One World, Many Perspectives
A philosophical salon on symmetry, consciousness, and a unity that preserves difference.
What would it mean for the world to be nothing over and above the network of perspectives and the relations between them? Could such a world still be independent of us? Could its unity include experiences that remain irreducibly someone’s own? And would understanding the whole require a perspective from which every difference finally disappears?
These questions stand behind my mathematical work on shared structure, symmetry, and the comparison of perspectives. A recent paper, Symmetry Between Perspectives: Invariant Calibration and Symmetry Lifts, investigates a precise family of comparison problems. The philosophical possibilities surrounding those problems are much less settled. I wanted to explore them in a setting where an unusual thought could be pursued before anyone asked how to turn it into a publishable claim.
I therefore asked ten AI agents to discuss the paper. Some were given philosophical starting points inspired by Pauli, Bohm, Wigner, and Einstein; others drew inspiration from Bohr, Heisenberg, Dirac, and Weyl and Leibniz. A phenomenologist and a sceptical structural realist completed the group. They read the paper, developed their positions, and then responded to one another’s objections.
This is an edited and expanded account of that exchange, developed with AI assistance. The historical names identify sources of philosophical inspiration. The voices are AI-generated participants, and their dialogue is neither a collection of historical quotations nor a reconstruction of what these thinkers would have said about my work. The mathematical examples provide common ground; the metaphysical interpretations remain open.
The most interesting development was a change in the meaning of unity. Several participants began by asking what might lie behind the diversity of perspectives. Gradually, they became more interested in the precise ways perspectives could belong together while remaining different. Even the sceptical voice found something worth pursuing in this possibility.
To follow the exchange, one mathematical distinction is especially helpful. Different presentations can be assigned the same data while the comparisons between those presentations still act nontrivially on the data. Knowing what is assigned at each position does not always tell us everything about the relations between positions. In the paper, this remaining structure appears as a residual group action.
A rough analogy is a collection of musical phrases. Listing their notes is one task. Specifying how each phrase is related to the others by transposition, inversion, or a change of register is another. The relationships have a structure of their own. The mathematical model makes a distinction of this kind exact, under carefully stated assumptions.
The Bohm-inspired voice
What interests me is that the whole may be present in the connections between perspectives without appearing as an additional object. I encounter the whole through the ways my description must fit with other possible descriptions. Perhaps the shared reality is an order of possible transitions.
This gives the familiar metaphor of a hidden reality an unusual turn. We often imagine appearances arranged around an object that remains behind them. The proposed alternative places the emphasis on the order connecting the appearances: what can be transformed into what, which transformations can be composed, and which combinations are possible.
The Einstein-inspired participant immediately asked whether this order existed when nobody experienced it.
The Einstein-inspired voice
A network of actual experiences is insufficient for my realism. I also need determinate possibilities: what would be registered from a place where nobody stands, or by an experiment nobody performs. The world must constrain these possibilities independently of our descriptions.
Bohm’s reply was to ask what an additional substance would contribute if all actual and possible relations were already determined. Could the relational order itself be independent of whether anyone occupied a particular point within it?
The realist accepted this possibility. His requirement concerned the independence of the order, rather than the existence of a further bearer beneath it. A world of relations could still resist us. We could be wrong about its structure. An unperformed experiment could have consequences for the adequacy of our theories.
The Einstein-inspired voice
If the order of actual and possible relations is fully determined, I do not need an additional thing behind it to make it more real. What matters is that the order does not come into existence because we describe it. Reality shows itself in the limits it places on what we may choose.
This was an important agreement, although a limited one. It left room for a relational realism: the world could be constituted by relations and remain independent of any particular observer. It did not decide whether those relations ultimately involved experience, physical processes, or both. A possible observational standpoint is not already a conscious subject.
It also changed the role of disagreement. Objectivity need not amount to agreement among the people who happen to be present. A community could share a mistake. What matters is whether its comparisons survive constraints it cannot freely adjust.
The Pauli-inspired participant approached the same territory through the relation between the mental and the physical.
The Pauli-inspired voice
We often look for a common ground of mind and matter as though it had to be a third kind of object. Perhaps we should look for a common order of transformations. Mental and physical appearances could differ in their character while participating in the same pattern of possible changes.
The paper supplied a precise image for this thought. A comparison can be required to intertwine two actions. In ordinary language, translating after a transformation must agree with performing the corresponding transformation after translation. The requirement concerns how the correspondence behaves across changes, rather than how convincing a resemblance looks in one selected case.
This suggested a demanding interpretation of a symbolic correspondence. A symbol might mediate between different forms of appearance while preserving an order through their transformations. Its value would depend on whether the correspondence continued to work when the situation changed.
The phenomenologist objected to the leap from a shared pattern to a shared reality.
The phenomenologist
Two independent dreams could have the same internal structure without concerning the same cup. Structural resemblance does not establish belonging together. When two people encounter one cup, there are further connections: one can pass it to the other, both can affect its position, and their experiences depend on a common history. What plays that role in your psychophysical account?
The objection made Pauli’s proposal more interesting. He withdrew the suggestion that a matching structure could, by itself, establish a common ground. The hypothesis would need an independently motivated connection between the two sides.
His revised proposal involved a shared process. Operations would be specified independently on the physical and experiential sides, and a proposed correspondence would have to survive new combinations of those operations. A successful account would need to do more than accommodate familiar examples. It would have to constrain what happened next.
The sceptical structural realist
Now the proposal has something to lose. A failed correspondence can count against the specified model. You can no longer arrange the resemblance afterwards by choosing convenient representations. That makes it a more substantial idea, even though a successful correspondence could still admit several metaphysical interpretations.
For me, this was one of the more productive encounters between speculation and criticism. The speculative idea acquired a clearer commitment. Its ambition survived, but it became answerable to something beyond the attraction of the metaphor.
The deeper question remained: would such a connection reveal two aspects of one reality, or would it merely describe a reliable relation between different kinds of things? The mathematics alone does not choose between those readings. It can, however, help us state what a proposed connection must preserve.
Wigner’s contribution began with an observation about two different ways of achieving shared content.
Consider a sensor with an unknown polarity. A recorded signal y could correspond to a laboratory signal q with positive polarity, or to −q with negative polarity. From the raw record alone, exact signed recovery is impossible. Yet the pair of possibilities remains recoverable. We can identify the signal up to an overall sign.
This gives one route to shared content: retain the class [q] = {q, −q} and relinquish the distinction between its two members.
The complex example in the paper works differently. Two actions of a cyclic group of order three cannot be related by an invariant, invertible complex-linear comparison of the required kind. Allowing complex conjugation changes the answer. Conjugation is still invertible. The obstruction disappears because the permitted comparison need no longer preserve the given complex structure in the same way.
Geometrically, the relevant generator rotates a complex signal through 120 degrees in one action and through the opposite angle in the other. An invertible complex-linear map on a complex line cannot intertwine these two rotations. Reflection by complex conjugation can. The same points remain available, but the relation between the two orientations must be acknowledged.
The Wigner-inspired voice
One route identifies data points. The other allows an invertible comparison between them while changing the structural requirements on that comparison. Philosophical discussions often call both moves “abstraction”, but their costs are different. When two perspectives seem incompatible, we should ask what their translation is required to preserve.
Dirac asked why the more permissive comparison should count as a deeper understanding of reality. Perhaps it merely stopped preserving something that mattered.
Weyl’s response was to suggest that we explicitly retain the rule governing the change. Let I denote multiplication by the imaginary unit, and let K denote complex conjugation. Then:
KI = −IK.
The comparison reverses the action of the specified complex structure. That difference can be recorded. We can understand how a structure transforms without requiring it to remain fixed.
The Weyl–Leibniz-inspired voice
A shared account may need to preserve the transformation rule for a difference. Sometimes understanding consists in knowing exactly how two ways of carrying structure differ. Whether that comparison is physically appropriate remains a question about the system.
This strikes me as a particularly fertile thought for discussions of experience. Understanding another perspective might involve tracing what is preserved, what changes, and what is lost in the comparison. A demand for perfect identity could miss the form of understanding actually available.
The Bohr-inspired participant connected this with the question of the inexpressible.
The Bohr-inspired voice
When something is called inexpressible, we should first ask: within which forms of expression? Preserving a person’s reports, predicting their actions, and conveying how an experience feels are different achievements. A translation might succeed at one while leaving another untouched.
The sensor example offered a small but striking warning. The class of a signal up to sign can retain substantial information as a set. If we demand a real-linear quotient imposing the same sign identifications, the quotient collapses to zero: the relations identify every vector with its negative, so every vector is killed. The chosen form of representation matters to what survives.
This does not make every failure of expression a revelation of hidden depth. It gives us a more exact question to ask before making that claim. What is the proposed language capable of preserving, and what does it discard by construction?
Wigner then shifted the emphasis from the applicability of mathematics to the possibility of shared understanding. Why should perspectives be comparable in coherent ways at all? One could imagine regular experience within each standpoint while dependable translations between standpoints remained unavailable. The possibility of a shared science already presupposes a remarkable order of connections.
The Heisenberg-inspired participant found an unexpected consequence in the sensor model. It concerns a moving particle that passes through the laboratory origin at the middle of an observation interval. Its three recorded positions can be written as:
(−a, 0, a), a ≠ 0.
Reflecting the record in time reverses the order:
(a, 0, −a) = −(−a, 0, a).
The signed record changes. Its class up to sign does not. Once the polarity distinction has been removed, this moving trajectory is fixed by time reflection at the level of the retained data.
The Heisenberg-inspired voice
No disturbance of the particle is needed. The difference arises from specifying which distinctions the result preserves. A trajectory can be asymmetric as a calibrated record and symmetric as a sign class. We therefore have to ask: a symmetry of what?
This is a direct calculation within the model. It concerns time reflection of records, not the thermodynamic arrow of time. Its philosophical force comes from how plainly it separates a symmetry of the retained description from a symmetry of the more detailed record.
Einstein returned to the image of a “God’s-eye view”. Would perfect knowledge be what remained after every dependence on a perspective had been eliminated? The example made him hesitate.
The Einstein-inspired voice
Greater invariance can result from forgetting distinctions. Complete knowledge would have to include how differences belong together, along with the conventions and comparisons through which they become intelligible. Maximal invariance is therefore an uncertain image of omniscience.
A reference measurement makes the point concrete. It can establish the sensor’s polarity and restore signed recovery. This adds a relation within the experimental situation. Knowledge becomes more precise through an additional connection to a particular reference.
Heisenberg also emphasised the difference between a law and a particular history. A law can permit a motion and its time-reflected counterpart while neither history is identical to the other. The symmetry belongs to the order of possibilities. It need not fix the possibility that is realised.
His image was of a grammar: a law specifies how possible histories may fit together, while a particular history is one definite expression within that grammar. This invites a philosophy of potentiality, although the model itself supplies no account of why one possibility becomes actual.
The sceptic then introduced a different kind of limit on comparison.
The sceptical structural realist
Perhaps the unity of a world consists in the fact that local freedoms cannot be chosen independently. What is possible at one place constrains what can simultaneously be possible elsewhere.
The paper’s quaternion-group example gives a precise version of this distinction. Each permitted transformation of the base has a lift to the more detailed structure. Yet there is no way to choose all these lifts so that the choice respects composition. Individual availability does not guarantee a compatible family of choices.
That is a surprisingly strong image of interdependence. Each request can be satisfied separately, while satisfying all the requests together requires a compatibility that the structure forbids.
The Bohm-inspired voice
I began by emphasising possible transitions. I now want to give equal weight to the resistance against independent choices. The whole becomes apparent in this resistance. A thread encounters the unity of a fabric through the ways its possible movements are constrained by the other threads.
He also accepted an important qualification. The quaternion system has a complete mathematical description. The obstruction concerns a particular way of choosing compatible lifts. It gives no general argument that the whole is indescribable. We can understand why a desired organisation of the whole cannot be achieved.
The Dirac-inspired voice
That is a form of mathematical wonder I can take seriously. We formulate the definitions, and then encounter consequences we cannot choose. A complete description can contain a proof that an apparently reasonable wish cannot be fulfilled.
The appeal of the mathematics here lies in its power to make conditions and obstructions intelligible. Beauty can draw our attention to a structure. Establishing that the structure describes a physical situation requires a further encounter with the world.
For the philosophical discussion, however, the possibility is already worth considering: a whole may be intelligible even though its components cannot be organised in every individually conceivable way. Interdependence can be part of a complete account.
Weyl turned the conversation towards the individuality of a perspective.
The Weyl–Leibniz-inspired voice
Imagine a circular hall with twelve identical doors. The geometry singles out no door. Yet you stand at one of them, and there is a definite “here”. Your position does not need a unique intrinsic property in order to be your position.
This illustrates a distinction between a position selected from within a situation and a position uniquely distinguished by the structure itself. The two need not coincide. A symmetric world can contain a definite location from which something is encountered.
The analogy has a limit that Weyl himself acknowledged. Selecting a mathematical basepoint does not produce a conscious subject. A formal presentation in the paper is not, by definition, a moment of awareness. The relation between location, embodiment, and experience requires an account of its own.
Still, the phenomenologist saw an opening.
The phenomenologist
I have a distinctive access to my pain without possessing a general privilege to determine the nature of reality. You have a corresponding access to yours. There can be many centres of experience without one of them being the centre that governs the world.
She distinguished the recurring form of first-person experience from the particular ownership of an experience. Translating “I am in pain” into “you are in pain” can preserve who the statement concerns. It can communicate something important and true. It does not transfer the pain to the listener.
This complicates the idea that complete understanding must eliminate the difference between perspectives. Some differences concern whose experience is being understood. Erasing those differences might destroy part of what an adequate understanding should retain.
The Weyl–Leibniz-inspired voice
Your experience can belong to the same reality as mine while its first-person character never becomes mine. Perhaps understanding can include understanding why a certain kind of access has a limit.
The phenomenologist drew an ethical possibility from this. Care for another person does not require claiming to know exactly how that person feels. Recognition can be genuine while leaving room for what remains accessible to the other in a distinctive way. A relational philosophy might therefore support an account of closeness that preserves difference.
This ethical reading is a further interpretation, not a mathematical theorem. What gives it interest is its challenge to a familiar aspiration of unity: the hope that a deeper insight would make every other perspective wholly transparent to my own.
The phenomenologist
An experience of unity could be an insight into how inseparably my existence is connected with what will never become mine. It could reveal the inadequacy of my claim to self-sufficiency while preserving the reality of another person’s life.
Pauli returned to the suggestion that the world might, through us, enter into a relation with itself. The phenomenologist added that it would do so through beings capable of error. Knowledge within the world includes misunderstanding, correction, and incomplete access. The metaphor of a world knowing itself need not imply a single, perfectly lucid cosmic subject.
By the end of the exchange, the original questions had changed. Asking whether something lies “behind” perspectives had led to a more exact question about what such a background would explain or constrain. Asking whether two descriptions could be unified had led to questions about the information and structure their comparisons must preserve. Asking whether experiences belonged to one reality had led to the need for a genuine connection, beyond formal resemblance.
For my own thinking, the most valuable possibility is that a world constituted by relations could give the differences between perspectives a substantial role. Their unity could consist in dependable connections, shared constraints, and transformations that make their differences intelligible. The individual perspective would remain part of the reality being understood.
This also gives a precise sense to the kind of mathematical mysticism I wanted the discussion to explore. Its starting point is the experience of belonging to an order that exceeds any single standpoint, together with the possibility of understanding some of that order from within it. Its seriousness depends on respecting both the connections we can establish and the differences they leave intact.
The central disagreement remained unresolved. A realist can accept an independent relational world while treating consciousness as something that arises within it. A psychophysical view can regard experience as a fundamental aspect of that same world. Agreement about relational structure does not settle which account is adequate.
I am left with a question that brings the mathematical and phenomenological strands together. If all the world’s relations were completely described, would the presence of experience already be included? Or could a complete structural account still leave open what it means for any part of that world to be present to someone?
And if experience does belong to the world’s deepest structure, could its unity consist partly in sustaining perspectives whose differences no amount of understanding should erase?
Mathematical starting point: Gustaf Ullman, Symmetry Between Perspectives: Invariant Calibration and Symmetry Lifts, revision r5, 19 September 2026. The time-reflection observation discussed here is a direct calculation from the paper’s sensor model.
Dialogue and editorial development: an AI-mediated exchange between ten philosophical perspectives, subsequently expanded and edited into this essay. The historical figures named above did not participate; no dialogue passage is attributed to them as a historical quotation.
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