward he asks what we do when counting a set, and traces it back to the capacity to relate things to things, to let one thing correspond to another, or to map one thing by another. His theory of numbers is to be erected on this basis. In the same work he constructs the natural numbers as the structure of a simply infinite system with a distinguished initial element and a unique successor mapping. The number is thereby detachable from the concrete object counted and determined through its position in a generated structure.[121]

20Dedekind is therefore neither simply evidence that the basis was forgotten nor counterevidence against every drift. He expressly names the enactment of determination: counting rests on assigning. At the same time he shows how an abstract system can arise from this, in which the individual numbers are identifiable through successor, order, and induction. He hits the possible transition especially precisely in the preface: Practiced readers no longer repeat all the steps of initial spelling when reading; correspondingly, arithmetical truths can be acquired through practiced chains of inference and later appear self-evident. Dedekind even notes that complex concepts such as the cardinality of a set can falsely seem simple as a result. This is primary evidence for the structural possibility that an enactment is abbreviated in a learned operation—not for the claim that all mathematicians had in fact forgotten the genesis.[122]

21A specifically set-theoretical representation is shown by von Neumann in 1923 in the definition of ordinal numbers: An ordinal number is fixed as the set of all ordinal numbers preceding it. In this construction, 0 = ∅, 1 = {0}, 2 = {0,1}, 3 = {0,1,2}, and 4 = {0,1,2,3}; successor formation is expressed by α+1 = α ∪ {α}. The four is here a set with a particular role in the recursive ordinal structure. This presentation is not simply identical in meaning with four counted objects: Its determinacy results from set formation and successor relation. The construction remains reconstructible, although later proofs can use the already defined 4 as sign and operand. This is an example of relation-preserving formal construction and operative detachment; a back-projection of this set-theoretical role onto earlier concepts of number does not follow from it.[123]

16.5.1 Zero: empty place, number, and formal object

22Zero initially seems to be an especially clear piece of evidence for the movement of this book. Yet the historical juxtaposition “The Greeks saw in zero an origin; today it counts as nothing” does not withstand examination of the sources. For Euclid's arithmetic, determination begins with the μονάς, the unit: A unit is that by which something is called one; a number is a plurality composed of units. In these definitions, zero is neither the first numerical object nor the origin from which numbers are derived. If one asks in Greek mathematics after the starting point of number formation, the sources first point to unit and plurality, not to a zero.[124]

23This does not exclude a Greek use of a zero sign. In the astronomical sexagesimal calculation that Ptolemy uses in the Almagest, a special circular sign appears, often rendered as an overlined omicron. In tables it marks a missing or zero-valued place within a technical calculation, above all in sexagesimal fractions. This sign makes it possible to keep place values and astronomical magnitudes readable. But it does not show that Greek arithmetic generally conceived zero as a number or as a metaphysical beginning. The safest textual finding is narrower: A zero notation is functionally employed in a particular mathematical-astronomical practice; its relation to the modern concept of number is not simply identical.[125]

24For the relational question, this limited function is interesting. The zero sign is not itself an astronomical thing in the table. It keeps a place or a value open within a regulated representation and prevents the neighboring places from being read incorrectly. Its meaning arises through the place-value and calculation system in which it is used. This, however, is a modern conceptual description of the function; Ptolemy does not develop from it a general theory of zero as relation or process.

25Another step is documented in Indian mathematical literature. In the Brāhmasphuṭasiddhānta (628 CE), especially in chapter XVIII, verses 30–35, Brahmagupta treats śūnya in rules for addition, subtraction, multiplication, and division. Equal positive and negative magnitudes yield śūnya; the sum with śūnya leaves positive or negative magnitudes unchanged; the product with śūnya is also regulated. Here zero is not only a placeholder, but a value in whose calculation further operations are carried out. Brahmagupta's rule for zero divided by zero does not correspond to today's convention, which leaves the expression undefined. This too makes visible that “zero” does not possess a meaning that is identical across history.[126]

26These stations do not yield a simple line of transmission, “Greek origin → today's zero.” The Greek astronomical zero notation, the Indian positional notation, and Brahmagupta's arithmetical rules must first be treated as different practices. The paths by which astronomical and computational procedures were mediated between India, the Arabic-speaking world, and Latin Europe are historically important; but a direct, gapless origin of the Indian calculating zero from Ptolemy's sign is not thereby proved. The usual history is therefore not that Greeks invented a relational zero that was later forgotten. Rather, it shows that number signs, placeholders, and calculational values were connected with one another in different ways in different cultures.[127]

27Even “the zero of today” does not have only one reading. As a number, 0 can designate the additive identity: for every appropriate a, a + 0 = a. In a place-value system, the digit 0 marks an empty rank whose position nevertheless changes the value of the other digits. In a coordinate description, 0 can mark the chosen origin of an axis. In a set-theoretical representation of natural numbers, one can, as with von Neumann, identify 0 with the empty set ∅; 1 then becomes {∅}, and further numbers arise through the successor construction. These meanings are connected, but they are not the same definition.

28Above all, the empty set is not simply “nothing.” In the language of set theory it is a determinate object: a set that contains no element. Its role is determined by the relations of set theory; as an element of another set, it can itself occur in further constructions. Likewise, additive zero does not necessarily mean the absence of an object. It is a value with a determinate calculational function. The claim that modern mathematics treats zero as “nothing” therefore captures at most one particular intuitive interpretation — for example, “no apples” — and not mathematical use as a whole.[123]

29Zero is thus suitable as a touchstone, but precisely not as a smooth confirmation of the thesis. What can be demonstrated historically is a sequence of functional shifts: a sign for an empty or zero-valued table position; a calculational value that enters rules as an operand; finally, a formal representation as the empty set within an axiomatized number structure. The relational conclusion of this book is: a determination that is used within a notation or operation can appear in a later theory as an identifiable element. But the final step does not necessarily erase its relation. In von Neumann, zero is constructed precisely through set formation, the successor relation, and its position at the beginning of the ordinal series. Only when the object so determined is presupposed in later use without also reading its mode of construction does the question of reification arise.

30For that reason, the more precise thesis is not: “The Greeks saw zero as process; we see it as nothing.” The ancient sources do not support this opposition. More precise and historically more productive is the question: When does a sign that makes a missing place readable become an operable number value — and when is this value posited in a later systematics as an autonomous object? Zero then does not show a single, linear drift, but several modes of determination that can converge under the same sign: origin of a chosen scale, empty place, result of calculation, additive identity, and empty set. Precisely their shared notation can conceal differences of genesis that must be reconstructed separately in historical terms.

31The bone carver remains the early reading anchor of this volume: strokes are carried out, preserved, and read again as countable traces. The sources on Dedekind, however, document no direct historical transmission from the incised bone to modern set theory. The relation here is initially a relational reconstruction and a comparison of practices. In both cases, a trace or assignment can make a multiplicity available without every later use repeating the first enactment. The bone shows a material operation; Dedekind's text reflects a logical capacity for assignment and transfers it into an abstract number structure. The structural analogy is productive for the question, but a gapless genealogy would be unproved.

32This also makes it possible to specify the passage on Gödel more precisely. Gödel certainly proves a proposition that is not derivable within a certain formal system; relative to this system, his result refutes its completeness. But he does not provide an alternative mathematics outside axiomatization, and he does not lead calculation back to the act of counting or notching. His coding makes syntactic relations arithmetically workable: a number is assigned to a formula and to a proof, and the assignment is chosen so that the encoded sequence of signs can be reconstructed. Here the number functions as the code of a relation and of an enactment, not as their historical explanation. Gödel's proof shows what can be said within and about such a system; it proves neither that the abstract four is false nor that the natural number arose from a “false thought.”[128]

33As a relational reading, this investigation can retain this: in formal use, 4 can be the starting point of further operations without fourfold counting or engraving being repeated. But the role of four in the act of counting, in Euclid's definition of multiplicity, and in von Neumann's set representation is not automatically identical. The relation of emergence can remain known and reconstructible, as in Dedekind, while a concrete operation does not perform it again. Here “forgetting” initially designates only a possible non-actualization in the individual step, not a secured loss of meaning. Whether a new systemic role is read back as the original ontology of number is an additional historical and philosophical question.

16.6 Kant: Number, Categories, and the Unity of Self-Consciousness

34The question concerning the origin of a determination therefore leads to Kant. He is important for this section not because he founded set theory. He did not. Rather, what matters is that Kant investigates the conditions under which a manifold can be cognized as a determinate object and a sequence as number. His categories are not merely a subsequent collection of drawers for things that already stand fully fixed. They are pure concepts of the understanding through which the manifold of intuition is brought into rules of unity. Under the category of quantity, Kant orders unity, plurality, and totality; under relation, he orders inherence and subsistence, causality and dependence, and community or reciprocity. This order therefore also describes how determinations can belong together in an object.[129]

35For number, the text is still closer to the guiding question. In the transcendental deduction, Kant determines the pure schema of quantity as number: it is a representation that comprises the successive addition of homogeneous units to one another. For Kant, number is “unity of the synthesis of the manifold” in a homogeneous intuition, insofar as time is traversed in the apprehension of the manifold. The result of a counting can consequently be used as a determinate number; but Kant explicitly binds its possibility to an enactment of gathering-together. The four is readable because units are added one after another and apprehended in a unity of synthesis. Kant thereby supplies a conceptual anchor for the distinction this investigation pursues: between the production of a determination and its later use as a finished starting point.[130]

36Kant's “I think” is also not simply an empirical thing that lies ready behind the representations as their bearer. In Kant, the transcendental unity of apperception designates the condition under which different representations can be connected in one consciousness and determined as cognition of an object; it must be distinguished from the empirical self as an object of inner perception. The relational question is directed toward the status this condition receives in the argument. Kant lays the synthesis open and at the same time determines the unity of apperception as a necessary condition of possible cognition. This theoretical role is not simply the continued meaning of a single enactment of counting: it belongs to a transcendental argument about the possibility of objective experience. Here this investigation can examine a structural analogy to the number four, but it cannot claim that the two unities are the same determination. The question is whether a unity analyzed in enactment receives, in the later grounding, a new role as presupposition, and whether this role is reified in Kant himself or only in later readings. Kant does not conceive the I as an empirical substance.[131]

37In Hilbert, a direct textual reference can be demonstrated. The Foundations of Geometry of 1899 begins with Kant's sentence that all human cognition begins with intuitions, proceeds from there to concepts, and ends with ideas. Hilbert calls the choice of axioms and the investigation of their relations “the logical analysis of our intuition of space”; the axiom groups each expressed connected basic facts of this intuition. This is more than a loose similarity: Kant stands in the printed text as an explicitly cited point of reference. At the same time, Hilbert describes geometry as a system of points, straight lines, and planes whose geometrical relations are determined by axioms. In his letter to Frege from the same year, he explains that a theory forms a framework of concepts and necessary relations; the basic elements can be thought in different ways as long as the relations and axioms are preserved. Hilbert's axiomatic elements are therefore not simply isolated things with independently fixed identity. They are characterized as positions in a relational structure.[132]

38Hilbert's quotation establishes a concrete connection to Kant, but it does not by itself close a circle of meaning or grounding. Hilbert uses Kant's sentence to situate his own investigation of spatial intuition and the choice of axioms; the axioms subsequently receive a function in a formal relational structure. Kantian synthesis and Hilbert's axiom system are therefore not two occurrences of the same determination. The relational reading can compare how both texts explicate presuppositions and which inferential roles belong to them. A back-projection would occur only if Hilbert, or a later reader, presented Hilbert's axiomatic meaning as that which Kant had already meant by synthesis. The direct reference is documented; this stronger equation is not.

39In Hilbert's axiomatics too, points, straight lines, and planes are not simply isolated things with a completely pre-established identity. They are determined through their relations and the axioms. Nevertheless, an individual proof can operate with them without repeating the intuitive enactment of determination. The objection of the book is therefore not directed against the mathematical productivity of axiomatics, nor against every abstraction. It is directed against the possible reversal in which the once stabilized rule counts as a presupposed order and its production is no longer carried along.

40Riemann cannot simply be classified as Kant's opposite pole. In his habilitation lecture, he asks which assumptions about space underlie geometry and to what extent they are confirmed by experience. More recent research argues that his geometry does not necessarily reject Kant's pure intuition. The relation therefore remains a philosophical matter of dispute; what is securely documented is Riemann's explicit investigation of geometrical presuppositions and their empirical validity, not a simple replacement of Kant by a reified mathematics.[133]

41For Gödel too, three levels must be kept apart. His incompleteness theorems concern precisely formalized systems and their provability; they do not prove that mathematics has lost its origin in an enactment. Gödel later explicitly concerned himself with the relation between relativity theory and Kantian philosophy, as two unpublished manuscripts from the years 1946/49 show. This documents his interest in Kant, but not a Kantian origin of his incompleteness proof. Historically closer to the proof are the formal systems and the Hilbert program. Kant is an important conceptual background here for questions concerning synthesis, unity, and object; Gödel makes visible a determinate limit of formalization. The two must not be shortened into a straight line of influence.

42The relational reading of this investigation can now begin more precisely. Abstraction or operative reuse is not already reification. With a number such as four, one must ask which relations carry its determinacy in a given context: unity, counting rule, successor order, or set-theoretical construction. If the number is restabilized in a new system by further rules, it is not self-evident that its meaning there remains identical with its meaning in the act of counting. The critical movement would lie in reading the new role as unchanged original meaning and thereby retrospectively explaining the earlier practice. The mere fact that four can be used as an operand without renewed counting does not document this transition.

43Kant and Hilbert therefore stand in a documented textual relation, but not necessarily in an identical relational role. Kant describes synthesis and determines its unity as a condition of possible cognition; Hilbert quotes Kant while his geometry fixes concepts through axiom-relations and tests their consequences. In Hilbert's preface, the Kant sentence receives a new function: it frames the axiomatic analysis of spatial intuition. From this it follows neither that Hilbert's axioms continue Kant's synthesis with identical meaning, nor that Kant already contains Hilbert's formal systematics. The structural similarity is a relational interpretation; the direct reference to Kant is textually documented. Neither from this nor from Gödel does there follow a gapless genealogy “Kant → set theory → Gödel.” It remains to be examined whether later readers back-project Hilbert's axiomatic role onto Kant's concept of synthesis, or whether they explicitly preserve the difference.

16.7 Göttingen and the Atom: Return Flows between Mathematics and Physics

44The preceding chapters have followed several stabilizations: a construction becomes a theorem, a ratio becomes a calculable magnitude, a mathematical form becomes the object of further operations. For the next step, however, the direction mathematics → physics is not sufficient. A physical question can select and change mathematical structures; the mathematical representation can then produce new physical questions and objects. Whether Göttingen around 1900 forms a special condensation of such return flows must therefore show itself in concrete transitions. As a provisional search map: a physical relation is grasped mathematically; mathematics gives it a new position in a formal system; this position is read back physically and can determine the original relation anew. The name or the sign may remain the same; the role need not.

45Riemann's Göttingen habilitation lecture of 1854 is a necessary beginning, but not evidence that geometry became relational mathematics only through physics. Riemann asks about the general concepts of multiply extended magnitudes and about which presuppositions underlie the metric relations of space. He distinguishes relations of extension or domain from metric relations: the same kind of spatial extension can admit different metric determinations. For continuous magnitudes he names measurement as comparison by superposition and by a transportable measure; where this is absent, only weaker comparative statements remain. Only in the application section does he formulate possible assumptions about space and explicitly ask to what extent experience guarantees them. Geometric determinations in this text are therefore neither merely free stipulations nor direct readings from nature: mathematical relations are systematically separated, while the metric presuppositions of physical space can be empirically determined only imprecisely.[113] [134]

46The later mathematical tools for the theory of relativity also did not arise as a response to Einstein's theory. Between Riemann and 1915 there were independent works on differential geometry and tensor calculus, including contributions by Christoffel, Ricci-Curbastro, and Levi-Civita. These procedures became relevant for Einstein when the physical problem of gravitation entered into a general problem of coordinates and covariance. In the Zurich Notebook of 1912/13, Einstein and Marcel Grossmann searched for field equations; Grossmann pointed Einstein to the mathematical literature. Einstein examined geometric candidates, including candidates connected with the Ricci tensor, and rejected or changed them not for purely algebraic reasons, but because he found their physical scope, covariance, and connection to known limiting cases problematic. Mathematical possibility therefore did not by itself determine the physical theory; conversely, physical requirements restricted which mathematical forms Einstein initially accepted.[135]

47For Göttingen culture, what mattered here was not only a single mathematical theorem, but the shared work on physical theories. In the summer of 1905, Hilbert and Minkowski, together with the physicist Emil Wiechert and the mathematician Gustav Herglotz, led a seminar on electron theory. The participants read Hertz, Lorentz, Poincaré, and Abraham, among others. Hilbert's lectures and seminars on mechanics, continuum mechanics, and electrodynamics continued this engagement; the archive correspondingly records physical topics in his teaching and research activity. These sources document institutional proximity and repeated shared work, not an overall program led by Hilbert that already anticipated the later theories.[136] [137]

48Minkowski's work makes the reciprocal movement especially clear. In the Grundgleichungen published in 1908, he distinguishes between a mathematical invariance statement about Lorentz transformations, the relativity postulate arising from it, and the physical principle once covariance is asserted as a determinate connection among observable magnitudes. He thus does not simply confront nature with a finished four-dimensional geometry. A mathematical property of the equations is first bound to physical expectations; the physical theory then gives it the status of a principle. In Space and Time, Minkowski describes how a reference system can be derived from natural phenomena without being uniquely fixed by them, and how the laws can be expressed unchanged under certain transformations. His pointed conclusion that three-dimensional geometry becomes a chapter of four-dimensional physics marks a new role for geometry: space and time coordinates no longer appear as separate, independently given descriptive frames, but as parts of a shared mathematical-physical structure. At the same time, Minkowski introduces the speed of light, or the ratio of electromagnetic units, into this structure as a physically determining parameter.[138]

49Here the E5 test is already visible. Lorentz invariance as a mathematical relation arises in a context interwoven with electrodynamics and experimental physics. In the spacetime representation it becomes an organizing principle by which further physical magnitudes—such as proper time along a worldline—can be determined. This new role can act back on earlier electrodynamics: the theory can now be read as part of a four-dimensional structure. This is a productive reconstruction, not a demonstrated confusion of result and origin. Minkowski himself marks transitions and distinguishes mathematical theorem, postulate, and physical principle; in doing so, he also provides a counterexample to the assumption that every systematization makes its relational steps invisible.

50Einstein was not a Göttingen scholar in 1915, but the connection between Göttingen mathematics and physics and his work is documented. At Hilbert's invitation he gave six lectures in Göttingen at the end of June and the beginning of July on his gravitational theory at that time. In his letters he reported that the conversations had clarified his theory and convinced Hilbert and Felix Klein. This encounter belongs to the prehistory of the intense exchange in November 1915; by itself it proves neither a one-sided adoption by Hilbert nor the reverse. Einstein's own return to generally covariant field equations proceeded step by step and was determined by mathematical as well as physical considerations. Hilbert's first communication on the Foundations of Physics, submitted to the meeting of 20 November 1915, places gravitation and Mie's electromagnetic theory of matter within a shared variational and axiomatic program. The surviving proofs and later versions differ; the priority debate therefore cannot seriously be resolved into a simple story in which one of the two merely takes over the other's finished theory. For our question, the more important point is this: Hilbert grasps physical magnitudes and laws within a mathematical variational context, while Einstein's physical requirements help determine the selection and interpretation of the geometric relations.[139] [140]

51Even “geometrization of physics” would be too crude as a description. Einstein's field equations represent the gravitational field by the metric tensor; the metric thereby receives a physical role and can no longer be treated only as a mathematical measuring apparatus. Yet Einstein repeatedly objected to the claim that physics had thereby been reduced to geometry or that matter had already been completely explained geometrically. Riemannian geometry is therefore physically restabilized in relativity, but it is not simply taken over from Riemann with the same meaning: Riemann's question about possible metric presuppositions is translated into a theory in which the metric field itself is dynamic and linked with matter. The later physical role of gμν must not be read back retrospectively as though it had already been Riemann's original determination.[141]

52The question of when a mathematical determination becomes an object of the physical world emerges still more sharply with atoms and gases. In 1900 Hilbert's sixth problem had demanded the mathematical elaboration of those limiting processes that lead from an atomistic description to the laws of motion of the continuum. This formulation does not simply presuppose that atoms are already present as immediately observed things: it names the transition between two levels of description as an open physical-mathematical problem.[142]

53The decisive conceptual neighbor stands in the same Hilbert text. A few pages before the sixth problem, Hilbert describes the continuum of real numbers as “a system of things” whose relations are regulated by axioms; in the same section he relates its existence to the system of rational numbers and to Cantor's number classes. Problem 6 then demands that the kinetic theory of gases be developed mathematically and that the limiting processes from the atomistic to the laws of motion of continua be rigorously grounded. This is a remarkable textual compression: the same lecture places mathematical number systems and physical models of matter under the more broadly framed program of ordering their objects and relations mathematically in explicit form. The two passages document a proximity of the research program, but not a shared formal axiomatization of number theory and gas theory.[143]

54This proves neither an explicit equation “atom = number” nor that Hilbert derives gas from set theory. The break meant here is also not that a mathematical model is confused with a physical thing already fully determined alongside it. The stronger relational thesis begins with the direction of scientific determination: the physical thing itself is grasped mathematically within the procedure. In the kinetic theory of gases, a particle counts as that whose position, motion, collision, and statistical distribution can be determined in the mathematical relations. The mathematical procedure then does not merely represent an independently fixed atom; it helps fix what counts in this theory as an atom and as the physical state of a gas. In the evidential chains of this period, the individual atom is not seen immediately. Observable effects support a particle theory; at the same time, the inferred particles are counted, ordered according to states, and calculated in regions of space or velocity. The equation does not stand in the text as the doctrine “physical thing = mathematics.” It is read here as the enactment of the theory: what is physically identified as a thing is precisely what can be determined and further treated as a particle within the mathematical order. This reading is a relational interpretation of the practice, not a statement that Hilbert or Boltzmann explicitly make about their own method.

55In this limited sense, set theory is not proven to be an ontology imposed after the fact. What becomes visible is a shared formal readability: Hilbert describes mathematical objects as ordered by axioms; Boltzmann's procedure treats physically inferred molecules as countable and statistically determinable particles. The relational interpretation does not begin between a finished model and a finished thing separate from it. It asks whether mathematical determination itself becomes that by which what the physical thing is gets decided. If a particle appears in the theory precisely as a localizable, moving, and countable unit, then the thing is not subsequently confused with an external representation; it is identified as this mathematically determined particle. Reification here would be the transition in which the mathematical determinacy gained in this way is carried forward as a property of the physical thing itself, while the enactment of determination no longer has to be read along with it. The sources document the close practical coupling and the physical use of the mathematical ordering of particles. That the authors themselves understood this coupling as an equation is not documented; the thesis concerns the implicit reading direction of the practice and must be tested in its concrete applications.[144]

56A concrete case is the molecular-kinetic interpretation of Brownian motion. In 1905 Einstein derived from molecular theory predictions about the motion of microscopically visible particles suspended in liquid. He explicitly formulated the test in both directions: observable agreement would make it possible to determine atomic size; failure would count against the molecular-kinetic theory. Perrin later investigated the visible colloidal particles, their distribution, and their motion. He therefore did not directly measure a single atom. Through mathematical relations among observable motions, temperature, particle size, and molecular magnitudes, he obtained values for Avogadro's number; the agreement among different procedures was assessed as strong but mediated evidence for molecules. Perrin himself called it indirect evidence in his Nobel lecture.[145]

57Here the movement under investigation becomes historically more precise. The mathematical relations of molecular theory presuppose distributions of visible particles that can be observed and measured; these measurements make molecular magnitudes determinable. The agreement among different procedures helps establish molecules as physical objects with which further work is then done in the kinetic theory of gases and in further calculations. The first connection does not disappear from science: it remains reconstructible as a theory of measurement. But in a subsequent calculation, the entire derivation from Brownian motion need not be performed again; molecular number, mass, or velocity can already be used as determinations of the gas. Precisely here lies the possible drift of meaning: the molecule is first identified within the theoretical-experimental context as the mathematically determinable physical particle; in the next calculation, this particle appears as a presupposed thing that bears further properties and relations. Physical objecthood is therefore co-constituted by mathematical determination and then used as the starting basis of further operations. In this return, the role of what “molecule” means changes.

58The historical boundary remains essential. Neither mathematics alone explains that molecules are real, nor does a measurement metaphysically make them into things. What can be documented is a change in their scientific status: from explanatory postulate, through quantitatively testable predictions, to a physically assumed object of further theory. Nor is “the atom” a single, unchanged E5; molecular size, Avogadro's number, microscopic particle, and atomistic unit have different roles. For that very reason, the back-effect must be traced separately at each determination. The reification thesis here would not be that measurement produces a thing from nothing, but that the success of a theory-guided identification makes it possible to use what has been identified in later calculations as a starting object, while the enactment of its identification is no longer constitutively carried along every time.

59A second, especially clearly documented feedback arises in Göttingen between 1925 and 1927. Heisenberg's July paper started from a physical problem: classical orbital quantities of the atom were not directly observable, while spectral lines and transitions supplied observable information. He therefore sought a new kinematics based on relations among, in principle, observable magnitudes. The computational objects that he introduced for transitions between states were not initially imported as finished matrices from an abstract theory. Born recognized that Heisenberg's multiplication rule corresponded to matrix multiplication. Together with Jordan, he developed from this a systematic theory: magnitudes were represented by arrays with state indices, whose product is summed over intermediate indices; the canonical quantum condition received a matrix form, and from this, among other things, conservation of energy and Bohr's frequency condition were supposed to follow. In their own retrospective within the paper, Born and Jordan call the new theory a closed mathematical structure that nevertheless was supposed to take account of the special features of quantum phenomena.[146] [147]

60The chain of transition is unusually visible here: spectroscopic relations and the restriction to observable magnitudes first shape the mathematical objects; the mathematical reading as matrices makes a generalized multiplication rule available; matrix mechanics then produces new calculations and physical consequences, which are in turn checked against spectra and transitions. The matrices are not neutral containers into which already finished physical content would be packed unchanged. Their indices, multiplication, and noncommutativity carry a determinate organization of transitions. Conversely, the mathematical concept of the matrix is no longer merely what it was in abstract algebra: through its assignment to quantum-theoretical magnitudes and transitions, it receives a new physical function. M0 and M1, P0 and P1 are therefore not simply identical in meaning.

61The Göttingen share must nevertheless not be made the sole origin of quantum mechanics. Dirac developed in Cambridge, independently, a related but distinct algebra of q-numbers; Schrödinger formulated wave mechanics in Zurich in 1926. Jordan worked on links between these approaches, and Born’s probabilistic interpretation arose in exchange with further researchers. The Göttingen specificity does not lie in an exclusive invention, but in the close sequence of Heisenberg’s physical reinterpretation, Born’s and Jordan’s mathematical systematization, joint application, and subsequent teaching. Hilbert’s winter lecture course of 1926/27 was worked out with Nordheim’s help; the published text explicitly explains that essential parts of the mathematical execution came from John von Neumann. It places the new quantum theory alongside Schrödinger’s approach and Dirac’s independent work, and tries to ground the theory in general form. The formulation of the article therefore already differs from the local calculational practice from which Heisenberg’s matrix approach had emerged.[148]

62In the later mathematical stabilization, this role shifts again. Hilbert, von Neumann, and Nordheim arrange the new quantum-theoretical relations in a more general framework; von Neumann’s Mathematical Foundations of Quantum Mechanics of 1932 systematizes the use of states, operators, and Hilbert spaces. This presentation makes the theory more transferable and mathematically more precise. Yet it is neither a mere copy of Heisenberg’s first calculation nor proof that its entire physical meaning was already contained in 1925 in the later abstract framework. It restabilizes the early transition relations in a more comprehensive language. Whether this replaces, preserves, or sublates the original observation-related reading in a new theory must be examined separately for individual concepts — for example “observable,” state, operator, and measurement.

63Several movements from physics to mathematics and back are thereby documented. In relativity theory, electromagnetic invariance, the speed of light, coordinates, and metric are related to one another in changing roles; in quantum mechanics, observable transitions are translated into a multiplication structure that makes possible a new theory and further physical calculations. Göttingen makes these interactions institutionally available through seminars, lectures, joint publications, and the proximity of mathematics and physics. Hilbert’s sixth problem explicitly names the project of treating physical sciences axiomatically; however, it is neither a master plan that all participants followed nor the origin of relativity theory or quantum theory. Nor does the record show a single cycle that connects all the developments named into an unbroken chain. What is documented are several, partly connected feedback movements.

64Göttingen can therefore provisionally be described as a historical point of condensation, insofar as this means the unusually close coupling of mathematical systematization, physical problems, and institutional cooperation. A culmination in the sense of a final closure is not documented. The sources show countermovements: Riemann makes the empirical uncertainty of metric statements explicit; Minkowski distinguishes theorem, postulate, and physical principle; Einstein corrects mathematical decisions after physical objections; Heisenberg begins from observable relations; Hilbert and von Neumann mark that a later general presentation performs new conceptual work.

65The case of the atom and Brownian motion at the same time makes more precise what can be meant by “reification.” Here it is not documented that a measurement makes the atom immediately visible. What is documented is a chain in which mathematical relations determine observable motions, measurements of visible colloidal particles disclose molecular quantities, and the converging evidence stabilizes molecules as objects of further calculations. The break does not lie in later confusing a model with an independently given atom. Rather, the physical atom is identified as this thing through mathematical-experimental determination: as a unit whose size, number, and motion are fixed and reused in the relevant relations. In the same text, Hilbert’s lecture of 1900 brings together the axiomatic talk of the mathematical continuum as a “system of things” and the demand to work out mathematically the movement from the atomistic to the continuum. This is a strong historical indication of the close coupling between mathematical order and physical objecthood, but not proof of a direct derivation from set theory or of a literal identity of atom and number. Brownian motion remains a contemporary comparison here, not an exclusively Göttingen research line. Whether later physicists treat this mathematical determinability explicitly or practically as the atom itself, and whether they thereby reread the earlier evidence retroactively, is the next step to be documented.[137] [145] [143] [144]

66The material thus supports more than a one-sided history of foundations and applications: mathematics and physics mutually determine their further possibilities; in the atomistic case, the physically confirmed determination can also enter further calculations as a new object. But it does not support the claim that every such transition is necessarily reification. After each round one must ask which relations are preserved, which new ones are added, and what the next calculation already presupposes. Success in continuation shows the capacity of the new connection; it does not prove that its meaning or grounding has remained identical with the initial context. In this limited sense, Göttingen is a point of condensation of relational restabilization, and the atom case makes visible how a theory-guided determination can become a physical starting object for further operations.[137] [145]

16.7.1 The atom as a forensic guiding case: the same name, different determinations

67The atom case bundles the question of this chapter, because here the name persists over a long time while the modes of determination change profoundly. The following sequence is not a genealogy of an object that would have been present unchanged from Democritus to quantum mechanics. It traces points of connection at which older problems or results are taken up, and in each case asks what the “atom” means in the new order.

68For Democritus, the finding from chapter 7 first applies: ἄτομος means uncut or indivisible. The testimonies, above all Aristotle’s reports, attribute to Leucippus and Democritus full, indivisible bodies in the void; differences are described, among other things, by shape, arrangement, and position. The surviving sources contain neither electric charge nor elemental identity, atomic weight, nucleus, or spectral line. Our relational reading asks how a limit of division and a distinguishability of configurations are connected; it does not replace the textual finding of an atomistic body. The word “atom” therefore already carries two possible paths of reading: the historical concept of the indivisible and the later natural-scientific designation of a respectively more precisely determined constituent of matter.[8] [9]

69Gassendi, too, does not simply continue this gap. As section 15.1 showed, he reads Epicurus and Lucretius explicitly and takes from ancient atomism indivisible corpuscles and void. In his own natural philosophy, moles or magnitudo, figura, pondus, and soliditas are added as corporeal determinations; at the same time he distinguishes mathematical divisibility from physical matter and connects atomism with a theologically created natural order. The atom in Gassendi is thus neither merely the perspectival limit of our distinction nor the later particle of a chemical element. It is a physical, extended, and resistant body whose indivisibility and motion are grounded within an early modern natural philosophy.[62] [149]

Dalton: a chemical atom becomes determinable through weight ratios

70With Dalton, the leading question shifts. In A New System of Chemical Philosophy, the atom is not first determined by a cosmological theory of the void, but by the regular quantitative ratios in which substances react chemically. Dalton collects analyses, compares the weights of combined substances, and assigns relative weights to the assumed smallest particles. He states explicitly that the central tasks of the atomic system are to determine the relative weights of simple elements and to find the number of simple particles that are joined in compound particles. The experimental operation does not provide the weight of an isolated atom placed on a balance. It provides the weight ratios of reactants and products; from them, and from additional assumptions about the simplest numbers of combination, atomic weights and formulas are inferred.[150]

71The atom thereby becomes a computationally fruitful chemical unit. Its determination depends on analytical procedures, concepts of elements, and rules by which whole-number ratios are translated into formulas. Dalton also speaks of “compound atoms”: compound particles that we today would often call molecules. Thus “atom” here does not consistently mean the same thing as in the present distinction between atom and molecule. Even in Dalton, the word changes its range depending on whether the simple elemental particle or the chemically compound particle is at issue. The mathematical operation is not applied externally to a thing that had already been completely determined: the assumed object receives its chemical identity by carrying the constant mass ratios and numbers of combination that the procedure extracts from the reactions.

72The continuity of this result was contested and further determined. In 1811 Avogadro proposes that equal gas volumes under the same conditions contain equal numbers of “integral molecules.” This enables him to infer relative molecular masses from gas densities and compositions from volume ratios. At the same time, his text distinguishes the molecules of an elemental gas from the smaller elementary molecules of which they may consist: in a reaction, such compound gas molecules can divide. Avogadro explicitly criticizes Dalton and offers a different explanation of the volume relations.[151]

73A new relational order thereby arises: gas volume, density, temperature, pressure, and number of molecules determine one another within the hypothesis. In 1858 Cannizzaro makes the consistent application of Avogadro and Ampère a means of bringing chemical and physical weights together. He explains how molecular weights are obtained from densities, their composition from elemental analyses, and atomic weights from these results. In doing so, he states explicitly that in formulas the atom of an element in the free state may represent the whole molecular quantity or a part of it. This teaching for students shows that “atom” and “molecule” were not merely names found by observation in chemical systematization: their distinction was stabilized by measurement relations, formula use, and instructional rules. The unity of the name leads from Dalton to Cannizzaro, but the order of determination has changed.[151]

Thomson and Rutherford: from the chemical particle to internal structure

74At the end of the nineteenth century, the atom is determined differently once again. In “Cathode Rays” (1897), J. J. Thomson demonstrates electrostatic and magnetic deflections of the rays. From angles of deflection, field strengths, and radius of path he obtains the ratio of mass to charge, m/e. The value found is similar in different gases and with different cathodes; Thomson concludes that the rays consist of negatively charged material particles, which may be atoms, molecules, or still finer divisions. The experiment initially does not show a small body inside the atom. It establishes a relation between path, electric charge, and inertia, from which a common negative component is inferred.[152]

75In his mathematical investigation “On the Structure of the Atom” (1904), Thomson places this component into a proposed atomic structure: negatively charged corpuscles move in a uniformly positively electrified sphere. His calculation examines the equilibrium, stability, and periods of oscillation of rings or shells of such corpuscles. He connects the number and arrangement of the corpuscles with atomic weight, electrical valence, and periodic properties. Here a mathematical problem of stability becomes part of what describes a physical atom and distinguishes it as an elemental atom. The model is not only an image that illustrates an already independently determined interior world; the calculated equilibrium and oscillation relations determine which internal arrangements count as candidates for chemically different atoms. That Thomson treats these proposals as hypotheses and tests them against known properties remains equally part of the textual evidence.[152]

76Rutherford shifts the role of the atom again. Geiger and Marsden report diffuse reflection of alpha particles in 1909; in 1911 Rutherford compares measurements of deflection angles and scintillations with mathematical predictions for single and multiple scattering. His decisive alternative concerns the probability of large single deflections: it cannot readily be explained in Thomson’s model, but it becomes quantitatively intelligible with a very small central distribution of charge. Rutherford therefore assumes an atom with concentrated positive charge and spatially extended negative electricity. The nucleus is at first the physical determination that explains the scattering distribution under the assumed forces and geometries; it is not seen directly as a finished component.[153]

77Moseley’s X-ray spectra add a further relation. The frequencies of characteristic X-ray lines do not simply order elements by their mass, but approximately follow a lawlike relation to the whole number N. Moseley identifies this number with the number of positive electrical units in the nucleus. The atomic number thereby receives a role that newly orders atomic weight and chemical classification in relation to one another. The “place” of an element in the sequence is physically determined by a relation between spectral frequency and nuclear charge. The atom is here at once the bearer of a nuclear charge, the source of a characteristic spectrum, and an element whose chemical position is numerically fixed. These roles are connected, but they are not identical.[154]

Bohr: spectral lines determine states, states determine the atom

78In 1913 Bohr explicitly begins with Rutherford’s atom and its explanatory power for the scattering of alpha particles. He adopts the image of a small positive nucleus and orbiting electrons, but points out that classical electrodynamics makes this system unstable: an accelerated electron would have to radiate, lose energy, and fall into the nucleus. Bohr therefore posits stationary states and quantized transitions. The frequency of the emitted radiation is connected with the energy difference between two states; from these relations he derives the Balmer-Rydberg structure of the hydrogen spectrum. The orbital representation remains in his atomic model, but its physical meaning is changed by non-classical conditions.[155]

79Bohr’s atom is therefore neither merely Rutherford’s nucleus with ever more precise electron orbits nor a directly observed planetary body in miniature. It is a system whose possible states are selected by quantization rules and whose transitions become testable through frequency and energy relations. The observed spectral line is not only a property that is subsequently attributed to an already fully determined atom: it belongs to the relations through which Bohr’s stationary states are determined as states of this atom. Conversely, the theory lets a new reading of the spectrum arise from these states. Bohr thereby shows a concrete feedback from mathematical theory to atomic structure, without the earlier meaning of the Rutherford atom thereby continuing unchanged.

Heisenberg, Born, and Jordan: the atom as an order of transitions

80Heisenberg’s 1925 paper starts from a crisis of the old quantum theory. He states that the new quantum-theoretical mechanics should be based exclusively on relations between quantities that are in principle observable. Classical orbital quantities such as the position and period of revolution of an electron are especially problematic, because they appear in the theory as basic quantities even though they cannot be obtained from observations in the same way as spectral frequencies. In his calculation procedure, Heisenberg therefore does not simply replace a visible orbital image with a new invisible image. He orders quantities according to transitions between states: frequencies correspond to energy differences, and the associated amplitudes are connected by rules that do not allow them to be multiplied like ordinary numbers.[156]

81Born recognizes matrix multiplication in Heisenberg’s multiplication rule; Born and Jordan develop it further in 1925 into a systematic mechanics. The matrix elements carry state indices and represent transition quantities; product formation sums over intermediate states. The canonical commutation relation between position and momentum becomes a basic condition of this new computational order. The mathematics is selected and changed by the physical problem: spectral lines and transitions make a particular index structure relevant. The mathematics systematized in this way at the same time gives the physical problem new forms in which conservation of energy, frequency conditions, and further consequences can be calculated. The atom in this order is not necessarily a small mechanical body with a path fixed at every time. It is a quantum-mechanical system whose physical quantities are determined by states, transitions, eigenvalues, and non-commuting operations.[157]

82This is not proof that Heisenberg abolished every statement about atomic reality. His work begins with atomic spectra and aims to ground quantum-theoretical mechanics; Born and Jordan apply the new form to concrete atomic problems. At the same time, what the calculation immediately determines changes: in place of a continuous path that an electron traverses at every moment, discrete transition relations and algebraic quantities appear. Born’s later probabilistic interpretation of the wave function arises in another argumentative context, connected with Schrödinger’s wave mechanics. It must not be retrospectively equated with Heisenberg’s original matrix approach.[158]

83Hilbert and von Neumann belong to the subsequent mathematical systematization, but not as inventors of a single “quantum atom.” In 1928 Hilbert, Nordheim, and von Neumann attempt to formulate the foundations of quantum mechanics in a more general form; von Neumann’s Mathematische Grundlagen der Quantenmechanik of 1932 orders states in Hilbert spaces, physical quantities as operators, and measurement results within a general formal framework. This strengthens the transferability of the theory to many systems, including atoms. At the same time, the object of this formal representation is initially a quantum system in general, not the special structure of a particular atom. The mathematical systematics provides conditions for possible states and measured values; a concrete atomic interpretation still requires assignment to a physical structure and to experimental procedures.[159]

84Didactic transmission makes the shift visible without causing it by itself. Born’s Vorlesungen über Atommechanik of 1925 still systematizes the old quantum theory; the volume Elementare Quantenmechanik, published with Jordan in 1930, already teaches the new matrix mechanics. In Cannizzaro’s chemistry teaching, the weight unit of the hydrogen atom was an explicit means of making relative molecular and atomic weights understandable to students. The later textbooks, in turn, can presuppose Bohr orbits, spectral series, states, and operators as defined units. What was developed in a research paper as a difficult relation between experiment, formula, and physical interpretation can become, in teaching, the name, rule, and solution path of an exercise. This does not mean that the student necessarily learns a false atom; it means that a later relational determination becomes operationally available under the same name, while the earlier determination does not have to be reconstructed each time.[160]

Back-projection: when Democritus already “anticipated” the modern atom

85For the strongest form of our thesis, it is not enough that the concept of the atom changes, or that later calculations use earlier results. What is at issue is the reading back of a later role into an earlier source. There are concrete examples of this, but they are not universal. The 1911 Encyclopædia Britannica entry “Atom” begins with the meaning of the Greek word “indivisible” and defines it as the physically ultimately indivisible particle of matter; it then describes Democritus and Epicurus as if they had taught the granularity of matter in this scientific sense. A present-day overview by the Institute of Physics states even more explicitly that Democritus was “right”, that atoms are the basic units of matter, before it contrasts the ancient picture with a modern correction through empty space. In this account, the later scientific determination becomes the standard for what Democritus allegedly already meant.[161]

86This is a demonstrable case of didactic and popular-scientific back-projection, not proof that every history of science modernizes Democritus. The ancient term ἄτομος denotes uncut, indivisible bodies within a doctrine of fullness, void, and motion. Daltonian elemental weights, Thomsonian corpuscles, Rutherford’s nuclear charge, Bohr’s states, and Heisenberg’s transition matrices do not belong to this determination. The common family resemblance consists in the fact that each case asks for a unit that is not further reduced, or that is theoretically basic. This formal similarity does not establish identity of meaning. When a later introduction says that Democritus already “guessed” the modern atom, this difference is precisely what is concealed by the continuing name.

87The finding can now be separated into four levels. First, experimental relations are measured: mass ratios, deflections, scattering angles, or frequencies. Second, these relations are mathematically ordered: as relative weights, m/e, scattering distributions, frequency laws, or matrix products. Third, a theory interprets what this order says about a physical atom: chemical particle, association of corpuscles, nuclear atom, structure of stationary states, or quantum system. Fourth, the textbook and the popular narrative can present this interpretation as a thing already given, and can even carry it back into the ancient past. None of these levels follows automatically from the preceding one. In particular, a mathematical representation of measurement relations is not already, by itself, the ontological statement that the atom is “nothing but” mathematics.

88The relational thesis is narrower and at the same time more radical: in each theory, the physical atom is determined as this atom by the concepts, measurements, relations, and permissible operations of that theory. When the unit produced in this way then serves as the starting point for further calculations, its role changes. The Dalton atom of the chemical formula is not automatically the Rutherford atom of scattering; the Rutherford atom is not automatically Bohr’s system of stationary states; and the atom in matrix mechanics is not merely a more exact drawing of the same classical orbital body. Some earlier determinations remain as restrictions or approximations; others are replaced, reinterpreted, or limited to new domains of objects.

89The strongest evidence for relational closure therefore does not lie in the fact that modern science successfully calculates atoms or confirms them experimentally. It lies where the later meaning of the name “atom” is presented as the original meaning of the ancient source. The 1911 encyclopedia and present-day popular mediation provide tangible examples of this. By contrast, the primary texts also show counter-movements: Avogadro criticizes Dalton, Bohr names the breaks between classical mechanics and quantum assumptions, Heisenberg explicitly discloses his reinterpretation, and the Göttingen mathematical systematization distinguishes general formalization from concrete physical interpretation. The case therefore does not prove a seamless history of reification. It shows precisely how a name is carried through different relational orders, and how a later determination can read the earlier one retroactively as if it had been the same from the beginning.

90Thus the question of this section is not whether atoms exist. It is: which atom is produced in which theory by which relations, and posited as a physical thing? The physical evidence and the mathematical procedures are not interchangeable “containers”, but components of this enactment of determination. The atom that appears as a presupposition in a later calculation can be well founded scientifically; yet its later usability does not prove that its meaning is identical with that of the earlier determination. Success in continuation is not proof of origin. Only the concrete reading back—for example, when Democritus’s ἄτομος becomes the precursor of the Bohr atom or the quantum atom—makes the strong closure visible.

91The sequence can be condensed as a relational concept map. It does not mark a seamless transmission and does not presuppose identity of the objects:

92The map makes visible precisely what the name alone conceals: each line connects different kinds of evidence and operations, and no transition is proved by mere chronological sequence. The strongest back-projection is found here not in the specialist scientific primary texts, but in selected encyclopedic and popular accounts that present Democritus as someone who had already correctly guessed the modern atom.[161]

Textbook and exercise: how the atom becomes a determinable task

93A present-day textbook shows the next transition not as a mere shortening, but as a new order of learning. Chemistry 2e by OpenStax first introduces, in one section, Thomson’s deflection experiment and the charge-to-mass ratio determined from it. In the following section it defines the atomic number as the number of protons and declares this to be the “defining trait” of the element: it determines the identity of the atom. The mass number is introduced as the sum of protons and neutrons; charge results from the difference between protons and electrons. Learners then work through examples and exercises in which they calculate particle numbers from atomic number, mass number, and charge, or use a simulator to assemble protons, neutrons, and electrons into an atom.[162]

94Here, an earlier finding of physics and chemistry becomes a didactic sequence of operations: given numbers → symbol of the isotope → calculated composition → classification as element, ion, or stable or unstable atom. For each individual exercise, the student does not have to derive Moseley’s spectra or the development of nuclear physics again. In this task-space, “understanding the atom” means, among other things, correctly applying the rules of assignment. This can stabilize the meaning of the atom for the learner in a new way: as a composition identifiable by Z, A, and charge. Yet the textbook also preserves part of the genesis, explains Thomson and Rutherford, and offers simulations. Its presentation is therefore not evidence that origin and experiment have disappeared; it shows how an historically produced object enters, in teaching, into a manageable order of calculation and classification.[162]

95A study on physics teaching of the Bohr model makes the limit of this didactic re-stabilization visible. McKagan, Perkins, and Wieman report that learners without sufficient connections and comparisons between the atomic models often hold on to a Bohr-like conception; a curricularly integrated juxtaposition, by contrast, can lead them toward the quantum-mechanical description. This is not proof of a general psychological law, and still less an ontology of the atom. But it is empirical evidence that the teaching sequence does not stand neutrally beside the models: which relations between them are taught influences which determination learners treat as still valid.[163]

96The didactic finding therefore remains double. Exercise and symbolization make atomic determinations transferable and allow further work with them. At the same time, they can make a particular later role—such as “element identity = number of protons”—available as an immediate rule. Whether learners thereby learn only an efficient definition for chemical tasks or also read the atom itself as an independently given numerical composition cannot be concluded from textbook exercises alone. This would require classroom observations, student texts, and comparisons across several textbook epochs. The didactic re-stabilization is documented; a comprehensive thesis of reification remains open beyond that.

16.8 Codification and self-reference: when a result receives a new role

97After the Göttingen feedback loops between mathematical form and physical determination, the following cross-check returns to an earlier station of notation. It asks how equations and codes can become objects of further operations, and what of their original role remains in the process.

98Descartes’ Géométrie makes visible how a geometrical relation is transferred into a calculable form. A figure is not simply replaced by a modern coordinate system; Descartes develops procedures for expressing geometrical problems through dependencies of sought and given lines and for casting these in equations. The equation thereby receives a new role: it can be transformed algebraically without each transformation repeating the original construction. The sign relation remains connected with the geometrical task, but the operations on the expression now also determine what is treated, in the further calculation, as the same relation. The equation in the geometrical starting problem and the equation as operand of an algebraic transformation must therefore not be posited as identical in meaning merely because they have the same written form.[164]

99With Euler, the function becomes a central object of analysis. A dependency between variable quantities can be designated as an analytic expression, transformed, and inserted into further calculations. This analytic role is not automatically the same as the geometrical or physical relation from which an expression was obtained. In the case of sine and cosine, Euler shows both directions: he develops series and relations of the functions analytically and relates them to quantities that arise from the circle. The analytically workable function can then be used to determine geometrical quantities. The geometrical reapplication is therefore not a mere repetition of the origin, but takes place through the analysis that has meanwhile been expanded. This is a mathematical achievement; whether it is also a back-projection onto the earlier geometry must be examined in each case in relation to the proof and to the language used.[108] [109] [110]

100Hilbert’s sixth problem shows that such mathematical forms in Göttingen were not merely applied to an independently complete physics. In 1900 Hilbert calls for probability and mechanics to be treated axiomatically; at the same time, the method of mean values in mathematical physics and especially the kinetic theory of gases are to be developed rigorously. He also explicitly includes the limiting processes that lead from an atomistic conception to the laws of motion of the continuum. The transition between model descriptions is thereby itself a mathematical research question. Physical problems shape which mathematical relations are developed; mathematical systematization, in turn, shapes how these problems can be formulated and further worked on.[142]

101The Göttingen finding from §16.7 is important here as a distinction: Hilbert’s sixth problem of 1900 concerns the axiomatic treatment of the physical sciences. The foundational program worked out in the 1920s, by contrast, aims at the formalization of mathematical theories and at a finitary consistency proof. Both projects stand within Hilbert’s broader interest in axioms, rules of derivation, and the carrying capacity of formal systems, but they are not the same research project.[115]

102Gödel begins precisely from the amenability of such systems to formal treatment. In his 1931 paper he arithmetically codes signs, formulas, and proof relations. The system can thereby express statements about its own formulas and their provability. For sufficiently expressive, recursively describable systems with the required consistency conditions, Gödel shows that there are statements that can be neither proved nor refuted within the system; his second incompleteness theorem also limits the internal proof of the system’s own consistency.[116] [117] [118] [128]

103Gödel therefore does not simply say that mathematics cannot “explain itself.” His theorems concern precise limits of formal provability and do not prove that these limits were caused by classroom instruction or by forgetting the history of a concept. For the relational question, the decisive point is that a number that initially codes a sign or a formula receives a new role in the proof: it stands in arithmetically definable relations to other codes and represents syntactic properties. The formalization explicitly preserves the decoding rules, and thus part of its back-reference. At the same time, the proof operates with the code number within a new number system. The structural analogy to a didactic sequence A–B–C–D is therefore limited: it draws attention to presupposed intermediate steps, but it proves neither loss of meaning nor an identical historical dynamic.

104This also marks the limit of the thesis. Codification, mathematical operationalization, and teaching can make results available as presuppositions. But a result that had one role in R0 can be determined differently later in R1; its reuse does not secure identity of meaning. The following chapter therefore examines not only whether the derivation is shortened, but also which new relations definition, example, exercise, and examination give to the object of learning, and whether later readers back-project this didactically re-stabilized role onto the earlier scientific determination.

16.9 Publication as a form of transferability: from Gassendi through Newton to Hilbert

105The three stations do not lead to a straight chain Gassendi → Newton → Hilbert. Rather, they show that different forms of publication make different parts of an enactment of determination visible and bring other parts into a transferable form. The question whether early modern publications are reifications must therefore not be answered by claiming that their authors did not acknowledge their predecessors. In the texts examined here, sources and predecessors are often explicitly named. Instead, bibliographical recognition must be distinguished from relational recognition: a text can cite its origin without preserving, in every later use, the relations through which its concepts and results were determined.

106Gassendi’s publication history first shows that print does not transport an unchanged authorial will. His Animadversiones of 1649 translate and comment on Diogenes Laertius’ tenth book; the ancient source remains present in the text and is examined in detail philologically. The more comprehensive Syntagma philosophicum appeared in 1658 after Gassendi’s death. Henri-Louis Habert de Montmor and friends collected and ordered manuscript material for it; the printed treatise now presents the transmitted ancient doctrines more systematically in logic, physics, and ethics, and examines them within the horizon of what Gassendi considered philosophically, historically, and theologically defensible. Bernier’s French adaptation of 1674–75 again shortened, reordered, and paraphrased parts of the work. The origin is therefore not simply erased: it is a supporting part of Gassendi’s method. Yet each publication stage reorients the same transmitted voices toward the questions and orders of its readers.[149]

107With Newton, the form of presentation changes. The Principia of 1687 order definitions, laws of motion, mathematical propositions, and corollaries in a synthetic architecture of proof. In the preface, Newton does not claim to begin without presuppositions: he formulates the procedure of determining forces of nature from phenomena of motion and explaining further phenomena of motion from these forces. The history of emergence—Halley, earlier calculations, Kepler’s and Galileo’s results, the revision of De motu—can also be historically reconstructed. In the printed work, however, the concepts quantitas materiae, vis, and motus can function as clearly defined inputs of mathematical operations. The proof does not have to pass again, at every corollary, through the whole path from observation, selection, and interpretation of the phenomena. This is the gain of the synthetic form. The possible drift arises only when a later reader takes the ordered presentation to be the complete emergence of the determinations and reads their definitional role as if it were the original constitution of the things themselves. Newton’s preface and scholia are, in this respect, important counter-evidence against the claim that he simply concealed this path.[165]

108Hilbert’s texts make the tension explicit. In the Foundations of Geometry of 1899, a system of axioms is meant to fix the geometrical relations in such a way that the meaning of the axiom groups and the reach of their consequences become clear. In the Mathematical Problems of 1900, Hilbert transforms research questions into a printed program for the coming decades. For the sixth problem he names Mach, Hertz, Boltzmann, and Volkmann; Boltzmann’s work prompts him to carry out rigorously, in mathematical form, the limiting transition from atomistic conception to continuum laws. Elsewhere in the lecture, Hilbert explicitly warns against interpreting mathematical rigor in such a way that concepts from geometry, mechanics, and physics, or the “influx of new material from the outside world,” would be cut off. New mathematical signs, Hilbert writes, should remind him of the phenomena that gave occasion to their formation. This is not a profession of forgetting the origin, but a programmatic attempt to hold systematization and back-reference together.[166]

109At the same time, with the scientific journal, what circulates as a transferable unit changes. The early Philosophical Transactions were at first a mobile medium for letters, reports, and communicated claims; only gradually did they become an archive of individual, clearly assigned contributions to knowledge. In the nineteenth century, specialist journals, catalogues, and bibliographical indexes strengthened the idea of the scientific article as a contribution attributable by name, original, and published serially. This development was neither linear nor fixed from the beginning. But it made individual results findable, citable, and connectable. A later article could take over an equation, a theorem, or a measured quantity as a known starting point and concentrate the new work on the next step.[167] [168]

110Here lies the narrower answer to the question of why such publications can be read as reification. Not because they appeared, in principle, without source references. Gassendi cites his ancient authors, Newton marks his methodological and mathematical predecessors, Hilbert names the physical works to which his program responds. A literature reference, however, initially preserves an address of origin; it does not automatically carry along the whole relation of determination that was effective in the earlier text. An author can be correctly named while their result receives a different function in the new contribution: evidence, definition, axiom, measured value, or presupposition of a calculation. The origin is then bibliographically recognized, but it is no longer constitutive for the current operational step.

111The category error at issue in this investigation does not arise from publication alone. Nor does it consist in mistakenly taking a mathematical unit beside an independently complete physical thing to be the same thing. The stronger thesis is this: in scientific practice, the physical thing itself is posited as a mathematically determined thing. The result of a measurement, calculation, or model formation establishes what is henceforth to count as the object; mathematical determinability is not merely its external description, but becomes the criterion of its physical identity in the system. Later work can then use this thing as already given. The relations through which it was determined in the earlier investigation are not necessarily denied; yet the new system role can displace them from what currently counts as the thing and replace them with other relations.

112The comparison therefore carries a narrower thesis than “modernity builds on predecessors without acknowledging their origin.” Historically more defensible is this: early modern publications develop forms in which determinations are documented together with their origin and at the same time made detachable from their original emergence. Reification is the possible conceptual consequence of this detachability when the later role is read as the original foundation of the object. Whether this inference actually occurs must be checked in each case by means of the later use. The publication form creates a possibility for relational drift of meaning; it does not prove it. The line Gassendi–Newton–Hilbert is therefore not a chain of increasing blindness, but a sequence of different procedures for connecting source knowledge, order of proof, and reusable results — each with its own back-bindings and its own abridgements.

16.10 Hilbert’s sixth problem: the return loop in the case of gas

113To explore this further, a single passage initially suffices. In 1900 Hilbert himself does not describe a one-sided relation in which mathematics merely maps an already finished physics. He sketches an “ever-repeating and changing interplay between thought and experience”: real phenomena press new questions upon us, open mathematical fields of knowledge, and can, once these have been taken up into the “realm of pure thought,” make answers to older problems possible and promote existing theories. The direction therefore leads from experience into mathematics and, from mathematical treatment, back to other questions of experience. This is an immediate textual finding for reciprocal retroaction, not only our metaphorical description.[166]

114Hilbert also explains how this transfer is supposed to occur. When concepts from geometry or natural science enter mathematics, mathematics is to examine their principles and fix the concepts for deductions through a complete system of axioms. At the same time, the newly formed signs are to recall the phenomena that gave occasion to the concepts. The result is a double demand: the concept must become precise enough within the mathematical system to carry consequences, and its origin in an extra-mathematical problem is not to disappear from view. Hilbert himself therefore formulates a possible counterforce to reification: formal determinability and recollecting back-binding are to work together.[166]

115Problem 6 makes this interaction concrete in the relation between atomistics and continuum. Hilbert takes Boltzmann’s Lectures on the Principles of Mechanics, published in 1897, as his starting point. According to Hilbert, Boltzmann had indicated limit processes from an atomistic conception to the laws of motion of the continuum; these transitions were now to be grounded in a strictly mathematical way. Hilbert then expressly adds the opposite direction: the laws of motion of rigid bodies, too, could be derived by limit processes from an axiom system for a substance continuously filling space. The “conversely” is decisive. Atomistic and continuous descriptions are not merely two names for the same object, and Hilbert does not say that one simply dissolves the other. He makes their possible reciprocal derivability and the question of the equal legitimacy of different axiom systems themselves into the mathematical-physical problem.[143] [166]

116The back-determination can therefore be formulated more precisely. The physical distinction between atomistic gas and continuum motion determines which mathematical limit processes are sought. The chosen axiom system in turn fixes which consequences can be drawn from the respective model and under which conditions two model descriptions count as transformable into one another. If a continuum result is then used again to describe a gas, its meaning has not simply been carried along unchanged from the atomistic starting point: it has meanwhile received a mathematically defined role as a limit result. Conversely, the atomistic also does not remain untouched when its physical task now consists in providing precisely those limit conditions that justify the transition to the continuum. In this way the relation between the models begins to co-form the determinations within both models.

117Hilbert’s own sequence of work offers a concrete second textual finding for this. In 1912 he published Foundations of the Kinetic Theory of Gases in the Mathematische Annalen; the essay at the same time designates itself as a reprint of the last chapter of his book on linear integral equations. He therefore brings the kinetic theory of gases into a mathematical context whose concepts and requirements of proof Hilbert develops in his theory of integral equations. In the preserved textual passage Hilbert criticizes Lorentz’s mode of inference: it essentially uses the unique determinacy of a solution without proving its existence; precisely the theory of linear integral equations is needed to provide this proof. The mathematical standard thereby changes what can count as a grounded solution of a physically relevant problem. At the same time, the kinetic theory of gases supplies the object on which the new analysis of integral equations receives its concrete task and its physical sense.[169]

118Here the return flow is not only programmatic, but visible in the object of publication itself: the mathematical theory of linear integral equations is transferred into a chapter on the kinetic theory of gases; conversely, its physical question makes the existence and determinacy of solutions stand out as central mathematical questions. This is stronger than “Hilbert applies axiomatics to physics.” The mathematical relation determines how the physical solution is justified; the physical problem determines which mathematical relations become relevant at all. Both systems change their task-setting and the role of their concepts in contact.[169] [170]

119The limit of this finding is just as important. Hilbert’s work of 1912 is not to be equated with a complete proof, rigorous for all cases, that the macroscopic laws of motion follow from atomistic theory. Hilbert developed a method for the approximate solution of the kinetic equation by means of series ansätze; later Chapman–Enskog procedures extended this line. The difference between a formal or asymptotic approach and a rigorous proof of the limiting transition remains essential. Recent research on the sixth problem shows that precisely the validity and limitation of such transitions themselves became an object of research. A mathematical result therefore cannot simply be read as proof that the physical determination on the microlevel and that on the macrolevel are identical in meaning.[171]

120What has thereby been gained concerning reification? Hilbert provides strong historical evidence that mathematics and physics determine one another reciprocally. His demand that mathematical signs should recall the phenomena that gave occasion to their formation shows that he explicitly wanted to secure the back-binding methodically. But this programmatic recollection does not remove the possible break. The mathematical determination can at the same time be the path by which a physical thing is identified at all: the gas-thing with which the theory calculates is the mathematically grasped particle in its states and relations. If this determination is later reused as the thing itself, then a model has not been confused with an independently given object; rather, physical objecthood has been fixed in mathematical determinability. Hilbert’s text documents the process of interaction, but not that he himself explicitly understood this identification in this way. Whether later applications treat the mathematically determined unit as the complete constitution of the physical thing and read the earlier evidence from it must be shown at the next level of transmission. Göttingen is thus not proof of a completed reification, but the clearest place so far where the mechanism of reciprocal restabilization becomes visible.

17. When Learning Carries the Reading Rule Forward

1The preceding chapters have followed how a determination can be named linguistically, stabilized in a mathematical form, and used as the starting point for further operations. Now a further question is added: What happens when not only concepts and procedures, but learning itself is organized? A derivation can become teaching content; teaching content can enter a sequence of levels; and one level can be presupposed as prior knowledge for the next. This didactic stabilization is initially an achievement. It allows people to continue working together beyond what can be derived again in each individual case. Taken by itself, it does not prove that what is learned is understood as an independently given thing.

2The history therefore does not begin with an opposition between free learning and school. Long before the modern mass school there were canonized texts, monastic and cathedral schools, universities, curricula, examinations, degrees, definitions, and firmly practiced forms of questioning. Medieval learning, too, was organized. What must be examined anew is how the range, pacing, comparability, and state responsibility of this organization change — and for which groups. Chapter 16.7 has also shown that teaching operations in the scientific center were part of the production of mathematical-physical order: seminar, lecture, and textbook transmitted procedures and gave them new disciplinary references. What is passed on therefore does not have to retain the same relational determination at every station.

3Johann Heinrich Pestalozzi offers a counterexample to the assumption that instruction must erase the emergence of a determination. In Wie Gertrud ihre Kinder lehrt (1801), he first describes arithmetic as an activity with things and with signs that stand for things: children count objects, dots, fingers, or small objects such as peas and little stones. They are not merely to hear number relations, but to carry them out through counting and recounting. Pestalozzi orders these activities into a method and a sequence of easy, mastered steps; after each step only one small further step is added. The enactment is therefore not simply left out. It itself becomes the teaching procedure that a teacher can repeat.[172]

4Here lies the tension that this chapter follows. The child learns not to equate different apples with regard to certain differences — color, size, condition. For the act of counting, these differences are at the same time set aside: each apple counts in this action as a countable unit. “Three” arises in this situation from bringing together three distinguished objects under a counting point of view. If “three” is later used in place-value calculation, set theory, or an examination problem, other rules and relations order the number. The sign can remain the same even though its role changes. Pestalozzi’s procedure keeps the counting enactment present in learning; it does not prove that later applications continue the same determination unchanged. It is precisely this difference between reuse and identity of meaning that must be examined didactically.

5The shift toward greater range does not begin only with the factory. Prussia’s General-Land-Schul-Reglement of 1763 required school attendance for children from the fifth to the thirteenth or fourteenth year of life and named Christian instruction, reading, writing, and answering questions on the basis of an approved schoolbook. Its justification lay to a considerable degree in religion and state-ordered education. The decree documents a claim and an administrative order, not the comprehensive fulfillment of this claim; moreover, no general introduction of arithmetic instruction may be derived from the wording relevant here.[173]

6Another step can be seen in the Prussian Abitur regulation of 1812. It required an examination and a certificate for students who wanted to move from Gymnasien and learned schools to the university. The regulation explicitly justified examinations also by the need to assess comparably the performance of the students and the condition of the schools. For mathematics it listed a determinate course of material: arithmetic with applications and proportions, algebraic calculation, equations, logarithms, parts of Euclid, and plane trigonometry. Here it is no longer only fixed which books are to be read. An examinable bundle of abilities is defined that regulates the transition into a next educational level.[174]

7This was not, however, education for all children to the same extent. The Abitur concerned a limited student body preparing for the university. Alongside it stood elementary schools, craft training, church instruction, and other paths whose goals and mathematical requirements differed from one another. The Prussian development thus shows an administrative and examination logic that preceded mature industrialization. It was not simply its consequence.

8In France, the connection between mathematical training and state need became visible in another way. The École polytechnique was founded in 1794 in a situation in which specialists were needed for engineering and state services. Its training was intended for a selected group. Admission took place through examinations; the program of Year VIII (1800) set out in detail the required mathematical material. Its authors wanted to specify the examination requirements, make instruction in Paris and the Départements more comparable, and thus create a shared standard for selecting applicants.[175]

9The program is an especially clear piece of evidence for didactic scaling. A national examination does not presuppose that all applicants had the same teacher or the same path of learning. But it does presuppose that the required knowledge is sufficiently determinable and comparable. Arithmetic, algebra, geometry, trigonometry, and applications of algebra to geometry appear as delimited fields of learning with expected procedures. The historical connection is concrete: the new examination regime fixed what an applicant in different places had to be able to present. It still does not follow from this that mathematical relations were now understood as things. It shows that they could be transferred into a transmissible curriculum and into a comparable examination result.

10Cauchy’s Cours d’analyse of 1821 allows such mediation to be observed in the textbook itself. Cauchy wrote the course as professor at the École royale polytechnique for his students and as a contribution to a stricter presentation of analysis. At the very beginning he distinguishes variable and constant quantities; he determines a limit by a quantity that variable values approach in such a way that their difference becomes arbitrarily small. In the case of the infinitely small, he describes a variable quantity whose numerical value decreases indefinitely and approaches zero. The textbook compresses a complicated mathematical controversy into a sequence of definitions, propositions, and proofs — but it does not simply omit the justification. Cauchy leads his readers through the arguments with which analysis is to be newly grounded.[176]

11This source limits, rather than confirms, a sweeping thesis of the disappearance of genesis. A definition makes a concept transportable; a course of instruction orders it so that a student can apply it later. Yet Cauchy at the same time explains how the concept is to be understood mathematically and what burden of proof it is meant to satisfy. Moreover, no direct path from this course to a particular elementary-school problem is documented. Scientific text, university textbook, schoolbook, and examination must be demonstrated as different stations; mere temporal sequence does not yet make them a chain of transmission.

12In Great Britain, a report of the Newcastle Commission of 1861 shows what an examination can make visible about the reading rule actually practiced. Among other things, the commission had pupils solve a practical ratio problem: What do five dozen eggs cost if five cost two pence? Of 1,344 pupils in the first class at 53 schools, only 256 solved the problem. The report names as a problem that many had mastered the mechanical steps of multiplication and division in firmly practiced blackboard problems, but could not translate the context into the necessary numerical relations: five dozen had to be recognized as sixty, and the price for sixty eggs as twelve groups of five, thus 24 pence.[177]

13This is not proof of a general property of school instruction; it is a time-bound inspection finding with a particular sample and a particular examination task. It shows a concrete difference between two relational roles: pupils can carry out the practiced multiplication or division procedure, but they do not reliably map the price-quantity relation of the new task. The examination therefore does not simply show that the same number or operation was learned in shortened form. It shows that the learned operation requires another assignment when the context changes, and that this new assignment often does not succeed. This is evidence of a gap between routine and transfer, not of reification as such.

14British elementary education broadened during the nineteenth century through the coexistence of voluntary and public sponsors, local initiatives, and later state regulations. The Education Act of 1870 established school boards where places were lacking; the Act of 1880 made school attendance compulsory for children from five to ten years of age. Even in the early 1890s, attendance was by no means complete. The parliamentary debate connected education partly explicitly with the competitiveness of British industry. This documents an industrial motive, but not a simple chain of cause and effect. Confessional education, literacy, administration, and political reforms had histories of their own.[178]

15The technical needs, too, were not simply fulfilled by the same school. Mechanics’ Institutes, which spread from the 1820s onward, offered instruction in mathematics, geometry, and technical subjects in many places to adults — often artisans and skilled workers. They were mostly local institutions organized in different ways and are not to be equated with state elementary schools or engineering colleges. The training of engineers, the further education of artisans, and the primary-school instruction of children belonged to different institutions and social paths. For each of these groups one must ask: who had to master what knowledge, and for what purpose?[179]

16At least four levels thereby become distinguishable. Conceptual stabilization gives a determination a reusable name. Operative stabilization makes it possible to continue calculating with a result without deriving it again. Didactic stabilization orders results as learning material, exercise, and prior knowledge. Only an ontological reading would be present if the stabilized determination counted as a characteristic of the object that exists independently of how it was ascertained. No level automatically proves the next.

17The textbook and examination sources nevertheless show how a society can portion the enactment of determination. A curriculum determines which procedures are to be available before an examination. An examination takes a section of these procedures and demands its reproducible execution. The next level of instruction can presuppose the acquired ability instead of repeating its entire emergence. This is a real form of didactic closure. It shortens the path of learning so that a larger number of learners can acquire abilities building on one another in limited time. At some points — as with Pestalozzi and Cauchy — the derivation or the activity remains explicitly the object of instruction. At others — as in the routine exercise described by the Newcastle Commission — the practiced operation can be decoupled from the interpretation of the situation.

18The sources support a limited but important conclusion. Modern educational organizations multiply and standardize procedures from science and calculating practice. Curriculum, textbook, example, exercise, and examination themselves form a new context in the process: they determine what counts as a definition, which cases serve as examples, which operation counts as the correct application, and what prior knowledge the next step presupposes. The pupil therefore possibly learns not only less about the same E5; he learns E5 under a didactically newly determined role. Whether this restabilization displaces the older meaning, expands it, or keeps it reconstructable must be shown in concrete textbook and examination sequences. Curricula alone do not establish an ontological reading in which what is learned is taken to be a property of the world.

19The sources make a possible feedback visible, but not an inevitably closed circle: lifeworldly differences are mathematically determined; procedures enter technical language and teaching material; exercises and examinations restabilize which relations count for a correct solution; the learned reading rule returns into new everyday situations. In the process, the meaning of the adopted quantity can change because new relations take the place of the old ones or weight the old ones anew. Only when this later role is subsequently read as the original determination of the quantity or as a characteristic of the world itself would the strong form of closure be reached. The teaching and examination sources gathered here show conditions and individual breaks of this possibility, not its general completion. The open question is therefore: When does a society learn not only results, but also an altered reading rule so successfully that this reading rule itself appears as the structure of what is read?

18. The Measurement Cycle of the Atom: From Mathematical Formulation to the Interpreted Trace

1The open question can be sharpened once more in the atomic case. So far we have followed how the object “atom” changes between ancient atomism, chemistry, scattering theory, and quantum mechanics. Now the issue is the enactment through which a particular atomic reading is experimentally stabilized at all. A single apparatus does not enact this cycle by itself; it becomes visible in a sequence of theory, experiment, evaluation, and renewed theory. The following can serve as a working map:

2Mₙ → Gₙ → Sₙ → Lₙ → Pₙ → Mₙ₊₁

3Mₙ designates a mathematical-theoretical expectation, Gₙ the experimental arrangement, Sₙ the registered trace, Lₙ its regulated readout, and Pₙ the physical determination that emerges from it. The signs describe our reconstruction, not a formula of the historical authors. The Rutherford/Geiger-Marsden complex is especially well suited for testing the transition concretely: the sources document theory, apparatus, counting, error correction, statistical relation, and interpretation in close connection. At the same time they show that the process does not begin with a presuppositionless measurement.

4Before the trace: theoretical expectation and apparatus

5In 1909 Geiger and Marsden had described a small but unexpectedly large number of strongly deflected alpha particles: with platinum, approximately one in eight thousand appeared to be diffusely thrown back. The finding did not fit the expectation that the deflection arose exclusively from many weak interactions in the Thomson model. In 1911 Rutherford treated the large deflection as the consequence of a single close encounter with an intense, strongly concentrated atomic center. He calculated the path of a charged alpha particle under a Coulomb force and obtained a hyperbolic trajectory; in the calculation the deflection depended on the impact parameter, the charge, and the velocity. From the distribution of possible impact parameters he derived a quantitative expectation for the frequency of deflections at different angles. For a thin scattering body, the theory predicted in particular a dependence proportional to \(\csc^4(\varphi/2)\), along with relations to foil thickness, to central charge, and to the velocity of the incoming particle.[153] [180]

6Here mathematics is not merely the subsequent description of a nuclear structure that has been seen. It determines which difference the researchers vary in a targeted way: angle, foil thickness, material or atomic weight, and velocity. In 1913 Geiger and Marsden explicitly write that their experiments were intended to test Rutherford’s theory, and they name these variations as the testing program. The theory therefore determined parts of the later apparatus and readout. Yet it did not precede the entire process: the striking backscattering finding of 1909 was an empirical disturbance to which Rutherford responded with a new mathematical theory. The historical course is recursive, not one-sided: earlier experimental practice → anomaly → new calculation → targeted follow-up test.

7The 1913 apparatus makes the selectivity of this test tangible. An intense, collimated beam of alpha radiation struck a thin metal foil in an evacuated metal chamber. A zinc sulfide screen, fixed to a microscope, could be moved around the foil on a graduated circular platform. The rotation assigned a scattering angle to the location of the observed scintillations. The vacuum chamber was meant to reduce scattering and absorption in air; diaphragms narrowed the beam; foils could be exchanged and varied in thickness. The geometry made angular distributions comparable. It thereby determined which particle paths could enter the measurement range at all and at which position an impact was looked for.[180]

8The apparatus was not a neutral window. For large angles the screen had to be brought close to the source; there, β- and γ-radiation as well as disturbing scintillations made counting more difficult. Because of the arrangement, the researchers could not measure the range from 150° to 180°. The source decayed during longer measurement series, which is why the count values had to be corrected over time. The zinc sulfide screen did not produce a visible flash of light for every alpha particle that hit it: in their procedure Geiger and Marsden determined its scintillation efficiency at approximately 85 percent and took it into account in quantitative estimates. They also discussed scattering at diaphragms and observed a background even without foil. At low count rates and at very high rates, the reliability of counting decreased. Their values therefore did not arise through simple reading-off, but through defined arrangement, comparison, correction, and limited readability.[180]

9From the trace to the measured value

10What the researchers initially saw were flashes of light on zinc sulfide, not atomic nuclei. Within this experimental context, a single flash counted as an indication of an incoming alpha particle. Only the counting rule — flashes in a particular visual field and time interval — made it into a frequency. Only the position of the rotatable chamber related the frequency to an angle. Only the known or separately determined quantities for beam intensity, source decay, foil thickness, distance, screen area, and velocity allowed different experimental series to be compared. The trace was neither the atom nor already the measured value; its readability depended on the apparatus and on the corrections.

11In the experimental series Geiger and Marsden counted more than 100,000 scintillations in total. For the angle test they compared the corrected distribution with the \(\csc^4(\varphi/2)\) dependence calculated by Rutherford; they report measurements from 5° to 150° and a spread of the observed numbers from about 1 to 250,000. They also varied foil thickness and material and investigated the dependence on particle velocity. These independent variations are important: merely finding the angular curve again would have tested less than a relation that at the same time held up for thickness, material, and velocity. The measurement was, however, not a completely independent confirmation in the sense of a theory-independent observation. Selection of the foils, geometry, corrections, and calculated comparison values already followed a theory. But the variation also produced results that could have made the theory fail.[180]

12The physical conclusion was not: “The nucleus is visible on the screen.” It was: the frequency of large deflections agrees with a theory in which the large single deflection comes from a strongly concentrated central charge, and cannot be explained by the Thomsonian explanation from nothing but small, cumulative deflections alone. In 1911 Rutherford had first shown that his scattering calculation made the large deflection possible. The follow-up measurements shifted the status of the assumption: a concentrated center became a strongly experimentally supported determination of the atom. Nevertheless the conclusion was bound to presuppositions — among them the assumed force, the geometry of the scattering, and the model of a rare single collision. Rutherford noted that the basic scattering consequences initially did not depend on whether the center was posited as positive or negative; he chose the positive sign for the calculation. The early scattering data alone therefore did not yet supply all determinations that later nuclear physics was to ascribe to the nucleus.[153] [180]

13The circularity is thus operative, but not logically circular. The mathematical theory guided the experimental arrangement; the apparatus generated and selected light traces; counting rules and corrections transformed traces into frequencies; the frequencies supported a new physical determination; this then became the starting point for further calculations and tests. The finding could, however, deviate from the original expectation — exactly that had happened in 1909. In 1913 the researchers tested several quantitative consequences instead of merely presupposing the nucleus as a finished interpretation. Operative closure here means mutual dependence and resumption, not tautology.

14The quantitative determination of the central charge also remained changeable. In 1913 Geiger and Marsden estimated it from the proportion of particles scattered under specified conditions and obtained, for heavier elements, a value approximately equal to half the atomic mass. This estimate was explicitly only approximate. Moseley’s X-ray spectra opened up another path of measurement and interpretation in 1913/14: the frequencies of characteristic lines ordered elements by a whole number that was connected with the nuclear charge. This changed again what emerged from the scattering experiment as nuclear determination. The earlier estimate did not disappear as a historical fact, but its position was newly determined within a new mathematical-spectroscopic order. The atomic nucleus that served as the starting point for later calculations was a result of this return loop and not an object fully read off in a single measurement.[154] [155] [156] [157] [158] [159] [160] [161] [181] [162] [163] [180]

15Heisenberg: when the trace does not draw a path

16With Heisenberg, the same test point shifts from spatial deflection to the spectroscopic transition relation. A spectral line first appears as a trace on a photographic plate or as a registered deflection of a spectrometer; after calibration, its position is assigned to a wavelength or frequency, its strength to an intensity value. Heisenberg’s work of 1925 did not itself carry out this experimental practice again with its own spectrograph. It drew on known spectra and on the theoretical order of the old quantum theory. Heisenberg explained that his mechanics should be grounded on relations between quantities that were in principle observable. This was a methodological reordering, not the claim that every theoretical quantity could be read directly as a single deflection.[156]

17The old quantum theory connected stationary states with discrete transitions; spectral frequencies were read as differences of state energies. Heisenberg now shifted the role of the classical path coordinates: instead of retaining a continuous electron path as the basis, he ordered quantities according to transitions between states, for example \(q(n,n-\alpha)\), and connected them with transition frequencies. The mathematical multiplication rule for such transition quantities required a sum over intermediate states. Born recognized matrix multiplication in this; Born and Jordan developed from it a systematic, noncommutative mechanics. Matrix mechanics took up known spectral relations and gave them a new mathematical form from which energies and spectra of concrete systems could then be calculated.[156] [157] [182]

18The apparatus cycle is partly mediated here. The spectral line itself comes from the work of spectroscopists and their instruments; Heisenberg takes over known frequency data. The transition amplitudes with which his calculation works are not simply directly read measured values, but theoretically ordered quantities. His paper therefore does not rebuild the measurement chain from the detector onward. It changes the reading rule for such results and the mathematics that operates with them. The new mechanics in turn opens further calculations and new comparisons with spectra. The atom is thereby not simply defined as a “matrix.” What is immediately stabilized mathematically are state and transition relations; what ontological description of the atom follows from this remains a further question. Wave mechanics, matrix mechanics, and their later unification also show that an empirically fruitful theory does not necessarily fix a single intuitive construction drawing of the atom.[157] [158] [159]

19Heisenberg’s case is therefore a second, sharpened form of the cycle, but not an identical repetition of the Rutherford experiment. In Rutherford, spatial impact locations and frequencies provide the trace to be read; in Heisenberg, already evaluated spectral data enter a mathematical reordering that moves away from classical path representations. In both cases, physical determinacy is not gained from a single unmediated look. In the Rutherford case, the primary sources show the experimental arrangement and error corrections with particular completeness; in the Heisenberg case, the operative return loop between spectral data and mathematical form is visible, but not every individual measurement apparatus is part of the theoretical work. This gap must not be filled with an invented complete chain.

20What operative closure asserts — and what it does not

21The atomic case thus permits a precise thesis: the atom as a scientifically determinable object is brought forth through the systematicity of measurement. Theory, experimental arrangement, registered trace, readout rule, and mathematical order together determine what counts as atom and as nucleus in a concrete research situation. “Bring forth” here does not mean creating matter from nothing; it means generating the object as this measurable and calculable object. This enactment is not arbitrary: the scattered particles and their frequencies offer resistance to the theory and can force it to revise itself. The measured flash of light, taken by itself, is not yet a nucleus. Only the regulated assignment to angle, count, and mathematical relation makes it evidence for a nuclear atom. “The atom has a nucleus” is the powerful short form of this enactment of determination.

22The next step comes when this determined unity enters into further calculations, is determined again in later experiments, is made available in the textbook through formulas and exercises, and is finally related retrospectively to older concepts of the atom. Then the short form can cover over the whole enactment. Yet the individual stages must be kept apart: repeatable measurement is not presuppositionless; mathematical stability is not identity of meaning; didactic availability is not already ontological reification; predictive success does not prove that the starting object was given completely independently of its order of determination. Strong closure arises only where the later determination is read back as an originally presupposed constitution — as in the examples documented earlier that let Democritus “guess” the modern atom.

23Here the title is fulfilled. “The things that do not exist,” in this historical argument, are neither atoms nor other physical phenomena. What is meant is the imagined thing, devoid of relations and already completely determined, which is supposed to exist ready-made behind every apparatus, every trace, and every theory, and whose constitution one then only reads off. The atom that the historical investigation actually finds is brought forth as a scientific object through measurement systems: rays are guided, flashes of light and photographic darkening are registered, angles and event frequencies are read out, spectral frequencies are ordered, and mathematical relations are tested. From these enactments arises a robust determination of the atom that withstands further experiments, but also remains exposed to them. The experiment does not arbitrarily create what it confirms; it participates in bringing forth the object by creating conditions under which it becomes distinguishable and determinable as an atom. What is not documented, by contrast, is a completely determined atom that would already lie ready independently of all such relations and whose properties would merely be read off. The title aims at this presupposed, relationless thing.

24From the incised line to the atom, a movement thereby becomes visible: differences are brought forth and preserved as trace; traces are read according to rules; their public readability is stabilized mathematically; this stabilization acts back on apparatuses, experiments, and new readings. The enactment does not close because it confirms itself, but because every achieved state enables new operations and at the same time remains correctable. What persists is not necessarily an unchanged object behind all stations, but a history of its determinability. At this point the historical volume can lead into Volume I: not “What is the thing independently of its relations?”, but “What must happen so that something becomes readable as distinguished and thereby as this determinate thing?”

19. The Neutrino: From the Missing Quantity to the Measured Particle

1The atomic case began with a name that was carried through changing theories and measurement orders. In the neutrino case, the starting point is different. In the historical transmission, what stands first is not an already fully determined particle, but a finding that does not balance within a physical account. Only afterward is an additional element proposed, taken up into a theory, and finally made the target of an experiment designed specifically for it. The neutrino case therefore tests another movement: how does a deficit within a mathematical-physical order become an experimentally readable particle determination?

What is missing before it is called neutrino?

2The trace along which the question became sharper was the continuous energy spectrum of β-rays. In 1914 Chadwick's magnetic spectrometry made visible that the electrons from certain radioactive decays do not all have the same energy. In 1927 Ellis and Wooster, using a calorimeter, supported the interpretation that this distribution belongs to the decay radiation itself and does not merely arise through subsequent energy losses of the electrons. The measurement delivered a distribution of electron energies; it did not directly show where the energy missing from the balance went.[183]

3Why could the distribution appear as a problem? For the decay of a system at rest into exactly two final products — daughter nucleus and electron — conservation of energy and momentum determine the energies of the products up to the recoil. A single decay should then deliver a definite electron energy. The observed continuum could not readily be reconciled with this two-body order. One had either to change a presupposition of the balance, consider a process not yet taken into account, or posit a further final product. The continuum was therefore not already itself a particle, nor already an unambiguously determined “missing energy value.” It became readable as a deficit because measured values were compared with a balance that presupposed conservation laws and a specific description of the final products.

4This order, too, was disputed. Niels Bohr considered whether conservation of energy might hold only in a restricted way for individual nuclear processes; the experimental and theoretical interpretations of β-radiation were not yet closed. The gap was therefore relative to a contested, although strong, conservation order. The exact source finding is not: “Nature showed an empty compartment.” It is: the observed electron distribution did not fit the initially considered balance of a decay with only the known final products. Only the choice to retain conservation of energy as a continuous condition made a further element into a possible explanation.[183] [184]

Pauli adds an element

5In his letter of December 4, 1930, to the “dear radioactive ladies and gentlemen,” Wolfgang Pauli formulated a “desperate way out.” He referred both to the continuous β-distribution and to statistical difficulties with the nuclei nitrogen and lithium-6. His hypothetical, electrically neutral particle was to have spin \(1/2\), obey the exclusion principle, and have a mass on the order of the electron mass, in any case no greater than \(0{,}01\) proton masses. In β-decay it was to be emitted together with the electron, so that the sum of their energies could be constant. At the same time, Pauli asked whether such a particle, with penetrating power approximately similar to γ-rays, would be experimentally detectable at all. In this letter he called it a “neutron”; the word “neutrino” and the later difference between neutrino and antineutrino must not be projected back into the wording of his initial hypothesis.[184]

6Thus \(D_{M_0}\), the deficit within a balance, is not yet identical with \(\nu\). Pauli carried out an inference: if energy and momentum remain conserved and an additional energy share escapes the possibilities of detection, a neutral, hardly ionizing particle could carry it. The particle thereby received its first determinations — charge, spin, rough mass limit, participation in β-decay — but neither an experimentally secured existence nor an elaborated theory of interaction. It was more than a name for a numerical difference, but less than the later measurable and mathematically systematized neutrino

Fermi makes the hypothesis calculable

7Enrico Fermi took up the hypothesis in 1933–34 and changed its theoretical role. His paper begins with the continuous spectrum and with the conclusion that, if conservation of energy is retained, part of the decay energy escapes the observational possibilities of the time. He treats the electron and the neutrino as particles that are produced in the β process together with the transition between the proton state and the neutron state. With operators and a transition Hamiltonian, he calculates mean lifetimes and the distribution of β energies and compares them with the measured data. In his presentation, particle production is not a mere image: it is a mathematically formulated operation that determines which transitions are allowed and which spectra follow from the theory.[185]

8Fermi's original theory must not be translated into the later standard form of the weak interaction as if he had already written it. The modern four-fermion formula with its present fields, indices, and particle names does not appear in his paper in that form. Fermi explicitly concedes that the form of the interaction term does not follow uniquely from the basic assumptions and that, at first, a criterion of simplicity guides the choice. He tests his construction against invariance conditions, conservation of momentum, and agreement with spectra. Historically, this stabilizes an operative role: the neutrino enters a theory from which decay rates and energy distributions can be calculated. At the same time, it remains open which theoretical features of this early determination of \(\nu\) will later be changed.[185] [186]

The theory sketches a possible detector

9If the neutrino is electrically neutral and interacts with matter only extremely rarely, one cannot simply place a source and a counter next to each other. Fermi and others provided the theoretical quantities with which Reines and Cowan could estimate a possible detection reaction. Their 1953 proposal aimed at the inverse β process on a proton:

10The expected cross section was very small. Therefore, an extremely strong neutrino flux, a large number of proton targets, and a signature that filtered out background events as well as possible were needed. The β decays of the fission products in the reactor supplied an intense flux of electron antineutrinos; water provided many protons. The theoretical reaction determined which secondary products and temporal relations the apparatus had to search for. Thus \(M_1\) was already involved in \(G_1\) before the measurement: it determined what the detector had to be made sensitive to.[186]

11The reaction produces a positron and a neutron. The positron rapidly loses energy and annihilates with an electron; the two 511-keV photons produced in this process can be registered almost simultaneously in two scintillation detectors. The neutron is slowed down and finally captured by cadmium; the capture produces a sequence of high-energy γ quanta with a measurable delay. What was sought, then, was not a trace that by itself meant “neutrino”, but the theoretically expected coincidence: a prompt signal in spatially separated scintillators, followed by a delayed signal whose energy distribution and time distribution matched cadmium capture.

12The Savannah River arrangement of 1956 made this selection material. Two tanks of water containing cadmium chloride served as target volumes; three layers of organic liquid scintillators around them formed the “club-sandwich” detectors. Photomultipliers and electronics transmitted the signals to amplifiers, pulse-height selectors, and coincidence circuits. Lead and paraffin shielded against γ rays and neutrons; the underground room further reduced cosmic disturbances. The apparatus did not register everything that happened in the water. It selected signals that could be ordered spatially, temporally, and by pulse height in the way that the inverse β reaction led one to expect.[187] [188]

What was actually measured?

13Reines and Cowan documented several tests of the reading rule. For the events selected as signal, they examined the pulse heights of the prompt and delayed pulses, their time interval, the dependence on reactor power, and the rate in comparison with the calculated expectation. Events with short delays were compared with events with longer delays, which estimated the random background. Calibrations with a positron source made of copper-64 and a plutonium-beryllium neutron source tested the efficiency; standardized pulses and cosmic events served to control drift. The two independent detector units were evaluated in parallel.[187]

14The controls directly affected the interpretation. When the proton density in the target was approximately halved by adding heavy water, the reactor-dependent rate was also halved, while the separately tested neutron efficiency remained essentially the same. An increase in the cadmium concentration shifted the delay distribution as expected; removing the cadmium made the reactor signal disappear. The researchers also tested whether neutrons or γ radiation entering from outside could simulate a similar sequence. Their result was not a single flash of light, but a statistically evaluated, reactor-power-dependent excess of events whose spatial, temporal, and energetic structure corresponded to the expected reaction products.[187]

15In the report published in 1956, they give, for the combined detector, a signal rate of about three events per hour with a signal-to-background ratio of about three to one; the reaction rate they derived corresponded, within the uncertainties, to the expected order of magnitude. Their own conclusion is that the results confirmed the existence of the neutrino proposed by Pauli and incorporated by Fermi into a quantitative theory. Historically more precisely, one can say: scintillation pulses and their coincidence distributions were measured; through calibration, background determination, reactor-power comparison, and reaction theory, they were read as events of the inverse β reaction; only this conclusion stabilized “neutrino detected” as a physical statement. The detector did not catch a neutrino and did not show a particle image. It registered the products of a rare interaction and tested whether their joint distribution matched the theory.[187] [188] [189]

A closed but resistant enactment

16The neutrino case lets the operative circle be seen especially clearly:

17The first order \(M_0\) made the deficit visible; Pauli read it as an indication of an additional member; Fermi gave this member a calculable role; the theory determined the expected interaction and signature; the apparatus generated and filtered traces; the readout rule ordered these traces as coincidences; the reactor-dependent excess was interpreted as evidence for the neutrino and then entered into a new theoretical and experimental practice

18This is not a logical circular proof. The theory determined the test, but it did not guarantee its result. Without a reactor-power-dependent excess, without dependence on protonic target density or cadmium, or if there had been a convincing explanation by background events, the hypothesis would not have received the same experimental support. The apparatus could disappoint an expectation. The experiment was resistant, but not without presuppositions: only because interaction, target material, time window, calibration, and coincidence rules were theoretically and technically fixed could its result count at all as an experiment concerning a neutrino.

19Nor does the transition from Pauli to Reines and Cowan mean that exactly the same \(\nu\), with a finished list of unchanging properties, was present at every point. Pauli posited a neutral, light particle with spin \(1/2\) in order to solve energy and statistics problems; Fermi made the particle a component of a quantitative theory of β decay; in 1956 Reines and Cowan investigated the corresponding inverse reaction with a reactor antineutrino. The determinations are historically connected, but their roles are not identical. The successful detection tested a connection in which theory and apparatus were already working together; it does not prove retroactively that all later properties were already fully determined in Pauli's letter

The already found neutrino becomes an object of testing again

20The history does not end with the detection. In the solar neutrino experiments, a different deficit later arose. Raymond Davis's chlorine experiment determined the flux of electron neutrinos through the reaction with chlorine-37 and the subsequent extraction and counting of generated argon-37 atoms. Compared with the predictions of solar models, the measured rate was smaller. This “solar neutrino anomaly” was not the same problem as the continuous β energy distribution: here it was not an energy share within a single decay that was missing; rather, fewer electron-neutrino captures were counted than expected from a model of the solar flux and from a particular detector reaction. This difference too was relationally determined — by solar model, reaction cross section, efficiency, and the special sensitivity of the chlorine method to electron neutrinos.[190] [191]

21The oscillation hypothesis changed the reading: a particle marked out at production by an electron-neutrino reaction can show a different flavor assignment at detection, because the phases of the superposed mass states advance differently during propagation. Thus “neutrino” was not only a particle name with a mass or a decay role, but a connection among production flavor, propagation, energy, path length, and detection reaction. The result of the first detection remained preserved in the experimental history; its theoretical meaning was expanded by new relations.

22The decisive test came in 2001/02 at the Sudbury Neutrino Observatory. Heavy water there made it possible to count charged currents, which responded only to electron neutrinos, and neutral currents, which responded to all active neutrino flavors. The charged-current data yielded an electron-neutrino flux of approximately \(1{,}76\times10^6\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\); the neutral current yielded a total flux of approximately \(5{,}09\times10^6\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\), compatible with solar models. The non-electron-flavor component was therefore not merely a new explanation of the old deficit: it was positively determined through another reaction channel and another readout order. KamLAND tested the related idea with reactor antineutrinos over long baselines. In the first analysis, the ratio of observed inverse-β events to those expected without oscillation was \(0{,}611\pm0{,}085\) (statistical) \(\pm0{,}041\) (systematic); the energy distribution and the baseline dependence supported the oscillation interpretation.[191] [192]

23Here too the conceptual chain must remain exact. SNO measured Cherenkov light from charged end products and neutron signals from the reactions on deuterium; KamLAND measured prompt positron light and delayed neutron captures. None of these detectors read a flavor directly from a neutrino. The physical statement arose from different reaction channels, calibrations, background models, flux comparisons, and statistical fits. The new findings determined the neutrino as a particle whose observed flavor could not simply be equated with an unchanging identity during propagation. This is not a refutation of the early detection, but a new determination of its role.

From detection to the systematics of particles

24With the neutrino, electron, proton, neutron, and the further particles found in accelerators and cosmic radiation, what grows after the Second World War is not simply a list of things. Researchers now have experimentally stabilized particle determinations with comparable information about charge, mass, spin, decay, and interaction. This gives rise to a new problem: how can these findings be brought into a common order? Systematics first uses the particles as given inputs of its work. In our work-immanent language one could call this “level 5”; historically, that is our analytical concept, not an expression used by the physicists of the time.

25A concise case is Gell-Mann's “Eightfold Way.” In it, the known baryons and mesons are not merely lined up by similarity, but treated as members of symmetry multiplets. In the symmetry-limit case, the members of a family have equal masses; the observed differences are taken up into the systematics as breaking of the symmetry. The relations between the already known particles thus generate new conditions for what would still have to be missing. The order does not only describe an inventory; it also determines which place in it is unoccupied.[193] [194]

26The Ω⁻ hyperon case shows the return loop. The symmetry order predicted a particle with determinate quantum numbers; in 1964 Barnes and collaborators reported a bubble-chamber trace that they evaluated as a hyperon with strangeness \(-3\). The particle order had thus arisen from experimentally known relations and now directed the search for a member not yet observed. The finding was nevertheless not tautological: a trace had to be reconstructed as a decay chain, made compatible with momenta, charge, and conservation laws, and tested against alternatives. Here the feedback can be seen without dissolving the experimental resistance: known particles → symmetry order → predicted place → new trace → confirmation or correction of the order.[194]

27This order, however, was not a systematics of all known matter. The Eightfold Way concerned above all baryons and mesons, that is, hadrons; electron and neutrino did not belong to these hadron multiplets. Gell-Mann could draw on leptons as a mathematical analogy or in the discussion of interactions, but his particle octet did not unite atom, electron, neutrino, and nuclear building blocks in a single order. The point of condensation here is therefore local: a subfield of particle physics turns an empirical inventory into a relational classification that generates new search expectations. A later unification of different interactions would be a further historical step.

28In 1964 Gell-Mann's schematic baryon and meson model shifts the basis once more. Hadrons are now treated as combinations of hypothetical quarks: baryons, for example, as \(qqq\), mesons as \(q\bar q\). The known particles thereby appear at the same time as empirical data of the model and as composite results of a deeper mathematical-physical order. Precisely here, “particle” must not remain a uniform concept: the observed hadrons are determined differently from the postulated quarks, whose role first arises from the classificatory and compositional schema. Gell-Mann himself left the ontological question open; he wrote that a search for free quarks could help one become convinced of the existence of “real quarks” or rule it out. The mathematically fruitful role in 1964 was therefore not equated with an experimental detection.[195]

29This makes the connection examined here more precise. The new systematics no longer retraces, in every operation, the entire path through which electron, neutrino, or hadron were determined as objects of measurement. It works with the stabilized particles and their ascribed quantum numbers as its current starting material, places them in common relations, and lets new predictions arise from these relations. In this limited sense, the result becomes the basis of its own further order. The origin is not necessarily unknown or deliberately forgotten; it merely does not have to be enacted anew in every systematic step. Reification would be more strongly established only if the systematically determined role were subsequently read back as the complete constitution of the particle, independent of its experimental and theoretical production. The Ω⁻ case documents the productive feedback; Gell-Mann's explicit reservations about quarks at the same time show that systematization and ontological fixation could historically be kept apart.

What the second case contributes to the title

30The atom case showed how a persisting name is determined anew through different orders. With the neutrino, what stands first is a difference in a balance. It is interpreted as a possible additional particle; a theory makes the particle calculable; the theory helps fix which interaction a detector is to register; apparatus and readout rule transform their traces into a statistical finding; this finding becomes a presupposition of later experiments that determine the neutrino again.

31This does not mean that the neutrino would be “only a calculation” or first materially arises in the measuring device. It means: the neutrino as precisely this physically determined something is, in the historical reconstruction, not first fully given and then neutrally read off. What the sources show is a connection of conservation laws, measured values, theoretical positing, interaction calculation, target material, coincidence signal, calibration, and renewed interpretation. The experimental finding stabilizes the particle determination and at the same time limits what may be projected back onto its early hypothesis.

32The gap itself was not the neutrino. It was a difference within a particular balance. The neutrino was proposed as a physical answer to this difference, then worked out theoretically, and finally identified experimentally through its effects. As soon as it was identified, it became the condition of new calculations and measurements, which in turn made new properties and relations visible. What was found therefore did not stand outside the conditions of its finding. It itself became part of the conditions under which further finding took place.

33The historical point is therefore neither “atoms do not exist” nor “neutrinos do not exist.” It is: the thing already fully determined without relations, which is supposed to be presupposed as finished before every enactment of determination, is not what this sequence of sources finds. It finds resistant physical traces and rule-governed ways of ordering these traces into object-statements. The second case thereby sharpens the question with which Volume I can begin: not only how a thing is determined anew again and again, but what must happen so that a difference becomes readable as a determinate physical something.

20. The Field: When the Connection Itself Becomes the Physical Carrier

1After the neutrino, the trace leads to the question of how a larger physical order arises from already determined particles. One possible next step is the concept of the field. It connects local measured quantities, forces, particles, and interactions into a connection that no longer exists only between named bodies. The word sounds spatial: a field seems to be a surface or an extended region. But what is determined mathematically and physically as a field? And when does the systematic framework begin to treat its own connection as that which it was originally supposed to describe?

2First, the word-image must be separated from the mathematical role. A field is not necessarily a geometrical surface. In mathematical physics, it can be an assignment that assigns certain values to every point of a region: for example, a direction and strength of magnetic action, or several components of an electromagnetic field. Conversely, a finite set of measurement points is not yet a field. Between the measured places and beyond them, a field determination arises only through a theory, equations, boundary conditions, and evaluation procedures. The field is therefore neither simply the surface on which something lies nor the mere sum of observed points. The historical question concerns the transition through which such a mathematical assignment is read as a physical connection and finally as an independent carrier of effects.

3Back to Descartes: Geometry Becomes Coordinatable

4For this transition, we must go back once more to Descartes. The modern image of a coordinate grid laid uniformly over space, however, must not be projected back unchanged into his Géométrie of 1637. Descartes does not develop there a finished Cartesian grid surface in today's sense. Rather, he connects geometrical construction and algebraic calculation: lines receive names, a line chosen as a unit makes it possible to express geometrical operations as arithmetical operations, and the dependencies between known and unknown lines are transferred into equations.[196]

5Descartes describes this procedure strikingly explicitly as a regression from an assumed solution. One should first “consider the problem as already solved,” give names to the required lines, and then pursue their mutual dependencies until the same magnitude is expressed in two ways and an equation is thereby found. Only when the relations are complete enough can the construction be rebuilt from the equations. This analysis is therefore not simply circular reasoning: the assumed solution serves as an order of search, while the dependencies and the subsequent geometrical construction test whether the problem is actually determined. But it shows an important early step: the result can be posited as a provisional form in order then to work out, backwards, the conditions of its production.[196]

6Later coordinate geometry generalizes this practice. Instead of fully constructing a concrete figure each time, one can mark its points by ordered numerical values and treat geometrical relations algebraically. A coordinate grid is at first a reference system: it establishes how differences of position are transferred into numbers. For a field, this possibility is expanded in that not only individual points are designated, but a value, or a structured magnitude, is assigned to every point—or, in spacetime, to every event. In this sense, the image of the net laid over space captures a decisive feature: it makes spatial differences readable as comparable places of a common order. Historically and mathematically, however, we must keep the levels apart: coordinate grid, field assignment, and physical measurement process are not the same. The grid determines the mode of representation; the field equation determines permissible relations among the assigned values; the experiment tests which of them can be maintained in its signals.

7From Faraday's Lines of Force to Maxwell's Mathematical Determination

8Faraday's lines of force did not arise as an abstract sequence of points. They belonged to practices in which the effects of magnets, conductors, and induction were investigated through iron filings, movements of conductor loops, and changes in currents. The lines made directions and distributions visible; they were a way of ordering spatial effects. In 1846 Faraday even considered whether oscillations of the lines of force themselves could be connected with light. That was a speculative physical interpretation, not an already mathematically secured description of a hidden object.[197]

9In 1855 Maxwell translated Faraday's lines into a mathematical representation. He describes a state at every point of space by a quantity with direction and magnitude, and compares the distribution with a flow in an imagined incompressible medium. Yet Maxwell expressly designates this mechanical representation as a kind of artificial notation which, by itself, does not yet establish a physical theory. He allows competing approaches to stand side by side and ascribes to the provisional theory above all the value of guiding experiments without blocking later explanations. Mathematics is therefore not merely supposed to connect measurement points; at the same time, Maxwell does not claim that the mathematical representation is already the exhaustive constitution of what it represents.[198]

10In the works of 1861–62 Maxwell developed mechanical models with vortices and idler wheels in order to make the transmission of electromagnetic effects imaginable. In 1865 he formulated the theory of the electromagnetic field in a more developed form. His own definition remains bound here to a physical assumption: the field concerns the space in the surroundings of electric and magnetic bodies; the theory assumes moving matter there, through which observed electromagnetic phenomena are produced. According to Maxwell, the energy too is supposed to lie literally in the field, while he distinguishes between a description by electric and magnetic polarization and a mechanical hypothesis about motion and stress of a medium.[199] [200]

11Here a tension is already visible that is decisive for our investigation. The concept of the field orders effects that were previously described between bodies or through lines of force as states of a spatially distributed connection. At the same time, in Maxwell it remains open what physical status belongs to this connection: mathematical representation, mechanical mediator, state of the medium, or carrier of stored energy. The equations and the ontological interpretation are not the same. That later physics adopts individual equations does not prove that it also adopts Maxwell's original mechanical interpretation; just as little does the disappearance of a mechanical image prove that the concept of the field had become meaningless.

12Maxwell's Treatise makes the empirical linkage explicit. He wants to make phenomena measurable, define measurement methods and standards, connect the measured quantities mathematically, and then test how this mathematical form is connected with dynamics. He even emphasizes that the mathematical linkage of different classes of phenomena can secure the experimental laws more strongly. At the same time, the theory opens further questions and provides means for investigating them. This is neither a presuppositionless reading-off of a finished field reality nor a mere calculating within a closed symbol system: measurement, mathematical linkage, and new experimental possibilities interlock.[201]

13When the field itself becomes the foundation of further physics

14In the further course of physics, the concept of the field is stabilized in a new way. Maxwell’s equations can continue to be used in changed notation and without Maxwell’s concrete vortex mechanics. Hertz’s experiments do not simply give the theory a later confirmation. They make visible how a mathematically determined relationship helps shape the conditions of an experiment — and how the experiment, in turn, decides whether and in what form this relationship holds.

15Hertz’s experiments show not only the expected wave, but also how an unexpected order of measurement can bring the theory into crisis. In his retrospective account of the work of 1888, published in 1894, he reports that an initial plan provided for comparing effects on a wire with effects propagated directly through the air. The theory led one to expect that, with the same finite propagation velocity, both would interfere in the same phase at different distances. Yet the phase sequence first observed seemed rather to fit an infinitely fast propagation through the air. Hertz abandoned the experiment in discouragement; when he resumed it, he examined different readings and even allowed for the possibility that unknown causes slowed propagation in the wire. He explicitly emphasizes that he did not read the results in such a way that they conveniently confirmed a preconceived view.[202]

16This deviation is neither an immediate proof against the theory nor a free pass to save it by arbitrary auxiliary assumptions. It first forces the whole chain of determination to be examined again: What does the primary conductor produce? Which direct and wire-guided effect reaches the secondary conductor? How are phase and interference read from the spark pattern? Which assumptions about the wire, space, and the apparatus enter into the calculation? In Hertz’s evaluation at the time, the velocity in the air space had to be taken as finite, but greater than the velocity in the wire. Thus “more mathematics” was not simply added to an unchanged initial order; the physically relevant relation between two paths of propagation received a new articulation. Hertz’s report also contains an instructive feedback between calculation and experiment: an incorrectly calculated period of oscillation deviated by a factor of √2; because the false value appeared to agree with the measurements, he trusted the calculation. According to his own statement, a contradiction would have led him to give the calculation less weight. Here, then, a neutral measurement does not simply confirm a finished mathematics. The apparent agreement arises from mutually dependent procedures of measurement, calculation, and interpretation; the uncovering of a calculation error changes how the earlier comparison is to be read.[202]

17In the supplementary experiment on waves in air space, the wave-like effects were to emerge more vividly and make a direct measurement of the wavelength possible. Hertz compares this directly determined length with the value previously obtained indirectly; their nearness is also meant to support the earlier conclusion. The arrangement already presupposes a mathematically readable relation: an oscillating primary conductor produces an effect, a neighboring resonator with a spark gap reacts to it, and the position of the receiver is systematically varied. Hertz had previously observed weak sparks at places where, for reasons of symmetry, a direct effect should be small, and therefore suspected reflections from the walls. For the main experiment, he has interfering metal objects removed and places a large zinc plate on the wall in order to make the reflection stand out in a targeted way.[202] Theory and mathematics therefore do not stand only at the end: they help shape which arrangement is built, which distances are compared, and which recurring differences are read as a test of propagation.

18What the receiver immediately provides, however, is not a visible label “electromagnetic wave,” but a spark in a small interruption of the secondary conductor. The strength or absence of the spark depends on the orientation and position of the resonator, on its tuning, and on the reflections at the walls. In the experiment with a reflecting wall, direct and reflected effect are brought together in such a way that stronger and weaker sparks occur at different places. From the ordered places of these maxima and minima, a wavelength is inferred. The trace, then, is the spark; the measurement finding is a reproducible dependence on place and orientation; the wavelength is a mathematically obtained determination; the statement about electromagnetic waves is the physical interpretation that places these relations within Maxwell’s theory.[202]

19Here the mechanism that we previously examined in the cases of the atom and the neutrino appears in another form. The mathematical system specifies which differences the apparatus is to be made sensitive to and how registered differences can be compared. Insofar as the expected nodes, antinodes, and length relations appear, the experiment makes the relations presupposed in the theory publicly testable. But it thereby does not confirm every presupposition individually, and still less does it prove that the concept of the field was already fully fixed independently of this experimental-mathematical order of determination. The agreement first concerns a network of theory, setup, calibration, rule of readout, and measured value.

20The way back is just as important. Hertz’s unexpected phase sequence was neither simply booked as a refutation of the Maxwellian order nor bypassed by an unchanged expectation. It forced him to consider competing relations between propagation, phase, and path of transmission, and then to examine calculation, setup, and readout again. A different experiment could have forced a revision of the mathematical relations or of the physical reading. Then it would not be merely an additional value inserted into unchanged mathematics: the new order could change which quantities count as decisive and how the previously used concept of the field is determined within it. Conversely, such a reconstruction presupposes existing concepts and procedures; even a revision does not begin from a theory-free zero point. The experiment is resistant, but not without presuppositions.

21Hertz’s example therefore shows both the power and the limit of feedback. The mathematical expectation is translated into apparatus and rule of reading; the trace can support or disappoint it; the result returns into the theory as a new determination. The theory that emerges from this, in turn, becomes the foundation of later apparatuses and measurements. In this way a determination can partly leave its original relations behind and take on a changed role in new contexts. Whether this operative stabilization finally loses contact with the relationships described is not yet proven by that. What can first be historically demonstrated is that the experiment does not neutrally image an already fully determined field, and that mathematics is not merely laid retrospectively over a finished finding: both enter into a regulated feedback whose next round can help determine the meaning of its quantities.

22Einstein’s general relativity sharpens the case and at the same time corrects a possible misreading. In the theory, the metric structure of spacetime is dynamic; it is not simply a rigid stage on which bodies lie. But it does not follow from this that Einstein takes spacetime points to be small, independently determined things. He describes coordinates as arbitrary numberings and holds that physical determinations amount to coincidences of events. A point therefore does not receive its physical role already through an isolated coordinate number. Geometry and matter appear in a coupled field order; the equations determine how metric structure and energy-momentum cohere.[203]

23This makes the comparison with quantum physics more precise. In an ordinary quantum field theory, a quantized field is described on a spacetime whose geometric structure is typically fixed as background. General relativity, by contrast, treats geometry itself as a dynamic quantity. Both theories use points, fields, and relations mathematically, but these expressions do not necessarily have the same role. In relativity, the metric field structure is part of what is dynamically determined; in quantum field theory on curved spacetime, a classical geometry first serves as the condition within which quantum fields and their states are determined. Hawking, for example, was able to calculate particle creation by a classical gravitational field; this kind of success is a physically productive connection, but not a complete quantization of spacetime.[204]

24The quantum-mechanical “field,” too, is not simply a refined surface with measured values at every point. In the axiomatic formulation of Wightman and Gårding, quantum fields are treated as operator-valued distributions. This means that, mathematically, a field operator is not without further qualification a well-defined measured value at a single point; the theory works with quantities smeared over test functions and with relations between expectation values. The mathematical systematics thus becomes more abstract, while its empirical readability is produced through scattering processes, counting rates, correlations, and detector signals. In this theory, the field is not identical with the measurement points, but orders possible measurement relations in a formally precise way.[205]

25The culmination: when the systematics produces its own presuppositions

26Here the culmination we have been following comes into view. A mathematically determined field makes new physical calculations possible. Their results are used to build experimental arrangements and to read measurement data as field effects or particle excitations. If findings agree with expectations, they stabilize the relations under which the experiment was set up and evaluated. If they do not agree, they can force a revision of the theory; this revision changes which relations further experiments are supposed to test. What emerges is not a logical circle in which the result is already fixed, but an operative feedback: mathematical presuppositions structure measurability, measurements limit the admissible mathematical continuations, and the changed mathematics becomes the starting point of new experiments. The systematics continues to describe physical connections, but the determinations with which it does this are themselves also further determined within the cycle.

27At this point, “self-enclosure” must be stated precisely. It does not mean that physics no longer performs experiments or that every measurement necessarily confirms what the theory presupposes. A prediction can fail; a signal can be absent; an anomaly can force an extension, alteration, or abandonment of the model. Nor is the mathematical correctness of a field theory the same as its empirical success. Rather, the possible reversal lies in the fact that theory increasingly recognizes its objects only in its own formally stabilized roles: field values, particle excitations, states, and interactions become inputs for the next calculation step, and their connection to the original enactment of measurement and distinction is no longer carried along in every application.

28This is more than a mere forgetting of the prior history. When the concept of field appears in new relations, it can be determined anew there. A classical field as distributed intensity, a metric field as dynamic geometry, and a quantum field as an operator-valued distribution are not simply three names for the same unchanged something. The formal continuity of the word “field” must not hide its different roles. Nor may the later role as a fundamental physical quantity be projected back onto Faraday or Maxwell without examination. The methodological point is not that mathematics always remains right or can absorb every deviation into itself. It is this: an experiment is possible only when it operationalizes certain differences; a deviating finding is itself legible only when one can examine which presupposed relations fail to meet it. Depending on what changes in that process, not only the prediction is corrected, but possibly also the relational meaning of the mathematical object.

29The strongest suspicion, therefore, is not that modern physics no longer describes “anything at all.” It demonstrably describes mathematically ordered, experimentally testable connections and delivers successful predictions. The open question is stricter: Does this also clarify what the field is independently of the determining relations that make its values, equations, and observable consequences legible? If the theory provides complete justifications only within its own systematics, but no longer shows its connection to the conditions from which its objects emerged, then operative success can appear as the justification of its own beginning.

30The pattern now resembles the neutrino case, but the starting point is different. In the case of the neutrino, a difference in a balance preceded the hypothesis of an additional particle. In the case of the field, a mathematical-physical order itself can become the starting point from which particles and effects are newly determined. In both cases, the result returns into the conditions of further research. The difference is important: in the case of the field, theory does not merely systematize a single suspected member, but can provide a comprehensive framework in which measuring apparatus, measured value, particle, and spacetime are made legible together.

31Hilbert’s program, to order the physical theories axiomatically and in particular to bring gravitation and electrodynamics into a common variational connection, forms a historical test point for this. It documents an explicit claim to systematize physics, but not an unbroken direct line from Hilbert to every later quantum field theory. The actual connections must be demonstrated separately for each step—from Hilbert’s foundational work through relativity to the axiomatic programs of quantum field theory.[139] [137] [205]

32This still does not prove the strong thesis that the systematics had ceased to describe what it wanted to describe. What first becomes historically visible is something more precise: an order that emerges from measurements and mathematical relations can operationalize itself to such an extent that it uses its own concepts as presuppositions of new measurements and calculations. It then continues to describe successfully—but increasingly through the structures that it has itself stabilized. Whether and when these structures count as the complete constitution of what is described is the next source-critical step. The title here receives its sharpest, but still open, meaning: no physical object is denied; what is questioned is the presupposed thing that allegedly already stands fully fixed independently of the history of its determination.

21. Closing Movement: How Correct Results Become Presuppositions

1The results examined do not become problematic because they are false. Atomic physics, the experimental detection of the neutrino, the mathematical formalizations, and the measurement procedures based on them have produced reproducible and technically usable results. Precisely this confirmation makes the following question necessary: What happens to a correct result when the enactment through which it was determined no longer has to be carried along in its further use?

2In the cases examined here, a result did not arise without presuppositions. Its determination depended on relations: on that with which it was compared; on the mathematical order in which differences became calculable; on the apparatus that produced certain traces; and on the rule according to which these traces were read as measured values. For the experimental case, the dependence can be written as a working map:

3Here \(R_0\) denotes the preceding order of determination, \(M_0\) its mathematical-theoretical elaboration, \(G_0\) the experimental arrangement, \(S_0\) the registered trace, \(L_0\) its rule-governed readout, and \(E_0\) the stabilized result. This sequence of signs is not a general law of nature. It only holds together the dependencies that were examined in concrete sources in the atomic and neutrino cases

4The result \(E_0\) can be completely correct. Its correctness does not erase the relations under which it was determined. Rather, it permits the next step: a result can be fixed and reused without its whole genesis being repeated every time. No one has to perform Rutherford’s experiment again for every calculation; a proof does not have to be carried out completely anew in every application; a later neutrino experiment does not have to repeat the entire history from Pauli’s balance problem to Reines and Cowan. Scientific work needs such stabilization. It is a condition of cumulative research, not a failure

5The operative sequence is therefore shortened. What first appeared as \(R_0 \rightarrow \cdots \rightarrow E_0\) can, in a later step, simply be inserted as \(E_0\). It now operates in a new context \(R_1\):

6The result becomes the presupposition of a further calculation or measurement. It has not thereby become false. But its role has changed: in \(R_0\) it was a produced finding; in \(R_1\) it is a usable starting point. Whether both roles carry the same determinacy must be examined. The later use can preserve the older relation, shorten it, or restabilize the result anew in a new environment

7Here teaching is added. Definition, textbook, example, exercise, and examination establish a further relational environment. A scientific connection can become learning material in a textbook; there a formula can serve as the presupposition of the next exercise. This is neither bad teaching nor the concealment of a truth. Instruction must relieve results of the complete repetition of their genesis if learners are to continue working with them. At the same time, the teaching examples examined show that the order of learning practically co-determines what counts as the correct use of a concept: in the atomic exercise, for example, atomic number, mass number, and charge become inputs for the calculation of a composition; in the case of the neutrino, instruction can begin with the stabilized particle determination without re-deriving the original balance problem in every application.[162] [163]

8Thus the same signs can appear in two different movements. For the first generation of research, determination leads to the result. For the next generation of research or learning, the result can stand immediately at the beginning. “Under these experimental and mathematical conditions, \(E_0\) was determined” becomes, in a legitimate short form, “\(E_0\) holds.” “The measured reaction products are read under these rules as a neutrino effect” becomes “Neutrinos are particles with these properties.” The short form can be factually correct. Yet it places the result in a new role, from which further properties, experiments, and explanations are determined

9The sequence does not close by simply returning to the same point. It produces new conditions:

10The result is stabilized, taught, presupposed, and built into new calculations and apparatuses. The next measurement can confirm, limit, or change it. A deviation can require a new mathematical relation; with this relation, the roles of the magnitudes involved can also change. The new determination in turn becomes a result and a possible presupposition of another round. Hertz’s work makes visible precisely that an unexpected trace is not automatically adapted to the theory: it can force an examination of the entire chain of calculation, apparatus, measurement, and interpretation.[202]

11This feedback is not necessarily a logical circular proof. The theory does not guarantee its result; an apparatus can fail to deliver an expected trace. And a result does not become false merely because its enactment of determination is not repeated in a single operative step. The decisive difference is therefore:

12A system can calculate correctly with a stabilized result and at the same time not make that result’s original determining relation visible again in the current calculation. Likewise, successful continuation does not imply that the result was already the complete presupposition of that connection from which it itself emerged. Success in continuation demonstrates the capacity of the continued order; it does not retroactively prove the identity of meaning of its initial concepts.

13This also names the reversal into reification more precisely. It does not already occur when a result is reused or taught. The stronger movement begins where a result determined in \(R_0\) receives a new role in \(R_1\), and this later role is then read back as the original, independent constitution of the object. The direction of determination is then reversed:

14is read as:

15Teaching can reinforce this reversal because, for didactic reasons, it may sensibly begin with an already stabilized object: object, properties, formulas, exercises. But instruction does not produce the movement by itself. The atomic case shows how the name can persist despite changing orders of determination; the neutrino case shows how a balance deficit becomes an experimentally stabilized particle through hypothesis, theory, apparatus, and readout. In both cases, the decisive back-projection would be demonstrated only if a later object-role were in fact presented as the original meaning of the earlier source. The mere shortening in an exercise is not sufficient for that.

16The gap toward which the historical volume runs is therefore not necessarily an empirical gap. It is the possible gap between the stabilized readout and the relational enactment that makes it determinable as this measured value. A science can make its measurements more precise, improve predictions, and extend its models while the relations through which its objects became legible as determinate objects in the respective context recede into the background in ongoing use. More empirical knowledge does not automatically eliminate the question of the enactment of determination; it can even multiply the number of stabilized presuppositions.

17This does not mean that modern physics is ontologically false or describes nothing real. The historical cases support a narrower thesis: mathematically and experimentally determined results can become presuppositions of further systematic work; new relations can change their role; teaching can reproduce this later role; and a back-reading can finally make it appear as the presuppositionless constitution of the object. Whether this last step has actually taken place must in each case remain a matter of source-critical examination. Reification here does not name the success of the systematics, but the unjustified transition from “This is how the object is determined in this order” to “This is how it was always already given independently of this determination.”

18The path of the historical volume can thus be read backward without equating its stations. A notch preserves a trace of an enactment. Numerals make differences legible according to a rule. Greek texts determine through distinction, construction, ratio, and measure. Translations arrange determinations in new linguistic relations. Manuscript, commentary, and instruction make such orders repeatable. Mathematical procedures stabilize results as reusable forms. Experiments translate such forms into apparatuses, produce traces, read them out according to rules, and lead measured values back into new mathematical-physical orders. Teaching finally hands over their results as starting points to further learners and researchers.

19As a historical working map, this progression can be noted as follows:

20This is a sequence of historical practices and changes. It is not identical with the work-immanent orders of the Tractatus. The signs \(|\), \(A|B\) and \(|(|)\) are not temporal stations through which history would have passed. Rather, the map of relations allows the historical sequence to be read under a comparable relational question: Which distinction makes a trace legible? Which determination is stabilized? Which new relation takes up the result, and does this new role change what is carried forward under the same name? Shared legibility does not mean identity of the historical and work-immanent movements

21The long historical path thereby brings into view a presupposition that its own successful stabilizations can increasingly make dispensable. Scientific practice does not have to repeat the entire emergence of a determination in every calculation, measurement, or teaching exercise. Precisely this relief makes cumulative research and teaching possible. At the same time, it can cause the fact to recede from view that the result being used had been determined within an earlier context. The historical volume needed many pages to show how it becomes possible to read the result without also having to read the enactment that produced it. Not because the ground would have been historically unknown, but because its omission can be an achievement of transferability.

22However, the historical explanation is not therefore simply to be extended by one more cause. We can ask why a student thinks of a particle when hearing “neutrino”: the answer leads to textbooks, disciplinary systematics, and experimental stabilization. We can ask further why textbooks determine it in this way: that leads to the theories, measuring procedures, and readout rules on which the determination rests. But the question of by what means anything at all can be determined as distinguished from something else is not simply the next historical event in this chain. Here we change the kind of question. What is missing is not an archival document still to be found; the question concerns a condition for historical objects, sources, and differences being readable at all as determinate.

23The historical volume therefore does not end with the claim that it has historically explained the origin of all distinction. It reaches the limit at which historical reconstruction raises a foundational question. This limit is not proof of an answer; it marks the transition that Volume I takes up. There the work-immanent reading does not begin with two already finished sides between which a separation later occurs. It begins with difference as an enactment through which sides first become determinable at all:

24The historical movement can now become readable within this work-immanent order without being identified with it. In the historical material, difference becomes a trace, trace becomes a reading, reading becomes a determination, determination becomes a stable and publicly transmissible result. In the Tractatus, by contrast, \(|\) does not stand as the first event of a historical series, but as the work-immanent starting point for the question of determinability. Only \(A|B\) designates the determination of sides within a distinction; \(|(|)\) records that enacted distinctions can in turn enter into distinctions. These relations are a reading anchor, not a retrospective proof for the historical narrative

25Here the title receives its precise meaning. “Things That Do Not Exist” does not mean that there are no atoms, neutrinos, or matter. What is meant is the fully determinate thing that supposedly, independently of the relations of its determination, already is exactly what it is later described as being. Historical reconstruction does not find such a relationless thing. It finds traces, measures, relations, apparatuses, readout rules, mathematical forms, resistant findings, results, and renewed determinations. The “thing that does not exist” is the thing detached from its conditions of determination and nevertheless presented as their original ground.

26We have gone a long way in order to return to a question that Volume I already places at its beginning. We have traced how differences can become traces, traces readings, readings determinations, determinations results, and results presuppositions. Science did not become less exact in this narrative; its results did not therefore become more false. They became more robust and more reusable. Precisely for that reason, the enactment through which they had been determined could disappear from their next use. At the end, therefore, we do not stand before a lack of determinations, but before the question of their presupposition:

What must happen for anything at all to be readable as something determinate?

27And then, for the transition, almost a page turn is enough:

28The historical volume is therefore not redundant. It shows how a cultural history could arise in which this stroke itself again appears to require explanation.

Notes and Sources

Reading Basis and Method

29Work-immanent reading anchor: Volume I places difference “|” before number; number is a stabilized or readable order of difference, not an information container that stores within itself, independently of relations, the enactment through which it is produced. Volume IV sets | → A|B → |(|). These orders are not temporal stages. Self-referentiality means that distinction acts upon distinction. Δ := |(|) remains exclusively a work-immanent reading anchor: not historical proof and not an additional operator or object.

30Guiding question for the historical examination: A name, sign, or symbol can persist while its relational role changes. For every use, one must therefore ask: by what is the expression determined here, from what is it distinguished, to what is it related, which operations bring forth its determinacy, and what function does it have in the argument? For later occurrences, the same questions about the respective relations of determination arise anew. Recurrence of the name is not evidence for identity of meaning.

31Working model of relational semantic drift: R0 determines E5; E5 is taken over into a later context R1; R1 brings new relations with it and can re-stabilize E5 anew. E5\(R0\) and E5\(R1\) must not be equated without source examination. If R1 subsequently reads R0 and presents its own determination as the original sense, the strongest form of back-projection is present. It is historically demonstrated only when later readers, teaching texts, or arguments actually interpret the earlier source with this later role.

32Three cases of stabilization must be kept apart:

331. Relation-preserving stabilization: A result becomes reusable, while the relations that determine it are preserved or remain reconstructible.

342. Operative detachment: The enactment is not performed anew at a particular step. This is an achievement of scientific and mathematical work and by itself proves neither loss of meaning nor reification.

353. Relational restabilization: The result enters into new relations that change or extend its role. A drift of meaning is possible here; it is not proven solely by the fact that the original derivation is missing.

36Reification as a strong thesis to be tested: A determination is taken over from its original context, restabilized in a new order, and finally read as if its later role were the original, independent constitution of the object. The strongest proof would be a documented back-reading: R1 determines E5 anew, and a later reader attributes E5\(R1\) to R0. Mere continued calculation, teaching, or successful application is not enough.

37Fallacy of continuation: That E5 is presupposed in E6, E7, and E8 initially proves only that the continuation proceeds from E5. It does not prove that E5 is the complete grounding of E1–E4, from which E5 emerged. Success of continuation documents operative capacity, not the complete origin or the identity of meaning of the presuppositions.

38Textual finding and historical reconstruction: The Greek passages show concrete linguistic and mathematical relations of determination: Heraclitus’ enactment of separating and coming-together; Democritus’ transmitted differences of shape, arrangement, and position; Euclid’s number as a plurality composed of units and his constructions; Plato’s examination of divisions; Aristotle’s investigations of limit, division, measure, and mathematical mode of consideration; Archimedes’ quantitative determination of floating behavior through comparisons of body, volume, and fluid. The Latin sources show translation decisions, category orders, and changing uses of substantia, essentia, subiectum, qualitas, quantitas, accidens, and relatio. Copying, commentary, teaching list, and instruction reproduce these concepts, but at the same time also variants, corrections, and competing readings.

39Mathematical and physical finding: Chapter 14 traces how construction, ratio, numeral, calculation, algebra, diagram, and measurement enter into new working contexts. Scheiner and Galileo show that a published observation can receive a new evidential function in another cosmological argumentation. Gassendi reads ancient atomism through a system of his own, revises interpretations, and at the same time preserves relations such as location and order. Newton connects relational measurements with theoretical magnitudes such as absolute time and space; Leibniz shows that symbolic operationalization can preserve relational justification; Euler gives functions and sine/cosine new analytic roles that can be applied back to geometry, without therefore being identical in meaning with their geometrical point of departure.

40Foundations, number, and zero: Riemann makes the presuppositions and empirical reach of geometrical systems explicitly visible. Hilbert determines geometrical concepts through axiom relations; several models show that names alone do not guarantee fixed meaning. Gödel encodes signs and proof relations by numbers; Gödel numbers receive a new syntactic role and are not identical in meaning with counted pluralities. Euclid’s unit and number, Ptolemy’s place signs, Brahmagupta’s operable śūnya, and von Neumann’s empty set are different modes of determination of zero. A Greek zero as ontological origin is not documented. Dedekind reflects both assignment in counting and the construction of a successor structure; his remark about shortened chains of inference shows the possibility of skipping steps, not the general loss of relational meaning.

41Kant, Hilbert, and Göttingen: Kant describes number synthesis and unity of apperception; Hilbert cites Kant, but gives the quotation a new function within his preface to axiomatics. This documented textual relation proves neither identity of meaning nor a genealogy from Kant through set theory to Gödel. §16.7 reconstructs feedback loops between physics and mathematics in Riemann, Minkowski, Einstein/Hilbert, and early matrix mechanics. Hilbert’s lecture of 1900, in the same text, describes the mathematical continuum as an axiomatically ordered “system of things” and calls for the mathematical elaboration of the transition from the atomistic gas theory to the continuum. Together with Boltzmann’s counting of molecules, this shows how mathematical determinability helps decide what counts in the theory as a physical particle. §16.7.1 traces this mechanism in the case of the atom, from Democritus and Gassendi through chemical weights, Thomson, Rutherford, and Bohr to matrix mechanics. The changes in the order of determination documented there are not an unbroken genealogy; the explicit back-projection of the modern atom onto Democritus is found in selected encyclopedic and popular accounts. Not documented are a derivation of the concept of the atom from set theory or a literal equation atom = number. The relational thesis concerns the possible category error of fixing the physical thing itself through its mathematical determinability and then treating the object so determined as an independently given presupposition of further calculations; this is an interpretation of the practice, not an explicit statement of commitment by the historical authors. Einstein and Perrin further show how measurable motions of visible colloidal particles provide theory-mediated evidence for molecules, which then function as objects of further calculations. §16.9 connects this with Gassendi and Newton through the change in publication forms: the authors often explicitly acknowledge sources; the possible reification lies in the fact that a cited address of origin does not automatically carry the original relations of determination along into the next use. §16.10 reads Hilbert’s sixth problem closely: he describes an interplay of thought and experience, explicitly places atomistic→continuum and continuum→rigid bodies side by side in the gas problem, and in 1912 works on kinetic gas theory with integral-equation methods. This documents reciprocal determination of mathematical task and physical problem; the stronger claim that later readers equated the limit result with the original determination of the thing remains to be documented separately. Göttingen is thus an institutional point of condensation, not a linear or completed culmination.

42Education and didactic restabilization: Pestalozzi keeps counting as enactment present in instruction; Cauchy guides students through definitions and proofs; the Newcastle examination shows a difference between a practiced arithmetic operation and the new price-quantity assignment of a word problem. The atom section adds Cannizzaro’s explicit teaching of relative weights and a concrete current textbook: OpenStax leads from experimental history to definitions, calculation problems, and a simulation in which learners construct atoms through particle numbers. A study on teaching the Bohr model also shows that linking or separating model sequences can influence the concept of the atom used later. These pieces of evidence establish neither a general process of reification in schooling nor a uniform causal chain from industrialization to mass instruction. They show that definition, formula, example, exercise, examination, and next learning step themselves form a relational environment. A pupil can therefore learn a determination under the same designation in a new role. Whether this role abbreviates, extends, or overlays the older meaning must be clarified by comparing concrete teaching texts, classroom observations, and examination practices.

43Neutrino case and beginning particle systematics: The continuous β-energy distribution became readable as a balance problem because it was compared with a specific conservation order and a description of the known end products. Pauli’s hypothetical neutral particle was an interpretation of this deficit, not its mathematical identity. Fermi gave the particle a calculable role; Reines and Cowan designed an inverse β-reaction for it and in 1956 evaluated prompt-delayed coincidences, proton density, cadmium dependence, calibrations, and reactor power. The finding is operative circularity, not logical self-proof. Solar-neutrino and KamLAND/SNO results show a second pass: the already detected neutrino is newly determined through flavor, propagation, and detector channel. Gell-Mann’s particle systematics then shows how experimentally stabilized particles themselves become inputs of an order of classification from which new particles can be predicted. The Ω⁻ case documents the feedback; Gell-Mann’s reservations about free quarks limit any equation of mathematical function and physical existence. Pauli’s energy-balance deficiency and the later solar-neutrino deficit are different kinds of difference and must not be equated.

44Levels of knowledge in each section: 1. Textual finding – what the source actually says or does. 2. Historical reconstruction – which chain of transmission, teaching, and application is documented. 3. Relational inference – which role the determination has in the respective context and whether it changes. 4. Still unproven hypothesis – whether a later determination is projected back onto an earlier source. Every proposed local instance of drift remains testable by asking what source evidence would weaken or refute it; if relations explicitly function only as access to an independently determined magnitude and their role remains stable, the relational inference for that case must be withdrawn. The Tractatus sharpens these questions; it replaces none of the historical evidence.