Where is the Platonic Realm?

I think it's all in our minds.

The five Platonic Solids: (left to right) the tetrahedron, the octahedron, the cube, the icosahedron, and the dodecahedron. These are the only 3D polyhedron that are both convex and regular.

In math philosophy there is a long-standing debate about the reality of the Platonic Realm. I’ve heard it said that most mathematicians believe in its real — albeit intangible — existence.¹ But if it is real, where is it? What is its nature? What exactly is in the Platonic Realm?

The sense that it must be real comes from how we seem to discover math rather than invent it. Aliens on other worlds might not invent unicorns or baseball, but it’s hard to conceive they wouldn’t discover numbers and basic geometry. Math seems “out there” to be found — its discovery open to any intelligent mind.²

In fact, many see math as being a priori. A decade ago, I wrote an essay about how a disembodied, environmentally isolated intelligence would inevitably discover math [see Inevitable Math]. That said, the only truly a priori recognition is the famous proclamation by Descartes. As explored in that essay, math is discoverable through only the experience of mental content, so it is not entirely a priori.³

There are big questions here — questions that have been in play since they were recognized literal eons ago (and likely will remain so).

1. Is the Platonic Realm in any sense real?⁴

2. If it is real, where is it located?

3. And what is the nature of its reality?

4. Why do math and geometry seem so prominent?

5. What else besides math and geometry is Platonic?⁵

6. What makes something “Platonic”?

One more burning question burns bright enough to stand alone:

Why is math so ‘eerily effective’ in describing reality?

The observation that math is so effective — besides being a bit eerie — accounts for much of the belief in the reality of the Platonic Realm. Eugene Wigner, famous in both theoretical math and theoretical physics, wrote a well-known paper, The Unreasonable Effectiveness of Mathematics in the Natural Sciences (1960), that delves into its astonishing — almost eerie — effectiveness.

What do we discover when we discover math? (I think an appropriate term might be “recognize” — we recognize apparent absolute truths as axiomatically true.⁶)

Roger Penrose’s Three-World Metaphysics

Mathematician and theoretical physicist Roger Penrose explores the Platonic Realm to varying levels of detail in several of his books. His metaphysics on this comprise three connected worlds: the physical, the mental, and the Platonic. He uses this diagram to visualize it:

Roger Penrose’s Three Worlds and their circular relations.

Note how the relation between these worlds is circular. Penrose begins with the Platonic Realm, with its physics math describing (and it seems Penrose might go so far as to say defining) reality. Note that the P.R. contains more than physics math, so only a portion of that world is related to the physical world. This is why the relations are shown with cones — only a portion of each world is related to the next.

Our collective human mental world is only a portion of the physical world. This is the second relation. The physical world encompasses our mental world (and much more).

The mental world includes all our thoughts. Most of which are concrete day-to-day thoughts: what to have for dinner; how the kids did in school today; what’s going on at work, etc. Quotidian thoughts. But some of what we think is abstract. Typical examples are math and geometry, and these are certainly prominent in the Platonic Realm. There is more to human abstract thought than math, though. Examples include Justice, State, and Money.

So, only a portion of our mental world is related to the Platonic world. An important point about the relation between mental and Platonic is that in contrast to fiction, to stories, we don’t invent Platonic objects. The usual term is “discover” — we discover (or recognize) the Platonic forms of, for example, numbers and spheres.⁷

As a mathematician, Penrose seems to fit the description of seeing the Platonic Realm as distinct from the other two worlds and — in some important sense — real. A world waiting to be discovered and explored. To be recognized.

My Version of Three-World Metaphysics

I generally agree with Penrose’s thinking; this is no exception. My version of his diagram makes some minor changes, though:

Firstly, my starting point is the physical world. We’re physical beings and a part of it, so I do not consider the mental world separate from the physical world. (Nor, I believe, does Penrose. This is not a version of mental dualism.) What is distinct about the mental world, worth calling out, is the deep mystery of consciousness — the Hard Problem. How can clay have opinions?⁸

More importantly, how can clay devise abstractions? This seems a unique property of an intelligent mind. Our ability for abstraction gives us access to fiction — and to the Platonic Realm. (There is a relation in that fiction also often involves a search for recognized truths.)

Another difference is that I question the relation between the Platonic Realm and the physical world. If, to me, the P.R. is an abstraction, how can the physical emerge from it? Penrose’s circularity here bothers me. The dotted parallel lines suggest something I’ll get into below.

There is also that it’s not clear to me how distinct Penrose considers the Platonic Realm. He has the physical world emerging from (a portion of) it. This seems to imply a separate reality and accounts for the sense that we discover it. In that context though, it’s not clear to me how actually distinct Penrose sees the mental world. He certainly doesn’t believe it to be computational. It’s the main topic of at least two of his books.⁹

Regardless, I view the Platonic Realm as part of the mental world, and therefore as part of the physical world. I suspect Penrose would agree, though this leaves hanging the question of how the physical emerges from the Platonic.

I see Platonic objects as things we abstract and idealize from our experience of the physical world. The Platonic Realm is thus as real as the rules of baseball.¹⁰ The key for me is that the P.R. is a subset of our mental world (which is in turn a subset of the physical world — this is reified in the diagram’s cone-shaped relations).

The Platonic Realm is the enduring collective of our abstractions. It is not “out there” somewhere but is based on our collective intellect over time. It is based on a convergence of what we experience and abstract from the natural world and is made manifest in our science and math texts.

The real world therefore isn’t an approximation of the Platonic Realm. If anything, the arrow points the other way. The Realm is a distillation of physical reality. While it is true that real world objects are rough versions of their ideal forms, it is our collective abstract view of them that elevates them to the Platonic Realm. We recognize the purity and universality of the distillation.

We Need a Bigger Metaphysics

My starting point, the physical world, requires an origin story.

Our reality has proven to be so lawful, so regular, that science — the study of those laws — is possible. Indeed, a key axiom of that science is that physical law is identical throughout the universe. This universal lawfulness does seem to point to a real existence for Platonic objects.¹¹

Or at least for Platonic laws behind physical reality. I suspect there are, in a sense, two Platonic Realms. The diagram might look like this:

The first (leftmost) is the lawful metaverse that provides a context for the Big Bang. If the Big Bang is a thing that happened, then it would seem some lawful metaphysics must responsible (unless one declares the Big Bang axiomatic and without cause).

The second (rightmost) is the Platonic Realm we discover through idealizing and abstracting the physical world. It is the recognition of the lawful physical reality in which we exist. Kant (and others) wrote about how our minds are structured to perceive reality in terms of time and space. I’d say this is because our brains are products of the physical world, and it is the physical world that is structured in terms of time and space. So, of course we perceive the world this way.

And now the metaphysics is no longer circular. The Platonic Realm of our intelligence is a distillation of our experience, the recognition of the underlying lawful metaverse.¹²

It is possible that our recognition of physical laws gives us some vision of the metaverse context for reality. That’s what the dotted red lines in my Three-Worlds diagram were meant to suggest. That there is a kind of one-to-one emergent relationship between the Platonic and the physical. But the implied circularity bothers me, and I prefer the Four-World version.

I think that’s enough for this time (perhaps even too much). When I pick this up again, I’d like to explore the Platonic Realm in terms of content. In particular, the implications of abstract concepts such as Beauty or Justice being Platonic. Especially in light of some recent results about the “good/evil axis” of LLMs, is there a basis to believe these concepts might be absolutes?

Until next time…

1. Because otherwise their life’s work would be based on something not real.

2. Which is why many believe that math is the best way to begin to communicate with alien intelligences. Or at least to demonstrate intelligence.

3. It remains true that discovering math does not require physical empirical support.

4. Which opens a Costco-sized can of “What Is ‘Real’?” worms.

5. As I understand it, Plato placed the idealized notions of “Good” and “Beauty” in the Platonic Realm.

6. And if it turns out that we can place higher concepts, like morality or justice, in the P.R., this suggests they must have absolute qualities.

7. And therefore, presumably Justice, State, Money, etc.

8. It’s also a challenge to study because it’s the one problem we confront with both an inside and an outside. Most problems we only face from the outside.

9. The Emperor’s New Mind (1989) and Shadows of the Mind (1994).

10. Which is to say, very real.

11. And by the way, for me, falsifies philosophical idealism.

12. Of course, now it’s the metaverse that needs either an origin story or to be taken axiomatically. But I think it’s easier to swallow axioms that are just laws.