The Mathematics of Idealism

Metaphysics & Epistemology
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What follows is a synoptic overview of my essay on Charles Pinter’s book ‘Mind and the Cosmic Order’, in which I argue that his work provides a compelling, modern cognitive framework in support of philosophical idealism—a stance I view as implicit throughout his work. I briefly corresponded with Charles Pinter about his book a year before his death; as I say in the essay, I think it’s an important but overlooked book.

Naturalism is often conceived in terms of a “bottom-up” physicalism: the universe consists fundamentally of material “simples” (particles, fields, physical laws), while the macro-world of forms, shapes, and conscious agents is an epiphenomenon or secondary byproduct of these.

In Mind and the Cosmic Order, mathematician Charles Pinter challenges this assumption by examining the cognitive foundations of form. He poses a simple thought experiment: Imagine all life vanishes from the cosmos, leaving matter, stars, and galaxies undisturbed. Does an unobserved mountain still have a shape?

Pinter argues that without the animal sensorium to carve boundaries into the continuous matrix of quantum fields, the mind-independent universe is an undifferentiated flux. It possesses matter and energy, but no inherent boundaries, shapes, or individual objects. Form and structure are biologically superimposed.

Pinter’s core argument centers on the Gestalt—an organized whole grasped by a mind as a primary unit of perception.

  1. The Primordial Glance: Cognitive science demonstrates that perception is top-down rather than bottom-up. When you open your eyes, your mind encounters an undivided panorama (a Gestalt) before it ever parses individual objects. Context and meaning are the starting points of experienced reality, not the endpoints (a point which finds support in phenomenology).

  2. Relational Reality: Consider three coins arranged on a table. Are there three things or four? Materially, there are only three coins; no new matter is added to create the triangle. Yet we also see the form of a triangle. Pinter argues that such Gestalts are relationally real. Like “north,” which only exists relative to a magnetic pole, formal structures are objectively real, but only relative to an observer’s cognitive apparatus and intentionality.

  3. Implications for Physics: Applied to quantum mechanics, the wave function is not a physical cloud floating in space, but a mathematical Gestalt describing a reality not yet carved into distinct states. The act of observation does not create anything, but it actualizes a specific outcome from a matrix of possibilities.

If form and structure require an observer to be actualized, physical science cannot treat its mental models as truly mind-independent. Matter provides the ground, but the mind is intrinsic to the cosmic order.

Questions for Discussion:

  • Are formal structures (like the triangle formed by the coins) ontologically real, or are they subjective?

  • If context and intentionality are required to define an “object,” can a purely physicalist ontology account for structure?

Original essay published by Philosophy Today | PDF version

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As I read you, this is a version of dualism. The ground is G. We can’t say anything about it. We live f(G), which is the ground after processing by “mind.” So the function f is “mind” that “processes” a “raw ground reality.”

But “mind” is given a biological basis, so that somehow we have the ground processing itself. “Experience” can start to look like an epiphenomenon. Why does the “ground” give off these strange “fumes” ?

Personally, I think subjectivity goes all the way down. So everything is subjective or “from a point of you.”

You contrast subjective with “real.” If we demystify objectivity and understand it as merely ideal subjectivity, the problem dissolves.

Triangles are as “real” as marriages and protons and prime numbers. We might consider that a person is a towering “abstraction.” A vortex of metabolism and sign-trading is a “you” or a “me.”

I think we agree that a certain wholesale “physics-driven” ontology is naive. Democritus is great as a scientist but terrible as an ontologist, because he didn’t account for the possibility of ontology in the first place. He ignored the central issue of the “significance” of human signs. He ignored the normativity implicit in the very notion of rationality. The world is a “forum.” This ontology tacitly presupposes, because it aspires to determine reality “rationally.” Rationality is not a single-player game.

Walker Percy is great on this stuff. Once we have language, we have a world that does not cleanly “reduce.”

I’m with you on this. A “purely physicalist” ontology is, in my view, a toy for noobs. Its own “disavowed” context — in many cases — is a dramatic-heroic nihilism. More generously, it makes sense to try for an economic explication. Simple is good, where simple works. In this case, “physicalism” is too simple and even simple-minded.

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Yes but…not any form of ‘substance dualism’ ::face_vomiting: . We need to be very clear about what Pinter is saying. I think it maps quite well against the Kantian distinction between noumenon and phenomenon: that the universe unobserved is an undifferentiated flux of energetic relationships (Kant is mentioned around 30 times in his book). But we can only ever perceive and understand the universe in terms of the gestalts, the cognitive wholes, which are meaningful to us, which are not inherently existent in the world, but without which we would know nothing.

But glad to have established some common ground!

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For my own part, I recognized the limitations of the substantialist approach quite some time ago. In my view, a thing never exists in and of itself.

Take coins, for instance: they rest on a table; gravity keeps them from levitating; and in the prevailing environment, the metal alloy from which they are cast remains in a solid state (rather than a liquid one, thanks to the temperature).

Thus, the coin itself—as presented to our minds—is possible only as a tension sustained over a sufficient duration within the ongoing interplay of all forces.

For this reason, I have always carefully avoided substantialism in its radical form. Yet pure relationalism is equally impossible: after all, the coin does exist as something—something that interacts with something else.

Philosophy has yet to thoroughly sort out these questions.

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All of that is perfectly true, but a rather different point. It is the subject of another book I’m studying called Dynamics in Action by Alicia Juarrero, which is about how constraints are also causal. Pinter’s gestalts are more cognitive in nature.

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Yes, I grasped your question right away. But it seemed crucial to me to pin down the ontology before discussing the epistemology.

Although… over the past year of research, I still haven’t fully figured out which is actually more important. =)

Is a triangle real? As an entity within our imagination, it certainly is. But can we equate reality outside our consciousness with the reality inside it? That is a point where I wouldn’t rush to conclusions.

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Boy that’s a leading question!

This is the example given in the essay:

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Plainly the shapes comprise a triangle — but is it really a triangle? Pinter’s comment is:

“If existence is limited to the material, then you have only three things, because no new material is added when the threesome is formed. However, if reality were limited to what is material, there would be no such things as structure or form, because they neither add to, nor take away from, matter. The threesome of coins is a separate reality for us because it has a separate quality in perception.” (Mind and the Cosmic Order, p. 44, bolds added)

A little further down he gives another example:

“The atoms of a teacup do not collude together to form a teacup: The object is a teacup because it is constituted that way from a perspective outside of itself.” (Mind and the Cosmic Order, p. 44)

His point is, we cannot avoid seeing such forms, as they are part of the way the mind makes sense of its world. He also demonstrates that this is by no means limited to human minds, as it is also basic to the way even insect cognition is organised. So the question as to whether the form is in the mind or in the world, whether it is inside or outside, is rather ill-posed. The form is constitutive of the world as experienced, which is the world we know.

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Let me digress slightly from this specific topic to offer another example. Take the law, for instance: it is impossible to conceive of it without structure. If we tried to do so, we would end up with a chaotic jumble of isolated cases. The same applies to construction: without a degree of idealism in our thinking, we simply could not design anything before actually building it. Idealism is an excellent tool for engaging with the world.

In fact, it is precisely that idea—that image or abstraction acting as an intermediary between the mind and the world—that made us human in the first place. After all, we do not manipulate the actual objects within our minds; rather, we operate using mental representations of them.

On the other hand, there is reality itself, which can only maintain a stable state thanks to a specific structure of connections and relationships. Therefore, I would not simply dismiss the notion that structure is essential.

But this raises a logical question: is the structure of the world (as we conceive it) identical to the structure constructed within our consciousness?

I have come to the conclusion that we lack the necessary data to make a definitive judgment on this matter… And I admit that, in this regard, the concept of “irreducibility” proposed by the Frankfurt School appeals to me—at least on a psychological level.

And here is something else that is very important. I would like to lay out my personal stance on all this, so as to spare the reader a million links that substantiate it.

My position is this: I sincerely want the world to always be something more than what we think it is. I feel comfortable living with that.

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I think your question about whether “the structure of the world (as we conceive it)” is identical to “the structure constructed within our consciousness” gets to the root of the matter, but I wonder whether it still assumes a representational picture that is foriegn to classical philosophy.

The idea of a form or essence is not necessarily a mental representation of something external. To grasp a form is, in the classical understanding, for the intellect to become conformed to what it knows — the conformity of thinking and being. The triangle apprehended by the mind is therefore not simply an image constructed inside consciousness which we then have to compare with a triangle “out there”. The mind apprehends the intelligible form instantiated in the particular instance.

Here I refer to another of my favourite ‘potted quotes’ from a lecture by Plato scholar, Lloyd Gerson:

Aristotle, in De Anima, argued that thinking in general (which includes knowledge as one kind of thinking) cannot be a property of a body; it cannot, as he put it, ‘be blended with a body’. This is because in thinking, the intelligible object or form is present in the intellect, and thinking itself is the identification of the intellect with this intelligible. Among other things, this means that you could not think if materialism is true… . Thinking is not something that is, in principle, like sensing or perceiving; this is because thinking is a universalising activity. This is what this means: when you think, you see - mentally see - a form which could not, in principle, be identical with a particular - including a particular neurological element, a circuit, or a state of a circuit, or a synapse, and so on. This is so because the object of thinking is universal, or the mind is operating universally.

….the fact that in thinking, your mind is identical with the form that it thinks, means (for Aristotle and for all Platonists) that since the form ‘thought’ is detached from matter, ‘mind’ is immaterial too ~ Lloyd Gersion Platonism versus Naturalism

This is why I find Pinter’s example so interesting. He is not referring to classical philosophy nor to Platonic forms, but it’s not a stretch to see the likeness. The three coins do not contain an additional material object called “the triangle”; nevertheless, their arrangement has a recognizable and intelligible form. Its intelligibility is actualised in cognition, but that does not make it merely subjective. It is the way the mind must construe the form.

And that, I think, is where Pinter’s discussion of form diverges from what we’re inclined to think about “ideas as mental representations.” (Also don’t loose sight of the fact that Pinter’s previous books were on mathematics and algebraic geometry, in which he was an acknowledged expert.)

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I don’t understand how I accidentally deleted that message…
I’ve roughly reconstructed the text using a translator.

I fully understand your point of view. However, the fact remains that—regardless of whether this is the case or not—we have no verifiable means to prove or disprove it.

Consider this: the very way we acquire knowledge and structure our thoughts has not always been the same; it changes from generation to generation and from culture to culture. I wouldn’t simply call it “discourse,” but neither would I claim that the structure of our cognition is identical to the structure of the world—after all, differences in cognitive approach exist even between two people from the same family, let alone across different cultures.

There is certainly something to all this, but what exactly—and to what extent?

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I suggest a review the essay linked to the OP as it provides a more detailed exposition of Pinter’s ideas.

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And I do agree with that although I don’t know what bearing it has, unless it’s presented as a rationale for flat out relativism.

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I appreciate the analytical method of testing theses. I will attempt to follow an analytical chain of reasoning (please correct me if I make a logical leap).

Let us consider the argument:
If the intelligible structure of the world is grasped directly and unequivocally by the intellect, then that structure must be necessary—and the only one possible—for any mind encountering that aspect of reality.

Take the pure abstract disciplines: geometry and logic. If the world’s structure were rigidly fixed and directly transmitted to the mind, we would possess a single, invariant mathematical and logical system.

In reality, however, we have Euclidean and non-Euclidean geometries, as well as classical and many-valued logics—all of which are internally perfectly consistent.

This demonstrates that our mind does not merely “resonate” with a single, static form of the world; rather, it is capable of constructing diverse, mutually irreducible, yet logically flawless languages ​​of description.

If idealism insists on the existence of a single, absolute identity between mind and world, it must confront the fact that multiple such ideal systems exist.

I am willing to acknowledge that the idealistic approach contains a profound intuition. Yet, a crucial point remains: in my analysis, I am drawing upon something outside of idealism—specifically, the historical fact that various geometric and logical concepts exist. And perhaps there is a logical flaw here. Now, to verify the genuine truth of my assertions, we once again require an external yardstick or an external perspective. But where are we to find one?

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An interesting thought occurred to me while reflecting on all this:

Why, when speaking of the mind, do we assert that the mind constitutes the structure of the world? Why not the other way around—why, when speaking of the world, do we not say that the world is structured in accordance with the structure of our mind?

Does it follow, then, that proving the first thesis does not refute the second?

To be honest, I don’t like either thesis in its exclusive form. Perhaps I simply feel that things shouldn’t be that way—and that’s all there is to it. =)

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I am on the side of this “conformity.” But your version of dualism discussed above seems to reject this conformity.

As I read you, you are an indirect realist of the Kantian flavor. There “is” an “objective” or “external” reality that offers us a “raw material.” Then this raw material is “cooked” by the nervous system. So the “forms” in experience look to be “only mental.” “Being” is “projected” so that it is “outside” of all possible experience.

I read Plato as “not infected” by the Cartesian dualism. “Forms” are real because we live them constantly. To be the “same” Socrates over many years to “perform” a “form,” and others performed this form with Socrates, by taking him to be the “same” Socrates.

If you make experience a functional output of something else, then that seems to make “form” merely “mental.” I think (?) you want to say something “stronger” than the usual indirect realism.

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Right. Good stuff !

But this plays into “ontological perspectivism.” “Consciousness” is the “presence” (being) of an “aspect” or “moment” of the object itself. We sometimes thematize this moment of the object as a “perception,” but this is the exception that only has life via the rule. This is the neutrality of Mach’s “elements.” For the most part, we just live “the primacy of perception.” We can switch into a psychological mode that “converts” a “moment” of the object into a “perception” that is then used to predict the perceiver rather than the perceived object. Our ability to do this tempts people to stuff “world-from-POV” inside “pineal glands.”

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I sense a resistance to the suggestion of philosophical absolutism. But I don’t presume any such thing. The bottom line of my interpretation is that the idea of the mind-independence of empirical objects is mistaken. Whereas the overwhelming conviction of scientific realism is that the real is precisely what exists independently of us: the vast universe, with h.sapiens as mere blips within it. But notice that this already is posited as if from some point outside ourselves.

I think this is a Kantian type of insight. The order we find in the world is inseparable from the way the mind orders its perceptions. It is not necessary that this grasp be absolute. But this is precisely why we are able to apply logic and mathematical reasoning to the flux of experience, which is essential to scientific method.

I question that there is an ultimate separation between “the structure of the mind” and “the structure of the world”. From a phenomenological perspective reality is not something we stand outside of and subsequently construct a representation of. We are already in the world, and the world is disclosed to us through the structures of our subjectivity. But this does not mean that the world is simply constructed by the individual mind. That structure is trans-personal — or if you like trans-subjective — characteristic of not this or that subject, but of subject-hood generally.

So I would be inclined to say that neither “the mind constitutes the structure of the world” nor “the world is structured according to the structure of the mind” quite gets it right. Both still presuppose a separation that may itself be at the root of the philosophical problem of modernity which is imposed on us by individualist philosophy. Classical philosophy, by contrast, was participatory. It retains a sense of our participation in the intelligible order of things rather than treating the knower as a separated spectator.

I acknowledge that I am introducing ideas from classical metaphysics that Pinter himself, in his book Mind and the Cosmic Order, does not suggest. But I’ve made the case for the similarity between Pinter’s ‘cognitive gestalts’ and ‘ideas’ in the sense understood in classical philosophy. They’re by no means identical, but I think the similarity is defensible.

I take it as unarguable that the Cosmos pre-exists h.sapiens. But the way in which exists outside our intellectual and somatic grasp of it is incomprehensible, plainly. That is what I take ‘the in itself’ to mean. As for ‘living the forms’ - yes, precisely what I mean above, in the ‘participatory’ aspect of classical metaphysics.

In fact, to digress, maybe the issue with classical metaphysics arises when the ideas are no longer lived, but become stale dogmatic recitations or textbook formulae, absent the living discipline which was a fundamental aspect of the original tradition (per Pierre Hadot).

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We ended up coming back to this idea after all.

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Fair enough. That position is the rule rather than the exception. But that position seems to take an “absolute” time for granted. It posits a singular universal past, “the” past.

Are you not applying a “human form” (the human concept of absolute time ) to what it posited as “independent” of those forms ?

Above, you say to someone else that:

But you insist, it seems, on a quasi-Kantian separation. Granted, you strip off all the structure of X except this absolute time. No problem for you perhaps, but for me it “occludes” the identity of consciousness and being. For me “consciousness” is not a stuff that emerges from another kind of stuff. Being is an entity or a stuff that entities are made of. As you probably already understand, I reject idealism. But my radical pluralism via ontological perspectivism will sound like idealism to many.

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No flavor of Kantianism as such, is supported by indirect realism. Any realism is direct in its origin and manifestation, regardless of the kinds of its objects.

“…..Thus the transcendental idealist is an empirical realist, and grants to matter, as appearance, a reality which need not be inferred……” (A371)

The real need not be inference, because it is given, and all that is given is so directly, without any other necessary condition beyond some sufficiently receptive capacity.