Asking a Theoretical Physicist About the Physics of Consciousness | Roger Penrose | EP 244

[Music] I'm Stephen Blackwood, and I have the great honor today to be here with Sir Roger Penrose and Dr. Jordan Peterson. Let's get right down to it. Jordan, I know you have questions you're keen to pose to Sir Roger. Over to you.

Yeah, well, I've wanted to talk to a theoretical physicist for about 30 years, and so I'm pretty happy that you're the theoretical physicist that I get to talk to. I'm probably not representative.

Well, that might be even better. So I want to jump right into it. A colleague and friend of mine is an AI engineer and a computer engineer, and he's built a lot of the world's great chips — iPhone chip, first 64-bit chip, the Alpha back in 1985. We were having a conversation. I said I was coming to meet you and that you... I don't want to put words in your mouth, believe me, but that you believe that consciousness is, in some fundamental sense, non-computational. I asked him what he thought about that, and part of the reason I asked him is because he's, of all the people I've ever met, and maybe of all the people in the world, he's the person who's done most to build arguably brain-like algorithmic systems.

So I asked him if he thought that there was a distinction between the algorithmic computation of cognition per se and whatever consciousness might be, and he thought it was algorithmic all the way down. I understand that you don't believe that. I also went with him a couple of times to a consciousness conference in Tucson, where Hamlet spoke, so we got familiar with that line of reasoning. And I also understand, I believe, that part of the reason that you think that consciousness is necessarily non-computational is because of Gödel's theorem. So maybe we could enter there. I'm very curious about your proposition that consciousness per se is non-computational, and I'm curious about why you came to that conclusion and if you think that's a warranted conclusion.

What do you think about that in relationship to these complex AI systems and also in relationship to Gödel's theorem?

Well, I've never seen the argument refuted. I've just talked to people who've never really understood it. As far as I know, no. The argument goes back to when I was a graduate student, and I was doing pure mathematics — algebraic geometry. I went to three courses which were nothing to do with what I was supposed to be doing. One of them was a wonderful course by Hermann Bondi on general relativity, which had a big influence on what I did later on. One was a talk by the great physicist Paul Dirac, and that taught me about quantum mechanics. And the third one was, of course, by a logician called Steen, and he taught me about Turing machines, the notion of computability, what it is and how you understand that, and the Gödel theorem.

I had heard vaguely about the Gödel theorem previously and had been rather worried because it seemed to show that there were things in mathematics that you couldn't prove. What I learned was that it's not like that at all. Well, it is like that in a sense if you lay down the rules of what you call a proof, and if those rules are such that they could be checked by a computer— checked whether they've been correctly applied by a computer—so computation rules, in that sense, then you can construct a sentence.

This is what Gödel did — which, by the way, it's constructed. You can see that if you trust the rules, let's say if you believe that the rules do, if they say yes, you've proved it, tick, then you believe it's correct. Let's say if you have trust in the rules, that trust extends beyond the rules. In other words, you can see that a certain statement is true by virtue of your belief that the rules only give you truths. Yet that statement is underrivable, unprovable using the rules — that statement of faith about the rules.

It's not a statement of faith. I'm sorry, I didn't understand.

Well, the faith is — it's not a faith. You understand the rules, you check them, you say yes, that's okay. If that rule is correctly applied, I agree. It does — you know, it's a lot — it — it's a rule which is within something that you believe to be appropriate.

And this rule, it's built up out of things like this, which nobody would dispute. You say, "Okay, if you follow those rules and it says yes, that's a proof," then you believe that this thing that it says yes, it's a proof is actually a true statement.

So does a proof really mean that it's true? If you believe that, that conviction that the proofs actually do what they're supposed to do gives you something beyond the rules themselves.

Okay, that's — sorry, that's what I was referring to with this — with the word faith, is that the statement of belief.

Well, I guess I'm wondering what you think it is that constitutes that belief. Okay? And why the word understanding specifically? Because that's the thing in some sense that's outside the system — the understanding.

Yes, it is because you can see it is because it's the understanding that the rules give you only truths that enables you to understand that this Gödel statement is actually true.

And so is that belief in that truth of that proof? That is one of the things that Gödel pointed out would be necessarily outside any system that's both — what is it — formal, logical, and coherent.

What shows? It shows that — I mean, I read it in this particular way. I don't think he said it quite like this, but I read it in the following way that understanding, whatever that word means, is not computational.

Okay, okay, it's not that that is what it's — not the following of rules, it's something else.

Okay, so let me ask you a question about that. So this is a three-prong question. Okay? Yes, it seems to me that there's a high probability that the future is actually indeterminately different than the present and the past, that it's actually unpredictably different.

Oh, this is a different question. Now you're talking about determinism.

Yes, yes. But I think it's tied to this idea that computation can be complete and algorithmic. I don't think it can be because if the future differs in a fundamental manner, an unpredictable manner from the present or the past, then a deterministic algorithmic system can't maintain a grip on the horizon of the future.

And I have another part of that question, but it's a different question. So I think it's important to distinguish these things.

Yes, because up to this point, I was not talking about indeterminism.

No.

No, I was talking about rules.

Well, just yes or no, and I mean it's not a question of maybe or — I mean it isn't even talking about the laws of physics at this stage, that's the second step.

I feel like I was — I guess I looked forward to something like the potential necessary function of consciousness. So because one of the things consciousness seems to do, from a neurophysiological perspective for example, we tend to become conscious of our procedural errors. And so consciousness becomes alerted to the errors and then zeros in on the source of the error in some sense and corrects it.

And so it looks to me like it's something like a correction system for underlying algorithmic systems. So for example, if you practice a motor routine for a long time, you build specialized algorithmic machinery in your brain that runs it, but maybe you put in an error. You're playing a difficult piano phrase for example, and you stumble over a note you've automatized that you play it. You listen and you hear the anomaly, which is the error.

Your consciousness focuses in on that. A large brain area will activate as a consequence of becoming aware of that error. Then when you practice the new routine that's corrected, the brain area will shrink and shrink and shrink until it's a small part of the brain, usually in the back of the left hemisphere. And now you've built another automated machine to play out that phrase.

Consciousness, I think it was Whitehead who said that at least the purpose of consciousness, although he might have used thought, was to increase the number of things that we can do without consciousness or thought. But it seems to be this horizon phenomenon.

And the reason I was asking about the indeterminacy of the future was twofold: if the future is deterministic, then an algorithmic system could in principle adapt to it. But it doesn't seem to me that the future can be predictable, and I think that that might be grounded in something like quantum indeterminacy because there isn't a fundamental determinism that propagates all the way up.

Well, you see, I mean, the things I was talking about up to this point — but not do they — weren't even to do with the laws of physics. So that's a separate question. I mean, it did relate to that, which was my own views certainly depended on that.

But the question of determinism is a separate issue, and the normal way we look at quantum mechanics is it does involve an indeterminism, which you can have a theory which does that too, but that's different.

You see, the Gödel argument is to do with things where you have definite rules. You can check whether these rules have been followed or not, and the question is whether it coincides with your understanding about what things are true or false in mathematics. So that's what it's to do with.

Now, you see, you can question how you move from that into other aspects of what consciousness does. And also the question you were referring to is when something's automatic. Your penis can play things — and obviously, where the little finger goes next is not something that he or she decides to do. It's all largely controlled by the cerebellum, probably, which, as far as we know, is entirely unconscious.

So the greater number of neurons in the brain, which are in the cerebellum, seem not to be acting according to conscious actions at all. It's something completely unknown, right?

Yes, that's a very strange thing that, you know, people make the case as well that there's some simple relationship between neuronal function and consciousness, but as you pointed out, the cerebellum does — the cerebellar activity doesn't seem to be conscious at all. And then you're — there's a tremendous amount of neurons in your autonomic nervous system distributed throughout your body. And there may be some consciousness associated with that, but it's not particularly acute.

And most of the time, it's entirely unconscious, and the autonomic nervous system is running your digestive system and your heart and all of these inner automated systems. And it's interesting too because often becoming consciously aware of a highly functional unconscious system actually impairs its function rather than improving it.

That could be, yeah, sure. I'm not quite sure what this tells us about consciousness; it just tells us certain things are not conscious, which are controlled by neurons in the brain. And so it's a different issue.

Right, but I might just jump in to ask you if you would say a word or two more about why it is that consciousness cannot be reduced simply to mechanistic processes.

Well, you see, I'm very careful to say I'm not talking about consciousness in all its aspects. Yes, for example, I mean, I have nothing to say about the perception of the color green, for instance. I mean, sure, there's something going on which makes green have a certain impression on one, but this is not what I'm talking about.

And probably most of the things that we think about when we talk about consciousness are not what I'm talking about. So I'm only talking about a very specific part of what consciousness does, and the argument is that if this is something which is not a computational process, then it sort of sheds a question mark on the whole thing.

But it's only very specific to the question of understanding. So I tend to make that point clear, and understanding is something which in the — certainly in the normal usage of the word implies conscious, and you wouldn't say of a device normally that it understands something without it being aware of something, and aware means being conscious of it. So that's just normal usage, and I'm going along with that.

So I don't know what most of these words mean, but I would say that understanding is something which requires consciousness.

Yes, there's one way into this to speak about — I mean, so much of our thinking, of course, is calculative. There's a goal there; we're calculating how to get to it. And so there's a huge amount of life that is like that.

But to then ask the question about why this is a goal or why this is worthy of being a goal or what would make it worthy of being a goal or what would make that worthy of being a justification for that to be a goal — the kinds of thinking that you have to engage in in order to reflect upon the nature of the ends and purposes is distinct from the kinds of thinking you engage in to calculate your way to a goal.

And that seems to point towards a realm or a kind of thinking or awareness that is clearly distinct from simply mechanistic calculations.

Yes, I mean, I certainly wouldn't disagree with that. It's just — it's hard to know whether those things could be put into a computational system. The reason for concentrating on this very specific area is that I can say something about it, that's all.

So the particular area is mathematical proof.

I see most people don't bother themselves with mathematical proofs, and they're conscious too. So I'm certainly not saying that's an indicator of consciousness.

I mean, I'm saying it is something which requires consciousness, but I completely accept there are all sorts of other aspects of consciousness which are going on all the time and which are much more important. I've gone along with that too.

But it's just that if you can find something in what consciousness seems to do which is not demonstrably not computational, that's saying something, and that's the limited little thing I'm trying to say.

Now, you started working with Hameroff, as I understand it, to try to provide something approximating a localization or a new neurophysiological account of what this non-deterministic process might be in…

Ah, but I didn't say that's usually non-deterministic. That's different. Okay, it's very easy to confuse it.

Well, and I am confused about them apparently.

Now, you see, non-deterministic means that rules don't have a clear statement about what happens next, and maybe there is a choice about what happens next, and that choice might be random or maybe choice in some more personal sense that you have a reason… I don't know, but usually one talks about randomness.

There you say that the theory does not have a complete description of what it tells you happens in the future because there isn't a random element in it, and the way quantum — huge random — yes, in the future, that's what normally the way in which one talks about quantum mechanics normally.

I mean, I have — and that's a truly random feature in not predictable in current quantum mechanics, that's correct.

Yes, okay, so that's what I was referring to when I was referring to the indeterminacy of the future horizon.

It was — sorry, it was that randomness that I was trying…

Yes, well, you see, but this is another question. You could have a random device which is otherwise computational. I mean, just you put in at certain points, okay, do something randomly. The thing is that — don't think an interesting question there; I don't think that gives you anything in the way of establishing results which seem to be a non-computational process like with the Gödel thing.

Okay, so, okay, so there's an evolutionary answer to the problem of emergent randomness, and the answer is — so a mosquito, mosquito is a good example — or fish, any animal that lays a tremendous number of eggs that could conceivably march to maturity.

So there’s genetic mutation in all… So maybe, let's say just for the sake of argument that a given mosquito lays a million eggs, fertile eggs, in its lifetime.

Now there's variation in those mosquito patterns, and at least a certain amount of that variation is random. That's a consequence, and it's actually a consequence I would say of events that are actually manifesting themselves in some sense at a quantum level because at least some of the mutations are caused by solar radiation.

And so there's disruption at a molecular level, and so evolution seems to be able to use the admixture of randomness into structure as a means of dealing with the indeterminacy of the future. And to some degree, it does that through death, right? Because of those million mosquitoes, on average, only one managed to propagate itself to reproduction, or we'd be knee-deep in mosquitoes, and like No, and Sam was saying…

Yes, but what I'm trying to say is that what's going on with consciousness is different from that because I don't see how this — you know, you're putting randomness in the way you're suggesting, and clearly that is an important aspect to evolution and so on; I certainly wouldn't deny that at all, but it's not the same thing.

So that consciousness isn't producing randomness in response to indeterminate factors? Yes, it's — when I say non-computational, I don't mean that it's random at certain times. I mean something quite different.

So what, okay, let's zero in on that.

So well, because I'm very curious about what you do mean. I mean this is obviously a tremendously important distinction between algorithmic computation and the computational algorithmic domain and something that's in some sense outside of it, and I'm struggling to understand at the most detailed level, let's say, what you envision the structure and function of consciousness or maybe just the function.

It's not producing mere random variants, and I—that can't be because random is too widespread. So at least at the very least, so for example, if you study creative people— we've done a lot of this—there's in some sense more randomness in their speech because imagine that if you utter a given word, there's a certain probability that another word will emerge in the field around that.

The creative people use lower probability concepts and words in their approach, so there's a kind of randomness. They go farther out into the word association field, and that does help them generate more creative solutions.

But that's not — if that becomes unconstrained to too great a degree, you get, well, maybe like a manic creativity that's counterproductive.

And too random people are jumping too much from disconnected point to disconnected point, and so consciousness doesn't seem to be creative. Consciousness doesn't seem to be a mere random walk.

So that's a psychological take on that. But so what do you think is—what do you think? I'm still struggling to understand what you think consciousness does.

It does understand.

Yes, you see, I think probably you're trying to make me be more specific than I can be because I don't know how to make a device that can understand something.

So I'm just trying to say that whatever understanding is, it's not a computational process, and that's the argument.

Okay, okay, it's not — so you're not trying to specify what it might be; you're just saying it has to be something that's known computation?

Yes, that's right. Yes.

Oh, is there — okay, is there a fundamental link? I mean, when we say non-computational, does that mean, or does that not mean by definition that consciousness is some deep level is free?

No, that's right. I'm not — I mean these are open questions; I'm not saying that.

I mean, maybe there is an aspect of indeterminism in it, and that could be, but that's not what I'm saying. The trouble is I think it's not a concept which people usually appreciate.

So I can give you examples of non-computational things, and one of the examples I often give is if you take an imagine a pattern of squares, equal squares — or just a normal square array — and you can consider a finite shape made out of squares. I think it's called a polyomino shape made out of squares.

If you're given a finite set of these polyominoes and the question is can you cover the plane with those shapes, only those shapes, no gaps, no overlaps, now that question, the answer yes or no—the answer is definite yes or no; either you can or you can't, but it's not an algorithmic process.

It's shown mathematically that there is no algorithm which can tell you yes or no whether these shapes will cover the plane.

Okay, so when I was talking to my brother-in-law, I was talking to him about these AI systems that learn how to recognize, let's say, cats from photographs. He told me there is no way of algorithmically determining the program that the machine learning systems will eventually apply to the problem of identifying cats in a photograph.

But if you let the AI neural networks run and then you analyze their output, you often get something that resembles an algorithmic program as an output that you could have hypothetically calculated if you could have specified the search space.

It's something like that, but there's no way of—there's no way of doing that without letting the program do its walk through the domain of cat photographs with its differentially weighted neural network architecture. You can't a priori predict it.

Yeah, well, I still don't think it's the same thing.

I mean, I could certainly give it different shapes and you can say, "Tell the machine which of these will tell the plane which won't." Now, will that learn to give you correct answers? Well, probably usually it does, I suppose.

Once it's tiled, you could formalize the process by which it was tiled, right? Because you could describe the mechanisms or the order in which the tiles were located, then the rotation of them — you could specify it after it had all been laid down.

And I think that's analogous to what the AI systems seem to be doing when they're learning to perceive, but the trouble is that it's not the message whereby you can tile the plane.

I mean, just the theorems tell you that they — you can't put — you can't put them on the computer. I mean, you might get the thing which works most of the time, right? Quite possibly.

Well, so in those tiling problems that you're describing, yeah, and so that is—I see, I guess I kind of see why you're interested in the tiling issue.

So that has to do in some sense with the ability to map a surface with a certain representational form. And so if you have these tiles that you described, and you're trying to completely cover a given surface, that's a mapping problem.

I see what you're doing with that. Are there different ways that you could conceivably solve that problem?

So, okay, so even if you do converge on a solution, you haven't converged on the only solution.

Oh, absolutely. I mean, there was an earlier result which showed that if it were true that any way of tiling that plane with these shapes, with say some given set of shapes — if it tiles the plane, it can do it periodically with a repeating pattern.

If that were true, then there would be an algorithm. But it's not true because there are certain ways of tiling a plane which do not have a repeating pattern.

Repeating patterns? So now that's so cool because I was wondering today, I was wondering why in the world is he obsessed with tiles? What's going on?

Well, the fact that you're— that the tiles are—you're essentially mapping an area with a predetermined concept in some sense—that's the tile shape, and you said it can be non-repeating and still solvable.

But you see, the one in front of the math building, that's an example. That's an example of a tiling set, only two different shapes there. I mean, these aren't polyominoes, but never mind about that.

Those two shapes will tile right out to infinity, but there is no way of doing this which is periodic.

And you can see that this sort of piece—so how do people actually do it then? Like, because there's another way of telling you how to do it with the tiling. So do you—so you designed the tiling in front of the building, so do you actually tell the workmen how to start the tile? And then how do they figure out how to do it?

Or you had a plan for the whole thing and you devised the plan for the whole thing?

Yes, so you provided the map.

Yes, okay.

How in the world did you get interested? Do you have any idea how you initially got interested in the tiling problem?

Yes, would you say it certainly was this computability question that is the connection?

Yeah, I had learned. I think I'd seen an article in the Math Reviews — this reviews mathematical papers and so on — and I'd seen that there was somebody who produced a set of tiles which would tile the plane only in a non-periodic way.

And I hadn't seen what they were like, and there was a conversation. I think it was just after I'd been appointed to my chair here, the Raspberry chair but before I'd taken it up, and I had a conversation with an American mathematician.

And he had told me in detail about— I think there's a mathematician called Raphael Robinson who got the number down to six, and he got a set of six tiles which would only tile the plane in a non-repeating way.

And he said that Raphael Robinson — this was Simon Kochin who's an American mathematician — and he said that Raphael Robinson was somebody who liked to get the number to the smallest number. He was sort of a perfectionist in this way, and he said he's got this wrong; it—he started out with several thousands, you see, and he got it under six, and he was pretty pleased with that.

And I said, "Well, I can do it five." I happen to know I had a set with six, you see, but I knew that I could reduce it to five.

How did you know?

How did I know that you could reduce it to five? Because the way that — I mean, it's just a technical point. There was a certain shape for matching, and this shape only fitted into one other tile. So that I could glue that—the ones with — it's just a slight detail point.

Let's get glued, some of the tiles together to make them five.

When you're mapping the plane, do you map it to the precise borders of the plane, or can there be overlap? You know what I mean? Can it be messy on the edges, or are you trying to precisely cover, let's say, a rectangle?

It keeps on going beyond the edge and then you cut it along the edge?

Yeah.

Okay, okay, that's right.

Okay, yeah. Yeah, yeah. Okay, so now you also had some interaction — interactions, at least at arm's length, with Escher.

Oh, yeah.

Yeah, so what I read was that you and your father had been interested in Escher's work, and you worked out with him the ever-ascending staircase, which, by the way, seems to me quite similar especially to the music in Bach's third Brandenburg concerto, which — and I talked to a musician this week about how Bach managed to make this continual ascending spiral that never really goes up.

That's true. There is a thing like that, yeah.

Yeah, right, yes. So, and then you sent the drawings of the staircase to Escher.

Well, the story was a little bit longer than that, because I had been at this—I was a graduate student, I think in my second year. I can't quite remember. And I and the colleague went to Amsterdam to go to the International Congress of Mathematicians, which happened every four years.

At this congress, I happened to see one of my lecturers, and he had a catalogue which had one of these Escher pictures. I thought, "What on earth is that?" You see? And he said, "Well, there's this exhibition in the Van Gogh Museum by this artist M.C. Escher." Never heard of him before. I went to see the exhibition. I was absolutely blown over by these pictures. One in particular, I think was called Relativity, and I came away thinking, "Gosh, that's amazing!"

I went — whether I could do something a little bit different that I hadn't actually seen in the exhibition, and there I tried to make a construction with bridges and roads going into possible ways, and I simplified it down to this thing that people refer to as a tri-bar.

I've seen the tri-bar, and I showed my father.

I mean, I didn't know that there's a Swedish artist called Oscar Reutersvärd who had done things very similar earlier, but Escher didn't know about him either. But anyway, there were other artists who've done things like — if you look carefully in the old — there's a Breughel, which has a picture of gallows, and they're joined up differently in the top.

Yes, I've seen that. I've seen that picture.

So there are other people who had played with these ideas. But I hadn't quite seen it in Escher. And so my father and I wrote an article. He developed the staircase; that was his actually. He was designing buildings.

And then he produced the staircase which went round and round, and we decided to write a paper on this. We had no idea what the subject was or what journal did we send it to; so my father said, "Well, I happen to know the editor of the British Journal of Psychology, so let's call it psychology."

So we sent it to them, and they accepted it. He said he thought he could get the attitude to accept it; they did. And this was — we gave reference to Escher's — the catalogue to Escher's exhibition, and then my father had correspondence with Escher, with letters going backwards and forwards.

And then I think I was driving in the Netherlands for some other conference, I think, and I was curious. And I, when I was reasonably close to Escher, I phoned him up. I got the phone number from my father, and he was very nice and he invited me and my then wife to tea, and I just had a chat with him.

He sat at one end of a long table; I was at the other end. He had two piles of prints, and he said, "Well, this pile I don’t have many left, I'm afraid." And he pushed the other part of me, "Choose one."

So I sort of went through these things, and I picked one out. Pretty hard to choose one out of all that, and I chose one called Fish and Scales. He was actually rather pleased because he said, "Well, most people don't understand that one." So I felt a bit flattered by that.

But this — I then gave him a set of little pieces of just one shape. And I gave him a set of them and said, "Well, can you tile with those?" And then a little while later he wrote to me and said he'd seen how to do it, but he wants to know what the underlying principle was. So I did — I wouldn't pray that I was a very bad correspondent; it took me a little while before I got back to him.

But I showed him what it was based on, and on the basis of that he produced what I believe was his last watercolor, maybe his last picture, as far as I know, a thing called Ghosts, which is based on this.

It's the only tiling as far as I know that he ever did which is — what's called non-isohedral. You see, he usually did periodic ones, but they're periodic in the strong sense that if you find a shape, the next time you see it, it has the same relation to the whole pattern.

As a whole, so you could move this one into that shape and the whole pattern goes with it into itself. But the one I showed him was what's called non-isohedral, that you can have different instances of the shape.

So this one has a different relation to the pattern as a whole from that one. And so if I move this one into that, I can't bring the whole pattern along with it.

So you have two different roles that the shape plays. And in the last one of his pictures showed this.

So I'm curious too about two things now. I'm interested in why you're so fascinated by the relationship of a geometric shape that can be arrayed in a variety of different manners to this underlying problem of mapping. Like, so you're reducing or establishing a relationship between the problem of mapping a large terrain to the utilization of, like, very stringently defined— what would you call them—representational systems or that? That's a geometric form.

What is the geometric form conceptually in relationship to the problem of mapping?

Well, you have a shape, and then you have certain rules about which fit pieces will fit next to it, but there's certain freedom in that rule. You could put this one that way or another way, you see?

And you know, if it's just — if it's a shape which very clearly has to fit that way next to it, then it just repeats, you see?

But if there's some freedom as to what the next one will do, then you might have to make that choice, and certain choices will run you into difficulties later, and other choices maybe will allow you to continue.

Is there a relationship between that and what composers do with music? Because, I mean, there's a certain—well, there's a certain repeating determinacy in music, but obviously a composer just doesn't take a pattern and repeat it indefinitely.

They take a pattern, and the pattern seems to allow for some choice in movement from that pattern forward.

Well, maybe. I don't know. I mean, it said this, I mean, what makes a piece of music into a good piece of music? I mean, I have no idea; that's a much deeper issue.

Well, we do know a bit about it. We know that if it's too simple and repetitive, your interest gets exhausted.

Yes, exactly, it gets stale very rapidly. And then as it moves towards purely unpredictable, it becomes indistinguishable from noise.

So there's some place in between there, and you could probably move on that place where you get some ultimately harmonious relationship of predictable form and—well, something like the play of novelty.

That seems to me to be analogous to that possibility of shifting the shapes in this tiling problem. I mean, I think music is tiling something; it's a representational form.

No, that's probably some connection. It's just that music — I mean, there's so much more freedom as to what you do. You see with these tiling shapes, it's forced on you that, you know, either it fits or it doesn't, you see?

With music, it's much more subtle.

Right, I would — I would hate to make too much of a leap.

Yeah, yeah. Fair enough. One more question along that line. Now that triangle you made, yes?

Now what's the relationship between those paradoxical forms and the tiling problem?

Not much, because they seem to be — they're — I mean, there's a play of representation, an image there.

I mean, one of the things I've been wondering — I looked at all your diverse contributions, and I thought, "Wow, there's a lot of things happening in a lot of different places."

But there must be some — there's something that's not random. There's something at work that's kind of a uniting principle that might be — I don't know, it might be the problem that you're trying to solve in some deepest way that's uniting all these elements of exploration and interest.

I don't know; you're asking too hard a question. I don't know. I mean, sometimes I don't see any overriding principle.

I mean, there's a sort of thing, you know, something feels right. Now, why does it feel right? I mean, that could be something very subtle.

Yes, yes, maybe wrong too, or maybe they are wrong.

Yes, yes, it seems to me that that also is related in some important sense psychologically to that notion of understanding.

You know, the feeling that it's right — it's interesting that it can be wrong, but it's also interesting that it can be a predictor of— I got a student, she was very creative, and she would come up with hypotheses that were damn good, but she was more creative than the typical psychologist.

And I don’t say that in a denigrating way; I mean she was more like an artist than a researcher. And then what she would do is spend like six months writing out the algorithmic pathway to that conclusion, even though that is not how she derived it.

But she had a pretty unerring ability to jump forward to the right place with her intuition. And it's something like — I think it's something like a deep form of pattern recognition.

You know, you don't need the full pattern to infer what the pattern might be. You can have a sparse representation of it, leap to what might be analogous to a tiling solution.

I suppose, and that seems to be something related to the accuracy of intuition. I know when people become schizotypal, for example, and they're paranoid — that also happens in paranoia.

They have a lot of intuitions about patterns that might be there, but most of them are wrong.

And so it's like their pattern recognition system has become— well, it's exceeded the limits of its capacity for accuracy and is starting to see patterns in what's truly random — you know, prediction error.

Yeah, well, may I ask there's, I think, one important distinction here from what I can understand, Sir Roger, is what the nature of understanding is here is not — it's not to be simply reduced to belief or intuition, though it may be related to them.

When one understands something, let's say the simple equation that two plus two equals four, it's not a belief that that is true. Understanding is operating at a level that is beyond belief; it has a certainty, an inner certainty that is not subject to doubt fundamentally.

And what I was wondering, if it might be helpful just for the sake also of the people who may subsequently watch this conversation, if you would be willing, Sir Roger, to give us a sense of your— your way you describe the three spheres of matter and mind and mathematics, as that might give us a basis for some quite rich conversation subsequently.

Well, maybe you want me to describe the picture; I mean please.

Well, this was just a way of thinking about the relationship between mathematics and the physical world and the world of conscious perception. I was regarding each of these as a sort of world.

I mean, whether that's a useful way or not, it was just helpful to me. And there is the mathematical world, and I take a very Platonic view here that the mathematical world exists independently of us.

And so when we find a mathematical result, it's more like a discovery than intervention. So you — it's there already, and you find it.

So this is certainly a feeling that, as far as I'm aware, most mathematicians have, and the truths are there, and they're there independently of us.

And if we're lucky, we can find one of these truths and see why it is a truth. Now, that's one thing.

Now then there's the physical world, and the physical world — the more we learn about it, the more we find that it operates according to very precise mathematical laws.

But yet it's very small, you see, if you look at a mathematical journal, you find it's almost entirely full of things which have nothing whatsoever to do with the physical world.

They’re playing around with mathematics for its own sake. That part of the mathematical world, which actually does have direct relevance to the way that the physical world operates, is a small part of it.

So I have a picture of this mathematical world, and a tiny bit of that comes and imposes itself or whatever you'd like to say explains or whatever you'd like to say the physical world.

So the more we learn about the physical world, the more we see it is driven or acts according to these very specific tiny parts of the mathematical world.

And the second thing is in that physical world, which seems to be operating according to mathematics, there are entities which seem to be able to perceive and understand and have consciousness.

So the acquisition of consciousness in whatever way is a small part, you see? The world consists of rocks and things like that which don't seem to have any of this quality, but there are certain creatures, things such as people in this room and elsewhere, and probably other animals which may have less of it than humans.

But on the other hand, I'd certainly think they have consciousness, some of them. But still, it's a tiny part of the physical world which seems to have access or whatever the right thing is, seems to be able to — in a certain sense, this world of consciousness.

So it's, again, a small part of that, but there's only a tiny world part of the conscious activity which is concerned with mathematics.

So I had this picture which is sort of meant to be slightly paradoxical that each world in a certain sense comes from a little bit — a bit of the world preceding it, and so it's drawn in a way which is like this impossible triangle, which looks like a paradox.

That's only a little joke in a way.

I don't know how much depth there is to that.

And I rather like to depict it in that way. So this— this— so stop me when I'm wrong, okay?

All right, so it seems to me that the mathematical reality is something like the observation of the pattern regularity between things.

It's not the things themselves, or the — the physical — which, well, I'm thinking about the mathematical representation of the physical world.

So because things — there are things, obviously, but there are things in relationship to one another.

Yeah, and the relationships between the things, like the pattern that your tiles compose, is just as real as the tiles, right?

But it consists of the relationship between the tiles, and is it the representation of the relationship between things that's part of that mathematical world rather than—

I know it could be — you see, powerful. It's a physical thing that's just sitting in front of the mass building. So that's a physical thing, but it is represent — it represents a mathematical idea which only gives you the idea.

I could see, oh, these tiles fit together in such such a way, and these are parts of the Euclidean plane. And the Euclidean plane is a concept we don't actually have physically, but you can see by looking at the tiles carefully enough and see how they fit together that this is a mathematical thing you're looking at in a way.

And that this mathematical thing would allow you to continue if you understand what's going on indefinitely.

So the entire Euclidean plane could be covered according to the rules of those shapes, you see?

Okay, so let me ask you another question about that then.

So is the physical world one tiling solution to the plane of mathematical possibility?

I guess in a sense, I mean it's slightly— you see, it's not really talking about the laws of physics there.

It's only in the sense that Euclidean geometry is a pretty good approximation, that's all it's saying.

I mean, there’s not much physics going on there. You might say, “Well, what makes these tiles — some of them shine and some not shine or something like that?”

I mean, it's more like physics, but the actual design that's being used there, it's been put there by human beings according to what another human being said they should — where they should lay them down.

And that was driven by a certain mathematical concept, but it's different from the way that mathematics underlies the laws of physics, and that's quite different.

It might be if you took one of those tiles and threw it across, or how — you’d be pretty hard to do because they're quite heavy.

But the way that would move in the air before it came down and crashed — that would be a clear indication of a physical law where in which gravitation behaves.

And then the way the thing holds together, the law that holds the — makes these tiles solid be something to do with the, well, quantum mechanics to do with the ways that the atoms are constructed and how they connect with other atoms and what makes them solid rather than a fluid or something like that.

So that would be the way that mathematics drives the physics. It's general laws rather than a specific thing, I think that's what it was.

That's what I said—don't do it because in some sense, and I could obviously be wrong about this, but the physical reality seems to constrain the mathematical possibility because there's only some mathematical rules that govern the behavior of actual objects, even though there's all sorts of possible mathematics that could govern the action of all sorts of hypothetical objects, right?

So it's — so imagine if there's an underlying — I can't help but think this is associated with this many worlds idea, but if there's an underlying metaverse of mathematical possibility, you get the emergence of something like a—what would you say?

One concretized exploration of that possibility, and that now establishes a relationship between one element of that mathematical possibility space and—

Well, in reality itself, it doesn't exhaust the search space, but it's — and it seems to me that that's analogous to this tiling problem in some sense.

I think it's— I don't know, I can't help saying I think it's really very different from what one is trying to do in mathematical physics.

So in mathematical physics, you're looking for general laws which seem—individual instances agree with those laws.

So that an object like one of the tiles that are being used outside math's building — and the trouble is that it depends on detailed laws about the atoms which construct the tiles and so on, which has nothing to do with what we're talking about here, I don't think it is.

Well, I guess I was wondering partly because there's these fine-tuning arguments, you know, and the question arises, well, there's lots of ways these phenomena could be interrelated, but in reality, it turns out that there is a very finite and constrained number of ways that they actually are related.

And those—those are the fundamental laws. And then the question arises: well, why that set of constraints and not, you know, some other set of constraints that seems equally probable statistically, you know, if it was a sample of the mathematical domain?

Yeah, I guess I'm—I have to understand what you're saying a bit better.

I mean, you could say, “Okay, there's this building, well, the one we're in now, which has in front of it a certain tiling.”

I mean, that's — if you're going to explain that, I mean, that's very different from what mathematical physicists do, and they're just looking for general principles.

Right, and as far as we're aware, those general principles are not violated in what's been going on in this building.

However, that's not entirely what I would think because what's going on in this building and so on is an implication of what's going on in people's heads, and this does have to do with consciousness.

And what's going on in consciousness, in my view, is not yet part of current physics.

So do you allow your imagination to wander into the domain of metaphysical speculation about that?

I mean, because you're—you're making a case — yeah, I mean, I was talking to some divinity scholars the other day, and they were laughing, I suppose, about physicists who say with regard to the big bang and the hypothetical emergence of everything out of nothing that give us one free miracle and we'll proceed from there.

And so, I mean there is speculation among physicists that the laws of physics don't apply to whatever the state of existence was before the universe emerged into being, and you're making a case now as well that consciousness itself may not be able to be encapsulated within the realm of our current physical theories.

So what do you think the metaphysical — or do you? Or let me try and get—

I'd have to unpack something here because we're venturing on the different topics.

Yes, I know, which is the question of the big bang. Yes, which I have a different view on that from what you normally — okay, so...

But that's what that'd be fun; we can talk about that if you want to.

Yeah, that's an interesting topic to talk about, but that's really different, and then as far as—I don't even see a connection, as things stand, from what I'm worrying about in consciousness both — they stand outside the laws of known physics in some sense.

Let me say something else which outside the laws are known physics.

And this is not something that people normally even recognize as a problem. I mean, they do, but they shove it under the carpet, which is what's known as the collapse of the wave function.

Now, you see, current quantum mechanics, strictly speaking, is an inconsistent theory. That's rather a brutal way of saying what Einstein and Schrödinger and even Dirac said — that quantum mechanics is incomplete.

And the way to explain this is — okay, there's a wonderful equation which tells you how things — a state evolves in quantum mechanics called the Schrödinger equation.

Now, the Schrödinger equation tells you, if you know what the state of a system is now, the Schrödinger equation tells you what it will be tomorrow, if you like.

The evolution of that state is governed by this wonderful equation due to Erwin Schrödinger. The trouble is that it doesn't—

That's to say the way physicists usually use the Schrödinger equation is to work out certain probabilities of what an observation on the system would tell you.

So what you have to do is you wheel out of the cupboard a measuring device, and in this measuring device, you set it on the system which is evolving according to the Schrödinger equation, and it measures it.

And the process of measurement does not follow the Schrödinger equation. It gives you a probabilistic answer — this or this or this.

That's another outside the system problem; it's certainly outside the Schrödinger equation, right?

Right, right, right, and Schrödinger was terribly worried about this. I mean, he produced his cat in the box and all sorts of things, you see he clearly realized there was a problem, as did Einstein; there's no question about that.

Some others didn't— well, they took a different view. They said, "Look, we don't understand the theory well enough," and that's more that we're saying we're—Schrödinger's not saying that.

You're saying we understand it well enough to see that that's not the way the world operates.

When you make a measurement on a system, it does not follow the Schrödinger equation, and that's what people understand about quantum mechanics.

But it's a sort of vague set of rules about it. It doesn't tell you what constitutes a measurement, right?

It's a problem.

Right, yeah, that's a big trouble.

Yeah, yeah, they say if you do a measurement, then it just becomes probability for what this or that or the other, but it doesn't say what kind of a device makes a measurement.

Now there’s one school of thought which has been going on from way back till the early days of quantum mechanics. Wigner in particular promoted this point of view that it's a conscious being observing the system.

And that means—does that mean that's what Wheeler believed? I believe—I really might have believed quite a lot of people believe that.

I think von Neumann had a similar sort of view. I'm not quite so sure about his view, but certainly, Wigner— and I talked to Wigner about this, and I got the feeling from him that he wasn't quite as dogmatic he was made out to be on this issue.

He just thought this was a possibility.

I think, but anyway, that's—people often refer to it as the Wigner view that it is a conscious being who makes a measurement.

That's not my view; my view is that it's almost the opposite of that view—that there is an objective physical process which deviates from the Schrödinger equation in which the state does collapse so that it becomes one or the other or the other with certain probabilities.

And that this has to do with when gravity is brought into the picture, and there's reasons for believing this.

I don't want to go into that, but there is reason—I’d like you to go into it if you would be willing to because I'm…

Well, it's a very clear mathematical calculation; there's not a question about it.

It's quite what you do with it, you see, and what you do with it, according to me, is to say, "Okay, it tells you that this system has a lifetime, and it will, in that lifetime, become one or the other without a measurement."

It sort of — that's right, yes, without —

What's so interesting to me that you're interested in consciousness, and you see that consciousness in this Gödel theorem sort of manner.

And I would think the most predictable thing for you to believe as a consequence of that would be that it is conscious measurement that collapses the quantum indeterminacy, the waveform, and yet you don't think that it — that statistical vagueness will collapse into something that's essentially— is it either or? Is it binary?

Was it zero one, the collapse?

No, you mean probably — no, there's a probability it'll do — well, right, right, but when the probability collapses mean, well, if it's a two-state system, you see, you might have an object which is in a superposition of here and here.

Yeah, that was Dirac's first lecture, I remember, and he took out his piece of chalk and said — and he was talking about atoms, and you see, according to quantum mechanics or particles, a quantum particle can be here or it can be here, or it can be in a state which is partly here and partly here at the same time, and then he took out a piece of chalk.

And people tell me he used to break it in two; I can't quite remember because my mind was drifting away from what he was saying, and I was looking out of the window and thinking about something completely different.

And unfortunately, it only came back after he'd gone on to the next topic, so I missed the explanation, which was probably a good thing as I think back on it because probably the explanation was something to calm you down and stop worrying about the problem I suspect it was something like that.

So you don't think a conscious observer per se is necessary to collapse the wave?

Absolutely that is what I don't. I mean, I'm agreeing with you; I don't believe that.

Yes, but you do think that, if I'm not mistaken, that the presence of an observer in the universe — that is to say, the observation of the universe by us — is that true to say is fundamental to the universe?

Not really; that's an interesting question, but it's not part of my view. The world would be there quite independently of whether there were creatures, yes, of consciousness walking around on them.

Yes, so can I ask you a question about that? So it's related to this. So it's my understanding, and I could be wrong about this too because I'm way afield here; I'm out of my depth an area of specialization.

But my understanding is that in some sense, as far as a photon is concerned, that the universe is two-dimensional perpendicular to its direction of travel.

I don't see that now, but go on.

Well, I thought — and I thought — I thought that this mayan—it's a consequence of the contraction of things as the speed of light is approached.

Oh, I see, no, no, no, that's — that's something you're talking about, the Lorentz contraction —

Yeah, yeah, yeah. Well, I remember I thought as part of that, that the— that part of the reason that no amount of energy can propel something past the speed of light is because in some sense, the light beam is already where it is and at its destination at the same time, and you can't get flatter than flat.

Now, the reason I asked you that, though, was because it pertained to this other question, which was if you could imagine what the universe might be like phenomenally from the perspective of a light photon, that's very unlike the universe that we perceive.

Well, I see, I mean, if you were riding on — I mean, Einstein, yes, I know, riding on a light.

The trouble is that you can't sit on a light.

Yes, that's a problem.

But if you do — if you were nearly going, you know?

Yeah, very, very fast like that, the passage of time — you would think it hadn't taken any time at all, right?

Except, well, and that's the same as being at the starting point and the destination.

Yes, okay, okay.

So that — now, for us, we perceive things with duration and distance and so — but the photon is in the universe, and we're in the universe, but the universe looks very unlike each of those situational positions.

And so you said that there would be a reality independent of consciousness, but I'm curious, when you think of a reality independent of consciousness, what are the attributes of that reality?

Like is it a field of quantum potential?

Is it—

I'm not quite sure I understand the question, but — but I mean classically, there’s no problem.

I mean, this thing about the contraction and all that stuff with us going close to the speed of light and so on, this is classical physics, so we're not worrying really about the problems of quantum mechanics there, but they're already there in classical physics.

But if you had a particle traveling at the speed of light, let's say just less than the speed of light, then if you could sit on that particle, it would seem as though you got to your destination almost instantaneously.

That's correct, but that's nothing to economic — well, not directly.

Right.

Right, right, yes.

But the phenomenal universe at that speed is radically different than the phenomenal universe at our speed.

Yeah, but the universe is there. It's just a question of — I'm not quite sure I understand this.

You see, I'm trying to understand how it can be all of those things simultaneously.

Like, no, it's— what that means, that's not a problem.

What I mean is that there is a way of looking at relativity which means special and general relativity which is completely coherent and doesn't really worry about who measures what, it's just there.

You have a space-time which is this four-dimensional structure, maybe hard to understand and visualize and so on, sure, but it's the thing which is there.

People call it a block universe view.

Well, I think about it as a whole symphony at once, in some sense.

Well, if you like—but it's all there, and what something measures in that system, you have to go and ask the question if you had a body traveling with a great speed and there was a clock on that body, you'd ask, "For how many ticks does it happen for one end and the other?"

That's perfectly well defined.

If it was sitting stationary, you would have so many ticks between starting a thing; you could have one which goes out and comes back.

You might say only about four ticks. Where's this one that had a thousand ticks?

Well, that's the answer.

Right, in time.

So it's susceptible to all those interpretations simultaneously?

Yeah, because each one is just — it's only measuring it with — it carries the clock with it, and that clock ticks at a certain rate, and that's fine.

There's no problem, but I said there's no problem.

What I mean is that it's not a philosophical problem.

There's a little bit of a problem getting used to the ideas, sure.

No, I agree with that; yeah, that's not the issue.

You can, once you got used to the ideas, and you think, "Oh, yes, you can see time is something which depends on how you're moving."

And the clock which is moving —

Is there any difference between that statement and rate of change depends on how fast you're moving?

Like, is there any difference between time and rate of change?

The rate of change is because I think of time as the average rate of change, and so when you say that time slows down as you move faster, you're not saying much more than as you move faster, you're the—

That's just a question.

You see, I think the mistake here is to think of time as an objective thing which is attached to this model, and it's not.

Right.

There is no concept of when such an event happens. You say you might say, "Well, is this event later than that event?" Well, if they're what's called space-like separated, that is to say you'd have to go faster than light to get from one to the other, it's a meaningless statement because there is no universal concept of time in this model.

Right.

It's just not there.

I think you see that goes against what we normally feel about time.

You think about time, it's progressing, and somebody on the Andromeda Galaxy — we experienced duration, so yes, but you see, what about when is now?

I often use this — I think I use this example of two people crossing the street, and they're walking, just walking speed, crossing the street.

And the question is, according to one of these people, there is a con — at the same time as they cross each other, there is an event on the Andromeda Galaxy where a space fleet is, has been launched, and they're going to invade the earth.

According to the other one, the decision has not even been made yet as to whether they are going to invade the earth or not.

Now, this is only because you're trying to transfer your local notion of what you mean by time to the Andromeda Galaxy, and this depends on what frame you're using.

So if you're using one moving frame, it hasn't happened yet, and the other one, it has.

You just have to get used to that idea that there is no universal notion of time ticking away independently.

Independent frame of reference.

Yeah, that's right.

Yeah, yeah, okay, okay. Can we get you—can I go sideways one more time?

Because I'd like to ask you— like I said, I've been wanting to talk to a theoretical physicist forever.

I'm really curious about black holes, and so I have this idea. So tell me what you think about this.

So when a star collapses past the neutron stage into a singularity, is—and let's say there's multiple black holes; are they all the same singularity?

I don't know.

Okay, okay, well, I mean, you can link up them somewhere, but no, we don't — apart from saying we don't know, I would say no, they're different singularities.

Well, I was — I was inclined to that statement, and we don't really know what we're talking about here, but go on.

Okay, well, I was trying — I was enclosed that statement, and I — and we don't really know what we're talking about, but I was trying to wrestle with the fact that you get this unbelievably intense, not even single point gravitational field, and there are strange effects of—there are strange effects of time inside the event horizon of a black hole from the perspective of an observer.

Now if I remember correctly, if you were watching someone descend into a black hole from outside, don't they go slower and slower?

Then you would — you would see them hovering on the horizon and then fading away very quickly, actually.

Okay, just fade?

Yeah, they would fade.

What happens—what happens to their sense of time once they pass the event horizon compared to the sense of time the framework?

Well, they would go right through, and they wouldn't notice anything at the horizon.

Right, and what would that look—and you wouldn't be able to see a very big black hole?

Yeah, there's a little one. They would have been wrecked by the tidal forces, but yeah, if it's a big enough black hole, you could imagine going through it; you wouldn't even know you'd gone through the horizon.

If you could see someone descending into it, how long would it take them to arrive at the surface? Is that forever?

Not for them, no, but for you watching them.

But you just see them; you don't ever see inside the horizon. The light can't get out, right?

Right, right.

So you never see that?

That's right.

So they could be watching their watches and thinking, "Whoops, we've gone through now," and they would do that. But you'd—if you could see their watch from outside, you'd see that the hand slowing down and getting closer and closer to the moment when they cross the horizon, but fading out.

But it would be slowing down, yes? You'd see it slowing down?

Yeah, okay.

Then if you could see inside, would you see that continuing to slow?

No, no.

So I'm not sure what you mean inside—once they pass the event horizon, you can't see them anymore.

But as they approach the event horizon, if you were watching them, you'd see their clock slowing?

Yes, so if you're outside—if you're outside, yeah.

So then I'm wondering—you can't tell this, but their clock is going to slow the same way as they continue moving forward to the black hole, and that's the trouble, you see?

It's the wrong way to think of it.

Okay, that their clock is—is there a time which their clock registers that's saying there is a universal time which everybody is supposed to respect in some sense?

Relativity says no; there is no notion.

But I'm assuming their clock would continue to slow relative to you.

I'm not trying to assume an absolute time in the question; I’m just—I'm wondering is that as they approach—

I know, I know the problem that you can't detect it is the problem here.

But as they're moving towards the star relative to someone who's watching them, their clocks are slowing according to this frame of reference.

Signals that you would receive, maybe that clock emits a little flash of light.

Yes, yes, that exactly, and you'd see, "Look, those flashes are slowing down and getting farther apart."

Yes, that's right.

Okay, so then from the external perspective, I was thinking that it would take them forever to reach the singularity.

And if it takes forever, then that would be the same amount of time that it would take everything in the universe to collapse back into the initial singularity if the collapsing universe theory is correct.

And so the reason there's infinite gravitation in some sense at the point of the singularity is because that's a point at which the end of the universe is already manifest in the current universe, and that seems to me that that would be—

What would you say?

In keeping with the idea in some sense of a block universe?

Yes.

I'm not quite sure I see the problem.

You're thinking about the whole universe, a collapsing model?

Yes, you could certainly have an entire universe which is collapsing inwards, yes, and then you would hit the singularity before you see somebody else hitting it in those models.

You would find you're in trouble, and that your curvature is getting too big, and you could be killed by it as you were watching somebody else see, "No, they're happily not nearly there yet."

Right, not nearly there yet, that's what you would—

That's what you would— okay.

Okay, all right. Then your model isn't a big bang model with an initial emergence out of nothing and then eventually a collapse back to that?

No, no, okay, so how do you conceptualize that?

Well, first of all, it is a big bang model; otherwise, there is a big bang, but the big bang was not the beginning — the model—the reason people have trouble with this model, I think, is you’re probably gonna have trouble with it as well, you see.

People tend to think that if you have a model in which it keeps on going in some sense and your big bang is not the beginning that you've got to collapse back. So it expands and then it comes back and then you're back with—

But this seems simpler that way?

Yes, but this model is not like that, and that's where you've got to get your mind around.

And it's — people have trouble with this; I agree with it; it's a crazy idea, and I admit it's a crazy idea. The trouble is it seems it's quite likely it's true from certain observational things, but it's crazy too; it can be crazy and true at the same time.

Yeah, that's like a definition of light, yes.

But you see, in this model, the universe expands, and it expands and expands— this exponential expansion.

We seem to see the stars seem to be going starting to go away from us, these very distant stars that people look at with an increasing speed, right?

Right, right, yes.

It seems to be this exponential expansion, and that's what's driven the dark energy hypothesis; that's what they call it.

It's really — I claim it is absolutely nothing inconsistent with Einstein's 1917, was it in modification?

If you see that he regards his biggest mistake, but he's probably actually right.

That's to say the introduction of a cosmological constant is —

Right, right.

He introduced it for the wrong reason, that's true, but he was right to introduce it, even though he regards it as his biggest —

There's a real error; he needed to make things work, but he didn’t have any real practical reason for assuming that it was true, apart from he wanted the static universe; he didn't like the expansion.

No, that he didn't.

No, this was a time before I think Hubble had already seen the expansion, but he hadn't quite got through to Einstein how convincing these results were.

So he wanted the universe was just static and stayed there forever, right?

Right, and then he needed the cosmological constant to do that.

That's correct; he would need that.

However, he was wrong to assume when he got convinced, "Oh, the universe is expanding."

Sorry, he said, "Oh, that was a mistake. My biggest blunder," he said.

The trouble is his biggest smell that turned out to be true apparently.

I mean, this is an argument people don't necessarily think it was — people might not think it's the cosmological constant; I think it was; I think it's right; I have recent, you know, internal reasons for that.

But let's say that this is right; it's a cosmological constant; the universe expands and expands exponential expansion.

Now you might ask, “Who’s in this universe?”

Eventually, not us; the black holes will all have evaporated away by Hawking evaporation, swallow galactic clusters.

What's left in the universe pretty well photons now, giving the simplified version of the theory, because — because there are some questions about it still, but let's say it's dominated by photons, which is pretty well true, but not.

Let's take that okay.

Now the trouble with photons is that they don't feel the passage of time, right?

And more importantly, the equations governing light are the wonderful equations due to James Clark Maxwell, the Maxwell equations.

And the Maxwell equations have a very interesting property that they can't tell big from small; they're what's called conformally invariant.

But if you have a system in which you've got some electromagnetic field, and you could stretch this system to bigger or smaller, it doesn't notice the difference.

The equations work just as well, and you can squash them here and stretch them here.

Well, is that in part because space really doesn't mean anything to a photon in a sense?

Well, it's the scale, you see.

It's what we call—there's a term which I'll use here; it's called conformal. Conformal means big and small.

I very much like—you. We talked about Escher a minute ago; there are these Escher pictures called circle limits where he describes what's called hyperbolic geometry, but don't worry about that.

The most famous one is these angels and devils. You see, right, there's a circular boundary, and they look as though they get smaller and smaller and smaller as they get to the edge.

Yes, now as far as those angels and devils are concerned, the little ones are just the same as the big ones, right?

They don't know that they're smaller towards the edge.

And that, to them, is an infinite universe.

But to us, we can see, “No, there's this infinity which is just sitting there, and these angels and devils, if they don't know big from small?”

I'm not sure I have a bit of trouble using this to explain things because the angels and devils do have a size in the picture, but you see if they were made of massless material, that wouldn't know big from small.

So if they were made of just electromagnetism, then big and small are equivalent, and so you wouldn't know when you got to the edge of this universe.

So that infinity is just like anywhere else.

That's the difficult concept of this thing that the photons reach infinity without realizing—

Without realizing anything funny has happened, if you put it like that.

Infinity, in this conformal picture, is just like anywhere else.

It's only mass that knows the difference.

If you want to build a clock, you need mass.

And this comes from the two most famous equations of 20th-century physics, and the two most famous equations—one of them is Einstein's E=mc², of course, which tells us that energy and mass are equivalent.

And the earlier one was—it's Planck's E=hν, or E=hf, whatever you call the frequency, which tells you the energy and frequency are equivalent.

Put the two together: that tells you mass and frequency are equivalent.

So that means if you have a mass, it is a clock, it has a frequency simply determined by its mass, and this fact is really the basis of modern clocks, which are extraordinarily precise.

They don't directly give this because the frequency is much too high; you have to scale it down, but roughly, it's the same idea.

So a clock, a mass is a clock, but the other side of that coin is if you don't have any mass, you don’t have any clocks.

So you're not having time; you don't have any time scale; you don't have any distance measure.

So if the world is inhabited only by massless things, say photons, then it doesn't know big from small; it doesn't know hot from cold.

And so the idea is—and this is where you have to take a deep breath as opposed to all the other parts of this conversation—the idea is that the remote future is indistinguishable from a big bang.

So long as there is no mass around.

Now the remote future, the reason you have no mass around is basically—well, listen, there's a complicated part of the argument—let's say it's because it's mainly photons; that's good enough.

What about the other way? What about the big bang?

Well, there's lots of mass there, surely, but the thing is at the big bang, things get so hot, things are moving around so fast, if you like, that the energy or the mass — energy, mass, hyphen energy, the concept of mass according to Einstein — is almost entirely in their motion.

And then the mass becomes more and more irrelevant the closer you get to the big bang.

So again, you have a situation where mass is effectively zero.

So are you—is it your claim-belief theory that when things ground out in a universe that only consists of electromagnetic radiation that that is now a precondition for an event like the big bang in some sense?

Yes, I'm saying that the physics which is going on at the very remote future is extraordinarily like the physics going on at the very beginning.

And I'm going to end them when I say beginning; I only mean the big bang because it's not really the beginning.

Yes, yes, well, it's such a lovely place to end, and we have been going for an hour and a half, and I don't want to wear you to a frazzle.

I'm not frazzled.

Well, I think I might be. Would do — do you mind if I just asked?

Go ahead, and Steve, please.

One of the things that I'm very struck by in your account of the three realms of matter, mind, and mathematics, roughly speaking, is that the realm of mind cannot be reduced simply to the realm of matter, nor can the realm of mathematics be reduced simply to the realm of matter.

They each have their own existence.

Well, you see, it's a picture which I've used; I'm not sure whether it completely concurs with my current views, but go on.

What I wanted to ask you about is — and it's a two-part question, but I'll start with the first here — and that is that what is the relationship? It appears as though there's a fundamental intrinsic relationship between the realm of our consciousness, our thinking on the one hand, and the realm of mathematics, or let's say intelligible reality, that I'd like to hear you comment on.

I mean, just to maybe