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C. H. F. Warnke / English edition
SPACE TIME
A CONTINUUM OF ERRORS
188 pages21 chaptersDE / EN
DOI 10.5281/zenodo.24167274 ↗
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TRACTATUS PRINCIPIO SPACE TIME A CONTINUUM OF ERRORS by C.H.F. Warnke DOI: 10.5281/zenodo.24167274
Contents PART I — INTRODUCTION 1 PART II — SPACE AS A READING OF DIFFERENCE 8 1. The Confusion of Object and Position . . . . . . . . . . . . . . .9 2. The Self-Reference of Difference . . . . . . . . . . . . . . . . . 11 3. The Table as the Symbol 1 . . . . . . . . . . . . . . . . . . . . 13 4. The Difference Lies in the Reading . . . . . . . . . . . . . . . . 15 5. How a Spatial Object Appears from Reading Positions . . . . . . 17 6. The Third Dimension Is Not an Additional Substrate . . . . . . . 19 7. Photography as a Particular Reading Position . . . . . . . . . . 21 8. From Notation to Self-Relation . . . . . . . . . . . . . . . . . . 23 9. The Physical Reading of Number . . . . . . . . . . . . . . . . . 25 10. Number Is Not the Origin of Motion . . . . . . . . . . . . . . . 33 11. Measure as a Change of Reading . . . . . . . . . . . . . . . . 38 PART III — TRACTATUS NOTATION AND SEQUENCE 45
12. The Connection in Tractatus Notation . . . . . . . . . . . . . . 46 13. Physics as a Gradient Reading of Difference . . . . . . . . . . 53 14. From the Fundamental Trace to a Stable Reading of the World . 70 15. Archibald and the Error of the Inner World . . . . . . . . . . . 80 16. The Gradient: A Relation Read Back to the Observer . . . . . . 89 17. Light: The Limit of Self-Measurement . . . . . . . . . . . . . . 96 18. The Geometry of the Light Number: Mirroring, Breakpoint and Self-Reference . . . . . . . . . . . . . . . . . . . . . . . . . . 103 19. The Reading Is Determinate, Not the Continuation . . . . . . 119 20. The Universe We Never See Simultaneously . . . . . . . . . . 131 21. The Question That Forgets Its Premise . . . . . . . . . . . . . 157
SPACE TIME · PART I — INTRODUCTION
PART I — INTRODUCTION
Archibald and the Error of the Third Dimension
As a child I watched an animated series called Archibald the
Space Traveller. Together with his dog, Archibald travelled
through space and, in one episode, landed on a most peculiar
planet: its inhabitants were two-dimensional.
For a child who experienced his own world as three-dimensional
without giving it a second thought, the idea was fascinating. A
whole world without a third dimension! How could those beings
live and move without the spatial depth I had until then regarded
as an almost self-evident prerequisite of any movement?
Archibald arrived as a visitor from a three-dimensional world and
encountered beings whose reality was confined to two
dimensions. At least that was how I understood the story. I had
completely missed something remarkable: Archibald himself was
two-dimensional.
Not only the inhabitants of the strange planet but Archibald, his
dog, his spacecraft and the entire universe through which they
travelled appeared on a two-dimensional picture surface. I was
therefore watching a two-dimensional representation of an
apparently three-dimensional world, in which an observer
depicted as three-dimensional discovered another, supposedly
two-dimensional world. And it was precisely his
three-dimensionality that I had taken for granted.
1
SPACE TIME · PART I — INTRODUCTION
Yet the picture surface itself possessed no corresponding spatial
depth whatsoever. That depth arose only in my perception. But
even this was not quite the point.
The Film That Does Not Move
Consider an ordinary motion-picture film running at twenty-four
frames per second. Each image presents a particular, internally
complete arrangement. A person stands in a particular place, an
object has a particular position, a landscape takes on a particular
form. Nothing moves within the single frame. Only through the
perception of those frames in their ordered succession do we
experience motion.
We see a person walking and a spaceship flying, although every
single image presents merely a determinate appearance. We
experience continuous motion that is contained in none of those
individual images by itself. Yet we do not doubt for an instant that
Archibald really is travelling through space. Of course a film
requires temporal projection. Whether that principle also explains
reality itself is another question.
For if we understood the world merely as a succession of cosmic
still frames, we would already have to assume an external time in
which these images could succeed one another. Time would not
have been explained. We would simply have invented an invisible
film projector.
2
SPACE TIME · PART I — INTRODUCTION
The Observer and His World
Our investigation therefore begins not with motion but with
distinction: an observer distinguishes himself from his world. Yet
neither the observer nor the world need be presupposed as two
finished objects existing independently of one another.
They appear as the two poles of one and the same distinction.
Observer | World
This distinction has a remarkable property. The observer
distinguishes not only his world from himself but also his own
relation to that world.
The distinction itself is distinguished again.
Difference refers back to itself.
The Tractatus Principio denotes this operative self-reference by:
Δ := |(|)
This self-referentiality is not an event that would first have to take
place within an already existing time. It is the condition under
which different relational configurations can become mutually
readable at all. The Tractatus calls such a complete relational
configuration a Frozen Frame. Nothing moves within such a
frame. It is not an infinitesimal interval during which reality
briefly stands still. It denotes a complete order of distinctions that
presupposes no temporal duration of its own.
Only when the observer relates different complete relational
configurations to one another does the coherent order arise that
we experience as temporal continuation. Recurring differences
appear to us as enduring objects, altered relations as motion, and
3
SPACE TIME · PART I — INTRODUCTION
the coherence of those alterations, finally, as time. Time does not
bring distinctions into being. It is a reading of their self-reference.
The Surface of Reality
Let us return to Archibald.
His three-dimensional world appears on a two-dimensional
surface. His motion appears through the relation between
motionless pictures. Both appearances are immediately real to the
viewer, even though neither requires a third spatial dimension in
the picture surface nor motion within a single frame.
What, then, if our own three-dimensional perception were also a
geometrical reading of a more fundamental self-relation?
What if the depth we experience did not designate an independent
third fundamental dimension but a perspective on the recursive
depth of a topological-geometrical relation?
Then the surfaces of things would not be mere outer shells
concealing progressively smaller material objects. Each further
examination of a surface would bring forth distinguishable
surfaces once again. We would not discover a final content
existing without relations; we would repeat the original
distinction under altered conditions. Instead of countless mutually
independent spaces with concealed interiors, there would be one
connected relational order whose self-distinctions appear as
different surfaces, objects and depths. And now we can turn the
whole consideration around once more.
4
SPACE TIME · PART I — INTRODUCTION
Why a Photograph Is Possible at All
We are accustomed to treating a photograph as an image of
reality. The world is there, we point a camera towards it and
capture a fragment for a brief instant. This seems perfectly
obvious. But it already assumes what we ought to be explaining.
Why can a world appear in a two-dimensional image at all?
Why can spatial relations be recorded on a surface without losing
their spatial readability entirely?
And why can we recognise in a motionless photograph not only
forms but objects, distances and even indications of movement?
During exposure, a camera registers a particular light relation. Its
optics, orientation and recording surface determine which
differences can be stabilised as an image. What we obtain is
therefore not simply a miniature world but a particular, fixed
reading of its distinguishable relations. And yet we recognise a
spatial world in this two-dimensional representation. We see
nearness and distance, foreground and background, bodies and
objects.
Sometimes a single photograph is enough for us even to recognise
movement: a person jumping, an object falling, a treetop bending
in the wind.
We do not see movement itself.
We read a visible relational configuration in connection with other
possible configurations. Thus the photograph shows us not merely
how a three-dimensional world can be represented
two-dimensionally. It shows that spatial readability need not be
identical to the number of dimensions of its representation.
5
SPACE TIME · PART I — INTRODUCTION
And so our original question is reversed.
Perhaps photography is possible not because reality consists of
finished three-dimensional objects that can be projected onto a
plane. Perhaps we ought instead to investigate whether
photography is possible precisely because spatial objecthood is
itself already a stable reading of relational distinctions. In the
language of the Tractatus, a photograph would not be a piece of
reality lifted out of time. It would be the fixed appearance of a
particular measurement relation, readable as a complete,
distinguishable image-order.
And this is the deeper meaning of the Frozen Frame. We can
recognise a world in a motionless image because movement,
objecthood and spatial depth need not be present in that image as
finished contents. They become readable through distinctions and
their relations.
This does not prove the entire ontology of the Tractatus. But it
leads directly to the question from which our investigation begins:
If a two-dimensional representation is enough for us to recognise
a three-dimensional world, what, exactly, is the third dimension?
Is it an original property of reality?
Or a perspective on the self-reference of distinction?
And if time is experienced only through the coherent relation
between complete relational configurations, why do we
presuppose a four-dimensional spacetime substrate in order to
explain these appearances?
6
SPACE TIME · PART I — INTRODUCTION
Perhaps the error does not begin in the equations of physics.
Perhaps it begins when we mistake their geometrical readings for
reality itself.
Archibald believed he had discovered a two-dimensional planet.
Perhaps he should first have asked why his own world fitted onto
a two-dimensional surface at all.
7
SPACE TIME · PART II — SPACE AS A READING OF DIFFERENCE
PART II — SPACE AS A READING
OF DIFFERENCE
8
01 The Confusion of Object and Position
The Confusion of Object and Position
CHAPTER 01
1. The Confusion of Object
and Position
We ordinarily regard space as something in which objects are
located. A table stands in a room; one observer stands in front of
it while another looks at it from the side. We explain their
different perspectives by their different positions in
three-dimensional space. But this already presupposes what we
are trying to explain.
We assume that the table is a finished three-dimensional object,
that the observers are independent bodies, and that a space exists
in which their positions differ. On that basis their perspectives
can be calculated perfectly well. Yet we have not explained why
spatially different perspectives can arise at all.
For this we must begin with distinction itself.
In the Tractatus Principio,
|
denotes pure distinction.
It is not a line drawn in already existing space. Initially it denotes
neither length nor width, neither direction nor distance. It is the
condition under which something can appear as distinct from
something else at all.
Its polar reading is:
A|B
9
The Confusion of Object and Position
A and B do not designate two already finished objects.
They are the two distinguishable sides of the same relation.
Whether we call them observer and world, observer A and
observer B, or even hand and table does not alter the fundamental
structure.
It remains the same distinction.
Even if we speak of two observers, we have not introduced two
independent origins that must subsequently be connected. For
their distinguishability is already a reading of A|B.
The number of observers does not change the original relation. It
denotes different possibilities of reading that same relation.
10
02 The Self-Reference of Difference
The Self-Reference of Difference
CHAPTER 02
2. The Self-Reference of
Difference
The activity of difference does not end with making two poles
distinguishable. Difference can refer to its own distinction. Not
only can A be distinguished from B; the way in which A and B are
distinguished can itself be distinguished.
Difference distinguishes its own difference.
The Tractatus Principio denotes this self-reference by:
Δ := |(|)
This is not a process that begins after the original distinction in an
already existing time. Pure distinction, its polar reading and its
self-distinction operate simultaneously and recursively within one
another. That is precisely what allows different perspectives on
one and the same relational order. A perspective is not yet a
location in a finished space. It is a particular fixation under which
the relation is read. The possibility of another perspective already
belongs to the self-reference of that same distinction.
This requires no second observer as an additional cause. Another
observer can embody a different reading of the same order, just
as a single observer can alter his own reading.
In both cases the original relation remains the same.
11
The Self-Reference of Difference
Multiple observers do not create multiple original realities. The
self-reference of difference permits different readings of one
common reality.
12
03 The Table as the Symbol 1
The Table as the Symbol 1
CHAPTER 03
3. The Table as the Symbol 1
To make this clearer, consider a table. We designate the stable
relational form recognisable as that table by the symbol:
1
The one does not assert that the table is an indivisible material
unit. It is the chosen designation for the relation that remains
readable as the same table from different perspectives. We now
need a way to mark different reading positions without already
presupposing spatial coordinates.
Here the comma takes on a special meaning.
In our notation it does not merely separate the whole and
fractional parts of a numerical value. It marks the shift from the
designated relation to the position from which it is read.
Consider:
1,1
and
1,5
Both expressions contain the same one.
The table is still designated as the same relational form.
Only the reading position after the comma changes.
13
The Table as the Symbol 1
The first notation designates the table from reading position 1;
the second designates the same table from position 5. Thus two
distinguishable appearances can be read without having to
assume two tables. Equally, we obtain two reading positions
without first assuming a three-dimensional space.
The one carries the recognisable object relation.
The comma marks the change of reference level.
The number that follows designates the fixation under which that
relation appears. The object remains readable as a relation while
its perspective differs.
14
04 The Difference Lies in the Reading
The Difference Lies in the Reading
CHAPTER 04
4. The Difference Lies in the
Reading
Now let us assume two observers.
Observer A reads the table as:
1,1
Observer B reads it as:
1,5
Conventional geometry would say that A and B occupy two
different locations in an already existing space. But we can
describe the situation without that assumption.
Both readings refer to the same one.
They differ in their fixation.
The relation is:
1,1 | 1,5
What is being distinguished here?
Not two independent tables, nor two separate worlds, but
different reading positions of the same object relation. And the
difference between these reading positions is once again a
distinction within the same relational structure.
Thus difference refers back to itself once more:
15
The Difference Lies in the Reading
Δ := |(|)
We therefore require no additional space to relate the reading
positions. Their distinctness is itself a reading of the original
relation. It is important not to rush into subtracting the digits
after the comma as though they were ordinary numbers. Reading
positions 1 and 5 alone do not imply a spatial distance of four
units. That would already require a scale and a rule under which
different reading positions could be measured as distance.
So far we know only their difference.
Their spatial significance must emerge from the relation between
the two readings. The difference between reading positions is
more fundamental than the distance we later read from it.
16
05 How a Spatial Object Appears from Reading Positions
How a Spatial Object Appears from Reading Positions
CHAPTER 05
5. How a Spatial Object
Appears from Reading
Positions
Observer A sees the table under a particular fixation. Its front
may be especially clear to him while another side remains hidden.
Observer B views the same table under another fixation. A side
face may now become visible while the previously visible front is
foreshortened. The appearances differ, yet we can recognise them
as appearances of the same table. This requires the relation
between the readings to be preserved. Any two arbitrary pictures
do not automatically make a spatial object.
Only if their differences can be understood as mutually
compatible readings of the same relational structure does a
common spatial interpretation become possible. The table's
recognisability is therefore not simply an attribute of a finished
body that we have to presuppose. We can investigate it as an
invariance between different reading positions.
The one denotes this recognisable relation.
The places after the comma distinguish its modes of appearance.
It makes no difference here whether these readings are
performed by two observers or by one observer adopting different
fixations.
Difference remains self-referential.
17
How a Spatial Object Appears from Reading Positions
And this makes the emergence of spatial objecthood
understandable.
We regard a table as a three-dimensional body because we can
read its various possible views as appearances of one stable
relation. We never see every side of the table simultaneously. Yet
the other reading positions remain accessible as possible
continuations of our present reading. The object's spatial
completeness arises from this belonging-together. What we
understand as a body is the stable readability of a relation across
different perspectives.
18
06 The Third Dimension Is Not an Additional Substrate
The Third Dimension Is Not an Additional Substrate
CHAPTER 06
6. The Third Dimension Is
Not an Additional Substrate
We can now return to the question of space. In the Tractatus
Principio, the fundamental geometrical order is developed from
positing, distinction and the relations among distinctions. Volume
II describes these readings as radius, diameter and triangle.
Volume VI traces spatial appearance back to the observer-related
depth-reading of a two-dimensional relational network. Combined
with our notation, this provides a more concrete approach. The
one designates a stable relation within the structure; the comma
opens its reading under a particular fixation.
The number following it makes that fixation distinguishable.
Different fixations do not necessarily generate different objects.
They alter the appearance of the same relation.
And insofar as these appearances remain mutually compatible, a
depth of reference becomes readable—what we ordinarily call the
third spatial dimension. The third dimension would therefore not
be an additional original extension that first had to be added to a
surface, but the spatial reading of a relational order's
self-reference. An object's surface would then not merely be the
outer limit of a finished three-dimensional body.
It would be a particular appearance of that relation.
What we call the back, an interior or spatial depth would, under
different fixations, become distinguishable again as a surface or
19
The Third Dimension Is Not an Additional Substrate
partial surface. The connection need not disintegrate into several
independent geometrical planes. The same relational order
unfolds in its various readings. Depth is not necessarily located
behind the surface. It becomes accessible in the surface's other
possible readings. This suggests one topological-geometrical
self-relation from which differing spatial appearances can arise.
We still have to show which relations between reading positions
remain invariant and how their coupled variability can produce a
measurable three-dimensional order. But we need not presuppose
three-dimensional space as the foundation of that explanation.
20
07 Photography as a Particular Reading Position
Photography as a Particular Reading Position
CHAPTER 07
7. Photography as a
Particular Reading Position
Our photograph can now be understood more precisely.
Suppose we photograph the table from reading position 5. The
photograph shows a particular appearance of the relation:
1,5
It does not contain all the table's other possible readings. Yet we
recognise the depicted object as a table.
Why?
Because we can connect its visible appearance with other possible
reading positions of the same relation. We know that another
photograph would present the table differently. We could examine
a side, photograph the back or choose a different perspective.
The appearances change.
The recognisable relation is preserved, insofar as the readings
remain compatible. The photograph is thus not a miniature
three-dimensional world enclosed within a two-dimensional
surface. It is a stabilised appearance under a particular reading
fixation. The third dimension need not be contained in the image
itself. It is inferred from the relation between the visible image
and the other possible readings of the depicted connection. Here
we return to our introduction: Archibald appears as a
21
Photography as a Particular Reading Position
three-dimensional space traveller on a two-dimensional picture
surface.
His spatial appearance does not arise because the picture surface
itself has a third extension. It arises because the represented
differences can be read as spatial relations. A film additionally
connects different complete image configurations, which we
experience as motion. Whereas a photograph reveals the spatial
readability of a single registered appearance, film shows how
connecting different complete appearances can be experienced as
temporal continuation.
Both bring us back to the same basic question:
How does the self-reference of a difference become a reality
readable in space and time?
22
08 From Notation to Self-Relation
From Notation to Self-Relation
CHAPTER 08
8. From Notation to
Self-Relation
We now have a starting point that does not yet presuppose
finished space.
The one denotes the table as a recognisable relation.
The comma marks the change to the reading position.
The following number designates a particular fixation.
We can therefore write two different readings of the same table
as:
1,1 | 1,5
The two appearances remain distinguishable while their common
relation is preserved. Distinguishing between these readings is
again self-distinction within one order.
Thus we return to our origin:
Δ := |(|)
The mathematical task now becomes precise.
We must determine under which conditions different reading
positions preserve the same relation, how their differences relate
to one another, and when these differences yield a stable
geometrical reading of depth.
Only then can distance be derived.
23
From Notation to Self-Relation
Only then can a spatial position emerge as a measurable quantity.
Only then can we develop an unscaled geometry whose spatial
reading has not already been built into its assumptions. The aim is
not to use the digits after the comma as disguised coordinates. It
is to make visible the relation that permits a coordinate reading in
the first place. The table is not first situated at a position in space.
Its position is a reading of the relation in which it appears as a
table.
Space is not a container holding those different readings
together.
It is their geometrically readable self-relation.
24
09 The Physical Reading of Number
The Physical Reading of Number
CHAPTER 09
9. The Physical Reading of
Number
Let us return once more to the table.
We designated it by the one and distinguished two reading
positions:
1,1 | 1,5
Ordinarily we would treat this as a relation between two numbers,
1.1 and 1.5, whose arithmetic difference is 0.4. But that would
already be a mathematical interpretation without asking which
relation the numbers designate.
The Tractatus Principio offers another approach.
In Volume I, the decimal point is not understood merely as the
separator of an ordinary decimal number. The sign |. denotes a
scale flip: a change of reference level within the same numerical
relation.
Applied to our example, we write the two readings as:
1|.1
and
1|.5
In this geometrical working notation, the first one denotes the
table as a recognisable relational form. The scale flip opens
25
The Physical Reading of Number
another reference level. The following digit designates the
position from which the same relation appears.
The one itself has not been altered.
What changes is the way it is read within the structure.
The Observer as Fixation
Volume II designates the observer by ¤.
The sign does not name an independent substance standing
outside reality. It denotes the relational reference point from
which distinction appears as perception.
Volume II distinguishes the directions of unfolding and folding
back:
0|₊Δ – expansion
0|₋Δ – contraction
0|₌Δ – equilibrium of perception
These directions do not designate independent forces or spaces.
They are different readings of difference's self-reference.
One relation that Volume II formulates explicitly is decisive here:
0|(A|₊Δ) ≡ 0|(B|₋Δ)
The two sides display the same relational order under opposite
points of reference.
What changes is the orientation of the reading.
26
The Physical Reading of Number
What remains common is the self-relation preserved under both
orientations. This explains why differing observer perspectives do
not demand different original realities. The observer is himself a
fixation within the structure. A different fixation makes another
aspect of the same structure readable. The distinction between
reading positions is therefore not something we must first explain
through an existing space.
It belongs to the self-reference of the relation.
The Table Remains the Same—Its Reading Does Not
Consider again:
1|.1 | 1|.5
Up to now these have appeared as two readings, each beginning
with the one.
But the duplicated one is instructive.
We do not have two tables.
We have the same relation recognisable as a table under two
reading positions.
Why, then, should we write the one twice?
If both readings return to the same self-relation and can be read
together within it, we may contract their representation:
1|.1 | 1|.5
becomes:
1|.1|.5
27
The Physical Reading of Number
The one need not appear twice because it denotes the common
relational form. The reading positions remain distinguishable but
become readable within the same contracted relation. This differs
fundamentally from ordinary arithmetic. We have neither added
nor subtracted the numbers 1.1 and 1.5, nor produced a third
object out of two independent objects. We have made explicit the
common self-relation of different readings.
This contraction does not mean that arbitrary expressions of the
same numerical value are interchangeable. It is meaningful only
where the readings actually return to the same relational form
and are mutually compatible.
The expression
1|.1|.5
thus designates the table's joint readability under the two
distinguished fixations.
What remains invariant?
The one as the recognisable relational form.
What changes?
The reading position.
What is added?
Not another object, but an explicitly readable relation between
different perspectives. For an observer who perceives the table
from position 1, position 5 is initially another possible
perspective.
When he adopts that perspective, no second table appears.
28
The Physical Reading of Number
Instead another side of an already distinguishable relation
becomes accessible.
The unity of the connection is preserved.
And that is precisely what makes spatial objecthood possible.
The table can appear as a spatial body because its different
readings are not unconnected pictures.
They remain bound by their common self-relation.
From Scale Flip to Spatial Depth
We can now grasp the scale flip more precisely. Reading positions
.1 and .5 designate different fixations within the same relational
order. Changing between them is not the motion of a finished
table through an already existing space.
It designates an altered reading of the same structure.
The other reading position may reveal another surface.
A previously hidden side can come into view.
A previously visible face may appear foreshortened.
One orientation can be replaced by another. Yet these
appearances remain connected as long as they can be traced back
to the same self-relation.
That is exactly what our contracted notation expresses:
1|.1|.5
We no longer have to treat the two perspectives as separate
objects or positions inside an already given space.
29
The Physical Reading of Number
Their distinctness is readable within a common relation.
This opens a decisive possibility for our concept of space:
An object's depth can be understood as a geometrical reading of
the relations among its possible appearances. We need not
presuppose depth as an additional spatial extension. This also
gives greater precision to the idea of a single
topological-geometrical surface. The table's different sides are not
independent realities assembled inside a spatial container. They
are different readings of an order that distinguishes itself. Further
surfaces appear through altered fixations within that order.
The observer remains part of the same relation.
Whether one observer changes his perspective or two observers
read the relation differently makes no difference to the original
structure.
It remains:
A|B
And, in its self-reference:
Δ := |(|)
The different spatial readings therefore form no additional
origins. They designate different perspectives on the same
self-reference.
30
The Physical Reading of Number
The Decisive Reversal
Ordinarily we would say:
Because the table is three-dimensional, we can examine it from
different perspectives.
Our investigation reverses the direction of explanation:
Because the same relational order remains differently yet
coherently readable under different fixations, it can appear as a
three-dimensional object.
The contraction
1|.1 | 1|.5 → 1|.1|.5
makes this immediately visible.
The one is not doubled.
The reading positions are not declared independent spatial
coordinates.
They remain different determinations of the same relation.
Nor may we immediately equate them with metric distances. The
digits 1 and 5 alone do not imply a measured distance. Only after
the directed relations among their readings have been worked out
can a geometrical measure of distance arise. This is precisely why
the notation of the Tractatus is useful: it lets us read the
recognisable relational form, the reference-level change and the
different fixations together without presupposing them as
independent things.
Our original question now becomes concrete:
31
The Physical Reading of Number
How can the self-relation among different reading positions
within one relation give rise to a stable, three-dimensionally
readable geometry?
This is where the actual calculation of space begins.
32
10 Number Is Not the Origin of Motion
Number Is Not the Origin of Motion
CHAPTER 10
10. Number Is Not the Origin
of Motion
Before continuing we must rule out a possible misunderstanding.
We designated the table by the one and distinguished its reading
positions by digits following a scale flip:
1|.1|.5
This might suggest that we presuppose numbers and then
construct the spatial world out of them.
That would be a fundamental error.
For in the Tractatus Principio, neither mathematics nor physics
begins with number.
They begin with difference.
|
Its polar reading is:
A|B
And its operative self-reference:
Δ := |(|)
These three orders do not operate as successive steps in time.
They are different orders of the same distinction, acting
simultaneously and recursively. Only through this self-reference
33
Number Is Not the Origin of Motion
do recurrence, orientation, depth of reference and stable readings
become possible. What we subsequently call a number is one such
reading. We therefore do not use numbers to bring relations into
being. We use numbers to make readable a relational order that
has already emerged from distinction.
The One Is Not a Presupposed Table
When we designate the table by one, we do not claim that the
number 1 creates an object. Nor do we claim that the table is
originally a mathematical unit.
The table is a recognisable mode of appearance of a relation.
Only afterwards do we denote this recognisable form by the
visible mark 1. The one is not a prerequisite of the table's
existence but the selected reading of its relational recognisability.
The same applies to the digits used to distinguish its reading
positions. They are neither independent places nor ready-made
coordinates.
They are visible addresses of distinguishable fixations.
What the Scale Flip Means
The sign |. must not be misunderstood as an additional
fundamental operation.
Volume I of the Tractatus Principio develops the decimal point as
a scale flip: a shift in perspective to another reference level of the
same numerical relation.
When we write:
34
Number Is Not the Origin of Motion
1|.1
and
1|.5
we are using this previously developed possibility of reading.
The common one denotes the recognisable relational form.
The places after the scale flip mark distinct fixations of its
reading.
Since there are not two independent tables, we may contract the
shared relational form in our working notation:
1|.1 | 1|.5 → 1|.1|.5
However, this contraction is permissible only if both readings can
actually be traced back to the same relational order.
The common digit alone is insufficient.
Two arbitrary appearances do not become identical simply
because we label both with a one. The contraction must express a
preserved self-relation, not substitute for it.
The Intermediate Readings Do Not Create Time
Consider a possible continuation:
1|.1|.2|.3|.4|.5
Here a second danger arises.
35
Number Is Not the Origin of Motion
One might imagine that the numbers 1, 2, 3, 4 and 5 already
contain a temporal sequence and therefore bring motion into
being.
But this, too, would be wrong.
The numerical sequence does not generate continuation. A
directed continuation can be made readable through a numerical
sequence. The mere arrangement of digits does not prove that
these reading positions are connected in precisely that order. The
self-referential operation must sustain a corresponding directed
relation. Its trace must preserve which reading is connected with
another and in what manner. Only then can the succession be
read as motion. The digits are visible marks of this order, not its
cause.
Here we encounter the Frozen Frame from Volume II again.
Every complete relational configuration is motionless in itself.
What appears as motion is the coherent connection of different
complete relational configurations through the observer's
self-measurement. When that connection is read in a directed
manner, we experience temporal succession. Yet the directed
connection requires no external time to set it going.
Time is its experienced reading.
36
Number Is Not the Origin of Motion
Spatial and Temporal Readings of the Same
Self-Relation
We can now distinguish two modes of appearance.
Under a spatial perspective, different fixations of the same
relational order appear as different views and locations of an
object. Under a temporal perspective, the directed connection
between those fixations appears as change. We therefore need
not first draw a number line and assemble motion out of its
points. Nor do we require a three-dimensional coordinate scheme
in which reading positions are already fixed as places.
Our investigation begins prior to both representations.
It begins with distinction and its self-reference.
Only afterwards do numbers serve to represent the different
readings of this self-relation.
This also changes what we mean by calculation.
We are not seeking the number to assign to an already existing
position. We seek the relation among distinctions that remains
preserved when the reading perspective changes, and the
directed continuation that becomes readable from it. Only after
that may a numerical difference appear as a measure.
Only after that may spatial distance be calculated.
Only after that may an ordered continuation be read as a measure
of time. We do not explain space and time through numbers. We
investigate how number, space and time become different
readings of the same self-reference.
37
11 Measure as a Change of Reading
Measure as a Change of Reading
CHAPTER 11
11. Measure as a Change of
Reading
We represented the different perspectives on our table as reading
positions of the same relation.
Their common connection is:
1|.1|.5
Until now we have merely identified their different fixations. We
do not yet know what measures the change in reading position.
Ordinarily we would measure the distance between two positions,
using an existing spatial scale such as a metre. But that would
again presuppose the very space whose emergence we wish to
explain.
We therefore have to determine our measure differently.
Initially our measure is the accomplished change of reading itself.
Measuring an As-Yet-Open Position
Suppose an observer reads the table under fixation .1. We want to
investigate its relation to another reading position that has not yet
been determined.
We denote this open position by x.
Our question is:
1|.1|x
38
Measure as a Change of Reading
The one still designates the common relational form.
.1 designates the present reading fixation.
x is not a position already found in space. It denotes the open
determination of another reading to which the present one is to
be related.
Now we perform the measurement.
The observer determines his position anew within the relation to
his world.
The Tractatus Principio describes this original measurement as
the observer's self-measurement:
¤|World
Here the observer is not measuring an object independent of
himself from somewhere outside the relation.
He determines his own place within the structure of differences.
Suppose the measured reading configuration yields:
x = .5
Then the determined relation is:
1|.1|.5
A difference has now been accomplished.
The initial reading .1 and the measured reading .5 are determined
within the same relational form.
But what is the measure of this change?
39
Measure as a Change of Reading
Four Steps of Difference
We can consider the measured relation under a directed
continuation.
If we choose the reading in which the positions are connected
through distinguishable intermediate configurations .2, .3 and .4,
we obtain:
1|.1|.2|.3|.4|.5
Here we distinguish four transitions:
.1 → .2
.2 → .3
.3 → .4
.4 → .5
The number four was not introduced as a presupposed physical
length. The 4 is a symbol for 1 + 1 + 1 + 1. Each one denotes a
distinguishable enactment of difference. Nor is the plus sign an
original arithmetic operator that first produces the steps. It
makes visible their repeated continuation, here read expansively.
We therefore did not start with the number four and manufacture
four steps out of it.
We have four distinguished enactments whose joint reading we
denote by 4. The number designates the reading of the recurrence
of difference, and must not itself be assumed as the prerequisite
for that recurrence. We do not first measure a spatial distance
and then explain motion by it. We read the change of the relation
40
Measure as a Change of Reading
under a chosen, continuous order of distinction. Only afterwards
do we name that change four.
One condition is decisive: the intermediate readings must
genuinely belong to the same directed, mutually compatible
continuation. The digits .1 and .5 alone do not yield this particular
trace. Number counts the read enactment. It does not generate it.
The Retrocausality of Measuring Position
Here the significance of retrocausality in Volume II comes into
view.
We ordinarily picture the process as follows:
An observer occupies a starting position, passes through several
intermediate positions and finally reaches a destination.
The sequence seems to bring the destination into being.
But such a description already assumes that positions succeed
one another along an existing time axis.
The Tractatus Principio reads the relation differently.
Every Frozen Frame is a complete relational configuration.
The measured position .5 does not come into being only when an
external time has played the preceding positions one after
another. The measurement determines a complete relational
configuration in which the previously open reading position
appears as .5. From that determined configuration, the
connection back to the initial reading .1 becomes readable as a
coherent continuation. The present determination thus orders the
relations we subsequently understand as the path travelled.
41
Measure as a Change of Reading
It is not merely the path that produces the measured reading
position; rather, the determined reading position establishes
which connection to it can be coherently read as a path.
That is the retrocausal reading.
It does not mean that an effect travels backwards through an
existing time. It means that a determined relational configuration
makes its own history readable as a connection between complete
configurations. Our four transitions are not four instants that first
had to elapse before .5 could exist. They are the directed
continuation that we read from the measured configuration as
belonging together. Another permissible continuation might
connect the same endpoints and yet carry a different trace.
The measured result thus determines the conditions of coherence
without uniquely prescribing every intermediate relation through
its digit alone.
Observer and World Remain the Same Distinction
We therefore need neither an independent observer nor an
independent world as two original things.
Self-measurement remains a reading of:
A|B
And because distinction refers to its own distinction, its operative
origin remains:
Δ := |(|)
The measured x is an open determination within that self-relation.
The reading position .5 is its determined appearance under the
42
Measure as a Change of Reading
chosen fixation. The four transitions designate the directed
continuation connecting both reading configurations in our
example. Different modes of appearance can be gained from this
one connection. From a geometrical perspective we read it as a
positional difference; from a directed perspective as motion;
through the coherent connection of complete Frozen Frames as
temporal succession.
And from a numerical perspective we designate the four
distinguishable transitions by the number four. These are not four
different original processes.
They are different readings of one and the same self-reference.
The Original Measure
This changes the significance of measurement.
The conventional picture of space asks:
How far did the observer travel from position 1 to position 5?
Our investigation asks an earlier question:
What change in his reading-relation to the world has become
recognisable in the determined relational configuration?
The first question already presupposes spatial distance.
The second begins with self-measurement.
Measure is initially the recognisable, traceable change of relation.
Numerically counting its transitions is one possible reading of this
measure. An ordered change can thus become readable from an
unscaled self-relation before we attach an external unit of length
43
Measure as a Change of Reading
or time to it. We have thereby gained a first approach to tracing
position and motion back to the same origin.
And the decisive consequence is:
We do not measure the motion of an object through a
presupposed space. We measure the alteration of the reading
under which a relational order appears as spatial and in motion.
Time, then, is not the measure that first makes this change
possible.
It is the experienced coherence of its directed connection.
44
SPACE TIME · PART III — TRACTATUS NOTATION AND SEQUENCE
PART III — TRACTATUS
NOTATION AND SEQUENCE
45
12 The Connection in Tractatus Notation
The Connection in Tractatus Notation
CHAPTER 12
12. The Connection in
Tractatus Notation
The sequence of signs that follows traces the argument of our
derivation, not a temporal sequence in which difference came into
being. We distinguish the signs of the Tractatus from their
application to our table. The arrows indicate changes of reading
or acts of determination, not the transport of an object through a
presupposed time.
I. Origin: Difference and Self-Reference
|
A|B
Δ := |(|)
| denotes distinction, A|B its polar readability, and Δ its operative
self-reference. These orders operate simultaneously and
recursively, not as things generated one after another. A and B
are not independently presupposed objects.
46
The Connection in Tractatus Notation
II. The Physical Reading: Observer | World
A|B —reading→ ¤|World
(¤|World)|¤
In Volume II, ¤ designates not a person standing outside the
world but a point of fixation within the relation of perception. The
second expression makes readable the return of measurement to
the observer's own position. Observer and world remain the two
poles of one distinction.
0|₋Δ ↔ 0|₊Δ
0|¤|₌Δ
0|(A|₊Δ) ≡ 0|(B|₋Δ)
Contraction, expansion and their perceptual reading are different
orientations of the same relation. Changing the reference point
introduces no second origin.
III. Object and Change of Reference Level
Table —designation→ 1
1|.1
1|.5
1 is the visible mark subsequently assigned to a recognisable
table relation, not a number from which the table comes into
being. |. is the scale flip from Volume I. In our working reading,
the characters following it designate fixations of the reading.
47
The Connection in Tractatus Notation
1|.1 | 1|.5 —contraction→ 1|.1|.5
Contraction is permissible only insofar as both views belong to the
same recognisable relation. It is not a universal algebraic
cancellation rule.
IV. Open Self-Measurement and Determinate Reading
1|.1|x
¤|World
1|.1|x —self-measurement→ 1|.1|.5
x stands for an open determination of reading, not a presupposed
unknown spatial coordinate. .5 is the measured determination in
the relevant perspective. What is measured is the change of the
observer–world relation, not a distance within an already existing
coordinate grid.
V. Frozen Frames Are Complete
|_| ≡ 0|₌Δ
¤: |1_|2 → |1_Δ2
These notations are from Volume II. Subscripted 1 and 2 in |1_|2
designate two complete relational configurations; they are not the
table-one or reading positions .1 and .2.
In the words of our example:
complete frame with table-reading 1|.1
48
The Connection in Tractatus Notation
complete frame with table-reading 1|.5
The table-reading is merely one distinguishable reference within
the complete configuration in each case. There is no motion inside
either frame; only the coupling of complete configurations makes
a change of position readable.
VI. Directed Continuation
1|.1|.5
—selected coherent continuation→
1|.1|.2|.3|.4|.5
.1 → .2
.2 → .3
.3 → .4
.4 → .5
These are four distinguished transitions under an unchanged
reading rule. In the expansive perspective of Volume I we can
read their recurrence as 1+1+1+1. The symbol 4 denotes this
fourfold reading of an operation of difference. The number four
was not its origin.
1 + 1 + 1 + 1 —numerical reading→ 4
The endpoints .1 and .5 do not uniquely dictate this intermediate
sequence. The claimed four steps measure precisely this
compatible, directed trace—not every possible path between the
same endpoints, and not four presupposed metres or seconds.
49
The Connection in Tractatus Notation
VII. Coherence and Retrocausality
Volume II also formulates the coherence of the frame relation as:
|Σ = ((|_|₁|_|₂) ⇔ (|_|₂|_|₁))
The determined reading 1|.1|.5 allows us to read complete
relational configurations compatible with it as its history. If the
chosen four-step continuation is coherent, it becomes readable as
a path from the result backwards. The result does not
subsequently create past things or transmit a signal backwards
through finished time. Present fixation determines the backward
readability of a sequence of complete configurations.
1|.1|x —determination→ 1|.1|.5
1|.1|.5 —backward reading of the selected trace→ 1|.1|.2|.3|.4|.5
Thus the second arrow designates an examination or reading of
coherence, not the compulsory reconstruction of a unique
sequence from the digits .1 and .5.
VIII. Spatial Reading, Motion and Time
Spatial reading: 1|.1|.5
Directed reading of the trace: 1|.1|.2|.3|.4|.5
Measure of the trace's enactment: 1+1+1+1 → 4
Frozen-Frame coupling: ¤: |1_|2 → |1_Δ2
Temporal reading (Volume II): Δt = (Δ₂ |₋ Δ₁)
50
The Connection in Tractatus Notation
The spatial reading distinguishes fixations; the motion-reading
orders their alteration; the temporal reading coherently binds
complete relational configurations under the observer's
perspective. They are different readings of self-reference, not
three independently presupposed containers or dimensions. The
number 4 is not already a value of Δt.
IX. The Complete Argument
|
A|B
Δ := |(|)
¤|World ; (¤|World)|¤
0|₋Δ ↔ 0|₊Δ ; 0|¤|₌Δ
Table ↦ 1
1|.1|x —self-measurement→ 1|.1|.5
|_| ≡ 0|₌Δ ; ¤: |1_|2 → |1_Δ2
1|.1|.5 —coherent backward reading→ 1|.1|.2|.3|.4|.5
1+1+1+1 —numerical reading→ 4
Our concluding figure is this: no number measures a movement
independent of itself within a presupposed space. In the
observer's self-measurement an open reading-relation becomes
determined; from that result its compatible, directed continuation
becomes coherently readable. Its repeated differences can appear
51
The Connection in Tractatus Notation
as number, its differences of fixation as space, and its
frame-coupling as time.
The sequence of notation is now complete: the four transitions of
our chosen trace are already determined under its reading rule.
Chapter 13 does not ask once more how this determination is
possible; it asks what appears as distance, motion, resistance or
heat within the same relation under different physical rules of
measurement.
52
13 Physics as a Gradient Reading of Difference
Physics as a Gradient Reading of Difference
CHAPTER 13
13. Physics as a Gradient
Reading of Difference
13.1 The Unit of Measure as a Stabilised Relation of
Comparison
Having developed in Chapter 12 the route from distinction to
countable change, we now shift our question. In physics we must
determine under which rule of comparison such a change
becomes readable as distance, duration or some other measured
quantity.
A metre is a stabilised spatial relation of comparison; a second is
a stabilised temporal one. Neither is an additional origin of
difference, and they are not freely interchangeable names for the
same quantity. Each unit determines what kind of relation is
being compared and how its recurrence is to be read.
We can employ units repeatedly because their determination
remains stable. Apparatus makes the relation technically
accessible; it does not first produce its distinguishability. This is
the role of measure in our physics of gradients.
53
Physics as a Gradient Reading of Difference
13.2 Every Physical Statement Has an Observer
Reference
“The stone is three metres away” makes sense only within a
particular reference configuration: away from what, under which
spatial reading rule, and against which fixed comparison? A
measured value does not name an attribute of the stone floating
freely in space. It determines a relation. Even when we delegate
measurement to an instrument, this reference does not disappear.
The instrument records a distinction under its technical order of
comparison, which can be read back into the observer–world
relation.
Dependence on the observer in determination does not make
measurement arbitrary. Different observers can relate their
results when they preserve the relevant comparison rules and
reference configurations. What they jointly establish is a relation
that remains compatible on being read back, not a measure
existing outside every relation.
13.3 What Becomes Measurable Between Complete
Frames
A single Frozen Frame is complete. It contains neither intrinsic
duration nor a distance travelled. To specify physical motion we
therefore compare different complete relational configurations
under a stable observer reading. The spatial rule of measure
distinguishes the relation of position; the temporal rule
distinguishes its directed, recognisable continuation. Only their
determined connection can appear as velocity. Neither reading is
an independent container within which the other takes place.
54
Physics as a Gradient Reading of Difference
Nor does measuring mass, gravitation or temperature denote an
additional fundamental substance. Each selects a different
direction of comparison within the complete relational order. The
measured quantities remain distinct even if the same difference is
their origin.
13.4 The Table's Trace as a Transition to Physics
The table's trace from Chapter 12 has five reading positions and
four transitions distinguished under the same rule. As a count of
transitions, four is already determined. But we cannot simply
turn it into four metres, four seconds or four joules; these require
different, separately determined readings of measure.
Nor do the same endpoints compel the same intermediate
sequence under every kind of change. Which relation is readable
backwards under which rule belongs to the physical statement
itself.
The question of this chapter is therefore: How does the binding
of a relation change when the observer determines another
complete relational configuration? We shall call its graded
reading a gradient.
55
Physics as a Gradient Reading of Difference
13.5 What Is a Gradient?
In conventional physics a gradient describes the spatial change of
a quantity. That explanation is not yet fundamental enough for us,
however, because it presupposes space. In the Tractatus, we
begin by asking how a relation changes under a particular
orientation.
Volume I introduces several modes of difference. Three matter
especially here:
0|₊Δ
0|₋Δ
0|₌Δ
They mean:
|₊ – expansive reading: opening or unfolding.
|₋ – contractive reading: binding or folding back.
|₌ – equilibrium reading: the settling of the relation.
Expansion and contraction are not two independent forces. They
are the two fundamental directions under which the same
operation of difference appears. Equilibrium denotes its
corresponding stabilised relational configuration. A gradient in
our investigation is therefore initially a directionally
distinguishable change in a relation's binding or unfolding. Only a
specific physical rule of measurement turns it into a gradient
expressed, for example, in metres or seconds. Thus physics begins
to emerge as a multitude of readings of one common fundamental
trace.
56
Physics as a Gradient Reading of Difference
13.6 Gravitation: The Gradient of Folding Back
Volume II, Chapter X, describes gravitation as a gradient of
degrees of freedom.
Its fundamental relation is:
0|₋Δ ↔ 0|₊Δ
The sign ↔ makes the mutual relation of these two directed
readings visible.
The contractive reading designates the binding of the relation; the
expansive reading its opening. Volume II describes gravitation as
the folding-back reading of this common operation. Under one
fixation the relation appears as concentration, under another as
unfolding. It remains the same original self-reference.
Another notation from Volume II is:
0|Δ_g ≡ 0|(¤|₋Δ|₊Δ)
Here three signs require explanation:
¤ – the observer as relational fixation.
Δ_g – the relation of difference determined under the
gravitational reading; g marks this particular physical reading.
( … ) – the relation enclosed by the parentheses is considered as a
whole.
The formula does not designate gravitation as an additional thing
outside the self-relation. It describes it as a particular reading of
the relation between contraction and expansion with reference to
the observer. Consider again our table between .1 and .5.
57
Physics as a Gradient Reading of Difference
The gravitational question is no longer: Which unknown force
must move the table between two finished points?
It is: How does the binding of observed relational configurations
change under the gravitational reading?
The answer may later have a numerical value. But that value adds
a scale to a relation whose origin already lies in the folding back
of difference. The gradient has thereby been approached
relationally. No numerical gravitational strength has yet been
determined by our fourfold trace alone.
13.7 Heat: Change at the Boundary of Perception
Consider the same table from another perceptual perspective. We
touch it and can experience it as warm or cold. Ordinarily we
would say that its surface has a temperature, which can be
measured with a thermometer. But a thermometer, too, operates
under a particular rule of comparison; the unit in which we
express its result is itself a stabilised reading of measure. Volume
II, Chapter XVIII, examines the skin as an original boundary of
perception.
Its fundamental relation can be read as:
Inside | Skin | Outside
This does not describe three independently existing things. It
describes a boundary relation within which inside and outside
first appear as distinguishable sides. The observer experiences
change at that boundary, which he can read as warmth or cold.
Volume II, Chapter VIII, also explicitly traces heat back to an
58
Physics as a Gradient Reading of Difference
appearance of the directed drift of difference. Thus the thermal
reading returns to the same foundation.
The question is: How does the thermally perceived boundary
relation differ between the configurations under examination?
Care is needed here. From the notation
1|.1|.2|.3|.4|.5
it does not follow that temperature rises by four degrees. The four
transitions initially belong to our selected positional reading. A
temperature difference arises only after we determine a thermal
reading rule. Yet even that rule presupposes difference's original
self-reference. Heat is not identical with spatial displacement; it is
another physical reading of change. Its particular measuring rule
still needs elaboration.
13.8 The Speed of Light: The Fundamental Reference for
Spatial-Temporal Physics
The relation among different readings of measure becomes
particularly significant in the case of the speed of light.
In established physics, c has a special status: the speed of light in
vacuum is the fundamental reference linking spatial and temporal
measurements. We ordinarily express it in metres per second:
m/s
Here:
m – metre, the spatial unit.
s – second, the temporal unit.
59
Physics as a Gradient Reading of Difference
/ – the relational reading “per.”
But both units must themselves have come into being as relations
of measure. We can therefore expose the underlying distinction:
Metre | Second
We have not yet performed division. We have first made explicit
that measuring velocity relates two already distinguished
readings of measure.
Division itself requires explanation. Volume I denotes its reading
as:
|÷
It is the disentangling reading of difference. It determines a
perspective in which the two related measurements are
compared. Yet the light relation has a deeper meaning in the
Tractatus than merely the connection between two units.
Volume II does not describe the observer as a point outside a
finished world. The observer, denoted by ¤, is a fixation within his
own distinction. His perception arises through the relation
between the folding-back and unfolding directions:
0|₋Δ ↔ 0|₊Δ
The signs |₋ and |₊ denote contraction and expansion; ↔ their
mutual relation. The observer's equilibrium reading is written:
0|¤|₌Δ
¤ – observer as relational fixation, not an independent thing.
|₌ – the stabilised equilibrium reading.
60
Physics as a Gradient Reading of Difference
0 – the potential reference.
Δ – the operative self-reference of difference.
From this relation emerges the spatial reading of what we call the
sphere of perception. That sphere is not a presupposed
three-dimensional container or an additional original dimension.
It is the spatial appearance of the self-referential relation of
perception. The light relation is the particular measurement
relation in which this spatial appearance is fixed as a limiting
relation and coupled with temporal readings.
Volume II, Chapter IX, formulates the corresponding
speed-of-light reading as:
0|c ≡ 0|(Δ |÷ Δt)ₘₐₓ
The signs are:
0 – potential reference, not merely arithmetic zero.
| – distinction as relational positing.
c – the physical speed of light.
≡ – correspondence between relational readings.
Δ – operative self-reference of difference.
|÷ – disentangling or division reading.
Δt – temporal difference-reading of complete relational
configurations.
max – the limiting reading described in Volume II.
( … ) – the relation considered together.
61
Physics as a Gradient Reading of Difference
Thus the formula does not derive the speed of light from two
things already independent and named space and time. It reads c
as a particular limiting or fixing relation under which spatial and
temporal measurements can be coupled at all.
Nor is Δt an originally running time. Volume II describes the
temporal difference as:
Δt = (Δ₂ |₋ Δ₁)
Δ₁ and Δ₂ denote relations distinguished in two complete
relational configurations. The digits are indices, not presupposed
instants of time. Here |₋ denotes the contractive reading of
difference.
Consequently, the speed of light is the fundamental reference for
spatial-temporal measurement in physics. In our work-internal
derivation it becomes recognisable as the spatial reading of the
sphere of perception through the light relation. This does not
mean that the sphere of perception is itself a light particle moving
through a finished space. The measured speed of light is instead
the scaled physical appearance of an invariant reference within
that sphere. The speed of light does not generate the sphere of
perception; it is the physical reference under which the sphere's
spatial reading becomes measurable through the light relation.
The unit m/s can thus be traced back: metre and second are
already stabilised spatial and temporal readings of measure. Their
ratio is the public physical expression of a relation that does not
first arise from the units. The concrete numerical value of this
limiting reference does not follow from our table example alone; it
is fixed and scaled within physical measurement practice.
62
Physics as a Gradient Reading of Difference
13.9 Mass and Resistance: Bound Change
When we try to lift the table, we experience weight, physical
effort and resistance. Conventionally we would say that the table
possesses mass and that our body must exert a force. Volume II,
Chapter XVIII, instead traces the experience to the observer's
self-measurement in different complete relational configurations.
The observer directs himself towards another configuration and
experiences the change as bodily effort. Motion is not the
migration of unchanging substance through empty space; it is
read as the difference between complete relational
configurations.
Mass appears, in this view, as a particular bound reading of the
relation. The kilogram, too, is thus a later unit of comparison, not
the original cause of the perceived resistance.
The connection becomes clear when we consider the two
fundamental directions again:
0|₋Δ ↔ 0|₊Δ
In readings of mass and resistance, bound change is foremost. In
the gravitational reading, by contrast, we examine the folding
back and gradient of binding within the structure. Both readings
return to the same operation of difference. That does not make
them identical physical quantities. Each measuring rule
determines which appearance becomes distinguishable.
63
Physics as a Gradient Reading of Difference
13.10 Motion and Energy: The Difference Between
Complete Frames
How does energy relate to our change of position?
We established that every Frozen Frame is complete:
|_| ≡ 0|₌Δ
Volume II describes the coupling of two such configurations by:
¤: |1_|2 → |1_Δ2
The notation requires explanation:
¤ – observer as relational fixation.
|1_|2 – coupling of two complete frames; here the digits are frame
indices.
|1_Δ2 – the relation distinguished between them.
→ – a change of reading, not an external flow of time.
These frame indices must not be confused with the one of our
table relation. No movement occurs within a single frame. Only by
relating different complete configurations can the observer read
their difference as movement. Their coherent continuation
appears as time. The change so determined can be read under a
temporal rule of measure as energy.
Volume II writes:
0|E ≡ 0|(Δ |÷ Δt)
Here E is the physical quantity energy.
0| – puts the following quantity into relation with potential.
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Physics as a Gradient Reading of Difference
Δ – operative self-reference of difference.
|÷ – disentangling reading.
Δt – temporal difference-reading.
≡ – connects the physical appearance with its relational
description.
Energy is thus not an additional substance that sets difference in
motion. It is a particular physical reading of change in difference.
Nor is the joule an original building block of the structure of
difference. It is the subsequently stabilised unit of a particular
energy determination. Our four positional transitions from .1 to .5
are therefore not automatically four joules. That would require
deriving the specific energy-measurement rule from the relation.
The common fundamental trace explains why such readings can
be related at all; it does not replace their particular physical
determination.
13.11 Sound and Echo: Different Senses, One Table
Relation
We can not only see the table but touch it or infer its presence
and location acoustically from reflected sound. These perceptions
differ markedly. A visible tabletop is not a sound echo; neither is
the echo identical with the surface's resistance. Yet all three
perceptions can refer to the same table.
How is that possible?
The observer does not merely distinguish his world. He also
distinguishes his own readings of that world.
65
Physics as a Gradient Reading of Difference
Volume II formulates this return as:
¤|World → (¤|World)|¤
The first expression designates the observer–world relation; in the
second, that relation is itself referred back to the observer.
The reading becomes a subject of distinction in its own right.
The observer can experience his optical reading changing while
other perceptions remain in relations compatible with it. Thus the
recognisability of the table is not based on one sensory
impression. It rests on the connection between different
perceptions readable through self-reference. That is why we can
continue using 1 for the recognisable table relation—not because
the table was originally a one, but because we recognise it under
different reading rules as a connected relational form.
Acoustic measurement needs a rule of its own. Volume II supplies
the general basis of self-measurement; a complete quantitative
derivation of acoustics has not yet been supplied by it.
13.12 The Common Fundamental Trace and Its Various
Physical Gradients
We can now refer the different physical readings back to their
shared starting point.
Our determined table relation is:
1|.1|.5
Under a directed reading rule one possible continuation is:
1|.1|.2|.3|.4|.5
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Physics as a Gradient Reading of Difference
Four distinguished transitions are read numerically as:
1+1+1+1→4
This fourfold trace is already a determinate count of four
transitions under the selected stable reading rule. Its validity
comes from the backward readability of self-measurement, not
from a second certification granted from outside. Four counted
transitions are not yet four equal metres or four equal seconds.
Physical statements of that kind designate different, explicitly
defined readings of measure.
What appears as a physical quantity depends on the rule of
reading.
Under the spatial reading, a change of position appears.
Under the temporal reading, a coherent sequence of complete
configurations appears. Under the energy reading, change in
difference appears in relation to temporal difference. Under the
gravitational reading, the alteration of binding and folding back
appears. Under the thermal reading, a thermally distinguishable
change at a boundary appears. Under the optical reading, a
change in a visible relation appears. The light relation also has a
special role as the fundamental physical reference for
spatial-temporal measurement. Under the mass and resistance
reading, bound self-measurement appears as bodily effort.
These are not mutually independent original realities. They are
different physical readings of one common relational origin.
But the converse is just as important:
Common origin does not make the different physical readings
identical. Four positional transitions are not automatically four
67
Physics as a Gradient Reading of Difference
metres, four seconds, four kelvin or four joules. Such an equation
would merely introduce a fresh confusion of readings. Every
physical appearance requires its particular rule of comparison. Its
numerical value results from that rule. The unit names the
stabilised kind of comparison. Both remain traceable to the
original distinction.
13.13 The Retrocausality of a Physical Measured Value
A measured value denotes a determined relational configuration.
Its numerical sign alone does not decide which intermediate
readings belong to its history. The coherence explained in Part III
also holds for physical measurements: from the determined result,
those complete configurations are read as connected that are
compatible with the selected measuring rule.
This is the retrocausal reading of Volume II, not the transmission
of signals against an external flow of time. For physics it has a
concrete consequence: a measured temperature, position or
energy cannot be separated from the relation under which that
quantity was determined. Whoever adopts a different reading rule
must work out the corresponding connection of relational
configurations again.
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Physics as a Gradient Reading of Difference
13.14 Physics Reads Difference
In Volume II, gravitation, heat, light, mass and energy can be
traced back to different operational and measuring readings of
the same self-reference. They are not thereby the same physical
quantity: each has its own conditions of comparability. The light
relation occupies a particular role as reference for the
spatial-temporal measurement of the sphere of perception.
The chapter has thus answered what physical quantities can
mean within a structure of difference. What remains to be
explained is why an observer can bring forth a stable reading rule
across multiple changes in the first place. Only after addressing
that question can we develop a constructive model of space, time
and perception from the fundamental trace.
69
14 From the Fundamental Trace to a Stable Reading of the World
From the Fundamental Trace to a Stable Reading of the World
CHAPTER 14
14. From the Fundamental
Trace to a Stable Reading of
the World
14.1 What the Model Still Has to Explain
Chapter 13 treated physical appearances as different readings of
the structure of difference. One assumption remained open: how
can the observer connect his various perceptions and complete
relational configurations under a lasting rule at all?
We cannot simply delegate the answer to another instance, such
as a ready-made mind possessing its rules independently of the
world. That would separate observer and reality again. We seek a
rule of reading whose stability arises from the same self-reference
as the world it reads.
14.2 The Inference Backwards: Frame and Observer
Have the Same Structure
Our extended representation of a complete frame is:
Δ|Δ → Δ|_|Δ
The two Δ are not two things. Δ denotes the self-distinction of
difference; the notation illustrates its mutual folding back. The
enclosed |_| stands for the complete self-relation. The arrow
70
From the Fundamental Trace to a Stable Reading of the World
denotes the unfolding of a reading, not an event in presupposed
time. The whole line is our refined working notation. For the
closure of the Frozen Frame, Volume II expressly uses:
|_| ≡ 0|₌Δ
Now consider the observer relation from Volume II:
¤|World
(¤|World)|¤
¤ designates the observer as a fixation of a relation. The first
expression distinguishes observer and world; the second refers
this distinction back to the observer's position.
The decisive connection lies in the structural correspondence:
The complete self-relation and the observer's
self-measurement repeat the same arrangement of return.
We do not equate the formulas sign by sign. Their equivalence
concerns the structure of the operation: a relation is referred to
again, and its order of distinction remains readable.
The rule of reading therefore need not be invented by an observer
standing outside the relation. The observer is himself a stable
mode of appearance of the very operation he reads.
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From the Fundamental Trace to a Stable Reading of the World
14.3 How the Rule Stabilises Itself
Volume IV describes stability as an invariant self-relation. A
recurring order of difference is represented, for example, by:
OΔO
Here O stands for a particular order of difference, and Δ for its
renewed self-distinction. The formula describes the case in which
the order preserves its form through this distinction. Not every
recurrence meets this condition: an order may change or
alternate among different forms.
This allows us to specify the observer's reading. When the relation
¤|World is distinguished anew, nothing outside that relation must
decide what counts as recurrence. The ability to read an order
again as the same belongs to the stabilised self-reference of
observation.
Initially this is a rule of recognisability, not a claim that
distances are equal. It explains why different sensory readings
can concern the same table although sight, echo and touch are
not identical.
The further question of quantitative measure is: under what
conditions does not only the table remain recognisable, but also
the manner of its directed distinction become comparable from
step to step?
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From the Fundamental Trace to a Stable Reading of the World
14.4 Self-Referential Measurement at the Table
The table relation continues to be denoted by 1. That one stands
for a recognisable form, not an original numerical magnitude. .1
and .5 are distinct reading fixations; the dot denotes the scale flip
from Volume I.
1|.1|x → 1|.1|.5
x denotes the open other reading position. The result .5 by itself
determines no intermediate sequence. If we select a continuation
that can be read consistently throughout, we obtain:
1|.1|.2|.3|.4|.5
Under this selected rule four transitions can be distinguished.
Their numerical reading is:
1+1+1+1 → 4
The four ones here denote four accomplished distinctions. The 1
at the start of 1|.1, by contrast, is the stable mark for the table.
Under the maintained rule, the four transitions are already
determined as four differences. Their backward readability
belongs to the observer's self-measurement and requires no
additional external instance to grant permission to count them.
Nevertheless, four counted transitions are not yet four equally
long stretches or four equal durations. No unit grounded outside
self-reference is required merely to distinguish a continuation. An
expression in metres or seconds names the corresponding
stabilised comparative reading, not some later justification of
self-distinction.
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From the Fundamental Trace to a Stable Reading of the World
14.5 Expansion and Contraction in Numerical Reading
The two fundamental directions in Volume I can be read side by
side: |₊Δ denotes expansion, |₋Δ contraction. Repeated steps of
distinction appear in arithmetic reading as +1 and −1.
Expansion – |₊Δ
12345
1 + 1 + 1 + 1 + 1 + ...
Contraction – |₋Δ
54321
... - 1 - 1 - 1 - 1 - 1
The upper digits mark five readable positions. Between five
positions are four transitions; the five visible ones in the open
notation must therefore not be counted as five steps of our
four-transition trace. The string of ones illustrates repeated
distinction: in expansion the order is read in the +1 direction, in
contraction in the opposite −1 direction.
Neither series has a necessary starting or ending point. In
particular, contraction does not begin with a presupposed object
called “five.” Five is just a visible reading mark in an open
sequence. We obtain not two independent operations but two
directional readings of one operation. Reversing the numerical
reading does not mean that time flows backwards.
For the table, the same opposition is visible:
1|.1 → 1|.2 → 1|.3 → 1|.4 → 1|.5
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From the Fundamental Trace to a Stable Reading of the World
1|.5 → 1|.4 → 1|.3 → 1|.2 → 1|.1
Both sequences concern the same recognisable table relation.
They show opposite directions of reading, not paths of equal
length through an existing space.
14.6 How the Sphere of Perception Appears Spatially
Volume II uses the following relation for observer fixation:
0|₋Δ ↔ 0|₊Δ
0|¤|₌Δ
The first line displays the mutual directions of folding back and
opening. The second denotes the equilibrium reading of the
observer relation. The signs were explained in Chapter 13. What
matters here is what their common structure makes possible
for spatial perception.
When the observer stably recognises changes in his relations,
here and there, near and far, foreground and background become
readable as spatial distinctions. The depth of that perception is
not proof of an independent third fundamental dimension. In our
framework the fundamental dimensionality remains twofold;
depth arises as a perspective of self-referential reading.
The plus-one chain is not a spatial axis. It merely illustrates how
the same directed distinction is read in increasingly differentiated
form under a numerical rule. Only the observer fixation and
corresponding geometrical reading make a spatial appearance
possible from it.
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From the Fundamental Trace to a Stable Reading of the World
14.7 How Motion and Time Appear as Other Readings
Time, too, requires no second origin. A complete Frozen Frame
contains neither elapsed duration nor travelled distance. Volume
II represents the coupling of two complete relational
configurations by:
¤: |1_|2 → |1_Δ2
Here 1 and 2 are frame indices, not the table's mark or its reading
positions. The formula designates the observer's reading of
difference between two complete frames.
Their differences of fixation can appear spatially; their directed
relation can appear as motion; the coherence of this relation can
appear as time. Volume II already expresses temporal difference
as Δt = (Δ₂ |₋ Δ₁).
Backward readability is decisive: the determined result limits the
possible coherent histories. It does not create earlier frames
within a time running backwards. Once again it is the stable rule
of reading that operates, not an additional chronology outside the
relational configurations.
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From the Fundamental Trace to a Stable Reading of the World
14.8 What the Light Relation and Our Senses Contribute
The speed of light has a special physical reference role. As
Chapter 13.8 explained, through the light relation the spatial
reading of the sphere of perception becomes comparable with
temporal measurement. Metres per second already expresses two
derived readings of measure, not their origin. Optical, acoustic
and tactile changes associated with the table can be related to
one recognisable fundamental relation. This reveals the
self-reference of the observer's rule; it does not mean that all
three senses deliver the same information or that their particular
physical procedures of measurement are already identical.
We can now describe why different appearances can jointly be
read as one world. Their numerical calculation still requires the
specific physical rule of reading in each case.
14.9 Determinate Self-Measurement Needs No Second
Certification
We have deliberately not presupposed an instance of
measurement independent of the relation, which would first have
to allow the observer to make his own distinctions. Such a
demand would once again move the explanation outside the very
relation whose self-reference we have just derived.
The mirroring of a complete relational configuration and the
observer's self-measurement possess the same structure of
return. Under a stable reading rule, the observer can recognise
his differences as a connected continuation and read them
backwards from the determined result into his own relational
77
From the Fundamental Trace to a Stable Reading of the World
order. This is not a retrospective confirmation granted by
something foreign; it is the act of determination itself.
For our table's trace:
1|.1 → 1|.2 → 1|.3 → 1|.4 → 1|.5
1+1+1+1 → 4
We have five distinguishable positions and four transitions. Four
is determined as the count of those transitions under the stated
rule. No further proof is needed to permit a recurrence so
determined to be counted. The ones denote repeated application
of the chosen distinction rule, not four originally present,
independently calibrated lengths.
To express these transitions in metres, we add a spatial
comparative reading; to express them temporally, we use a
stabilised recurrence as the measure of time. We thereby alter the
kind of reading, not create an independent instance whose
permission first validates the original measurement. The same
notation of ones must not be confused with an assumed equality
of physical distances or durations. Their connection lies in the
preserved rule and backward readability within the same
observer–world relation.
Unambiguity lies in determining the relation, not in a
subsequent instrument and not in a permission granted
from outside to read it as a measure.
We have thus returned to the starting point of this chapter. The
stable reading of the world arises from the self-reference of
difference. Space, motion and time appear as different, mutually
compatible scalings of this self-measurement. The observer does
78
From the Fundamental Trace to a Stable Reading of the World
not stand outside the order he measures; his measurement is
itself a mode of appearance of that order's return.
79
15 Archibald and the Error of the Inner World
Archibald and the Error of the Inner World
CHAPTER 15
15. Archibald and the Error of
the Inner World
15.1 The Surface of Explanation
Archibald illustrates the shift once again. He believed he had
discovered a two-dimensional planet, failing to notice that his own
apparently three-dimensional world was depicted on a
two-dimensional surface. His error lay in taking a particular
spatial reading for the original constitution of reality.
A similar reversal occurs when we try to explain perception
through eyes, skin, nerves and brain. We examine the observer,
discover biological structures and then take them as the ground
of the very observer under whose perception they first became
distinguishable. Yet the brain, too, is something we distinguish
under a particular rule of observation. We end up explaining
observation through one of its own appearances.
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Archibald and the Error of the Inner World
15.2 The Table Sawed Open
Our table makes the same point. Initially we see its surface. If we
saw it apart to uncover its supposed interior, a new surface
appears. We recognise fibres and other structures which, under
greater magnification, unfold into further differences. The same
happens when we examine the body: skin, muscles, bones, brain,
nerve cells. We certainly acquire new knowledge. But none of
these readings leaves the distinction through which it appears at
all.
Every supposedly uncovered interior becomes a surface once
more under another distinction.
15.3 The Visible Signal Chain
A conventional biological explanation of touch begins with an
external stimulus: it alters the skin, producing an electrical signal
that travels along nerves to the brain.
This directed sequence is measurable.
But its measurability does not yet answer where the observer
relation comes from within which skin, brain and signal are
distinguishable. Volume II begins with the skin as a boundary of
self-measurement. The observer distinguishes himself from his
world at that boundary.
Our working notation is:
¤ | Skin | World
Here ¤ marks the observer fixation. The poles of the distinction
are `¤` and `World`; `Skin` denotes their bodily read
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Archibald and the Error of the Inner World
boundary, not a third independent pole. Nor are the two poles
presupposed as independently existing things.
The observer does not originally receive a message from an
outside world already determined independently of him.
He distinguishes himself at his own boundary.
What appears under a neurophysiological measurement rule as an
electrical signal chain is already a particular temporal reading of
this self-distinction.
15.4 Two Directions, One Relation
Imagine a wave. We follow its visible surface and read a
propagation to the right. Yet the visible direction of motion need
not describe the entire underlying operation. For our analogy we
therefore consider an opposing movement back beneath the
surface.
The two directions illustrate two readings of the same relation.
Volume II writes:
0|₋Δ ↔ 0|₊Δ
Here:
• |₊Δ – the expansive, outward-directed reading of difference.
• |₋Δ – the contractive folding back of difference.
• ↔ – the mutual dependence of both directions.
Under our temporal observation, the electrical signal chain from
skin to brain appears as an expansive reading.
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Archibald and the Error of the Inner World
We see a sequence of electrical registrations.
What this reading leaves out is the folding back through which the
sequence remains determined as part of the same self-relation.
The minus-reading is not therefore a second hidden electrical
signal that must anatomically travel in the opposite direction. It is
the closure of the structure of difference, referred back to the
observer.
Negative folding back conditions positive reading-out.
They are not causes occurring one after another but
simultaneously operative reading directions of the same
operation.
15.5 The Circle and Its Cut
We have returned to the beginning of our investigation.
Difference:
|
Its polar reading:
A|B
Its self-reference:
Δ := |(|)
What we have described as perception, electrical signal, warmth
or motion returns to the same self-distinction. This self-distinction
is circular: it refers back to the relation from which its own
distinction arises.
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Archibald and the Error of the Inner World
Volume I illustrates this geometrically in its diagram of the circle.
The 0 appears as a closed circle. The 1 is not entered as a counted
thing but as a stroke or radius cutting the circle, making a
particular direction readable within its closure. Mirroring this
stroke yields the diameter-reading of 2. Numbers thus appear as
geometrical positions of continuing distinction, not as finished
building blocks.
The cutting one plays a different role here from two other uses
of one. The table-1 denotes a recognisable relation; the
operational-1 in 1+1+... denotes a distinguished step. The
cutting-1, by contrast, illustrates how difference appears as a
stroke in potential read geometrically as a circle. We use the
same digit under three different reading rules, not as three
original units.
We do not begin with a ready-made circle in space. The circle is
the geometrical reading of closed self-reference; the stroke is
the geometrical reading of the cut |. Only within this relation does
a directed orientation arise.
We can express the connection as:
Closed self-relation | Cut | Directed unfolding | Folding back
This is an illustrative working reading, not a chronological
sequence of separate events. The cut does not produce folding
back afterwards; folding back already belongs to the
completeness of the relation within which directed unfolding
becomes determinable.
We have previously represented this self-relation as:
Δ|Δ → Δ|_|Δ
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Archibald and the Error of the Inner World
The two Δ are not independent things.
They denote the reciprocal return of difference.
The structure closes upon itself. The cut makes a particular
unfolding of that closure readable.
This is the geometrical meaning of our circle.
We can follow one direction without claiming that the whole
relation consists only of that direction.
The plus-reading unfolds.
The minus-reading folds back.
And observer fixation is the self-relation under which their
common order appears as perception.
15.6 Why the World Acts Back Upon Us
We can now understand why we experience warmth, pressure and
other effects of our world.
We distinguish ourselves in relation to our world.
That distinction opens an expansive reading under which an
outward appearance becomes determinable.
Yet the same relation is folded back upon the observer.
The world is not an entirely independent pole that must first be
connected to an equally independent interior through an
additional chain of signals.
It is the opposing pole of the same self-distinction.
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Archibald and the Error of the Inner World
Volume II denotes the observer fixation between expansion and
contraction by:
0|¤|₌Δ
Here |₌Δ denotes the equilibrium reading.
Perception is the determined appearance under which the
oppositely directed readings become operative within one
observer relation.
We experience a change in this boundary relation as heat.
Under a different reading rule we experience it as pressure or
resistance. Under a neurophysiological rule its directed
continuation appears as a cascade of electrical signals.
These appearances differ.
But they require no different original causes.
The contractive return closes the observer's self-distinction.
Expansive reading-out lets its change appear as a phenomenon of
the world. The seemingly separate interaction between inside and
outside is thus traced back to one relation returning into itself.
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Archibald and the Error of the Inner World
15.7 We Have Been Speaking About the Same Difference
All Along
Archibald saw a two-dimensional surface and believed he had
found an alien world with fewer dimensions. We observe electrical
signals and believe we have discovered the original mechanism of
perception. In both cases we mistake one direction of appearance
for the whole relational operation. Yet our investigation has
continually brought us back to the same starting point.
We considered difference.
We read it as number.
As spatial orientation.
As temporal continuation.
As motion, light, gravitation and heat.
As skin, brain and electrical signal.
Under none of these readings did we need a new original thing.
We examined only how difference distinguishes itself and how
that self-distinction appears under various rules. The circle makes
the return upon itself geometrically comprehensible.
The cut makes its directed unfolding readable.
Contractive folding back closes the order.
And the observer appears as a stable self-reading of the
connection. We see expansion and regard it as the cause, when it
is a direction of appearance of a self-relation closed within itself.
That is Archibald's real error—and ours.
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Archibald and the Error of the Inner World
We believe we can explain experience through objects that appear
within experience. Yet the original condition of these appearances
is not another thing hidden beyond the last surface. It is
difference referring to itself.
We have returned to |.
88
16 The Gradient: A Relation Read Back to the Observer
The Gradient: A Relation Read Back to the Observer
CHAPTER 16
16. The Gradient: A Relation
Read Back to the Observer
16.1 Three Metres from Whom?
What does it mean to say that a stone is three metres away?
Initially the statement denotes a spatial relation that we
distinguish between observer and stone under a particular rule of
measurement. Three metres is not an attribute attached to the
stone independently of this relation.
It denotes a scaled configuration of difference.
This also clarifies the meaning of gradient. A gradient is not
another thing acting between two objects. It describes how a
relation changes by degrees from the chosen observer reference.
A gradient is a relational statement.
When we speak of a declining gradient, we do not mean that an
invisible substance in space becomes weaker. We mean that the
binding or change of the relation appears in graded form along a
particular direction of reading. The gradient belongs to the
reading. It does not stand outside the relation to generate it.
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The Gradient: A Relation Read Back to the Observer
16.2 Scaling Difference
We previously considered different reading positions at our table.
We can perform the same investigation with a stone.
We denote its recognisable relational form by 1.
Two reading positions are:
1|.1
1|.5
Between them we can distinguish a directed sequence:
1|.1 → 1|.2 → 1|.3 → 1|.4 → 1|.5
This contains five positions and four transitions.
It does not presuppose that the transitions are equal in size or
that every position corresponds to a particular metre value.
Scaling arises only under the reading rule with which we compare
those differences. We can distinguish a relation coarsely or finely,
introduce additional intermediate configurations or capture
several transitions in a contracted reading. But that does not
make every arbitrary subdivision a valid measurement.
Correct scaling is that under which the distinguished
intermediate configurations remain compatible with the
complete relational order and coherently readable
backwards from the determined result.
The result 1|.5 limits which earlier configurations can count as its
connected history.
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The Gradient: A Relation Read Back to the Observer
A scaling that violates this belonging-together does not describe
the same coherent continuation. The rule of measurement thus
has a concrete task: to distinguish gradations in such a way that
their return to the preserved fundamental relation remains
possible.
16.3 Why the Stone Appears Heavy
If we try to move the stone, we would ordinarily say it has mass
and therefore requires a force to set it moving. Within our
derivation, however, the explanation begins with the alteration of
a complete relational configuration. The observer determines
another position within the structure. He cannot alter only the
stone's spatial reading position while ignoring all other relations.
The stone stands in relation to the ground, the observer and other
bindings within the entire structure.
The desired configuration must form a coherent complete
configuration under these relations.
Heaviness is here the gradient reading of the bindings that
must be taken into account in this self-measurement.
The object does not originally carry an independent property
called heaviness. We experience a graded difficulty in
determining another configuration while preserving backward
readability. What appears as bound order under the mass reading
is experienced as resistance to change under the motion reading.
The physical quantities remain different, but they must not be
presupposed independently of the observer reference.
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The Gradient: A Relation Read Back to the Observer
16.4 The Regress of Force
When we push against the stone, classical physics describes a
corresponding reaction force. Investigating the interaction more
closely leads to electromagnetic relations. Further investigation
opens up additional physical structures and descriptions.
We can thereby calculate relations with increasing precision.
But this need not answer the original question.
We have introduced another distinction.
Just as with the sawed-open table.
We open a surface and find another.
We examine that surface and find further differences. If we
declare each appearance the actual cause of its predecessor, we
enter a regress. Changing the word force to field, charge or
interaction does not remove the problem.
Those terms, too, denote particular physical readings.
Our gradient must not become the next hidden mechanism.
It denotes nothing beyond the graded relation the observer
distinguishes under his rule of measurement.
The question is not what invisible something really pushes the
stone. It is what alteration of the complete relational order can be
determined from the observer and read backwards as a connected
continuation.
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The Gradient: A Relation Read Back to the Observer
16.5 Energy and Heat as Different Gradient Readings
We can now situate energy and heat.
We move a stone across the ground.
Under a spatial reading, its position appears changed.
Under a mechanical reading, we determine a resistance and work
performed. Under a thermal reading, we register a warming of
the boundary relations involved.
These appearances are not identical.
But they are different evaluations of change in the same
structure.
Volume II writes the energy reading as:
0|E ≡ 0|(Δ |÷ Δt)
Energy is thus a measurement reading of change in difference
under its temporal differentiation, not an original substance that
must be added to explain difference.
Heat is similar.
When we move the stone and its boundary relation to the ground
changes, this change may appear as warming under a thermal
rule of measurement. Nor does heat possess a given value
independently of every reading. Its determination requires a
standard and a reference against which the relevant change
becomes comparable. This does not mean that spatial, mechanical
and thermal quantities can be translated into one another
arbitrarily.
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The Gradient: A Relation Read Back to the Observer
Each has its own reading rule.
What they share is not the unit of measurement but the
self-reference of the order of difference whose change they
scale in different ways.
16.6 Backward Readability Decides
We now arrive at the connection between gradient, Frozen Frame
and observer.
A complete Frozen Frame has no internal duration.
The observer distinguishes another complete relational
configuration and reads its relation to his previous position as
change. From the determined configuration that history becomes
readable under which it appears as a connected continuation.
This relation can be scaled in different ways.
We can distinguish larger and smaller steps.
We can determine a directed spatial distance.
We can evaluate its change temporally, mechanically or
thermally. But the chosen scaling must not destroy the
belonging-together of the complete relational configurations.
Backward readability is therefore the criterion for valid
scaling within the same self-relation.
It means that the gradations can be referred back from the
reached configuration to their own order of difference. This does
not mean that only one scaling is possible. Different scales can
validly describe the same relation. What matters is that
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The Gradient: A Relation Read Back to the Observer
comparison rules are not changed arbitrarily and that their
results remain compatible. This is the boundary between an
arbitrarily drawn intermediate sequence and a justified
measurement.
A gradient is thus the graded reading of a relation to the
observer.
A measure is the stabilised scaling of that relation.
And the correct scaling is one that remains coherently
readable backwards within the complete self-relation.
Accordingly, mass, force and energy need not be installed as
original causes of change. They can be understood as different
physical readings of bindings and changes that the observer
determines within the whole structure.
We have found no further hidden cause.
We have specified the conditions under which difference appears
as a measurable physical relation.
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17 Light: The Limit of Self-Measurement
Light: The Limit of Self-Measurement
CHAPTER 17
17. Light: The Limit of
Self-Measurement
17.1 Archibald and Outsourced Perception
Through his telescope Archibald discovers a planet that appears
to him as two-dimensional. The apparatus enables him to make a
distinction inaccessible under his previous perceptual reading. He
has not created an observer outside himself. He has technically
extended the boundary of his self-measurement.
The telescope changes what he can distinguish.
When Archibald subsequently flies towards the planet in his
rocket, he extends another reading: the alteration of his own
position within the structure. The rocket couples his
motion-related self-measurement to a machine that makes other
relational configurations accessible. Both devices change the
scaling of his measurement, not its origin. The sphere of
perception is therefore not an immutable spatial container
surrounding Archibald. It is the geometrical reading of the
current boundary of his self-measurement.
Schematically the relation is ¤|World. Telescope and rocket are
not third original poles but differently outsourced boundaries
within that relation.
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Light: The Limit of Self-Measurement
17.2 Two Geometries, One Measurement
To measure velocity, Archibald uses the familiar notation:
m/s
Metres per second.
Here we interweave two different geometrical measurement
readings of the same difference. The metre designates the
expansive spatial reading. When Archibald says an object is three
metres away, he determines his relation to it under a spatial rule
of measurement. Three metres is not an original property of the
object; it is a particular scaling of its relation to the observer. The
second, by contrast, denotes a temporal reading of comparison. It
does not unfold spatial distance outwards; it determines the
recurrence of an order of difference as comparable continuation.
Such recurrence can be read geometrically as a circuit. An
ordinary clock illustrates it: a hand goes around a circle, whose
revolution we divide into sixty second-positions. Spatial reading
appears directed, while temporal reading appears as a cyclically
stabilised comparative order.
We do not compare two independent things called space
and time. We interweave two geometrical readings of the
same self-distinction.
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Light: The Limit of Self-Measurement
17.3 The Hand and the Caesium Atom
A modern clock need not use a hand to determine a second. It can
employ a particular atomic transition frequency as its
comparative relation. What has fundamentally changed?
We have not left the recurring comparative order behind; we have
altered its technical realisation. With a clock hand we read a
position within a cyclic revolution. With an atomic clock we count
a defined number of periodic processes to realise the unit of time.
The physical mechanisms and attainable precision differ. A
caesium atom is not a tiny clock hand. Yet the fundamental
measurement task is the same: we distinguish a recurrence and
relate another alteration to that comparable order.
An atomic clock can realise our comparative rule far more
precisely and stably. It does not thereby create the
possibility of comparison in the first place.
Even atomic recurrence can be determined as recurring
identically only under a particular measuring rule.
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Light: The Limit of Self-Measurement
17.4 Unambiguity Belongs to Determination
We must distinguish determinacy from measurement accuracy.
Determinacy means that we have fixed which relation is to be
distinguished under which rule. Accuracy concerns how reliably
the instrument realises this determination. A finer dial division,
another decimal place or a more stable atomic clock does not
create an originally more precise time. It refines our reading of a
fixed relation of comparison. The sixty divisions of a clock
revolution are a particular, historically established scaling; they
do not mean that difference itself consists of sixty parts.
Scaling can refine our reading. It does not first create the
unambiguity of the relation being read.
17.5 Why Velocity Is a Reading of a Ratio
Archibald travels in his rocket to another relational configuration.
Under a spatial reading, this change appears as a distance
travelled; under a cyclically stabilised temporal reading, its
continuation appears as duration. Comparing the two yields
velocity. The ratio m/s thus designates the coupling of two
scalings. That coupling must remain valid within the same
observer relation.
If Archibald changes the spatial unit, the numerical value must be
convertible accordingly. If he replaces a clock hand with a more
precise timekeeper, this alone must not create a different relation
of motion. The rocket allows a different coupling of
movement-related self-measurement than his body without the
apparatus. That is more than a change of unit, but this technically
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Light: The Limit of Self-Measurement
mediated alteration, too, must remain readable backwards as a
relation between complete relational configurations.
Velocity is the read-out relation between a directed spatial
change of difference and its temporally determined
continuation.
17.6 Why the Light Relation Recurs Identically
Volume II writes:
0|c ≡ 0|(Δ |÷ Δt)ₘₐₓ
Δ designates operative difference, Δt its temporal
difference-reading and |÷ the relational disentangling reading.
The index max marks a distinguished limiting form of the
coupling.
From an external perspective we may call this repeatedly
identical light relation a natural constant. In our derivation,
however, its constancy does not designate a property posited
independently of measurement. We ask how a self-referential
measurement relation preserves the same limiting form under
different permissible readings.
Recurring equality is the stable self-relation of
measurement, not another original thing placed beside
difference.
This distinguishes its external designation from our relational
question. The Tractatus does not take invariance as a finished
answer but as the expression of a return whose conditions of
emergence must be derived.
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Light: The Limit of Self-Measurement
17.7 The Number Is the Read-Out Position
In today's International System of Units, the fixed numerical value
is:
299 792 458 m/s
Metre and second are themselves readings of measure. The
second is realised through a defined atomic frequency, and since
1983 the metre has been defined using the exactly fixed
numerical value of the speed of light. The decimal sequence is
therefore part of a chosen and stabilised order of measurement.
We do not measure some original numerical substance of light.
Far from removing the subject of our relational approach, this
shows how a repeatedly determined relation itself becomes the
basis of the unit used for further measurements.
The stipulation must not be confused with the physical light
relation. It is the way we express that relation numerically under
a particular measuring rule.
17.8 The Actual Task of Our Calculation of Light
Our starting question is therefore not why a supposedly
independent constant happens to carry a particular decimal
number. We must explain how the expansive spatial and cyclically
determined temporal readings of the same self-reference remain
mutually compatible when apparatus coupling changes. What
physics reads as an invariant limiting speed, we examine as the
limiting reading of a self-measurement recognising its own
relation across different geometrical orders of measure.
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Light: The Limit of Self-Measurement
This establishes the starting point for our geometry of numbers.
In the next chapter we investigate not three independently
existing numbers but the mutual relation among the geometrical
readings represented in Volume II through the light-number.
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18 The Geometry of the Light Number: Mirroring, Breakpoint and Self-Reference
The Geometry of the Light Number: Mirroring, Breakpoint and Self-Reference
CHAPTER 18
18. The Geometry of the Light
Number: Mirroring,
Breakpoint and
Self-Reference
18.1 The Question of the Number's Geometry
Chapter 17 showed what we interweave in measuring velocity: an
expansive spatial relation and a cyclically determined temporal
order of comparison.
Both are different readings of the same self-reference of
difference. Their distinguished limiting form appears under the
light relation:
0|c ≡ 0|(Δ |÷ Δt)ₘₐₓ
From this relation we can turn to the numerical reading in
Volume II. In the chosen SI order of measurement, the speed of
light is expressed as:
299 792 458 m/s
Volume II divides this numerical expression into 299 | 792 | 458
and reads the results 2|0 – 9 – 8.
An isolated interpretation of the digits, however, risks treating
pieces of a measured appearance as independent causes once
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The Geometry of the Light Number: Mirroring, Breakpoint and Self-Reference
more. We must first explain why the numerical figures in Volume I
acquire meaning only within one geometry.
18.2 One Is Zero from Another Perspective
Volume I represents zero geometrically as a circle. The circle
denotes a closed relation returning to itself, without a
distinguished beginning or end in that reading. One appears as a
line cutting the circle. Yet the line must not be introduced as
something alien, breaking open a finished closure from outside.
We would then have to explain where that alien line came from,
opening yet another surface.
One is zero under another geometrical reading.
Seen from the front, the circle displays its roundness. As a
conceptual side projection, the same closed form appears as a
line. We do not presuppose an original three-dimensional space
here; the comparison merely illustrates the different readability of
one relation. Apparent linearity does not abolish closure. The line
is already self-referential because it is a reading of the same
circular relation.
One does not inaugurate a self-reference that was
previously absent. It is its directed appearance.
This is exactly the structure of A|B: A cannot be determined
without B, nor B without A. Likewise, zero as circle cannot be
explained without one as its directed reading. And one makes
sense only as another reading of that closure.
We write:
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The Geometry of the Light Number: Mirroring, Breakpoint and Self-Reference
0|1
The further sequence
0|1|2|3|4|5|…
is neither a series of independently created things nor a process
occurring in an existing time. It is the recursive differentiation of
third-order difference. Each number distinguishes the existing
relation again and displays its cyclic recurrence under another
geometrical position.
Only the observer reads this recurring order as a temporally
directed continuation under stable self-measurement.
Time does not produce the numerical sequence. Its directed
reading presupposes a recognisable order of difference.
We can divide the same basic relation differently:
0|1|1 → 0|2
On the left, the recurrence of two one-readings remains explicitly
divided; on the right it is condensed into a two-reading under a
particular rule. The internal division of the representation differs,
but the common original relation remains readable backwards
under the appropriate rule.
Thus the double readability encountered in Chapter 17—directed
expansion and cyclic recurrence—is already present in 0|1. They
need not first be linked afterwards through the physical notation
m/s.
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The Geometry of the Light Number: Mirroring, Breakpoint and Self-Reference
18.3 Two: The Mirroring of What Is Already Closed
Volume I describes two as the mirroring of one: the geometrical
reading of a radius becomes a diameter. This must not suggest
that one needs a second half before it can be complete. As
another reading of zero, it was already self-referential.
Two makes the opposing relation explicitly readable as
polarity.
A|B does not designate a connection between originally
independent objects. The poles are determined solely through the
distinction they jointly sustain.
The same holds for observer and world. The observer does not
stand before a finished sphere of perception already read
independently of him. His observer position is determined within
self-measurement in relation to its opposing pole.
This supplies the connection in the reading 2|0: two makes
mirroring visible, zero makes its closed return visible. These
meanings are not things assembled retrospectively.
18.4 Three and the Breakpoint to Four
With three, Volume I introduces the triangle. An equilateral
triangle has three interior angles of sixty degrees:
60° + 60° + 60° = 180° = π rad
Three here denotes a closed relation among relations.
Four changes the geometrical order. A square has four
ninety-degree angles:
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The Geometry of the Light Number: Mirroring, Breakpoint and Self-Reference
4 × 90° = 360° = 2π rad
A diagonal divides it into two right-angled triangles. For each:
45° + 45° + 90° = 180° = π rad
The new figure thus contains two triangular orders, whose angle
sums add to 2π. At the same time the distribution of angles in
each triangle has changed from that of the equilateral
three-geometry:
60° | 60° | 60° as against 45° | 45° | 90°
The triangular relation remains distinguishable but is
referred back to itself under a new division.
The shift of angle sums from π to 2π illustrates half and full
circular angles. It does not mean that a square actually travels
around a circle, nor is it by itself a derivation of π. At this
breakpoint from three to four, Volume I situates the further
relation between division and the closure of the circle. Four
therefore appears not merely as the fourth counting step but as
another level of geometrical return.
18.5 Eight: Two Fourfold Orders in Relation
Volume I describes eight as an inner return to symmetry. Its
fourfold structure is mirrored.
Arithmetically, 8 = 4 + 4. For our geometrical use, 4|4 is more
revealing.
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The Geometry of the Light Number: Mirroring, Breakpoint and Self-Reference
The two fourfold orders are not independently given figures. They
are two readings of the breakpoint at which the existing
geometrical relation is distinguished anew under a further order.
In Chapter 17 we encountered a corresponding duality:
Metre | Second
Spatial measurement is expansive and directed; temporal
measurement is cyclically scaled and read as directed
continuation. Both are bound to the same self-relation.
In our application this double geometrical relation is read as 4|4
and hence as eight.
Eight here does not denote two alien geometries but the
mutual readability of two fourfold orders of the same
self-reference.
18.6 Two Already Operates Within Eight
What allows the two fourfold orders to become two
distinguishable readings at all? Their polar relation. And what
keeps them readings of the same order? Their mutual mirroring.
Two does not therefore stand as an isolated number beside eight.
It designates the mirror-relation in which eight first
becomes readable as 4|4.
Conversely, eight is a particular geometrical expression of the
polarity already determined by two. Without two, the fourfold
orders would appear side by side as unrelated magnitudes.
Without eight, two would remain a general mirroring whose
specific spatial-temporal reading had not been shown. Without
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The Geometry of the Light Number: Mirroring, Breakpoint and Self-Reference
zero, the closed return making both readings determinable as the
same self-relation would be missing. These are not separate
numerical causes in succession. The numbers support one
another as readings of a common geometry.
18.7 Nine: The Limit of Determinate Reading-Out
Volume I calls nine the keystone of single-digit numerical order.
With 9 + 1 → 10, the positional notation changes level.
This does not mean that difference itself ends. The previous
scaling of the numerical reading reaches a breakpoint and
continues under another level. Under our observer reading, nine
designates the limit of the determined sphere of perception. It is
not a wall at the edge of a given space, but the boundary of the
order under which self-measurement can be read as coherent
unfolding.
This limit, too, is not an independent third thing. It is the limit of
the polar relation read as two and of the double geometry
recognised in eight.
The mirroring of two, the double fourfold order of eight and
the scaling limit of nine are three mutually determined
readings of the same self-relation.
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The Geometry of the Light Number: Mirroring, Breakpoint and Self-Reference
18.8 The Light Number in Volume II: Determination and
Numerical Position
We return to:
299 792 458 m/s
Since 1983 the metre has been defined through the speed of light,
whose numerical value was fixed exactly for the SI system. The
present SI definition continues this binding to the chosen light
relation.
This matters greatly. It shows what we have meant by measure
since Chapter 13.1: The metre is itself a stabilised
comparative relation. We define the unit under a relation
measured as recurrently identical, then use that unit to read
further relations.
Someone who offers the resulting decimal number as independent
proof of the necessity of this very choice would mistake
self-reference for external justification. The digits are determined
under our order of measure; they are not an isolated numerical
substance of light. But their conventional status does not abolish
self-reference. It makes it visible: even the defining measure is an
enactment of self-measurement. An observer does not leave his
relation by declaring its recurrence a measure.
In the chosen representation, Volume II reads three groups:
299 | 792 | 458
First group: 299
2 + 9 + 9 = 20 → 2|0
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The Geometry of the Light Number: Mirroring, Breakpoint and Self-Reference
We deliberately preserve 2|0 instead of reducing 20 completely to
the isolated two through 2 + 0. Two denotes polar mirroring,
while zero explicitly preserves its closed return. The difference
between 0|1|1 and its contraction to 0|2 has already shown that
division into groups is part of the reading rule and cannot be
smoothed away without loss of relational content.
For the two following groups we carry the reduction of digits
through to a single-digit number. These differing representational
choices belong to the specified relational reading and must not be
advertised as one uniform digit-summing rule. In 299, nine
already appears twice. The limit of numerical reading is thus
contained within the group whose result is read as closed
mirroring.
Second group: 792
7 + 9 + 2 = 18 → 1 + 8 = 9
Here two is already present in the original group, eight appears in
the intermediate reduction, and nine becomes visible as the
limiting reading. The three roles cannot first be connected at the
end; they are already related within the same grouping.
Third group: 458
4 + 5 + 8 = 17 → 1 + 7 = 8
Here too we can distinguish 4 + 5 = 9. Nine belongs to the
reading of the group leading to eight. Eight, in turn, can be read
as 4|4: two fourfold orders in mutual relation.
The arithmetic steps can be checked. Their geometrical meaning
does not arise from arithmetic alone, but from the previously
developed, mutually determined roles of the numbers.
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The Geometry of the Light Number: Mirroring, Breakpoint and Self-Reference
18.9 The Complete Arrangement
The three groups yield the reading:
2|0 – 9 – 8
Or, in its extended geometrical representation:
2|0 – 9 – 4|4
This is not a new formula for a force or for the velocity of an
independently presupposed object. It is a summarising reading of
geometrical roles within a chosen numerical position. Two is the
mirroring relation of the two fourfold orders. Eight shows the
same relation as a double geometrical order. Nine is the scaling
limit of their common unfolding. Zero denotes the closed
self-reference already underlying every reading.
No element of this arrangement can bear its meaning
entirely independently of the others within our geometry.
Numbers are not construction blocks from which we first
assemble a light relation. They are different readings of the
relation in which they determine one another as differences.
18.10 The Return to the Complete Frame
The self-relation is:
Δ := |(|)
We have illustrated its complete folding back as:
Δ|Δ → Δ|_|Δ
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The Geometry of the Light Number: Mirroring, Breakpoint and Self-Reference
The arrow does not denote the temporal manufacture of two
independent differences. It makes the already reciprocal
self-relation visible under another reading.
The observer relation has the same recursive arrangement:
(¤|World)|¤
The observer determines himself in relation to the world and
reads the relation of difference back onto his own observer
position. The sphere of perception appears under this
self-measurement. The expansive spatial reading of measure and
the cyclic temporal reading of comparison therefore belong to the
same self-relation. We designate their common limiting reading
by the light relation.
One was already another perspective on closed zero. Two makes
its polarity explicit. Four shows a shift of geometrical division.
Eight shows two fourfold orders in mutual relation. Nine marks
the breakpoint of determinate scaling.
None of these numerical figures subsequently generates
self-reference.
18.11 The Test of Backward Reading: What Remains
Invariant When Units Change
We examine different numerical expressions of the same light
relation:
Measurement reading Numerical value
Metres per second (m/s) 299 792 458
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The Geometry of the Light Number: Mirroring, Breakpoint and Self-Reference
Metres per millisecond (m/ms) 299 792.458
Centimetres per millisecond 29 979 245.8
Millimetres per millisecond 299 792 458
Kilometres per second (km/s) 299 792.458
Metres per microsecond (m/μs) 299.792458
Kilometres per millisecond 299.792458
The conversions are straightforward.
One metre is a hundred centimetres; one second is a thousand
milliseconds. Thus:
299 792 458 m/s = 29 979 245.8 cm/ms
One metre is a thousand millimetres and one second a thousand
milliseconds. Thus:
299 792 458 m/s = 299 792 458 mm/ms
In both cases the designated relation remains the same. In the
second, even the bare numerical value remains unchanged
despite the change of units.
If we explicitly carry along the power of ten, we can return each
representation to the same sequence of digits:
m/s : 299792458 × 10⁰
m/ms : 299792458 × 10⁻³
cm/ms : 299792458 × 10⁻¹
mm/ms : 299792458 × 10⁰
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The Geometry of the Light Number: Mirroring, Breakpoint and Self-Reference
km/s : 299792458 × 10⁻³
m/μs : 299792458 × 10⁻⁶
km/ms : 299792458 × 10⁻⁶
Thus the selected grouping 299|792|458 remains accessible in
every decimal scaling.
But we must distinguish two different invariances here.
First, preserving a sequence of digits while explicitly retaining a
power of ten is a general property of decimal notation. It holds for
any number written in that manner. Reconstructing the digit
groups is therefore not a special physical proof about light.
Second, the recurring limiting relation of light measurement
under different observer and apparatus couplings is another claim
entirely. Simply changing metres to centimetres does not test this
relation; it only changes its scaling.
Nor may we turn this distinction into an external test bench
suddenly independent of every observer relation. Comparison
between different observer readings is itself determined,
stabilised and read backwards within a measurement order.
The question, then, is not which digits prove a primordial value
outside measurement, but what makes the complete relation
recognisable as the same under altered readings. Decimal shifting
illustrates the difference between numerical position and a
relation of measure. The light relation poses the more demanding
question of the stability of self-measurement under changed
observer reference. These are different levels of the same
relational investigation, not independent worlds.
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The Geometry of the Light Number: Mirroring, Breakpoint and Self-Reference
18.12 The Self-Relation Is Not an External Circle of Proof
Here we encounter perhaps the strongest objection to our
numerical reading. The numerical value 299792458 was
stipulated within a particular SI measurement order. The metre is
defined through this light relation. To present the resulting digits
as independent evidence that this stipulation was necessary
would turn one's own stipulation into its supposed external proof.
That would be a problematic circular argument.
But neither must we commit the opposite error. We cannot first
describe a physical measurement order as a positing by the
observer and then invoke it as an alien, observer-independent
authority against his self-measurement.
Self-measurement need not leave its own return in order to
become determinable.
The observer distinguishes a recurring relation, stabilises a rule,
establishes it as a measure and uses it to read further relations.
The measuring rule has not thereby been removed from the world
of relations. It remains itself a distinguished relation. A
problematic proof-circle would arise only if the observer
concealed the fact that he fixed the comparative rule and then
offered its numerical value as an independent cause of that very
stipulation.
The self-reference of difference, by contrast, is not the outcome of
this attempted proof, but its declared starting point:
Δ := |(|)
Difference distinguishes itself.
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The Geometry of the Light Number: Mirroring, Breakpoint and Self-Reference
Neither the clock nor the caesium atom, neither rocket nor
telescope, nor the definition of the metre takes us outside this
relation. They realise different readings, comparisons and
scalings within the same self-referential order.
The observer cannot detach his rule of measure from
self-measurement by declaring its recurrence exact. Unambiguity
of determination lies in the regulated relation; the exactness of
the defined numerical value lies in the stipulated numerical
representation. Technical measurement precision, by contrast,
concerns the reliability of its practical realisation. These roles
differ but are not independent origins.
The numerical reading is therefore neither an arbitrary
invention nor a view of numerical substance independent of
measurement. It is the determinate read-out of a relation
traceable back to itself.
This does not justify every arbitrary division of digits. The chosen
grouping must be disclosed and its relations must remain
readable backwards under the stated rule. Yet assessing them
requires no second instance outside all measurement to
legitimate difference in the first place.
The task of our geometry of numbers is not to certify a supposed
independent physical necessity after the fact. It is to make visible
which geometrical self-relation operates jointly in
determination, scaling and repeated reading-out.
Thus we have neither discovered a light-number as an original
thing nor installed an alien natural constant as a new origin. We
have shown how the light relation, in our geometrical reading,
expresses its polarity, scaling limit and interweaving of two
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The Geometry of the Light Number: Mirroring, Breakpoint and Self-Reference
orders of measure as a connected whole. Backward readability
decides whether the different appearances determine the same
self-relation.
It is not number that makes the relation necessary. It is the
relation that makes a particular numerical reading
possible.
And so our investigation begins and ends with the same
distinction:
A|B – 0|1 – Δ := |(|)
Difference distinguishing itself.
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19 The Reading Is Determinate, Not the Continuation
The Reading Is Determinate, Not the Continuation
CHAPTER 19
19. The Reading Is
Determinate, Not the
Continuation
19.1 Is Physics Deterministic?
Is physics deterministic? Within the distinction developed here,
the answer is yes—but not in the sense that a complete present
already fixes one unique future.
Physics is the determinate reading-out of a relation under a
stabilised measuring rule. The Frozen Frame is not a snapshot of
a process already underway, but a complete relational
configuration:
|_| = (0|Δ)
Within this closed relation a particular measurement is
unambiguous. If Archibald reads his distance from a stone as
three metres under his rule, it is three metres in this
measurement—not four or seven at the same time. That does not
mean that the stone possesses the attribute “three metres”
independently of the observer.
Determination arises in the measuring relation:
¤ | World
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The Reading Is Determinate, Not the Continuation
If we change the comparative rule, the number can change
without making the complete relational configuration
contradictory.
The reading-out is determinate within its posited relation.
The entire continuation of that relation is not thereby
determined.
19.2 A Frozen Frame Does Not Determine Its Successor
We previously wrote a series of complete frames as:
|_|₁ ¤ |_|₂ ¤ |_|₃ …
The observer reads different complete configurations as a
connected sequence.
Time arises as a difference between differences:
Δt = (Δ₂ |₋ Δ₁)
No single Frozen Frame contains motion through an
independently existing time. Nor does it contain a rule that must
produce precisely one next Frozen Frame. The first is complete;
the second is complete. Their order becomes readable as
continuation only within the observer relation.
Hence:
The determinacy of |_|₁ does not compel the unique
determination of |_|₂.
Yet arbitrariness does not follow.
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The Reading Is Determinate, Not the Continuation
19.3 Coherence: Why Not Everything Can Follow
Volume II already supplies the decisive notation for relating
complete frames:
|Σ = ((|_|₁|_|₂) ⇔ (|_|₂|_|₁))
Relational coherence—Stringenz—means the mutual readability
of their relation.
A second frame can belong to the continuation of the first only if
both form a connected difference within the complete relation.
That does not mean time must run backwards. It means that the
connection cannot be explained solely through the supposed
cause within the first frame. The history must remain readable
also from the determined result.
That is the sense of our retrocausality:
((Δt⁻ | Δt⁺) ≡ (Δt⁺ | Δt⁻))
Past and future are not two independent regions connected by a
force. They are different orientations of the relation in which the
observer determines his continuation. Coherence is therefore not
an additional force joining frames together.
It is the coherence under which different complete frames
can be read as a continuation of the same self-relation at
all.
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The Reading Is Determinate, Not the Continuation
19.4 Why We Still Remember Incorrectly
A remarkable consequence concerns memory. Memory is not the
unchanged storage of earlier Frozen Frames; it is their
reconstruction through present self-measurement. We read the
past under the relation of our present. This backward reading
may be unambiguous, but it can also permit several coherent
reconstructions or no longer make particular differences
accessible. We may recall a conversation yet forget who spoke a
certain sentence.
The earlier Frozen Frame was not therefore incomplete.
Only its present backward reading is no longer
unambiguous.
This also explains why a memory can seem coherent without
being correct in every detail.
Coherence is no guarantee of flawless memory.
Our memory evidently is not a video recorder but an archivist
whose computer is constantly glitching: sometimes the file
is missing, sometimes it opens the wrong one—and
sometimes it assures us with perfect confidence that this is
exactly the right memory.
What matters is that imperfect memory does not destroy our
observer invariance. We need not consciously reconstruct every
earlier distinction in order to determine ourselves as a
continuation of the same self-relation.
Memory is a reading-out of our continuity.
It is not its origin.
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The Reading Is Determinate, Not the Continuation
19.5 Only What Is Possible Is Possible
We arrive at an apparent tautology:
Only what is possible is possible.
But Volume I had already developed geometrically the relation of
possibility and impossibility:
☉|0|⊡
Possibility and impossibility are not independent domains.
They form a complementary distinction.
Zero denotes their common boundary and reference level. A
relational configuration is possible under a given reading rule
when it can be coherently determined within that relation. What
cannot be closed under the same rule appears impossible.
But we have determined only the limit of this reading.
Not the limit of every other perspective.
Impossibility is itself a relational determination.
Something that appears impossible from one perspective can
therefore become possible from another—not because a
contradiction within the same unchanged order suddenly
becomes true, but because the order under which the appearance
is determined changes.
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19.6 Archibald and the Impossible Dimension
The planet Archibald sees through his telescope illustrates this
once more. Its inhabitants appear to him as two-dimensional.
His observer relation is:
¤ₐ | Telescope | World
The index marks Archibald's perspective. The telescope is not an
independent third pole but the outsourced boundary of his
perception.
What follows from his seeing two-dimensional beings?
Initially only that his present self-measurement reads this
appearance as two-dimensional.
We have shown, however, that the two-dimensional order, in its
renewed self-reference, permits a three-dimensional reading. The
third dimension is not added as an alien spatial container. It
arises as another perspective on an already distinguished
geometry. We must therefore not confuse Archibald's reading
with the inhabitants' own self-relation.
For an inhabitant, another observer position applies:
¤ᵦ | World
The observers need not read their world under the same
geometrical perspective.
Archibald can perceive the inhabitants as two-dimensional
while they experience themselves as three-dimensional.
His perception is not thereby necessarily false.
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The error would be to infer that his geometrical reading must also
be the complete self-perception of the other observers.
Now Archibald boards his rocket:
¤ₐ | Rocket | World
The rocket extends his movement-related self-measurement, just
as the telescope extended his perceptual boundary. He changes
the relation under which he reads the planet. What previously
appeared to him as a mere surface may under his new position
show a depth relation. That does not mean Archibald first creates
a third dimension for the inhabitants.
He changes his perspective, not their original self-relation.
What seemed impossible becomes possible for Archibald because
altered self-measurement permits another geometrical reading.
This also explains why the impossible geometries of Volume I are
more than a thought experiment.
The Penrose triangle is possible as a two-dimensional drawing.
As a continuously connected three-dimensional body under one
unchanged spatial reading rule, it is impossible.
The same appearance: possible under one reading,
impossible under another.
The limit belongs to the particular determination.
But now the decisive question arises:
Can Archibald reach every other possible perspective and still
remain Archibald?
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19.7 The Observer's Capacity for Continuation
At this point the question of coherence leads directly to continuity
of the observer.
Volume IV defines stability as invariant self-relation:
OΔO
The order of difference changes in its reading-out yet remains
traceable back to itself.
Stability is not rest.
It is the recurrence of a self-relation under further distinction.
For the observer this means:
Not every possible relational configuration is a possible
continuation of himself.
Another configuration can be coherent in itself. But to belong to
the experience of the same observer, his invariance must be
readable again within that new complete configuration as its own
relation.
That is what the capacity for continuation means.
The observer does not remain the same because some unchanging
material core is transported through frames. He remains the
same because his self-relation is preserved through change.
We are reminded of the continued distinction:
1+1+1+1+1+…
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Here the ones stand neither for independently coexisting
observers nor immediately for seconds. They denote repeated
readings of difference whose connection is determined by the
same invariant self-relation.
The decisive issue is not addition.
It is whether the self-relation can be traced back to itself under
every further reading:
OΔOΔO…
If this return breaks off, we can still describe different complete
relational configurations. But we can no longer determine them as
the continuation of this observer.
Then you are no longer you there.
We can therefore define continuity precisely:
Continuity is the ongoing self-readability of observer
invariance within different complete Frozen Frames.
Continuity is thus not identical with uninterrupted memory. You
can forget something and remain yourself. But a relational
configuration that can no longer be traced back to your invariant
self-relation at all does not belong to your own continuation.
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19.8 Choosing a Possible Continuation
What role does choice play? It does not first turn an incomplete
condition into complete reality. It determines the orientation of
the observer's reading among possible complete relational
configurations.
Archibald can change his position.
He can extend his perception.
He can encounter possibilities previously unknown to him.
But each continuation must satisfy two conditions:
|Σ – the complete configurations must remain coherently readable
in relation to one another.
O Δ O – Archibald's self-relation must remain preserved through
the change as his own continuation.
The first condition concerns the coherence of frames.
The second concerns observer invariance.
Not every coherent configuration is therefore already a possible
future for Archibald. And not every previously excluded
appearance remains excluded from every other perspective.
Choice is neither the mechanical execution of a uniquely
predetermined path nor the creation of arbitrary realities.
It is the orientation of self-measurement within the
possibilities of its own continuity.
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19.9 Determination, Possibility and Continuity
These distinctions let us answer the chapter's initial question.
Physics is determinate insofar as a particular relational
configuration is read under a particular measuring rule:
|_| = (0|Δ)
Continuation is not thereby uniquely predetermined:
|_|₁ ¤ |_|₂ ¤ |_|₃ …
Yet it is not arbitrary, since it must form a coherent order:
|Σ = ((|_|₁|_|₂) ⇔ (|_|₂|_|₁))
Even the boundary between possible and impossible is
perspectivally determined:
☉|0|⊡
And if a possible continuation is to remain the experience of the
same observer, his self-relation must remain invariantly readable
backwards:
OΔO
We therefore obtain no linear chain of external causes carrying an
observer through finished time. We obtain a determinate
reading-out of complete relational configurations whose possible
continuations are bound to their own coherence and the
observer's invariance.
The reading-out is determinate. Continuation remains
open. Its relation must be coherent. And it is continuous for
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the observer only if his invariance can be read out again
within it.
Only what is possible is possible.
But not every possible relational configuration is a possible
continuation of yourself.
You do not remain yourself because you never change. You
remain yourself because, through change, you can still
return to yourself.
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20 The Universe We Never See Simultaneously
The Universe We Never See Simultaneously
CHAPTER 20
20. The Universe We Never
See Simultaneously
20.1 What Do We Actually See When We See a Star?
Archibald points his telescope towards the night sky.
What does he see?
A spot of light.
Not directly the surface of a star, not its mass and not its
distance. He registers an appearance of light whose brightness
and apparent position can vary under different conditions of
observation. Nevertheless, Archibald claims he is observing a
star.
How does he know?
We begin with our original distinction:
A|B
A denotes Archibald's observer position; B the opposing pole
distinguishable under that position. What appears as a star is
initially a particular light-reading of this relation.
With his telescope Archibald extends the boundary of his
perception:
¤ | Telescope | Spot of Light
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The telescope is not an independent third side. It changes which
differences can be registered within the observer–world relation.
The return of measurement remains:
(¤|World)|¤
Archibald does not leave his self-measurement by extending
perception technically. He changes its reach and resolution.
In Chapters 17 and 18 we examined the light relation as a limiting
reading of this self-measurement:
0|c ≡ 0|(Δ |÷ Δt)ₘₐₓ
In our derivation the speed of light does not first appear as the
motion of an independent object through finished space. It
denotes that stable ratio-reading under which spatial and
temporal determinations of measurement are coupled.
Thus Archibald initially registers a light relation. Only further
measurements let him determine this appearance as a
recognisable astronomical opposing pole.
Here the stability reading of Volume IV comes into play:
OΔO
An order of difference remains traceable back to itself under
further distinction.
The astronomical fixed point therefore need not be a motionless
spot of light. It is the recognisable relational position under which
various registrations can be read as appearances of the same
astronomical connection.
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As a recognised appearance, the star is a mirroring of
Archibald's own fundamental position `A|B`.
We do not mean an optical mirror reflecting Archibald's face. We
mean the opposing position within his self-measurement: what is
recognised as a star is B under the observer fixation A.
The physical description of a star as a luminous plasma body
emerges only through further, mutually coupled determinations.
And here we encounter a problem already familiar from the speed
of light.
We begin with a measured relation, determine an object of our
description from it, and then use that object as the basis for
further measurements.
20.2 The Atom We Must First Measure
Archibald needs no atomic physics to see a spot of light in the sky.
His eye distinguishes a light appearance. He need not calculate
electron transitions or know the structure of an atom to perceive
that appearance.
We can write the original relation as:
¤ | Eye | Spot of Light
Physics can of course describe the eye's operation in atomic and
molecular terms. But this subsequent explanation is not a
prerequisite for Archibald's ability to perceive at all.
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Now he wants to know more about the spot of light. He uses a
spectrograph and obtains a particular distribution of registered
light.
¤ | Spectrograph | Spectral Registration
That is another measurement relation.
We have not simply looked at the original spot of light in a more
immediate form. We have introduced another technical boundary
and thereby changed which differences can be registered.
To identify certain spectral lines, physics uses atomic references.
It compares registered wavelengths with known transitions and
characteristic spectral lines.
But how do we know these atomic transitions?
They, too, have been measured under particular experimental
arrangements.
We built apparatus, compared registrations and determined a
recognisable order of appearances that we describe as atomic
structure.
The atom appears to us as an object of physics because its
relations can be read out stably and repeatedly under these
measurement orders.
Now we do something remarkable.
We use this measured order as a standard for further
measurements.
The SI second is defined through the specified frequency of a
transition in the caesium-133 atom. We use the time
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measurement stabilised thereby to determine frequencies and
further couple spatial and temporal measures.
There is nothing wrong with this as a measuring procedure. We
can choose a stable relation as reference and use it to determine
others.
But we must not forget what we have done.
We read out a measurement relation as an atomic transition,
stabilised its recurrence as a standard and now use that
determination to describe further measurement relations.
The atom is therefore not the relationless origin of
measurement. It is itself a stabilised reading-out within a
measurement order.
The distinction becomes an ontological problem only when we
treat the atom thus determined as the entirely independent cause
of the measurability from which its physical determination
emerged.
Atomic physics can make correct predictions within its systematic
framework. The recurrence of atomic transitions can be used with
extraordinary technical precision.
But precision does not abolish the self-reference of measurement.
It stabilises it.
We thus encounter a familiar difficulty in physics. If the atomic
measurement order itself arises through particular apparatus
relations, we must ask what happens when we change the
arrangement.
This leads us to wave–particle duality.
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20.3 Two Experiments, Two Traces
Wave–particle duality is usually presented as a remarkable
property of light.
Under some conditions light behaves like a wave; under others its
particle character becomes apparent. Both modes of appearance
are described under the common term photon.
But what have we actually done?
We have performed different experiments.
Consider the double slit.
In one experimental arrangement we register individual detector
events. Over many repetitions they form a spatial distribution
showing interference.
We can describe this distribution as a wave-like order.
Now we change the experimental arrangement, introducing, for
example, a measurement that can make it distinguishable which
path the registered light took.
The accessible interference order changes accordingly.
Physics knows this relation and describes it, among other things,
through the complementarity of the relevant measured quantities.
Our question, however, comes one step earlier.
Did we even determine the same measurement relation in
both cases?
No.
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By changing the experimental arrangement we introduced a new
difference.
The measurement is no longer the same.
We can designate the two experiments as different technical
readings of the same original observer–world relation:
¤ | Experiment₁ | Registration₁
¤ | Experiment₂ | Registration₂
The experimental arrangements are not third original poles. They
are different configurations of the boundary of measurement.
The decisive point is that each registration receives its meaning
only within the complete experimental arrangement.
And this brings us back to our earlier counting sequence:
1+1+1+1+…
These ones are not originally existing light particles. They denote
repeated distinction under a particular reading rule.
As long as that rule is preserved, the repetition can be read as a
connected trace.
But if we change the experimental distinction, we change the
order under which those repetitions are compared.
The rhythm of our reading changes.
Initially this does not mean that a presupposed physical frequency
must necessarily change its numerical value.
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Rhythm here means the particular order of recurrence: which
differences we count, how they are related, and under what rule
their sequence appears to belong together.
In the first experiment we read a trace whose repeated
registrations jointly form an interference order.
In the second we introduce additional path information. Another
distinction now forms part of the measurement, and the joint
distribution of registrations is accordingly determined differently.
The number of individual detector events may be equal in both
experiments.
But that does not mean we have read the same trace.
Equal countability does not entail an identical relational
order.
This is the difference between mere accumulation and coherent
continuation.
The recurrence 1+1+1+1+… does not acquire meaning merely
because we write down ones. Its meaning comes from the relation
under which they become readable as repeated enactments of
distinction.
Change the experiment and we change that relation.
The new trace must be coherent within its own measuring
arrangement. It need not be identical to the trace of the previous
arrangement.
That both experiments can be described by a common physical
theory is therefore unsurprising. A theory can link different
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The Universe We Never See Simultaneously
registrations and calculate how they depend on their particular
measurement conditions.
The possible error begins with another conclusion.
We take different experimental readings, give them the shared
name light or photon, and then treat that name as designating an
object determined independently of those experimental
arrangements.
Then we ask:
How can this one thing be both a wave and a particle?
Perhaps the question was wrongly posed from the outset.
We did not directly observe one original object as a wave on
one occasion and a particle on another. Under different
experimental arrangements we determined different traces,
then ascribed their common readability to an object.
This does not imply that no stable physical relation can describe
both experiments together.
On the contrary, the success of quantum theory shows that the
results are not arbitrarily independent.
But a shared mathematical description does not prove that its
classical pictures—wave and point-particle—must exist as two
original properties of a relationless thing.
For our investigation, then, the duality is initially a duality of
particular modes of appearance.
The experiment changes the difference. The difference
changes the reading of the trace. Only the joint reading-out
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of different traces generates the question of one unified
object behind them.
Thus we have encountered the same reification that concerned us
in astronomical observation.
20.4 From Atomic Measure to Astronomical Distance
Let us return to Archibald.
He still sees his spot of light.
Now he studies it under various measuring arrangements. With
his spectrograph he determines characteristic wavelengths;
through further astronomical procedures he infers a distance.
We must keep these determinations distinct.
Spectral registration is a light-reading under a particular
apparatus. Distance arises under another comparative rule.
For nearby stars we can use parallax: the apparent direction of a
star is compared from different positions of Earth.
For very distant galaxies we may, for example, use calibrated
brightness relations of suitable astronomical sources.
Distance is therefore not already a visible property contained in
the spot of light.
It is inferred.
The same holds for redshift. We compare the registered
wavelength of a characteristic spectral line with its reference:
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The Universe We Never See Simultaneously
z = (λ_observed − λ_reference) / λ_reference
Initially we have obtained a ratio of two wavelength
determinations.
We have not yet directly seen spatial expansion.
Only by coupling redshift with a reading of distance do we obtain
the astronomical relation decisive for cosmology.
Light Reading | Distance Reading
These are different measurement orders made comparable under
a common observer position.
The atomic reference determines a spectral order of comparison.
Astronomical distance determines a spatial one.
Their common readability requires that the different registrations
can be related coherently under their respective rules.
We must not treat them as the same measurement merely
because we subsequently relate them to the same star or galaxy.
We examined this very error in wave–particle duality.
Bringing different readings together is legitimate.
Equating their original conditions of measurement is not.
The astronomical opposing pole is therefore not the sum of
independently discovered attributes.
It is a stable determination under which different readings of
light, distance and comparison can be connected.
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The distance so determined also affects our interpretation of the
registered light configuration.
A light registration attributed to a nearby astronomical source has
another spatial-temporal assignment than a registration
attributed to a very distant galaxy.
Calculating a distance does not change the spot of light we
receive.
But the distance we determine changes which complete
relational configuration we infer as the origin of the
present light registration.
This brings us to the Hubble relation.
20.5 The Hubble Relation: When Distance Becomes
Expansion
Astronomers observe many galaxies, determine their redshifts
and infer their distances.
At low redshifts, the Hubble relation is approximately expressed
as:
v ≈ H₀ · d
Here d denotes the determined distance, v the cosmologically
inferred recession velocity and H₀ the Hubble rate at the present
epoch.
For small redshifts we can write approximately:
v≈c·z
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and hence:
H₀ ≈ (c · z) / d
What has been determined here?
A displacement in light.
A distance.
And the ratio between these two measurement readings under a
particular cosmological description.
The number does not stand at the beginning of the relation. It
arises from its scaled determination.
We know this already from the speed of light. Metres and seconds
are stabilised comparative relations; the exact SI numerical value
of light speed is fixed through the definition of the metre.
The Hubble rate is different. Its value is inferred from
astronomical observations, not simply stipulated through the
definition of a unit.
Yet the fundamental connection is the same:
A number receives its physical meaning from the relation
under which it is determined.
Now we read astronomical distances under an expansive
direction:
0|₊Δ
Change in the spatially determined relation appears as expansion.
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At increasing cosmic distances we also register light assigned to
different stages of the inferred cosmic history.
Distance, light configuration and temporal assignment are
therefore coupled within cosmological measurement.
It does not follow, however, that the expansive direction alone
numerically fixes the particular Hubble relation, let alone its
acceleration.
We must take account of the determinate form of the
measurement relation.
In standard cosmology the time-dependent Hubble parameter H(t)
describes the development of the expansion rate, while cosmic
acceleration concerns the second time derivative of the scale
factor.
The Hubble rate itself may decrease even while the expansion
accelerates.
At large redshifts, therefore, the simple linear Hubble formula is
no longer sufficient. Relations between spectral shift, distance
and cosmic development must be determined in a more
comprehensive model.
Astronomical evidence for accelerated expansion comes, among
other sources, from the relation between the redshifts of distant
Type Ia supernovae and their luminosity distances.
Here we encounter a particularly instructive example: the
so-called Hubble tension.
The Planck mission's analysis of the cosmic microwave
background yields a present-day Hubble value of approximately
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67.4 km/s/Mpc, assuming the cosmological ΛCDM model. By
contrast, the SH0ES collaboration infers approximately 73.0
km/s/Mpc through a local distance ladder based on geometrical
references, Cepheids and Type Ia supernovae.
Both procedures have their own calibration chains, comparison
rules and uncertainties. Each produces a precise determination
within its respective analysis. But the values do not agree within
their stated uncertainties.
This is the Hubble tension.
What does the difference mean for our investigation?
Not that the two numbers must automatically express the same
physical truth from two equally valid perspectives. Whether the
discrepancy is due to unidentified systematic effects, model
assumptions or an incomplete cosmological description remains
an open question in physics.
Yet the very existence of the tension shows why we cannot
remove the underlying measurement order from the meaning of a
number.
The number 67.4 is not simply 73.0 written differently.
Both are results of differently structured chains of
determination intended to refer to the same cosmological
parameter.
Their disagreement makes the conditions of their common
backward readability a subject of research in their own right.
Here the significance of coherence from Chapter 19 becomes
directly visible.
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Coherence within a single measurement order does not guarantee
that two differently organised measurement orders are already
mutually compatible.
And that allows us to examine the Hubble tension's starting
question more closely.
The usual question is:
Why do two different measuring procedures yield different values
for the same Hubble constant?
But this already assumes that both procedures must
independently determine the same object in the same way,
regardless of their respective orders of determination.
We must not adopt precisely that assumption without
examination.
Within a shared cosmological description, both procedures are
referred to the same parameter H₀. Under that assumption,
agreement is a meaningful and testable demand.
But the shared designation does not prove that we have already
determined an identical reading-out relation in both cases.
Two individually coherent measurement chains are not the
same measurement merely because we designate their
results with the same symbol.
The question therefore shifts.
We need not first assume that nature has supplied two
contradictory values of an independently existing quantity.
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We can first ask under what conditions the two different
measurement chains may legitimately be traced back to the same
invariant relation.
The Hubble tension is not thereby resolved numerically.
But the supposed self-evidence of its original question
dissolves.
The empirical difference remains. Within the standard
cosmological model, the demand to explain its cause remains
meaningful as well.
What we have exposed is the ontological assumption that the
common name H₀ already guarantees an identity independent of
the respective measurements.
Our actual question is therefore:
Under what invariant self-relation may we expect two
differently determined measurement relations to yield the
same numerical value at all?
The coherence of the individual measurements does not yet
answer that question.
Only a coherently determined relation between the two
measurement orders could ground their common readability.
We have not defined the discrepancy away. We have made visible
that part of its apparent mystery which emerges when we confuse
a shared theoretical designation with an already unconditionally
secured identity of measurement relations.
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Thus the Hubble tension brings us directly to the geometrical
order under which such measurements can be understood as
statements about one universe.
20.6 Spacetime as a Reified Measurement Order
Standard cosmology describes the universe under a particular
spacetime geometry.
For a universe that is approximately homogeneous and isotropic
on large scales, it ordinarily uses
Friedmann–Lemaître–Robertson–Walker geometry. Its scale
factor a(t) describes the time-dependent order of spatial
distances.
The Hubble rate is expressed as:
H(t) = ȧ(t) / a(t)
Accelerated expansion corresponds to a positive second time
derivative of the scale factor:
ä(t) > 0
Within this mathematical order we can calculate light travel
times, distances, redshifts and their changes.
These calculations are not wrong because the geometry is used as
an assumption.
On the contrary, standard cosmology succeeds precisely because
it compares different astronomical observations within a common
mathematical framework and tests predictions.
But we must be careful what we conclude from that success.
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A mathematical description can be correct without already
explaining the independent existence of every concept it employs.
This is where reification begins.
We determine astronomical relations of light and distance.
We choose a geometrical order that permits their coherent
description.
Then we speak of that geometry as though it were an independent
substrate originally producing those measured relations.
The statement:
The observations can be described as an expansion of spacetime.
becomes:
The observations arise because an independently existing
spacetime expands.
The second sentence makes an ontological claim beyond the first.
The success of the description alone does not justify the additional
step.
On a cosmological scale, this repeats what we encountered in
atomic physics and wave–particle duality.
We determine an appearance under a measuring arrangement.
We stabilise that determination as an object or geometrical
quantity.
Then we use it to explain further measurements.
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As long as we remember that we are employing a particular order
of comparison, there is nothing objectionable in this.
The problem arises when the order of comparison itself is
declared an independent cause of the relations through which it
first became measurable.
With the atom, this happens when an object determined under
apparatus-based reading is treated as the unconditional source of
further measurability.
With wave–particle duality, it happens when different
experimental traces are attributed to a double-natured entity
already presupposed independently of them.
With cosmology, it happens when mathematical spacetime
geometry is no longer regarded as a determinate description of
astronomical relations but as their original substrate.
The error changes scale. Its logical form remains the same.
The Hubble tension illustrates this confusion especially clearly.
Both measurement chains aim at the same cosmological
parameter, yet a shared designation does not guarantee empirical
agreement.
Physics can and must investigate under its model framework why
the results diverge.
Our inquiry begins with the common relation that could
justify treating these different measurement results as
determinations of the same quantity in the first place.
The name of the quantity does not answer that question.
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We must avoid another confusion, however.
Atomic physics, quantum physics and cosmology can test their
respective models using multiple distinguishable measurements.
Their common physical description is not mere arbitrariness.
But even many mutually compatible measurements do not abolish
the original observer relation.
The coherence of their results shows that a stable order has been
determined.
It does not by itself answer why that order can appear as
measurable reality at all.
Here the Tractatus begins:
A|B
Δ := |(|)
We do not begin with a finished particle, a finished wave or an
independently existing space.
We begin with difference and its self-reference.
The apparatus determines the perspective of reading.
The complete relational configuration determines the
measurement result.
Repeated self-readability makes stable comparison orders
possible.
Only under those comparison orders do the numbers of physics
arise.
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The relation does not become necessary through its
number. The number becomes possible through the
determined relation.
We can thus take established physics seriously within its domain
without making its geometrical objects the origin of our
derivation.
It describes with extraordinary success how its determined
measured quantities relate to one another.
Our investigation asks how those measured quantities can arise
from self-referential distinction at all.
That is the difference between a correct calculation and an
explanation of the original possibility of its determination.
20.7 The Star That Is Not Where We See It
Let us return once more to Archibald.
He looks through his telescope and recognises a distant star.
Initially, however, all he has determined is a present registration
of light.
From further measurements he infers a distance. From distance
and the light relation he determines a travel time, assigning the
registered appearance to an earlier phase of the astronomical
opposing pole.
The light he currently measures therefore does not directly
describe a state of the star observed simultaneously with the
measurement.
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Astronomical observation links different complete relational
configurations.
For their coherence we wrote in Chapter 19:
|Σ = ((|_|₁|_|₂) ⇔ (|_|₂|_|₁))
The temporal difference-reading remains:
Δt = (Δ₂ |₋ Δ₁)
Archibald determines a present light registration and, under a
particular measurement order, reads the astronomical past
associated with it.
He need not claim that the present light registration itself travels
backwards through an existing time.
Instead, he determines the connection between complete
relational configurations that can be read backwards.
Distance now acquires its proper meaning.
It is not merely a number subsequently attached to a star.
It belongs to the particular spatial-temporal order of comparison
under which Archibald infers an astronomical history from the
light registration.
And thus we understand what it means that we never see the
universe simultaneously.
We do not observe an entirely present collection of stars and
galaxies in a finished spatial container.
We register light under our present observer position and derive
differing distances and past phases of emission.
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The entirety of these astronomical readings can be described
within a common cosmological geometry.
But that geometry arises as a determinate order of
measurements. Its successful application is no independent proof
that it must be the original substrate of all measured relations.
For Archibald, then, the universe is not a collection of objects that
can all be viewed simultaneously.
It is an order of light relations, distances and temporal
assignments inferred under his self-measurement.
The different instruments do not alter this fundamental point.
The eye determines an appearance of light.
The spectrograph determines another trace within the same
astronomical inquiry.
Atomic references enable further comparisons.
Determining distance opens a spatial reading.
Cosmological geometry orders these results within a shared
spatial-temporal description.
Every measurement has its particular question, its own
experimental arrangement and its own trace.
Coherence lets us compare these different traces.
But their common readability does not turn them
retrospectively into one original measurement.
This is the Continuum of Errors.
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We confuse the stability of our descriptions with the independent
existence of their objects. We introduce geometrical readings and
subsequently treat them as causes of the very relations we read
under them.
The error begins with the supposedly immediately found star,
repeats itself with the atom and the photon, and finally reaches
cosmological spacetime.
We need not discard the calculations.
We must understand the direction of their justification.
Physics describes what can be coherently read out under its
particular rules of measurement. The *Tractatus*
investigates why such readability is possible at all.
Archibald is still looking through his telescope.
He sees a spot of light.
Only the relation under which he measures it makes it a star for
him, with a distance, a history of light and a position in the
universe.
It is not the star that originally explains measurement. The
determined measurement relation first explains what
Archibald can recognise as a star at all.
And we return to our fundamental position:
A|B
We did not find the universe outside that relation.
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Through ever new measurements we extended our own sphere of
perception, eventually describing its distinguishable relations as
spacetime.
The possible error is not that we measure spacetime. It is
that we subsequently make the spacetime inferred from our
measurement relations the original prerequisite of those
measurements.
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21 The Question That Forgets Its Premise
The Question That Forgets Its Premise
CHAPTER 21
21. The Question That
Forgets Its Premise
21.1 What Does an Explanation Actually Explain?
We have now travelled a considerable distance. It began with a
table and led us through its surfaces and the different
perspectives of its appearance to the question of space, then to
motion, time and number, and finally into those regions where
gravitation, heat, mass, energy, light and electrical signals
confront us as physical quantities. Most recently, we looked
through a telescope with Archibald and had to admit that even the
star he thought he saw so obviously first appeared to him as a
present registration of light.
Quite a journey for a table that never really had to leave its place.
Yet this journey has repeatedly brought us to a peculiar difficulty.
Physics explains an appearance by examining its relation to other
appearances that have already been determined. Ask why a stone
falls and we arrive at gravitation. Ask why it feels heavy and we
may examine interactions between the stone and our body. And
when Archibald wants to know why he can see it at all, his route
leads through light, eye, nervous system and the corresponding
molecular and electrical processes.
There is nothing intrinsically wrong with any of this. Each
explanation may determine connections more precisely, enable
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predictions and extend our technical capabilities. Much of
physics' success derives precisely from its ability to divide a
question into further, more precisely investigable relations.
Still, with all due respect for that success, we might occasionally
allow ourselves the question that so easily gets lost under the
table. This time quite literally.
What have we actually explained if the explanation itself
consists of further determinations that also require
explanation?
There is something else we should not overlook. Perhaps we do
not always face problems set for us by the world. Perhaps we
produce some of them ourselves, through the way we express
what we measure in the first place.
We choose a geometrical figure to describe spatial relations, a
numerical value to record a comparison, or the image of a wave or
particle to make particular experimental appearances intelligible.
Such representations can be extraordinarily useful; much of
physics would scarcely be possible without them. But once we
forget that we have chosen a particular way of representing
something, its meaning can shift almost imperceptibly: the means
by which we describe a connection becomes a supposed
constituent of reality, whose nature we must now explain.
And when the images subsequently fail to fit together, we find
ourselves facing a mystery. Except that we may not have
discovered this mystery at all. We may have first posed it through
our own description.
We do not thereby create the measurement results. But we
can certainly create the problem we make out of them.
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A gravitational field, too, must become recognisable as a
gravitational field under particular conditions. Mass is not just a
number floating freely in the universe; it is determined within an
order of comparison. And what we describe as an electrical signal
in the brain appears only under a measurement relation able to
distinguish that signal from other possible registrations.
We can follow these dependencies almost indefinitely.
Increasingly sensitive instruments, finer models and more
extensive mathematical descriptions will open new
determinations. Scientifically this is highly productive. But
however many answers we obtain, they all already employ a
capacity whose origin they do not explain: the possibility of
distinguishing something as something determinate at all.
Physics explains its appearances through further
determinate relations. In doing so, it already presupposes
the determinability of relations themselves. And where it
confuses its means of representation with their origin, it
can create additional difficulties that it then tries to resolve
within the same representation.
This does not mean that every physical explanation is false. It
means that an explanation that works splendidly within an already
established order does not, through its success alone, become an
explanation of that order's origin. We can measure the floors of a
building with exquisite precision without knowing why it stands
on a foundation. Adding further storeys does not necessarily
answer that question—although the view undeniably improves.
The question accompanying us through this book, then, is not
which additional physical appearance we must still discover. It is:
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How does anything become readable as something
determinate in the first place?
Perhaps the surprising thing is not that no generally accepted
answer has yet emerged. More surprising is how readily we forget
to ask once the other calculations begin to work.
21.2 When the Result Becomes the Premise
Consider our table again. Archibald recognises it from one
direction, changes his position and sees another surface. His
perception changes, but the object remains the same for him.
The customary explanation is readily available: the table is
three-dimensional, so it can be viewed from several sides. What
could be strange about that?
Nothing at first—as long as the statement merely describes its
appearances spatially. But once we treat three-dimensionality as
the original cause of those appearances, we have already
presupposed the geometrical order whose determinability we set
out to explain.
The Tractatus Principio therefore begins elsewhere. We
represented different fixations of the same recognisable relational
form as:
1|.1 | 1|.5
And where their common traceability is actually preserved, we
can contract the relation:
1|.1|.5
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One does not designate a table manufactured out of mathematics.
It is the visible mark of the recognisable relation under which
different appearances become readable as views of one
connection. The places following the scale flip distinguish their
fixations. Only through the relation among those readings do we
seek to derive the geometrical depth we ordinarily presuppose.
The shift becomes clear when we set the two directions of
explanation side by side.
We can say: because the table extends through space, it has
different views.
Or we can ask: which relations must remain preserved among
those views for us to recognise them as spatial appearances of
one object at all?
The second question does not yet explain every geometrical
property of the table. It merely refuses to present the spatial
determination we seek as a ready-made answer.
The result of our description becomes a supposed premise
wherever we subsequently treat the geometrical order
obtained from appearances as their original producer.
A spatial body is not wrong as a means of geometrical
representation. The direction of justification becomes wrong
when we turn the image into an independent cause. We
presuppose spatial depth, explain the different views by it, and
then solemnly ask why those views happen to possess spatial
depth. The answer was already hidden in the way the question
was framed. The table has acquired no new secret; we have
invented one for it.
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Our numbers are no different. Under the chosen directed trace
we could read four distinguished transitions and express their
recurrence numerically:
1+1+1+1→4
The number four did not secretly organise those transitions. We
read a particular sequence of distinctions and assigned it a
number. Nor is the plus sign the origin of their continuation; it
denotes the mode in which we bring those repeated steps
together.
Distance does not arise when we award it a value in metres
either. Under a spatial comparative rule we determine a relation
whose result can then be expressed in metres. A unit is a
stabilised form of comparison; it did not invent what is being
compared.
This may seem almost obvious. Which makes the occasional
career of a standard of measurement all the more impressive.
First it is called in to help determine a relation. Once it succeeds,
we sometimes promote it to the supposed origin of that very
relation. An astonishing career for something that was only meant
to measure.
Using stable measures technically remains entirely legitimate.
Without them we could not build reliable orders of comparison.
The error begins not with measurement but with the subsequent
confusion of its different roles:
We read out a relation, stabilise its determination and use
that determination for further measurements. That makes
the standard reliable—not original.
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21.3 The Object We Recognise Through the Apparatus
The atom illustrates this especially well.
Archibald needs neither knowledge of atomic structure nor
calculations of electron transitions to perceive a star's spot of
light. First he sees a light appearance. If he wants to know more,
he can extend his perception technically, for example with a
spectrograph. He then receives another registration,
determinable under the comparative rules of spectroscopy.
This does not mean that we have simply uncovered some secret
object hiding directly behind the spot of light. We have first
chosen another measurement arrangement, thereby changing
which differences can be registered and compared.
To interpret characteristic spectral lines, we use atomic-physical
references. Those references are themselves based on
experimentally determined and repeatedly tested relations.
Apparatus lets us read particular registrations as atomic
transitions or structures; further investigations specify the
conditions under which those readings remain stable.
Eventually such a stable relation becomes a standard itself. The
SI second is defined through the specified frequency of a
transition in caesium-133. With that fixed temporal reference we
determine other frequencies and couple further spatial and
temporal measurements.
A remarkable and technically highly successful return:
measurement makes an atomic relation determinable as such; the
stabilised atomic relation becomes the reference for further
measurements; those measurements in turn open new physical
determinations.
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The Question That Forgets Its Premise
Nobody needs to abandon this practice. On the contrary,
repeatability is precisely what we demand of a useful measure.
But we should not forget where our concepts came from merely
because their application now works to so many decimal places.
We explain new measurements by an order whose physical
determination itself rests on measurements. Precision does
not abolish this self-reference. It makes it reliably usable.
A problem arises only if an atomic description obtained under
particular orders of comparison is made into a relationless origin
of measurability. An atom may of course function as an object and
a cause of particular processes within a physical theory. It does
not follow that its determinability has been explained
independently of every measurement relation.
We encounter the same difficulty under different conditions in
wave–particle duality.
In a double-slit arrangement we can register individual detector
events whose distribution over repeated trials shows interference.
Alter the arrangement so that path information becomes
distinguishable and the accessible interference changes. Physics
describes the dependence, among other things, through
complementarity.
At this point the decisive step has already occurred: we changed
the experiment and thereby introduced a different distinction into
the measurement.
Under one arrangement we read one trace, under another a
differently determined trace. Both can be described within a
common quantum theory. But their common theoretical treatment
does not make their experimental conditions identical.
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Our earlier counting sequence
1+1+1+1+…
therefore denotes neither an original collection of finished
photons nor the ticks of a cosmic clock independent of our
inquiry. It makes visible the recurrence of distinguishable
registrations under a particular reading rule. Which registrations
belong together and what rhythm is ascribed to their recurrence
depends on that rule. Changing the experimental arrangement
can determine a different order of trace even if the number of
events counted is the same.
Equal numbers of registrations do not necessarily yield an
identical relational order.
We might now bring the different appearances together under the
term photon and then ask how this one thing could be both wave
and particle. The question makes sense within certain
interpretations of physical theory. But perhaps we should first ask
what we have already done when we fashioned one object-concept
from different experimental determinations.
The photon, after all, has never personally promised us that it
would behave like a classical little ball or a classical wave under
every experimental arrangement. Those expectations come from
the pictures with which we describe its different modes of
appearance.
This is a particularly fine example of how a problem can be
constructed using the very means intended to explain it. We use
the image of a wave where it makes interference relations vivid,
and that of a particle where individual registrations stand out.
Then we superimpose both images on one classically conceived
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object and ask how it can fulfil the contradictory expectations we
have thereby created.
Measurement did not require us to turn both pictures into literal
properties of such an object. That demand comes from how we
summarise the results. The experimental finding remains; the
additional paradox emerges where we assign to our pictures an
obligation they do not possess as pictures.
So we have not necessarily made a mistake in the calculation. We
may have derived the wrong question from a perfectly correct
one.
The Tractatus does not contest the stability of quantum relations
or the success of their mathematical description. It asks about the
direction of explanation: what was measured in each case, what
was gathered under a common concept, and at what point was
that concept declared the unconditional cause of its own
experimental determinations?
The experiment changes the distinction. With it the
readable trace changes. Only their common theoretical
interpretation introduces the question of the one object
behind the different appearances.
For our investigation the duality is initially a duality of particular
modes of appearance. That they belong together under a common
physics is a further, demanding determination—not proof that our
classical pictures must be original properties of a relationless
thing.
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21.4 The Space in Which Time Is Already Running
Let us return from our instruments to Archibald's animated world.
Here we encounter the same confusion in a form a child can
understand and about which adults, apparently, still have a great
deal to think.
Archibald travels through space. We see him fly, land and marvel
at those two-dimensional beings whose spatial limitations he finds
so remarkable. But examine the individual frames and no
movement occurs within any of them. Each image shows a
determinate arrangement, completely readable on its own.
Motion is experienced only through the ordered relation between
these image configurations.
Of course we could say that the pictures move through time. For a
film projector this is a useful description of its technical
operation. But as an explanation of how time becomes readable at
all, it would be a rather peculiar trick: we would have switched
time on before explaining where it came from.
We may thereby have manufactured the apparent riddle of what
keeps time itself moving. We speak of time as though it flowed
and then find ourselves obliged to explain the motion of that flow.
Yet the flow was initially just our image for expressing ordered
change. Perhaps it would have been enough not to mistake it for a
river of its own.
The Tractatus Principio avoids precisely this assumption. It begins
with complete relational configurations, Frozen Frames. In the
present connection such a configuration is written:
|_| = (0|Δ)
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The Question That Forgets Its Premise
Within a Frozen Frame no temporal transition takes place; its
determinacy requires no independently running duration. Only
the relation between different complete configurations makes a
directed, coherent continuation readable, experienced by the
observer as motion and temporal sequence.
The temporal difference-reading is:
Δt = (Δ₂ |₋ Δ₁)
The notation does not insert an already finished time axis between
two events. It makes a particular relational difference visible
under its temporal reading.
The same problem arises with spatial depth. Explaining the
different views of a table through its already given
three-dimensionality can be perfectly serviceable geometry. But it
leaves unanswered the earlier question: through what relations
do those views become determinable as different spatial
appearances of the same table at all?
Once we describe space and time in a common spacetime
geometry, the shift merely acquires a larger stage. The
mathematical framework allows us to calculate motion, light
relations and gravitation together. Its predictions can be tested,
and many such tests are among the finest achievements of
modern physics.
Yet even the best description of its own geometrical order has not
necessarily explained the origin of geometrical readability.
We can determine the geometry of an appearance without
thereby having explained why that appearance can be
geometrically determined at all.
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Finished space is supposed to explain how distance is possible.
Finished time is supposed to explain how change can occur.
Finished spacetime finally undertakes to connect them. An
extraordinarily powerful construction—unless what we want to
know is why space, time and their common measurability can
appear as such in the first place.
The geometrical framework has already placed that very question
behind itself in its initial assumptions.
21.5 When the Number Becomes the Mystery
The reversal becomes particularly evident wherever numbers
achieve an almost awe-inspiring precision.
The speed of light links spatial and temporal measurements. Its
exact value in SI units is determined through the definition of the
metre, with the second already fixed. The value is therefore not
an additional physical origin that first produces the light relation.
It is a stable physical relation whose numerical representation
receives an exactly stipulated value under this measurement
order.
In Chapter 17 we examined the corresponding limiting reading of
the Tractatus:
0|c ≡ 0|(Δ |÷ Δt)ₘₐₓ
The light relation is not generated here by a decimal number
written down afterwards. Rather, its determinate relational order
is supposed to make intelligible the conditions under which
spatial and temporal readings can be coupled and then expressed
numerically.
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It would therefore be a rather remarkable trick of justification to
cite the fixed SI number as independent proof that the light
relation must generate exactly that number from itself. We would
first establish the standard and then let the standard confirm that
we established it correctly. Excellent for a definition; rather
modest as an ontological discovery.
Cosmology, however, gives us an example where the numerical
value is not supplied by definition: the Hubble tension.
Different measuring and calibration chains, under their respective
model assumptions, yield present-day Hubble values around 67
and 73 kilometres per second per megaparsec. These results are
intended to determine the same parameter within a shared
cosmological description. They do not agree within their stated
uncertainties.
That is a genuine empirical difficulty. It does not disappear
merely because we propose a philosophically more interesting
question. Thus far, the Hubble tension seems distinctly
unimpressed by elegant phrasing.
Precisely for that reason we must distinguish two difficulties. The
divergence of measured values is an empirical problem not
created simply by describing it. The broader mystery of why
values measured as the same quantity must necessarily agree,
however, depends on the shared model order within which we
compare them. That order is not arbitrary, but its assumptions
require justification and examination. Otherwise we risk declaring
our chosen comparison scheme a property of reality, then blaming
reality when it does not fit without further ado.
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And still the tension makes visible a premise of particular
importance to this investigation.
The measuring chains each have their own rules of comparison,
calibrations and theoretical relations. To compare their results
under the common sign H₀ is a meaningful and testable demand
within the chosen cosmological model. But their common
designation alone does not show that both chains already read out
the same invariant relation under identical conditions.
The name of a quantity is not evidence that its
measurement relations are identical.
We can therefore retain the physical question—why the values
differ—while placing a more fundamental question beside it: what
conditions must be satisfied for differently structured
measurement orders to belong together as determinations of the
same physical quantity?
Perhaps unidentified systematic errors exist. Perhaps model
assumptions need revision. Perhaps cosmology requires an
expanded description. The discrepancy itself decides none of
these possibilities, nor is it explained simply by pointing out that
the measuring chains differ.
What we can see is that coherence within one measurement order
does not guarantee compatibility between different orders. Their
common invariance must be determined and tested; it cannot be
inferred from their shared symbol alone.
The number, then, is not the real mystery. The question concerns
the relation under which we may expect two numbers to
determine the same physical connection.
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The point applies to other physical quantities determined through
different procedures and subsequently treated as properties of
one object. Comparability is an achievement of a measurement
order, not a gift bestowed by a name.
21.6 Where Physics Undermines Its Own Question
We can now specify the accusation so immodestly raised by the
title of this book.
One might be tempted to call physics an enormous edifice of lies.
But that would not only be a somewhat unkind description of a
science whose technical achievements we use every day with
remarkable ease. It would also miss the actual error.
Physics need not lie to present us with a reality whose origin we
misunderstand. It need not even get its calculations wrong.
After all, it is we ourselves who bring forth this reality in its
particular modes of appearance through our
measurements.
We distinguish, compare, delimit and determine. We develop
apparatus that allows further differences to be registered, and we
stabilise the results as recognisable objects, properties and
quantities. Under these conditions a world of spatial bodies,
moving particles, electromagnetic fields and temporal sequences
appears to us. The apparatus does not arbitrarily decide what
result it will deliver. But it belongs to the relation under which a
particular result can appear as such in the first place.
Because this world of appearances functions with extraordinary
reliability within its determined relations, we are tempted to
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equate it with the original structure of reality. But what we have
initially determined is its measurability and the forms in which it
appears. It does not follow that those forms are also the core of
things, independent of every distinction.
We confuse the reality that appears through our
measurements with the prerequisite from which its
appearance first emerges.
That is the real trick. And perhaps it is one we have mastered so
well that we have long since forgotten performing it.
Recall the atom. We construct apparatus, determine recurrent
relations under its measurement order and read them as atomic
structure. Later, an atomic transition becomes a standard for
further measurements. There is nothing technically wrong with
that; we may stabilise a particular relation and use it as
reference. But if we then declare the atomic order thus read out
to be the unconditional origin of the very measurability under
which it was first determined, we have reversed the direction of
explanation.
We now explain our measurements through a mode of appearance
that we brought forth through measurements. The procedure is
precise. Its circular justification is no less circular for that.
The same confusion recurs at another level in wave–particle
duality. We distinguish different registrations under different
experimental arrangements, describe their distributions through
different classical images and bring the results together within a
common physical theory. Afterwards, we sometimes treat wave
and particle as two originally present, mutually contradictory
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properties of a finished thing. Then we ask, with immense
seriousness, how that thing could be both at once.
Perhaps we have already constructed the difficulty through the
images that were meant to make our measurements
understandable. The success of quantum theory is not devalued
by this; it shows precisely that different modes of appearance can
be related under particular rules. But it does not oblige us to
declare classical images two independently existing constituents
of reality.
We have not necessarily discovered a contradictory thing.
We may first create the contradiction by turning different
modes of appearance into properties of a supposedly
finished thing.
And then, naturally, we search for the error in physics.
Where else?
That we might already have produced it through the form of our
question rarely seems the obvious possibility.
Space is no different. We determine different views of an object as
spatial appearances belonging together and then explain their
belonging-together through a three-dimensional body whose
three-dimensionality we had just inferred from those views. We
read the ordered relation of complete configurations as temporal
continuation and then explain this continuation by a time already
running in our explanation. Finally, we bring spatial and temporal
measurements together in spacetime geometry and are tempted
to turn that geometry into the independent substrate of the
relations it previously made so successfully describable.
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The error changes its appearance and scale. Its direction of
justification remains the same.
Here we encounter a distinction more important to this polemic
than any individual punchline. Measurement brings forth the
particular mode of appearance of a relation; it does not create the
universe at will. Different observers can compare results, discover
errors and recognise stable relations again. That is why the
readings are not arbitrary. Yet their stability does not imply that
their objects, independently of the relations under which they are
determined, have already been explained as original things.
Cut a surface and you obtain further surfaces. Investigate a
body's appearance under other conditions of measurement and
further determinate appearances emerge. Even atomic or
subatomic description does not release us from distinction; it
opens another order of readability. If we declare each new
appearance the true interior of the previous one, we have merely
moved the basic problem down another floor. The architecture of
the error remains unchanged.
Physics can explain appearances through further determinate
relations. A field is measured, a mass compared, an atomic
transition recognised as recurrent order, a duration determined
under a comparative rule. Such explanations are meaningful and
may produce excellent predictions. But if we use those
determined objects to explain the original possibility of their own
determination, we replace the question of the operation by one of
its results.
We must distinguish two forms of circularity. The problematic
circle of justification conceals its premise. It treats a read-out
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appearance as an independent cause of the measuring relation in
which it became readable.
The Tractatus Principio, by contrast, places self-reference
explicitly at the beginning:
Δ := |(|)
Difference distinguishes itself. Delta does not signify another
apparatus, a hidden force or an even smaller object behind the
particles. It denotes the operative reference of difference to its
own distinction.
If we turned that into a mysterious substance sitting somewhere
behind the universe and driving its remaining components, we
would not have overcome the error. We would merely have given
it an especially elegant symbol.
Difference is not the last thing we find behind things. It
denotes the condition under which anything can appear to
us as a thing at all.
Our criticism does not therefore claim wholesale that physics
rests on a mistake. Many physical theories expressly take account
of reference systems, apparatus and conditions of measurement.
Mathematical coherence, technical success and independent tests
are real achievements. None need be denied for our question to
remain meaningful.
There is no need to fire the equations merely to check their
jurisdiction.
The charge is directed instead at the unnoticed step that turns a
particular, successful reading-out into an ontology without
prerequisites. That step can indeed create riddles of our own: by
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extending images beyond the conditions in which they apply; by
ascribing to objects the origin of their own measurability; and by
turning spacetime into something that is suddenly supposed to
produce what we had first related under its geometry.
We do not create measurement results at will. But we can
certainly create the problem we make out of them.
Not every open question in physics arises this way. The empirical
Hubble tension remains a discrepancy to investigate even after
we expose the assumptions of the measurement chains involved.
But conceptual riddles created by confusing mode of appearance
with origin do not become more fundamental merely because we
seek ever more precise answers to a question wrongly framed.
Perhaps physics does stand atop a vast structure of premises
whose original connection it does not always examine. The
disquieting thing is not that the building must collapse at the next
measurement. The disquieting thing is how brilliantly it works
while we mistake its storeys for the ground on which it stands.
We measure ourselves a world, declare its appearances the
causes of their own possibility of appearing, and then
marvel at the riddles we thereby create.
Not every difficulty is our own work. But wherever we confuse a
reading-out with its origin, we have at least contributed to the
mystery.
And perhaps, before accusing the universe yet again of behaving
contradictorily, we should investigate the distinctions under
which we have made it appear that way to us.
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21.7 Archibald Asks a Different Question
Let us return one last time to the beginning.
Archibald lands on a planet whose inhabitants are
two-dimensional. He views their world from his supposedly
self-evident three-dimensionality and wonders how they can live
without the spatial depth he regards as the prerequisite of any
reasonable existence.
He overlooks one small detail.
He is a cartoon character himself.
His dog is a cartoon character. His spaceship is a cartoon
character. And the entire universe in which he parades his
superior three-dimensionality fits onto a two-dimensional picture
surface.
You would think that by now Archibald might have reason to
present his geometrical certainty a little less confidently.
That is the real comedy of the scene. Archibald has declared the
alien world's two-dimensionality a problem because he brought
his own three-dimensionality along as the original standard. The
problem is not initially with the inhabitants. It is with the question
he asks them. He has taken a particular spatial reading as a
premise and is now astonished that another world refuses to
conform to it.
A remarkably elaborate mystery for someone living on a picture
surface himself.
But are we really in a better position?
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We look at a table and explain its different views through
presupposed spatial depth. We experience motion and explain it
through a time already running. We measure light under spatial
and temporal rules of comparison and sometimes treat its
stabilised numerical values as though they had first generated
those relations.
We alter experimental arrangements and still expect to find the
unchanged appearance of an object supposedly determined in full
independently of the experiment. We infer astronomical distances
and past light configurations, order them within a common
cosmological geometry, and are tempted to regard that geometry
as the original substrate of all measured relations.
Yet every such investigation already begins with the same
fundamental position:
A|B
Not two finished things, but the poles of a distinction.
And that distinction can refer to its own operation:
Δ := |(|)
Under its polar reading, observer and world appear. Under its
geometrical reading, positions, surfaces and spatial depths
become determinable. Its directed, coherent continuation is read
as motion and time. Under stabilised rules of comparison we
obtain numbers and physical measures. Technical extensions of
the observer relation open further registrations, which we
determine as atomic structures, particles, fields or astronomical
opposing poles. And the common geometrical order of such
measurements finally becomes readable as spacetime.
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The Question That Forgets Its Premise
None of this is arbitrary. Nor may the Tractatus simply assert the
stability and mutual compatibility of these readings. It has to
make them intelligible within their respective relations.
For the coherence of complete relational configurations we wrote:
|Σ = ((|_|₁|_|₂) ⇔ (|_|₂|_|₁))
And for the invariance of observer continuation:
OΔO
The world described in this way is no arbitrary invention of
Archibald's. A measurement can confirm or contradict his
expectations. Different observers can compare results, correct
readings and rediscover stable relations. Precisely these
possibilities show why the order of difference must not be
confused with an individual's passing fancy.
But the world emerging under these measurements as the object
of our knowledge is not thereby the original connection itself. We
bring forth its particular mode of appearance under the
distinctions we choose. That those modes are compatible and
reliably measurable does not yet explain why we are entitled to
treat them as original things existing independently of the
operation that produces their appearance.
Perhaps, then, the real point of our inquiry is not that we have
found a new cause behind the causes familiar to physics. Another
cause would again be a determinate object of explanation, and we
would immediately find ourselves on the next lap of a journey we
thought had ended.
The point is that we finally pay attention to the direction of the
question. We do not merely measure properties of a finished
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The Question That Forgets Its Premise
world found waiting for us; through our measurements we
determine the form in which it appears to us as a world at
all. And if we subsequently declare those modes of appearance
their own independent causes, we can create mysteries arising
not from a stubborn reality but from our way of questioning it.
The fact that a question can be pursued scientifically does not
make its premise necessary. One can work with great precision on
solving a problem generated by one's own language. Precision is
not the problem. Unfortunately, it does not automatically protect
us from the error installed at the beginning.
Not: Which still smaller object hides behind the next surface?
But: Under what relation does something appear to us as a
surface, an object, or as spatially or temporally determinate at
all?
Archibald need not first marvel at how two-dimensional
inhabitants can live without his third dimension. He should ask
what makes their world appear two-dimensional to him in the first
place. Perhaps he would then notice that even his own spatial
self-evidence has a perspective.
The same applies to us. We need not abandon physics to ask this
question. Physics itself supplies the evidence of its practice: it
measures differences, compares relations, stabilises recurrence
and examines how registrations can be brought together under
common rules.
It must simply be allowed to ask about the conditions of that very
activity.
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The Question That Forgets Its Premise
It is not the relation that must justify itself before a
physical description, as though that description arose
independently of all relations. The description must show
under which relations its own concepts become
determinable in the first place.
The Tractatus Principio thus leads us not to some final object
hidden behind reality but back to what we already enact in every
new determination—and what we most easily overlook precisely
when a determination succeeds particularly well.
Perhaps physics might have been spared some of its self-created
riddles had it distinguished more clearly between the original
difference, the modes of appearance brought forth under
measurements and the concepts through which it describes those
appearances. Not every problem would disappear. But at least we
would no longer demand that reality reconcile our own pictures
for us—or search for the core of things where we have just
determined their next surface.
Archibald looks at his world once more. Perhaps he examines the
table, perhaps the star, perhaps only the surface of his own
cartoon existence. For the first time he does not ask what might
be hiding behind the next surface.
He asks what he is doing when he recognises something as a
surface at all.
Our journey ends where it began. Not with a thing, not with a
number, and not with a particularly impressive four-dimensional
container.
But with a difference:
182
|
And with a question that might have spared us a good deal had we
asked it sooner:
What am I actually distinguishing when I
believe I am explaining the world?
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