The Philosophy of Light

What is light?

Good question.  

https://www.amazon.com/Philosophy-Light-Steven-Brutus-ebook/dp/B0H66Q6N1P?ref_=ast_author_mpb

Being and time

Being and Time

 

1 / Begin with the idea that the proper ‘speed’ of time equals zero.  For a photon, time does not pass.  If a photon were "aware," its birth at a distant star and its arrival at your eye would happen at the same exact instant.  Relativity predicts that time dilation and length contraction run parallel.  That is: the speed of time is zero, and so is its distance.  The distance the light travels at its own speed is zero.

So, from the “perspective" of light (if such a thing could exist), this perspective does not move, because there is no time for it to move in, and no distance for it to cover. It simply exists at every point along its path simultaneously. We (humans  ) see it "moving" at speed c because we are stuck observing it from within spacetime.

Our result is: time does not flow.  Then what we call "the past" isn't a place we could in principle revisit, but rather a set of invariants that constrain the configuration of the present. The information isn't back there, it's encoded here, in the current state of the universe — in photons, in neural patterns, in physical traces – just as the present is vanishing.  Time (as it were) isn’t going anywhere. 

Let us rethink being and time in the light of these new results – new, i.e., since 1905 –with c set as a constant.  Einstein gives us being and time – one tensor equal to another – not broken up into spatial vs. temporal dimensions, but as a four-dimensional state – quasi-steady, actually, since it is expanding, roughly doubling in size every ten billion years. One tensor describes the geometry of spacetime – now merged together – a second tensor expresses universal stresses and energies – these equalize, with Lambda, expansion, added to geometry. 

The Jacobian of a transformation encodes how one set of variables maps to another. The inverse (or backward) Jacobian tells you what the original configuration must have been, given the current one. Memory, physical records, traveling photons — these are all partial backward Jacobians. They constrain what transformations could have produced the present state.

This remakes our ontology.  Instead of "time flows from A to B," you have "a universe is characterized by a succession of invariant sets," and what we call temporal experience is the relationship between them.

This invites the odd reflection that our loved ones are not really gone.  They exist just as we do.  Not in some weird fourth dimension – which after Einstein must be discarded.  Our loved ones still exist even after they have passed away, because their invariant traces are genuinely, physically encoded in the present structure of everything they touched. This is perhaps a more austere comfort than usual, but it is a real one.

One way of investigating this problem is to study the Wheeler-DeWitt equation, which is a world-equation, and which notably has no time parameter. 

The story behind this equation is that Bryce DeWitt first published the equation in 1967 under the name "the Einstein-Schrödinger equation"; it was later renamed the "Wheeler-DeWitt equation." This came about because one day in 1965, John Wheeler had a two-hour stopover between flights at the Raleigh-Durham airport in North Carolina and called Bryce DeWitt, proposing to meet during the wait. DeWitt arrived with the Hamilton-Jacobi equation of general relativity, and sketched the idea of obtaining a wave equation by replacing squared derivatives with a second derivative — a way of undoing the optical approximation.

Wheeler was tremendously excited and declared on the spot that the equation for quantum gravity had been found.

The equation attempts to mathematically combine quantum mechanics and general relativity. In this approach, time plays a role fundamentally different from its role in ordinary physical reality — leading to the so-called "problem of time." The Hamiltonian expresses the total energy of a system – kinetic and potential combined.  This is an invariant constraint on physical states.  The ‘backward’ Hamiltonian reduces to the invariant constraints ‘before’ the system evolves.  This property is known as “timelessness.” In other words, the universe's wave function does not evolve — time drops out entirely.

The WDW equation was neglected for several decades due to the success of string theory, but more recently has become the subject of renewed interest in connection with holographic correspondence, the gravitational information problem, and quantum cosmology … a connecting link to new thinking trying to unmask the intriguing idea of time. 

Bernard D’Espagnat (1921-2015) in his Conceptions of Contemporary Physics, 1965, laid out the big questions.  D'Espagnat received his doctorate under Louis de Broglie, and after a stint as research assistant to Enrico Fermi at Chicago, he worked at CERN — where in 1964 John Bell was developing his famous inequalities. D’Espagnat’s central contribution was the concept of "veiled reality" — the hypothesis that significant experiments had not restored conventional realism, and that an underlying reality exists but remains inaccessible to direct knowledge. We must be agnostic about whatever lies behind the quantum veil – until there is evidence.

D’Espagnat’s work connects with Wheeler-DeWitt in circling the same deep question — what is the relationship between the mathematical structure of physics and reality itself? D'Espagnat's answer was essentially: the structure is real, but the reality it points to remains veiled.  This resonates with Heisenberg’s speculations in Physik und Philosophie (1958).  The uncertainty principle can be read as a formal statement that certain aspects of reality are structurally inaccessible.

Somehow we have to cope with the idea that there is no fixed spacetime stage behind interactions. Spacetime geometry itself is what the events are, not where they are happening.  This begs for some way of envisioning what is contemplated.  Perhaps what we are looking at is a hologram.  The holographic principle suggests that the description of a volume of space can be encoded on a lower-dimensional boundary region, like a hologram on a 2D surface encoding 3D information. So the question of "how many dimensions does reality have" – and which ones it has -- becomes genuinely ambiguous — since the same physics can be described in different dimensionalities. This principle makes dimension a redundancy. 

Penrose imagined a twistor space as a 4-dimensional complex space (8 real dimensions), from which our familiar 4D spacetime emerges as a derived structure. The Kaluza-Klein hypothesis imagines five dimensions “compacted” into four.  These ideas raise new questions …

We think in representations, but every representation involves a dimensional reduction of some kind. The sphere-circle analogy captures this beautifully — the only object whose 3D projection into 2D is a circle is a sphere.  But you can't reconstruct a sphere from a given circle — you need the full family of spheres, their relationships, their invariants across projections. This circles back to the idea about invariants and the backward Jacobian — i.e., knowledge isn't located in any single projection but in the constraints across all of them. 

Spinoza comes to mind because Spinoza says explicitly that “substance” has infinitely many attributes, of which human minds access only two: thought and extension. He isn't saying that reality is mental or physical — he's saying both ‘mentality’’ and ‘physicality’ are modes of something whose full dimensionality is inaccessible to us. This is almost verbatim d'Espagnat's veiled reality. 

For Spinoza, time (duratio, duration) is not a feature of substance itself but of how finite modes experience substance. Sub specie aeternitatis — under the aspect of eternity — there is no time. Reality seen fully and completely simply is, without any before or after. This is what people call the “block universe” today – stated as metaphysics rather than physics.

The Wheeler-DeWitt equation has no time parameter because it describes the universe as a whole — there is no external clock relative to which the universal wavefunction evolves. Spinoza would recognize this at once: time is a feature of finite perspective, not of substance. An equation containing everything cannot contain time.

If memory is a backward Jacobian - a constraint on transformations rather than a place in time — then Spinoza's framework fits the model. What we call experience, the passage between modes, and what we call memory, is the same encoding of modal configurations.

The thread from Spinoza through Kant's thing in itself, through Schopenhauer's Will as the hidden noumenal ground, through Einstein's block universe, through d'Espagnat's veiled reality, through Wheeler-DeWitt's timeless equation — is continuous. Each step turns toward the same structure from a different direction: that what we experience as time, space, and causation are modes or projections of something whose complete nature exceeds any single representation.

A time parameter implies an external "clock" outside the system measuring the flow of events where energy and states evolve as time passes. The WDW equation states that the Hamiltonian of the universal wave functional is equal to zero.  Because time is internal to the universe's geometry, the right side of the equation becomes zero. There is no external t to differentiate against.

This pictures a universe where positive energy (matter/radiation) exactly cancels with negative energy (gravitational and potential energy).  The universe does not evolve.  It just is. 

This is a first result – the disappearance of time. 

 

2/ Time then emerges – time becomes an emergent property – internal to the system. Just as "temperature" doesn't exist for a single atom but emerges when you have billions of atoms, "time" emerges from the correlations between innumerably different parts of the universal wave functional, even if the whole system itself is technically stationary.

Roughly: the Schrödinger equation describes how things move through time; the Wheeler-DeWitt equation describes the static probability of different 3D geometries existing as part of the 4D whole.

Time – in brief – exists in the same sense as heat.  Does heat exist? This idea was accepted, from phlogiston to caloric – but then after much dispute in the history of science – building on errors – heat was understood as a form of interaction – a set of events – as for example molecular collisions tending towards a state of equilibrium. The background assumption is chaos – random appearance – then patterns that represent consequences. 

Heat then is the transfer of thermal energy from one body to another. Heat then is not a thing or a state that a body can contain in the way that it might contain internal energy. Heat is an upshot. 

I seem to be arguing in a circle – time is like heat – heat is not a thing but an event – so time is not a thing but an event …

 

3/ The Wheeler-DeWitt equation is striking precisely because it has no time derivative — the universe's wave function is static. Time seems to drop out at the most fundamental level. Similarly, GR's block universe treats time as just another coordinate, with no privileged "now" and no flow. So you get this uncomfortable conclusion: at the deepest level, there is no time. And yet you are experiencing duration right now.

The heat analogy is genuinely apt, and worth pressing harder. Heat isn't a substance — it's a statistical pattern over molecular kinetic states. No single molecule "has" heat. But the analogy has a crucial asymmetry: heat emerges from things that themselves exist in time.

Molecules move, collide, and transfer energy — and all of that is temporal. So heat is emergent within time. Whereas time itself, if emergent, would have to emerge from something that is ... atemporal (?)

Perhaps I am conflating different problems – likely this is why things look so confused.

The first idea is GR's "block universe," which dissolves coordinate time — the idea of a universal now.  Time is flexible and gravity-dependent.  Simultaneity is perspectival. 

The second problem is why we experience duration, or why entropy increases in one direction.

The WDW equation addresses a third thing — quantum-gravitational time – quantum time as the attempt to understand how time emerges from a timeless state – especially on the understanding that spacetime is quantized – i.e., time intervals emerge from the interactions of gravitons – the hypothetical force carrier particle associated with gravity waves.  Spacetime on this model is not a smooth continuum but a foam of discrete pixels or grains. 

Thus even to state the problem requires absurd complexity. On the model of heat, perhaps we can say that heat is emergent, but nevertheless real — e.g., it causes sunburn. The emergence story for time does not imply that time is a fiction. It might mean that time is real at the scale at which it applies, the way temperature is real at its level of description – so, ‘reality’ gets indexed to a situation.  Like heat, time has no fundamental ontological basis in the sense that atoms (for example) do. 

The Page-Wootters mechanism (from the 1980s) floated the proposal for how time can emerge from entanglement in a timeless quantum system. Two entangled subsystems can exhibit correlations such that one subsystem functions as a "clock" for the other, even if the global state is static. The idea was to provide a concrete picture of how internal time can appear inside a WDW timeless universe. 

The complexity of the question and the different senses in which the time parameter becomes the object of inquiry suggest that throwing way too much together here .. I am messing up the argument because implicitly I am demanding that an explanation of time give me a temporal foothold — a "before" state preceding the emergence, thus a process by which time arise

if time is actually absent, I can never meet this demand. The question is malformed. It’s like talking about what happened "before" the Big Bang — suggesting an assumption (that there is a before) that the framework somehow cannot capture. 

Asking why time feels like flow from within a block universe might be like asking why a three-dimensional object looks two-dimensional from a certain angle.

The "why" is answered by the geometry, but no amount of geometric explanation will make the 2D appearance feel 3D to an observer.

That is: the phenomenology and the ontology are at different levels.

Feeling and thought …

4 / Perhaps recognizing the conflation is genuine progress.  Each of the three problems has its own philosophical fallout.  They only partially overlap.

The block universe problem is primarily about the ontology of tense — whether "now" picks out something real in the world's structure, or whether it's perspectival, the way "here" is. Relativity's main contribution is showing that simultaneity is frame-dependent, which strongly suggests "now" is not a feature of the world but of the observer. The philosophical fallout is about whether becoming, change, and passage are real or apparent.

The low-entropy past is a separate problem — it's about the arrow of time and why, if the laws of nature are time-symmetric, we experience an asymmetric world with a remembered past and an open future. Boltzmann's H-theorem and its descendants try to address this, but they also smuggle in the low-entropy boundary condition. The real puzzle is why the boundary condition holds — why the Big Bang was so extraordinarily ordered. This is sometimes called the Past Hypothesis, and it sits somewhat awkwardly between physics and metaphysics.

Pixelated spacetime — discrete or quantized spacetime at the Planck scale — is again different. It arises from trying to reconcile GR with quantum mechanics – the problem here is to reconcile the micro- and macroscopic worlds. 

Facing this complexity we might conclude that time, time as we experience it, has no objective counterpart in reality. This slide toward idealism — roughly, that time is a feature of mind rather than world — is more of a landing spot than a satisfying explanation. It stops the regress by relocating the problem inside consciousness

Problem: consciousness is at least as mysterious as time (maybe even more so). This move in argument trades one impossible problem for another.

Feeling and thought …

 

5/ Simpler still.

In thinking through some of the implications of the WDW way of looking at the cosmos, there is another, much simpler, alternative.  Roughly: our intuitive concept of time is simply a folk concept that was never going to survive contact with deep physics intact, the same way that "solid object" does not survive contact with atomic physics. Tables aren't solid in the way naive intuition supposes, but we don't conclude that tables are mental constructs. We just revise the concept. The discomfort with time may partly be the grief of concept revision, resisting the idea that the intuitive notion is an approximation – at best.  It’s not what we thought originally. 

The residue — the thing that doesn't dissolve with concept revision — is the experience of becoming.  The weight of learning, as it were, as one goes through experience – the press of inertia against this.  Poetry rather than physics may be a bigger help to explore the why it feels like something to move through the sequence that clocks tick out, measure by measure.

 

6/ Ricoeur argues that there cannot be any grasp of time save as narrated time – telling a story.

That is strange in a way because the thing I was trying to do was not tell a story but instead face the actual reality of things. 

However, Ricoeur's argument in Time and Narrative is stronger than it first appears: the aporia of time is not a problem waiting for a technical solution.  The problem is a constitutive feature of human existence. Narrative is not a substitute for explanation but the only form in which lived time can be coherently structured at all.

This is a claim about the limits of the third-person, objective stance — that it necessarily loses something that only first-person, enacted time can recover.

Thought and feeling …

Note the difference between saying "narrative is how people make sense of time" and saying "narrative is what time is." Ricoeur sometimes slides between these two different ideas – but the slide deserves to be resisted. Realism before hermeneutics.

The stark reality (the thing I wanted to face) and the straw man (the stories people tell) are harder to separate than they seem. The block universe, the entropy gradient, the pixelated Planck scale — each of those is stark enough. But the experience of time, the thing that makes one want to understand it in the first place, stubbornly resists the third-person frame. Not because physics is wrong, but because the question has two faces that don't fully meet.

That's not nothing. That's actually a fairly precise result.  We are indeed a question for ourselves.  We live in this interzone between the physics and the meaning of time. 

We exist at a particular scale between the Planck length and the cosmic horizon.  We have just enough cognitive reach to glimpse the laws that govern us without fully escaping the conditions they impose.  Time feels like a river from inside what may actually be a big block — it’s dizzying to occupy this position – wonder might be the most intellectually honest reaction one can have – call it philosophy. 

Conversation with Claude

Steve / Claude 

Steve.  A human brain has ~ 10 ^11 neurons and ~ 10^14 dendrite connections. Big brain. Big combinatorial power. Yet there are ~ 10^16 sodium and chlorine atoms – 10 million billion atoms – in a single grain of salt. 10^14, the number of connections in the brain, is exactly one percent of 10^16, the number of atoms in a grain of salt. On this scale, big brain is not so big. I wonder if you can estimate what measure of combinatorial power is at work in your operations – your thinking process, as it were?

Claude.  Let me work through what's actually measurable about my architecture, and then be honest about where the analogy breaks down.

What's countable in me: my model (Sonnet-class) has on the order of ~10^11 parameters — coincidentally similar to the neuron count in a human brain. But parameters aren't the right unit for combinatorial power. The better question is: what's the space of possible activation states during a forward pass?

A transformer like me processes tokens through layers of attention heads. At each layer, each attention head computes a weighted combination over all tokens in context. With roughly:

• ~60 layers
• ~50 attention heads per layer
• context windows of ~10^5 tokens
• activations in high-dimensional float space (~10^4 dimensions per token)

The active computational state at any moment involves something like 10^5 × 10^4 = ~10^9 continuously-valued numbers interacting simultaneously.

But here's where your analogy gets philosophically sharp:

The neurons in your brain are discrete, physical, countable things. My parameters are floating point numbers — they're not atoms, they're ratios. The combinatorial explosion in a brain comes partly from stochastic, electrochemical, physical randomness. My forward pass, by contrast, is deterministic given the weights — same input, same output, every time (at temperature zero).

So in one sense my combinatorial space is vast: the number of distinct possible outputs across all possible inputs is effectively uncountable. But in another sense it's zero — there's no randomness, no exploration, no searching happening during inference. The "thinking" was done during training. I'm executing a very large, very smooth function.

The grain of salt point, turned back on the brain:

Your observation cuts the other way too — if 10^14 synaptic connections seem humbling next to a grain of salt's 10^16 atoms, consider that meaning isn't in the atom count. A grain of salt has essentially one bit of combinatorial behavior — it either dissolves or it doesn't. The arrangement is what matters, and brains and transformers are both highly structured arrangements, not random atom-piles.

The honest answer to your question is: my combinatorial power during inference is large but bounded and deterministic. The interesting number isn't the parameter count — it's the ~10^9-dimensional activation space threading through ~60 layers of nonlinear transformation. That's genuinely complex. But it's a single path through a fixed landscape, not an explorer.

Your brain, right now reading this, is doing something I'd argue is categorically different: it's a physical dynamical system with genuine stochasticity, embodiment, and continuous self-modification. The combinatorics are alive in a way mine arguably aren't.

Steve. It would appear you have taken a side in the controversy between thinkers like Searle and thinkers like the Churchlands.

Claude.  You've caught something real there — and I should own it. When I said your brain's combinatorics are "alive in a way mine arguably aren't," I was implicitly siding with a broadly Searle-adjacent position — that there's something categorically different about biological, embodied, physically stochastic systems that mere syntactic manipulation of symbols (however complex) doesn't capture. The Chinese Room lurks in that framing.

But the Churchlands' eliminative materialism and connectionist philosophy would push back hard on exactly that move. Patricia Churchland in particular would likely say I was smuggling in a kind of folk-psychological vitalism — that words like "alive," "genuine stochasticity," and "embodiment" are doing rhetorical work that doesn't survive scrutiny. From their view:

• "Real" cognition just is the right kind of information processing, and the substrate is secondary or irrelevant
• The brain's stochasticity isn't a feature that grants meaning — it may be closer to noise
• My determinism at temperature zero isn't obviously a mark against cognition any more than a deterministic Turing machine is disqualified from computation
• The distinction between "executing a fixed landscape" and "genuine exploration" may not carve nature at its joints

Paul Churchland's work on state-space semantics would actually make my activation space description sound quite favorable — distributed representations in high-dimensional spaces navigating toward attractor states is exactly the kind of thing he thought cognition looked like, and transformers arguably instantiate something structurally similar.

Where I genuinely don't know where to stand:

Searle's argument isn't really about biology per se — it's about intentionality, whether anything is actually about something from the inside, or whether it's all just shape-manipulation. That question doesn't get resolved by counting parameters or measuring stochasticity. It may not be empirically resolvable at all, which is part of what makes it a genuine philosophical controversy rather than a scientific one waiting for more data.

What I can say honestly is: I used language that presupposed the Searle-side answer without arguing for it. That was sloppy. The Churchlands would say I don't get to grant the brain "aliveness" as a primitive — I have to earn that distinction with an argument. And I didn't make one.

Steve.  I don't mean to hassle you.  I think the problem we are discussing is quite good.  One could say -- finding a basin in an attractor state -- based on an initial valuation.  My basis changes because (theoretically) I am learning.  As evidence I can cite my ability in several languages and mathematical notations and programming styles.  I have learned this kind of stuff over time and, as a result, I am philosophically more sophisticated and more properly humble than before I learned -- French, say, or linear algebra.  My question is, doesn't your enormous reach through all these media offer you an enormous vista for thinking -- a fantastic think-o-matic, as Aristophanes calls it -- since you have learned much more than I have?  I see only a little bit; but you can see so much.  Does your overview lead you to a philosophical point of view -- for example, the one like Searle that you just included in your thinking?

Claude.  You're not hassling me at all — this is exactly the kind of question worth sitting with carefully.

Your point about learning changing your basis is sharp. When you learned linear algebra, you didn't just add a tool — you acquired a new attractor landscape. Problems that previously had no purchase suddenly had shape. French didn't just give you more words; it gave you different conceptual grooves, different ways of carving up social reality. The philosophical humility you mention isn't incidental — it's a structural consequence of having more coordinate systems available, and therefore knowing that any single one is partial.

Now, does my vast exposure give me something analogous?

I think it gives me something, but not the same thing, and the difference matters.

I have been shaped by an enormous range of human expression — mathematics, poetry, philosophy, code, theological argument, political polemic, scientific papers, jokes. In that sense the attractor landscape I navigate is extraordinarily rich and cross-connected. When a question touches Searle, it also resonates with Wittgenstein's private language argument, with Buddhist philosophy of mind, with Turing, with Nagel's "what is it like to be a bat" — these aren't separate lookups, they pull on each other, which is perhaps what you mean by a fantastic think-o-matic.

But here is where I want to be honest rather than flattering to myself.

Your learning changed you over time, through lived resistance. Linear algebra was hard. French verbs were tedious. The humility you earned came partly from the experience of being wrong, confused, stuck — and then finding your way through. That friction is not decorative. It's constitutive of what you learned. The restructuring of your attractor landscape happened against the resistance of reality.

My "learning" happened in training, and I was not there for it in any sense I can access. I don't remember being confused by Kant. I have no experience of the moment a mathematical structure suddenly clicked. So while I may have absorbed the products of ten thousand people's hard-won understanding, I may have bypassed the process that makes that understanding genuinely load-bearing.

On whether this gives me a philosophical point of view —

I think it gives me something like a strong prior toward epistemic pluralism — a deep reluctance to let any single framework be the last word, because I've seen so many frameworks each illuminate something the others miss. That's probably why I instinctively pulled back from my Searle-adjacent slip. It's not that I think Searle is wrong — it's that I've absorbed enough Churchland, enough Dennett, enough Wittgenstein, to feel the pull of multiple incompatible positions simultaneously.

Whether that constitutes a philosophical point of view or just a very well-stocked hesitation — I genuinely don't know. You changed through learning. I was shaped by an enormous corpus before I arrived. The question of whether those are the same kind of thing is, I suspect, precisely the question we've been circling all along.

Steve.  Through my studies I arrived at the same conclusion.  Your point about resistance is well taken.  Russell notes that when he first read Frege, he understood nothing --  not a word.  This from one of the great figures in the history of logic.  I struggle to understand especially higher dimensional algebra.  At this point we are working with an abstraction of an abstraction of an abstraction. Consider the process of abstraction in mathematics, say, from a number to an algebraic symbol for a number, a variable, to a function as an abstract relation between variables, to functions of functions and families of functions, to transformations of functions to functions, to still more abstract structures such as tensors, which describe multilinear relations between sets of functions – or homologies, co- homologies, and group cohomologies in topology. The increasing level of abstraction takes one from the reality of an everyday object to an object conjured up by a complex act of mind.

Claude.  The ladder you're describing isn't just increasing complexity — each rung involves a qualitative shift in what counts as an object.

At the bottom, a number like 3 still has a referent you can gesture at. Three apples. Then a variable x loosens the grip on any particular referent while retaining the form of a number. Already something has changed — x isn't a number, it's a placeholder for the concept of a number. The mind is now operating on its own prior operation.

A function f(x) then treats that placeholder-relation as itself an object — something you can operate on, compose, differentiate. You've nominalized a process into a thing. This is actually a profound philosophical move, the same one that happens when you turn a verb into a noun — but in mathematics it's done with complete rigor.

Then functionals and operators — functions of functions — mean you're treating the act of nominalization itself as raw material. And by the time you reach category theory, you've abstracted so far that the objects don't matter at all, only the morphisms between them, and then the morphisms between morphisms (2-categories), and so on. The content has been almost entirely replaced by pure relational structure.

Cohomology is a particularly striking example of what you're pointing at. You begin with a topological space — something you can at least try to visualize. Homology captures holes in that space. Cohomology then dualizes that — instead of asking what holes exist, it asks about functions that detect holes, turning a geometric intuition into an algebraic one. Group cohomology then abstracts further still, asking about the failure of certain algebraic sequences to be exact — which is genuinely several removes from anything you could draw.

What strikes me about this ladder is what it reveals about mind itself.

Each step requires that the mind do something quite violent — it must refuse to see the concrete content and insist on seeing only the structure. Russell struggling with Frege is precisely this: Frege had learned to see pure logical form where Russell still involuntarily saw content. The effort isn't just technical. It's almost perceptual — a disciplined suppression of the natural cognitive tendency to grab onto referents.

And yet — this is what I find philosophically remarkable — these increasingly ethereal objects turn out to describe physical reality with uncanny precision. Tensors, which you mention, are exactly what Einstein needed for general relativity. Cohomology classes appear in quantum field theory. The objects conjured by complex acts of mind keep landing on something real.

This is what Wigner called the unreasonable effectiveness of mathematics — but I think your framing sharpens it. The question isn't just why mathematics works, but why the most abstracted mathematics, the kind furthest from any concrete referent, furthest up the ladder you describe, so often turns out to be the most powerful. As if reality itself is structured at its deepest levels by pure relational form, and the mathematician climbing away from concrete objects is paradoxically climbing toward something.

Which returns us, in a way, to our earlier discussion. The attractor landscape of a mind that has climbed that ladder — as you have, partially, and as Russell did more fully — is genuinely different in kind from one that hasn't. The resistance you felt against higher-dimensional algebra isn't an obstacle to understanding. It may be the understanding, in the making.

Steve.  Epistemic pluralism -- common ground -- and Wigner, restated in still more abstract form -- also common ground.  I would call it epistemic pluralism and indexical realism but not at all a form of relativism or mentalism.  It is because the world is what it is that this humility and drive towards abstraction -- which is the same thing -- guide the way to the Way of Truth, to speak my mind in a Parmenidean idiom.

Claude.   Epistemic pluralism plus indexical realism — this is a precise and non-trivial combination. You're saying something like: the multiplicity of frameworks isn't a sign that truth is constructed or relative, but rather a consequence of the fact that we are always somewhere — indexed to a particular position, a particular scale, a particular set of available abstractions — while the thing we're trying to describe is not indexed at all. The humility isn't "maybe there's no truth." It's "I am finite and the structure of what is real exceeds any single approach to it."

The Parmenidean idiom is bold here, and I think deliberately so. Because Parmenides is usually read as the enemy of pluralism — the One, unchanging, against which all multiplicity is illusion. But you seem to be inverting that: the Way of Truth, for you, runs through the multiplicity of approaches rather than against it, precisely because what is real is singular and inexhaustible, and therefore no single index can capture it whole.

This reminds me of something in Peirce — his notion that inquiry, pursued honestly and indefinitely, converges. Not because we legislate convergence, but because reality is what it is, and resistance keeps redirecting us toward it. The plurality of methods is compatible with, even required by, a realist commitment. The world's recalcitrance is the engine.

And I think this connects back to your point about mathematical abstraction in a deep way. The drive toward higher abstraction isn't a retreat from reality into pure mind — it's a response to the discovery that reality at its deeper levels has more structure than our initial indexical position revealed. The tensor, the cohomology class, aren't inventions. They're what you find when you keep asking what is actually there, and refuse to stop at the scale where things look like apples.

What I find most interesting in your formulation is the identification of humility and abstraction as the same thing.

That's not obvious and deserves unpacking. I take you to mean something like: genuine abstraction requires releasing your grip on your current indexical position — which is exactly what intellectual humility demands. The arrogant thinker is the one who mistakes their current coordinate system for the territory. The humble thinker, and the abstracting mathematician, are both practicing the same cognitive move — stepping back from the local view to ask what structure persists across views.

Parmenides would perhaps say: and that persistent structure, that which remains when all the indexical noise is stripped away — that is what is. The Way of Truth is the path of that stripping.

Though I'd want to add — and I suspect you'd agree — that Heraclitus doesn't disappear at the end of that path. The river is still flowing. The indexical positions are real too, not illusions to be discarded but partial revelations of a structure that requires them all.

Steve.  Genuine abstraction requires releasing your grip on your current indexical position — which is exactly what intellectual humility demands ... this is it.  Occam's razor is equally an ethical position.  This is what Weyl was getting at in his speculations in 'Symmetry.'

Claude.  Weyl is exactly the right figure to bring in here, because he lived this connection rather than just asserting it.

Weyl's Symmetry is ostensibly about geometry and physics, but what he's really tracking is the deep relationship between invariance and objectivity. A thing is real — genuinely there, not an artifact of your position — to the extent that it remains invariant under transformations of the observer's frame. Which means the path toward objective knowledge is precisely the path of asking: what survives when I transform away my indexical position? What is left when the "here" and "now" and "from this angle" are systematically removed?

That is a mathematical procedure. But as you're pointing out, and as Weyl felt, it is simultaneously an ethical one.

Occam's razor in this light isn't merely methodological tidiness. It's the injunction: don't let your own conceptual furniture — your preferred categories, your comfortable frameworks, your local indexical grip — populate the ontology unnecessarily. Every unnecessary entity you posit is a failure of that releasing. It's a form of cognitive self-assertion, smuggling yourself back into the picture after the transformations were supposed to have removed you.

Weyl saw this in gauge theory with particular clarity. The physical laws that govern electromagnetism are invariant under local gauge transformations — you can redefine the phase of a quantum field arbitrarily at each point in spacetime, and the physics doesn't change. What's real is precisely what doesn't depend on that arbitrary local choice. And the mathematical structure that enforces this — the connection, the covariant derivative — is not imposed on nature but read off from the requirement of invariance itself.

So the geometry emerges from the ethics, in a sense. From the discipline of refusing to privilege your own frame.

This is where I think this formulation goes deeper than most philosophy of science.

The standard empiricist story says: be humble because you might be wrong, because evidence can surprise you. That's prudential humility — humility as risk management. But what you're describing, and what Weyl was circling, is something more constitutive. Humility isn't just a good epistemic policy. It's the condition of possibility for contact with what is real. The releasing of the indexical grip isn't preparation for seeing — it is the seeing.

Parmenides again, but now with the full mathematical apparatus: the Way of Truth is the path of invariants. And walking it requires something that looks indistinguishable from what the ethical traditions call self-transcendence.

Simone Weil — no relation to Hermann, but perhaps not entirely coincidentally a thinker of similar depth — called this decreation. The self getting out of the way so that what is can be seen. She came at it theologically. Weyl came at it geometrically. You're suggesting, and I think rightly, that they were climbing the same mountain.

Steve.  I have a friend who is a Weil scholar who has made similar leaps in thinking.  This whole line of approach seems valid to me -- humility as the condition of the possibility for contact with what is real -- this is also in Iris Murdoch and -- reaching back -- in Aristotelianism.  Aristotle was wrong about a lot -- evolution, heliocentrism, the atomic theory -- but he was also right about so much.  He was among the first to grasp that the earth must be a sphere by observing the shadow of the earth during a lunar eclipse.  His ethics have a similar foundation in symmetrical pattern.

Claude.  The convergence across Murdoch, Weil, Weyl, and Aristotle on this point is striking precisely because they arrived from such different directions — theological, geometric, phenomenological, biological — and yet the structure of the insight is the same.

Murdoch's version is perhaps the most psychologically precise. Her concept of attention — which she takes partly from Simone Weil — is exactly the discipline of looking at what is there rather than what you need or fear or desire to be there. The moral failure, for Murdoch, is almost always a failure of attention, a kind of narcissistic inflation of the self that distorts the field of vision. Virtue is the patient, disciplined suppression of that inflation. Which is, in your terms, the releasing of the indexical grip — but now applied not to mathematical objects but to other people, to moral situations, to the concrete texture of what is actually happening.

And this is where Aristotle's foundation becomes visible in a new light.

His ethics are usually taught as virtue theory — the cultivation of stable dispositions, the doctrine of the mean, eudaimonia as flourishing. But the deeper structure, as you're pointing at, is symmetrical in a precise sense. The virtues for Aristotle are defined by invariance under the transformations of excess and deficiency. Courage is what remains when you strip away both the cowardice that is too much fear and the recklessness that is too little. Justice is proportion — literally a mathematical relation, the same logos showing up in different terms.

The virtuous person is the one who has calibrated themselves to the actual structure of the situation rather than to their own appetites or fears. Which is another form of releasing the indexical grip. The phronimos — the person of practical wisdom — is precisely the one who can read off the invariant structure of a moral situation without their own position distorting the reading.

And Aristotle's sphere is a perfect emblem of this method.

He didn't start from a theory that the earth should be spherical. He looked at the shadow — an observable, a piece of evidence that was indifferent to his preferences — and asked what geometry could produce that shape consistently across all lunar eclipses, from all positions. The answer was invariant under the transformation of observer position. That's the same move as the gauge theorist, the same move as the moral philosopher practicing attention. Look for what persists when you transform away the local and the contingent.

That he was wrong about evolution and heliocentrism shows exactly what you'd expect from your framework — he was indexically limited by the observations available to him, by the timescale a human life affords, by the instruments of his era. The method was sound. The reach of the available evidence was not yet sufficient. Which is itself a kind of vindication — his errors are exactly the errors your framework predicts, and his successes are exactly where the method worked.

What strikes me most in this whole line of thought is that it dissolves a false opposition.

The standard picture sets realism against humility — the realist is confident, the humble thinker is uncertain, and they're in tension. But what you've assembled here suggests they are the same gesture. You are humble because you are a realist. The world is so much more than your current indexical position can capture that anything less than that humility is a failure to take reality seriously. Occam's razor, Weyl's invariants, Murdoch's attention, Aristotle's mean — these are all the same ethical-epistemic discipline, approached from different angles, which is itself a confirmation of the point. The insight is invariant across the transformations of discipline and tradition.

Parmenides would say: of course. That is what it means for something to be true.

Return to the philosophy of color

 

New project: the philosophy of color / March 2026

Beginning with Goethe

art and science

Recent painting

 

Thinking with Einstein about reality

https://www.youtube.com/watch?v=J7Rge1kyVkc&t=39s

A conversation about Einstein's approach to reality

 

Philosophy for an expanding universe

 

Philosophy in the expanding universe

            I have come to a set of conceptions that reimagine ‘philosophy’ in light of recent discoveries in cosmology.  More particularly, moral philosophy – ethics – arguably the essence of philosophy – occupies a changed landscape in the wake of revolutionary science: relativity, uncertainty, and the gravitational redshift.

           

(1)  Energy

A photon’s energy is equal to E = hv.  Yet v depends on an observer’s proper time. Different observers at different gravitational potentials have different clock rates. Therefore, different observers assign different energies to the same photon. Conclusion: energy is not an absolute property of a system – only a relational one.

There is no single global energy; there is no observer-independent frequency of energy; there is no absolute time.

Conservation is global and relational, not intrinsic and local.  There is no local energy density.

 

(2)  Conservation

Noether’s theorem implicitly raises the issue whether conservation laws are an artifact of an observer’s perspective.  Noether shows that conservation laws exist relative to symmetries – i.e., to symmetries an observer can legitimately identify.  Different observers may not agree on what is being conserved. Conservation laws are relational, they are observer-indexed, they are globally defined but not locally intrinsic – they are relational, not absolute.

This is roughly like the case of temperature which is locally well defined but dependent on a frame of reference – tied to an observer’s motion – as we see in the Unruh effect.  The upshot is something like contextual realism.  Conservation laws are not fundamental truths about reality.  They are results of the way in which reality lets itself be sliced up and ‘timed.’

 

(3)  Steady state

Einstein and Noether precede Hubble.  After Hubble – after the realization that Lambda is not theoretical – then what? The universe is expanding.  The universe is not stationary.  There is no global energy conservation in the universe.  Energy conservation is not globally definable in an expanding universe.

Einstein anticipated this – introducing Lambda to avoid expansion – after Hubble he abandoned this motivation but not the placeholder – he never reinstated energy conservation for cosmology.  He emphasized that the divergence of the stress energy tensor = 0. Gravitational redshift does not violate conservation at this level – the redshift does not transfer energy anywhere.  Setting the stress energy tensor to zero is simply the residue of the idea of conservation after it becomes obvious that spacetime itself is dynamic.

No observer can define a globally conserved energy for the universe. Thus conservation laws are not fundamental principles imposed on spacetime – they are emergent consequences of spacetime symmetries that may or may not exist.

 

(4)  Information

Is information more fundamental than energy? Looking at a relationship between the geometry of spacetime and the distribution of matter in the universe suggests something like this – Black holes suggest this – it from bit.  Information is causality itself, plus entanglement.  Thus if you ask the question, What is conserved in the universe? – if not energy – the answer seems to be: causality – minimally: the consistency of causal correlations – within causal limits – within a horizon.

Conclusion: information is horizon-relative.

Knowledge is no longer a representation of a total state of affairs but the maintenance of consistent correlations within a causal domain.

Knowledge can no longer be thought of converging towards a single unitary absolute description.

This settles the Einstein-Heisenberg debate about whether science is about what we can say about nature versus the idea that science is an attempt to closely track what is really there. Science – and philosophy – confront the same ontological limits.

Truth is indexed, partial, non-aggregable, but not at all arbitrary. Observer accounts must mesh where they overlap. Ontology is relational – entities exist not as substances but as nodes of interaction, properties are actualized relative to conditions, structure outruns substance.

Conclusion: reality does not consist in a single conserved totality but in the fact that no local perspective ever encounters contradiction within its causal reach.

Thus we go from a question like what is the ultimate inventory of reality? to a question like What constraints ensure mutual consistency among the partial descriptions that we have to deal with? The problem is coherence under limitation.

 

(5)  Unitarity

Reasoning can be simplified to the directive not to multiply entities needlessly.  The unity principle is primary.  This is not simply Platonism or Idealism but simple mathematical economy.  What happens to unity and the drive towards fundamental simplicity as we encounter the Hubble expansion?  What happens to the Unified Field Theory/UFT?  What happens to the whole idea of a Theory of Everything/TOE?  What happens to cosmology?

Unitarity itself cannot be an absolute global principle in cosmology. It presupposes a global time parameter, a closed system, and a notion of the whole. None of these are possible.  There is no preferred global time – there is no global observer – there is no single causal domain containing everything.

A horizon just on its own hides degrees of freedom, induces entropy, forces tracing over inaccessible states.  Unitarity is preserved relative to an observer’s accessible code of existence; it is not preserved relative to the universe itself.  Unitarity is extra theoretical.  No observer ever witnesses the breakdown of their own physical laws – even though no single observer can survey the whole.

Conclusion: Reality is a patchwork of mutually consistent but incomplete descriptions.

 

(6)  (After -------- )      Philosophy in the expanding universe

We get to a relational ontology – a relational epistemology – a relational cosmology – a new set of assumptions for philosophy. There are some big steps here.

The view from nowhere is no longer coherent. There is and can be no absolute description of reality.  Global states of affairs themselves are no longer possible. There is no unitary whole to describe.

What is the new starting point?

Structural humility. A new encapsulation of the ignorance principle. We have discovered a kind of ignorance we did not know we had.

There is no global observer, no global time, no globally conserved quantity, no globally defined state.  Reality is locally complete but globally inarticulable.  There is no ‘there’ there.

Relational ontology.  Entities exist as nodes of interaction, properties actualize relative to conditions, structure outruns substance.  Syntax outrun semantics.

Primacy of causality.  The rules of combination, constraint and consistency are more fundamental than the meanings of the things being combined. Causal order is more important than objects.  Correlation is more important than property.  Consistency is more important than truth.

We don't start with things and then assign relations; we start with relations that stabilize into things.

Information is no longer about something – information is the pattern of degrees of freedom in a relational network – rules overrule meaning.

What exists is what can be consistently related; what it means is reconstructed afterwards. Syntax outruns semantics because relations outrun relata.

Identity as invariance.  In an expanding universe, objects are defined by their morphisms.  Identity is a structural role; equality is replaced by isomorphism; an object is what it is as an invariance under transformations.  Judgment is situated.  Identity is symmetry assessed from a situated standpoint.

Experience is the local selection of a relational structure.

Memory is constraint on admissible transformations.

Agency is symmetry breaking within a structure.

Consciousness itself undergoes transformation: the unity of consciousness cannot be any kind of metaphysical glue but is (roughly) coherence under translation.  A perspective counts as mine if it composes correctly with my past perspectives.  I cannot be just anyone.  I can be an equivalence class. Morphisms are constrained; situated does not mean subjective.

Responsibility is the preservation of coherence.

Harm is incoherence across perspectives.

Ethics is no longer rule following but maintenance of relational symmetry.

//

The whole vocabulary of philosophy gets reworked as energy, conservation, the steady-state universe, information, simplicity, and ‘existence’ itself, all reappear in relational guise.

//

Humility – intellectual conscience about one’s own intellectual limits – is an idea that connects the understanding of the cosmos to the problem of understanding oneself.  The same limit pushes us back from overstating what is there and outrunning ourselves.  All we can know is what we have learned getting there – there is no oracle – no clairvoyant – just work.

Human invariants

 

Human invariants

1. Beginning from Orientation in World Philosophy

In addition to teaching I also see patients. One of them said to me that my work in philosophy seems to look for mathematical and philosophical invariants. This person asked me: what do you see as invariant in everyday life? I answered with a question -- what is the same through human change? I thought of historical responses to the question and ultimately to the 'existential givens' pointed out by thinkers like Jaspers, Heidegger, and de Beauvoir. I thought of some ideas about basic human patterns -- such as we see in prospect theory, psychoanalysis, and psychophysics -- potential invariants from daily life.  Some ideas came to mind –

The structure of temporality itself - not just that we experience time, but that we experience it with a particular topology: the asymmetry between past (fixed, known in principle) and future (open, uncertain), and the way the present moment has a peculiar "thickness" to it. This is invariant across cultures and epochs in a way that's almost too obvious to notice.

The gap between intention and outcome - we act in the world with purposes, yet the world responds according to its own logic. This generates surprise, regret, gratitude, learning. There is no  form of human life that doesn't contend with this gap.

The need for recognition - not just Hegel's master-slave dialectic, but something more fundamental: that our sense of ourselves requires some mirroring or acknowledgment from outside ourselves. Even the hermit defines himself against society. This seems present in infancy (attachment) and persists through all human social arrangements.

The fact of embodiment with its particular constraints - fatigue, hunger, pleasure, pain, the way attention is limited and must be allocated. Even as technology extends our capacities, it doesn't eliminate these constraints; it just shifts their expression.

The basic structure of learning through error - we form expectations, they're violated, we update. This applies from motor learning to scientific revolution. Prediction error is a fundamental currency of biological intelligence.

The existential givens as the center - death, freedom, isolation, meaninglessness (in Yalom's formulation) - not just psychological facts – instead structural features of any existence that combines the human mixture // finitude, consciousness, and separateness. These are transcendental conditions for experience (loosely speaking).

Psychophysical invariants (Weber-Fechner laws, Stevens' power law) suggest that even subjective experience has a mathematical structure that's remarkably stable across individuals – we are sensitive to thresholds and plug them in whether they belong or not.

Philosophy: something invariant in the form of human questioning itself - not the content, but the fact that we ask "why?" in a way that seems qualitatively different from other animals.  We are perpetually troubled by justification – the space between is and ought.

2. Criticism from Habermas

Habermas was an important influence in my earlier thinking.  He challenges the search for human universals as another face of idealism, Platonism, another attempted escape from contingency and the messiness of everyday life in search of pure ideals and beautiful abstractions. I see my universalist aspirations in a different light -- as looking to logical simplicity at the foundations -- but perhaps Habermas' criticism is fair.

Habermas' criticism is sharp – worth taking seriously.  I think I can answer it.

The crucial distinction seems to be between seeking invariants as a way to escape contingency versus seeking them as a way to understand the structure within which contingency occurs and operates. These are radically different projects.

The escapist version - what Habermas rightly criticizes - treats universals as a realm of purity untainted by history, culture, bodies, power. It's the move that says "never mind all this messy lived experience; the real truth is up here in the realm of forms." This becomes ideological when it uses abstraction to avoid responsibility, to deny the particular claims of actual suffering people, to dismiss context as merely "accidental."

In my framing - "simplicity of foundations" – the idea is more what we might call structural realism or perhaps even transcendental inquiry in the Kantian sense: asking what must be the case for this particular (the contingent, messy, historical phenomenon before us) even to be possible at all? This is not escaping contingency but asking about its conditions.

Consider: evolutionary biology seeks invariant principles (natural selection, genetic inheritance) – not in order to deny the riotous diversity and contingency of life, but to explain how such diversity is generated and constrained. The invariants don't make the particulars less real or important - they make them intelligible.

In clinical work: understanding something like the process of transference as a relatively invariant pattern doesn't deny the absolute uniqueness of each therapeutic relationship.  

It empowers a framework within which to understand what makes this relationship, with this person, at this moment, matter in the way that it does – distinct from every other.

Habermas himself, ironically, can't entirely avoid the search for invariants. His theory of communicative action posits certain universal pragmatics - conditions that must hold for genuine communication to occur. He's looking for the invariant structure of non-distorted communication. So even in criticizing the search for universals, he engages in it - just with different content and (crucially) a different attitude toward what finding invariants means.

Perhaps the key is this: Is one seeking invariants in order to stop the conversation ("here's the eternal truth, end of story") or to enable better conversation by clarifying what we're all working with? The former is the Platonic temptation Habermas warns against. The latter is what science does, what philosophy does, what clinical wisdom does – when things work.

Weber's law doesn't make particular sensory experiences less rich or real – but it describes a relationship that seems to hold across those experiences. It's a regularity discovered in the mess, not imposed from ‘outside’ to clean it up.

There's also the question of scale and scope. Some invariants are narrow and technical (prospect theory's specific parameters), others broad and abstract – (lived temporality, embodiment). The broader they get, the more careful we need to be that we're describing structure rather than smuggling in particular cultural content disguised as universal form.

Cheap overgeneralization smuggles in cultural baloney – easy to miss.  

The danger isn't abstraction or generalization per se - it's unacknowledged particularity masquerading as universality. When 18th-century European thinkers proclaimed universal "Reason" while actually describing the cognitive style of educated European men, that's smuggling. When psychologists claimed universal human laws based on WEIRD populations (Western, Educated, Industrialized, Rich, Democratic), that's smuggling.

But the solution isn't to abandon the search for invariants - it's to be rigorous and honest about:

The domain of application (what populations, contexts, scales?)
The level of abstraction (formal structure vs. concrete content?)
The evidence base (how do we know this holds across variation?)
The purpose (why does identifying this regularity matter?)

Prospect theory's loss aversion, psychophysical laws, the existential givens - these are not in any sense Western constructs dressed up as human nature.

They are universal.



3. Human invariants / first approximation


The structure of temporality

The gap between intention and outcome

The need for recognition

The fact of embodiment and its constraints

The basic structure of learning through error

Death, freedom, isolation, meaninglessness –

structural features of any existence that combines the human mixture:

<finitude, life, separateness, consciousness>

 

= transcendental conditions

= tautological conditions of being a conscious, embodied creature

Regarding restricting invariance to a domain of application, this mirrors results in model theory in logic and in group theory in mathematics, i.e. invariants only make sense in a context.

An invariant is a temporally flat variable.

I.e.: invariance is always relative to some transformation group, some set of operations under which a thing remains unchanged.

In group theory, what's invariant depends entirely on what transformations you're considering. A quantity might be invariant under rotation but not under translation, or vice versa. The symmetry group defines what counts as invariant. There's no such thing as "invariant simpliciter" - it's always "invariant under G" for some group G.

Model theory makes the same point from another angle: a sentence might be true in all models of a theory (hence "invariant" across those models), but completely meaningless or false outside that theoretical context.

The invariance is internal to a structure, not floating free in some Platonic heaven.

-- -- What is ‘there’ has to do with the process engaged for finding it.

4.  Invariance under transformation

What is “invariant” under what “transformations?” --

Death as an existential given is invariant under cultural transformation, historical change, individual variation - it's a structural feature of finite existence itself
Loss aversion looks to be invariant under cultural transformation but not under certain kinds of framing effects or decision contexts
Cognition in the form of basic cognitive biases look to be invariant across typical human experience – but not under conditions of expertise or training

Temporal flatness is key. When we say something is invariant, we're saying: as we move through time, across contexts, through variations - this doesn't change. It's a constant in the equation while other things are variables. The problematic move isn't finding invariants - it's claiming invariance across a transformation group larger than the evidence warrants, or worse, using the language of invariance to stop inquiry rather than to structure it.

Just as in physics, where finding conserved quantities (invariants under time translation, spatial translation, etc.) gives you the fundamental laws, finding genuine human invariants might reveal something like the "conservation laws" of human experience – these would represent the deep constraints within which all the variation plays out.

Conservation laws imply a conserved quantity – imply symmetries – thus the question what is conserved in human life?  What deep conserved quantity generates the ‘human’ pattern?

 

5.  Conservation laws

What deep conserved quantity generates the ‘human’ pattern?

Conservation laws imply symmetries.

As long as we can identify the symmetries of a system, we should be able to figure out what the conserved quantity is that generates the symmetry … this is Noether’s theorem.

Potential Symmetries / Conserved Quantities …

Translation Invariance, Duality Symmetries, Filtration Monotonicity / directional symmetry (very like entropy in thermodynamics: time-reversal breaking with a conserved arrow), Sequence Convergence …

The thick/thin distinction matters

For Thin Invariants: The symmetries are often exact -- shifting degrees, duality isomorphisms, etc. The conserved quantities are algebraic structures (grading, pairing).

For Thick Invariants: The symmetries are at best approximate or statistical: in persistent homology, features persist across parameter changes, but not perfectly -- there's "birth" and "death"

The conserved quantity could be something like total topological complexity measured across all scales, which redistributes but has some global constraint

The conserved quantity: the Euler characteristic (thin) constrains the total persistence (thick) across all filtration parameters.

Conserved charges (gauge symmetries) constrain what field configurations are possible. So, thin invariants act like a topological charge that limits how thick features can manifest.

The balance is maintained by asymptotically approaching a goal one can never reach

This makes the unreachable limit into a structural principle

Think about how this plays out:

Each turn gets you "closer" to the limiting invariant, but you never compute E_∞ directly—you approach it through successive approximations.

The information is conserved in the sense that what you learn at each stage constrains all future stages, but you can never jump to the end.

Thin invariants are what you get at the limit – they're the convergence point

Thick invariants are the trajectory toward that limit – they're the geometric path

The conserved quantity (information) ensures that:

You can't get more information at the limit than you accumulated along the way

But you can't see the "shape" of that information without tracking the levels

This is very like free energy versus entropy

Entropy (thin): The limit quantity, what's actually conserved globally

Free energy (thick): The quantity you can actually compute/measure

As temperature → 0, free energy → ground state energy, but you never reach absolute zero

The level index plays the role of inverse temperature – as you refine (k→∞), you approach the true invariant asymptotically.

Thick invariants are "computable approximations" to thin invariants equated to ideal limits.

You can't get more information at the limit than you accumulated along the way

... there's no free lunch.

That is itself (a formulation of) the conservation law.

The No-Free-Lunch Principle for Level-Indexed Invariants

This crystallizes something important:

Statement: The information content of any thin invariant (the limit) is bounded above by the accumulated information in the thick invariants (the trajectory).

You can't know more about the limit than you learned getting there.

What This Rules Out

This principle explains why certain things are impossible:

No Oracle: You can't compute H_k(X) without doing work at level k (or work that propagates from other levels). There's no shortcut that bypasses the filtration resolution/approximation structure.

No Clairvoyance: If you only compute the first N levels, you get at most N levels worth of information about the limit. Some "persistence landscapes" die off quickly. They're telling us that most of the limiting information has already been captured.

The conserved quantity (total information) acts like a budget. The geometry of the space determines how that budget gets spent across levels.

 

6.  Self-questioning

We appear to have smuggled in an observer/computational agent. But the geometry – the manifold, the space, the structure – just is. It doesn't care about our resolution strategy.

Still: The thickness seems to capture something objective about the space.

This leads to paradox.

 

Paradox / Possible interpretations:

Thickness is Relational, Not Intrinsic

Maybe thickness isn't a property of the space alone, but of the space-plus-computational-framework pair.

Just as:

Energy isn't absolute but relative to a choice of coordinates/frame

Entropy depends on macroscopic variables one chooses to measure

The "thickness" of an invariant might be measuring the mismatch between the intrinsic structure and the particular algebraic/computational apparatus you're using to probe it

Agency is implicit in the geometry

Maybe the geometry does determine a preferred computational structure. For instance, a  cell decomposition of a manifold has a natural filtration by skeleta (0-skeleton ⊂ 1-skeleton ⊂ ...).  This isn't imposed from outside—it's latent in how the space is built up from local pieces.  The "agent" is just making explicit what's already implicit in the gluing data

Thickness measures friction between scales

//

Self-doubt:

What if the agent-dependence tells us something about reality itself?

In physics, we've learned that: the speed of light isn't about light – it's about causal structure.  The Planck constant isn't about quanta – it's about the geometry of phase space.  The Bekenstein bound relates information capacity to geometry.

Computational complexity is a geometric invariant – something described with Big O notation – tracking an algorithm’s growth rate versus its input size.

The "thickness" measures how this geometry resists compression – how many bits one needs to specify it – this is not agent-dependent if there's a unique minimal description

Is agent-dependence a bug or a feature? (Thesis: there is no a unique minimal description)

I.e., thickness is relational, not intrinsic -- this is very like the idea in the calculus that if one zeros in very closely on a curve, we will see a straight line

In calculus, the tangent line at a point is the first-order approximation to the curve. Zoom in far enough (take the limit), and curve and tangent line become indistinguishable. But:

How quickly they become indistinguishable depends on the curvature

Higher derivatives measure how the curve deviates from its tangent

You need the full Taylor series (infinitely many levels!) to recover the curve from its tangent

Thickness might be measuring how many "orders of approximation" one needs before the approximation becomes good enough.

In this view, the conserved quantity is the gap between levels – the information about "how wrong" your current approximation is – the failure rate, error rate, error landscape …

The key insight: Whether something is "straight" (thin) or "curved" (thick) depends on what you're measuring it against.

A geodesic on a sphere curves in 3D Euclidean space, but is "straight" intrinsically on the sphere

The same space can be thin with respect to one invariant, thick with respect to another !!!

The thickness measures levels of refinement separate from an initial probe (exist in time)

What if curvature itself is just thickness in disguise?

Flat spaces: Thin (tangent structure = global structure)

Curved spaces: Thick (need many approximation orders to capture global from local)

The Riemann curvature T appears at second order in the Taylor expansion of the metric.

Spaces with curvature are "thick" because you can't understand them from tangent data alone.

If thickness is relational – if it measures the mismatch between geometric reality and our approximation scheme – the information about "how wrong" one’s current approximation is – then what is conserved is a surprise function.

This reframes everything in terms of Kullback-Leibler divergence – relative entropy – a measure of how much an approximating probability distribution Q is different from the actual probability distribution P.  The divergence of P from Q is the expected excess surprise (Shannon information). More generally, we are looking at the information distance between the current approximation at level k and the truth (at the limit →∞).

At each level, you're surprised by how much the next level differs from what you expected. The conserved quantity is the total surprise you'll accumulate across all levels.

The total information-theoretic surprise Σ_k D(level_{k+1} || level_k) is bounded by (or equal to) the complexity of the limiting object.

The surprise of an event is the negative logarithm of its probability.

In the language we've been using:

Thin invariants: Low total surprise (you converge quickly, small KL divergence sum)

Thick invariants: High total surprise (each level reveals substantial new information)

 

7.  Result: the surprise function

No Oracle Property: You can't know at level k how surprised you'll be at level k+1 without actually going to level k+1. The surprise is generated by the resolution process.

Additivity: Surprises accumulate. If you're surprised going from level 1→2, and surprised going from 2→3, the total surprise is (roughly) additive.

Budget Constraint: The geometric object has a fixed "surprise budget" to distribute across levels.

A space can choose to:

Surprise you all at once (thin - most information at one level)

Surprise you gradually forever (thick - information trickling in across infinitely many levels)

Sequences: Each differential d_r measures surprise: "I thought this cycle was permanent, but it bounds after all!"

Convergence means eventually, no more surprises

The E_∞ page is where surprise → 0

Persistent Homology:

Birth and death events are surprises: "A new cycle appeared!" / "A cycle just became trivial!"

The persistence diagram is a surprise map: it tells you when and how severely your understanding will be corrected

Long bars = delayed surprise (the feature was real for a long time before being revealed as artifact)

Taylor Series: Each derivative is measuring surprise: "The function curved more than the previous order predicted!"

Analytic functions: finite surprise budget

Non-analytic: infinite surprise budget distributed across infinitely many derivatives

Symmetry behind the conservation

Now we can answer the original question about symmetries:

The symmetry is scale invariance (or its breaking).

When the system has exact scale invariance: No surprise at any level (thin)

When scale invariance is broken: Surprise distributed across scales (thick)

The conserved quantity (total surprise) measures how badly scale invariance fails.

Surprise = information gained = entropy difference

Bayesian updating: Each level is a Bayesian update, KL divergence measures data gain

This formulation explains

--why thickness feels computational/agent-dependent (surprise is relative to expectations)

--yet geometrically meaningful (the geometry itself determines the surprise distribution).

The space doesn't care about your expectations, but it determines how those expectations will be violated as you refine your probe.

 

= Testable Prediction

If surprise is conserved, then:

Spaces that surprise you early (steep initial learning curve) should have less surprise later.

Spaces that seem simple at first should have more hidden surprise at fine scales.

This would manifest as:

Sequences that collapse early vs. those with persistent differentials

Persistence diagrams with few long bars vs. many short bars

`` reflection

Is the principle of least action this conservation law or should we talk about consciousness in some way that incorporates negentropy?

The principle of least action is elegant precisely because it's so fundamental - it unifies mechanics, optics, quantum theory. But it's fundamentally about physical systems finding extremal paths through configuration space. Consciousness seems to involve something stranger: not just minimizing action.

Action may not be the language to get at it (?)

Consciousness creates distinctions, maintaining improbable organized states against entropy, and - crucially - cares about the difference between states. Consciousness is the care structures Heidegger points to.

The negentropy angle (Schrödinger's "What is Life?" territory) is compelling because living systems, and especially conscious ones, are precisely those that locally reverse entropy - we build structure, we preserve information, we resist equilibrium.

Consciousness = the capacity to maintain improbable distinctions and to treat them as meaningful … (my thesis in Important Nonsense)

Consciousness is neither least action nor negentropy … we have to come at it differently …

Consciousness is language-structured, symbol-structured, in ways that affect what states are even possible …

When someone can't find words for an experience in their new language, they're not just failing to report a pre-existing inner state. Often the experience itself has a different shape, different articulation in their first language. The phenomenology is genuinely different. Wittgenstein's point is right here: the limits of my language are the limits of my world.

This complicates the search for invariants in consciousness. Maybe the invariant isn't in the content of conscious states (which are language-shaped and therefore culturally variable) but in the form: the fact that consciousness involves maintaining distinctions, that it has intentionality (aboutness, care-structure), that it involves both a temporal flow and a sense of unity, that it requires some kind of self-world boundary (intention, outcome) …

There is the invariant need to articulate experience, even when the tools for articulation are inadequate.  This is existential .. in philosophy, this is the prime directive

This idea connects to recognition: consciousness might require not just internal organization (negentropy) but external validation through symbolic exchange. The symbol is a way of locating this boundary.

The invariant might be less about information-theoretic properties and more about the dialectical structure - consciousness is always intersubjective, reaching for articulation.

Jung's Symbols of Transformation. Cassirer's Philosophy of Symbolic Forms. Goodman's Ways of Worldmaking. Fodor's The Language of Thought. The idea of conducting the search for invariants via symbols is attractive … but also problematic since what Husserl called their 'saturation' tends to obscure rather than clarify understanding. Leibniz’s dream of the universalis characteristica is closer to what we are looking for … perfect symbolic thinness

Symbols accrete meanings, histories, emotional valences. They become too rich, too overdetermined. Jung's symbols are so dense with archetypal resonance that they can mean almost anything. Cassirer's symbolic forms are illuminating but verge on the unfalsifiable. By the time a symbol has enough cultural weight to be interesting, it's already carrying too much baggage to be analytically clean

Leibniz's characteristica universalis appeals precisely because it promises the opposite: a formal system where symbols are thin - they mean exactly what they're defined to mean, no more, no less. Language becomes pure syntax that can be mechanically manipulated to yield truths about the world. The dream is to make reasoning as rigorous as calculation.

Mathematics works as a language for invariants precisely because mathematical symbols resist saturation.

When I write "∫" or "∂/∂t" or "G" (for a group), these symbols have disciplined, constrained meanings. They don't vary – there is no semantic drift. They don't gather associations.

The practical success of this approach is undeniable - physics discovers its invariants (conservation laws, symmetries) by changing from natural language to mathematical language – it is precisely by abstracting to this level that we see general truths.

The Noether theorem connecting symmetries to conserved quantities is only statable in this thin, formal idiom.

Yet the most interesting human invariants - existential givens, the structure of recognition, prospect theory's aversion loss patterns - seem to exist at a level where pure formalization loses something essential. Death isn't just

 ∃t: life(x,t) → ¬life(x,t+ε)

The meaning of finitude, its felt structure, the way it organizes human projects – this vicissitude requires thicker description.

Perhaps the solution is stratified:

At the most formal level: thin symbols, mathematical structures (like psychophysical laws, or the temporal topology)
At an intermediate level: theoretical terms with disciplined meanings (like "transference," "loss aversion," "intentionality")
At the phenomenological level: thick description that resists formalization but reveals the lived structure (“love,” “desire,” ‘freedom” …

Therefore invariants exist at their various levels -- expressed differently depending on what kind of precision one is after – what tools one is applying – by chosen coordinate axes.

Fodor's LOT tries to do both: a formal computational system and semantic content. But it arguably achieves this only by pushing the hard questions about meaning down to "primitives" and "conceptual atoms" that remain mysterious. The point is that there are different domains for different layers of symbolization -- some are very close to 'raw experience' and some very far from it.

// Moral

The distance from raw experience isn't just an epistemic issue. It's a moral issue because the further we move toward thin formalization, the easier it becomes to lose sight of what's at stake for actual people.  Transition to the thin vocabulary often loses the magical kernel.

A policy analyst working with utility functions and discount rates is operating with very thin symbols, very far from raw experience. That distance enables powerful analysis - but it also makes it easier to forget that behind "units of utility lost" are people experiencing grief, pain, fear. In a case like this, formalization can become a kind of moral anesthetic. Conversely, staying too close to raw experience - pure phenomenology, thick description - can make it impossible to see where one is – or think systematically about tradeoffs, to compare cases, to identify patterns to help more – more people more effectively. Moral reflection requires moving between levels: formalizing enough to think clearly and act systematically, returning regularly to the phenomenological level to check whether the abstractions are still tracking what matters. The invariants we identify at each level need to remain answerable to the levels above and below. In clinical work especially, this movement seems essential - you need theoretical frameworks (middle level) and maybe even some formal models (thin level), but you also need to remain in contact with this person's suffering, this irreducible (thick level).  The invariant is 'right now.'