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Nobelist Roger Penrose Talks About His Impossible Triangle

At Closer to Truth, the mathematical physicist explains to Robert Lawrence Kuhn how he understands the relationship between mathematics, the mind, and the physical world

Last month, Robert Lawrence Kuhn interviewed eminent British mathematical physicist Sir Roger Penrose on the relationship between mathematics, the mind, and the physical universe (December 9, 2022/32:00 min). Penrose likes to illustrate the relationship between the three with an “impossible” triangle (see below).

Here are a couple of transcribed selections from the first part of the discussion in Part 1*, concerning the Penrose Triangle, with some notes:

Robert Lawrence Kuhn: Let’s start with your grand metaphysical framework, your three worlds — three mysteries: the physical world, the mental world, the platonic or mathematical world — each connected to the other two in your famous diagram of an equilateral triangle. What’s the origin of this vision of yours of foundational reality? (1:35)

Roger Penrose: Gosh, I don’t know. I mean the picture, of course, was influenced by that particular impossible triangle — which was deliberate in a way because, in the picture, you see a mathematical world at the top, which I regard as having its own existence independently of us and so on. A very small part of that has to do with the physics of the physical world we know. (2:06)

When we get our physics — the mathematics of our physics — right, it’s very, very precise. Of course we’re never quite right but it’s very, very precise. However, it’s only a very small part of the mathematical world which governs the operation of the physical world. So I represent this with a sort of beam coming down and physical world being encompassed in this very tiny part of the mathematical world. (2:29)

Penrose Triangle

Then in the physical world we have these conscious beings and these conscious beings are part — a very small part — of a physical world… So it’s a very small part of the physical world which seems to have direct relationship to consciousness. And I regard this consciousness as having a different kind of existence but it springs from that very tiny part of the physical world. (3:07)

But in the world of conscious experience, we also have understanding and we have understanding of mathematics. That again is a very tiny part of mentality. But nevertheless that tiny part of the mentality … in a sense encompasses or at least has the potential to encompass the top world, which is the mathematical world. And I sort of draw this as a kind of paradox because it’s a small part of each world which seems to encompass the totality of the next one. And it’s deliberately drawn as paradoxical just to emphasize the strangeness of this thing. (3:50)

Note: From Closer to Truth: Sir Roger Penrose is a mathematical physicist and philosopher. He is the Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute of the University of Oxford, as well as an Emeritus Fellow of Wadham College. He is a Nobel laureate for “the discovery that black hole formation is a robust prediction of the general theory of relativity”.

Robert Lawrence Kuhn: I think we would all agree with the strangeness. Does the ontological reality of the mathematical world — what you have on top — privilege that or have a a higher position in the hierarchy of ontological beings than the other two?

Roger Penrose: That’s an intriguing question. You might say, why did I put this at the top? And it is deliberately at the top. Whether I regarded it as somehow superior to the others I don’t know. I never quite thought of it that way but it has a more existence-in-itself. I suppose it has to be there. One could imagine that the physical world wasn’t there. I mean, philosophers argue about that sort of thing. But at least one can imagine that it wasn’t there. One could imagine that consciousness wasn’t present at all. But to say that mathematics isn’t there is somehow almost a contradiction. It conjures itself into existence in a certain sense so perhaps that’s my best answer to your question (4:53)

Robert Lawrence Kuhn: Now, many scientists — your colleagues, my colleagues — reject the ontological reality of both the mental and the platonic worlds. They say the mental is entirely derivative on the physical or supervenes on the physical. And the platonic world is just either a fictionalist view or a derivative from human language with no substance in reality. I assume you’ve had conversations along those lines. What are your responses to that? (5:26)

Roger Penrose: Yes, well, I don’t know. I suppose my mental world in a sense is a concession to those people — a different sort of people — who regard everything as mentality. (5:38)

Note: Idealism is a philosophical position according to which “reality itself is a form of thought and human thought participates in it” — Stanford Encyclopedia of Philosophy In other words, nothing is “merely” physical.

Roger Penrose: I’m not going to take a view really to say that anything is more real than another there. I’m just presenting the picture and the interrelationships between the different worlds and to say … I mean, I’ve been slightly talked into saying that the mathematical world is more fundamental — which in a certain sense I do think, because it has to be there even if the other two didn’t exist. (6:02)

However, the mental world… I’m not really saying it’s got a more powerful kind of existence than any of the others. I just think mentality has some kind of existence and to say it’s supervenient, I don’t know quite what that means exactly. (6:22)

Note: Supervenience means that everything about A depends on B: “there cannot be an A-difference without a B-difference.” — Stanford Encyclopedia of Philosophy. When applied to the mind, it means that everything mental is the outcome of purely physical processes (physicalism). Penrose knows what the term means, of course, but he doesn’t agree. If he did, his impossible figure would not be envisioned as a triangle.

Robert Lawrence Kuhn: Well, I mean, the idea is that the mental could not exist without the physical … (6:28)

Roger Penrose: … we might imagine having just mentality and having no relationship to physical matter. I can conceive of that view. It’s certainly not my view. So I don’t quite see why there is greater objection to the mental world from some folks anyway than the physical world. It seems to me one can argue a case for any one of them to have an existence. It’s not … I think one has to be careful about what my view is rather than just a view which I’m presenting… (7:11)

Robert Lawrence Kuhn: I think it’s very important to distinguish views that you present and your own personal views. I think sometimes they are kind of allied together, as people would view it. Because you’re so good at presenting different points of view, people think that everything you present is your view and that’s not necessarily the case. (7:30)

Note: Kuhn appears troubled here by the hesitancy with which Penrose presents his own position. In a world dominated by physicalist, Penrose begs to differ but he also wishes to avoid fruitless battles.

Roger Penrose: … To say that it supervenes, or whatever the word is, on the physical world is rather the presumption that we know what in the physical world produces mentality — which is a bit of a stronger statement than I’m making. I don’t claim to know but I have some ideas about it. But why do we say that mentality sort of comes about just because of the physical world being there? [It] doesn’t seem to me there’s any argument there…

People say, well, it comes from the physical world. Well, maybe it does. But we have to know in what way. What kind of physics produces mentality? I think there is this view — a sort of computationalist view — that when things get very complicated, somehow mentality swoops into existence I don’t see that at all. (8:45)

Note: Famously, consciousness is labeled a “Hard Problem” because no one has any idea how it relates to physics or to physicality generally. Computationalism is a physicalist theory that “the mind itself is a computational system,” like a computer, – Stanford Encyclopedia of Philosophy

Robert Lawrence Kuhn: And then, as you alluded, there are really an increasing number of people [for whom] — in whole, idealism or in part, panpsychism — where mentality has fundamental existence. And that’s not your claim. Your claim is just a visual representation of these three worlds that exist in some relationship to each other, to present that as a fundamental fact of the world. How people interpret it is their business. (9:18)

Roger Penrose: I think that’s right. That is the way I’m thinking about it, yes .

Next: How surrealist artist MC Escher influenced physicist Roger Penrose

*Note: The discussion continued over several segments of Closer to Truth. Here are Part 2 and Part 3 of Kuhn’s discussion with Penrose. More interviews with Penrose here.

Denyse O'Leary

Denyse O'Leary is a freelance journalist based in Victoria, Canada. Specializing in faith and science issues, she is co-author, with neuroscientist Mario Beauregard, of The Spiritual Brain: A Neuroscientist's Case for the Existence of the Soul; and with neurosurgeon Michael Egnor of the forthcoming The Human Soul: What Neuroscience Shows Us about the Brain, the Mind, and the Difference Between the Two (Worthy, 2025). She received her degree in honors English language and literature.

Nobelist Roger Penrose Talks About His Impossible Triangle