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ORAL HISTORIES

Interviewed by

Alan Lightman

Interview date

Location

Oxford, England

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In footnotes or endnotes please cite AIP interviews like this:

Interview of Roger Penrose by Alan Lightman on 1989 January 24,Niels Bohr Library & Archives, American Institute of Physics,College Park, MD USA,www.aip.org/history-programs/niels-bohr-library/oral-histories/34322

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Family background; influence of older brother Oliver; career of father; early interest in mathematics, particularly geometry; building 3D models out of cardboard; influence of Fred Hoyle's radio talks in the late 1940’s; questioning of some of the statements by Hoyle in his radio talks; meeting Dennis Sciama; early sympathy toward the steady state model; importance of aesthetics in scientific theories; early preference for closed universes; history of Penrose's ideas on the application of the second law of thermodynamics to cosmology; preference for a very mathematical structure of the origin of the universe; work with Hodge at Cambridge in pure mathematics; influence of Sciama in turning to physics and cosmology; history of work on the singularity theorems; influence of a lecture by David Finkelstein; early work on spinors; work on first singularity theorem for black holes; Penrose's preference for working on problems for which he has a different angle from everyone else; belief in a quantum era in the early universe; influence of Sciama in abandoning the steady state model; work with Stephen Hawking on the singularity theorems for cosmology; view of twistor theory; beauty of complex analytic structure; inseparability of mathematics and physics; importance of complex numbers; importance of visual images; reaction to the horizon problem; pre-eminence of the entropy problem; inadequacy of the inflationary universe model to solve the entropy problem; reasons why the inflationary universe model has been so influential; dislike for the inflationary universe model; particle physicists' lack of appreciation of the problems of general relativity; attitude toward the flatness problem; importance of quantum gravity in solving all these problems; the anthropic principle; reaction to de Lapparent, Geller, and Huchra's work on large-scale inhomogeneities; outstanding problems in cosmology: physics in the very early universe, galaxy formation, missing mass, nature of dark matter; ideal design for the universe and the unity of mathematics and physics; question of whether the universe has a purpose.

Transcript

I wanted to start with your childhood. Maybe you could tell me a little bit about what your parents were like.

My father was a scientist. Both parents were. They both were medically trained. My father became a professor of human genetics, but with a considerable interest in mathematical subjects. He was sort of an amateur mathematician, but with an interest in science generally. My mother also had a definite interest in science. My father really didn't let her practice medicine, though. He found it too threatening, I think. Neither of them was terribly personal, but my mother was much more so. It was very hard to talk to my father about anything personal. We talked about science. We used to go for long walks. He would describe how things grow. It was fascinating. He was certainly a great inspiration to me on the scientific side.

Did you get any inspiration from your mother on the scientific side?

Not so much, but to some degree. However, I think I got a lot of understanding from her in other ways.

Was she supportive of your interest in mathematics and science?

Yes. But he was much more dominant. He worried when I elected to do mathematics that this was not an appropriate subject. Science was all right, but mathematics was something you did only if you were fanatical and weren't interested in other things. So when I wanted to go to the university and study mathematics, just mathematics, he got one of the lecturers there to give me a special test to see whether I could do it properly.

Hoping that you might not…

I think perhaps he was hoping that I might fall down on this test. Unfortunately I did do quite well!

Besides discussions with your father in science and mathematics, do you remember any influential books that you read?

I never read very much. Let me mention something else. I think I got a lot from my older brother too. We were three brothers, and I was the middle. I also had a sister who came along much later. She was born when I was 13. My older brother was very precocious and he was way ahead at school. I think I learned quite a lot from him. He would sometimes explain mathematical things to me. My younger brother was more a threat in a different way. He was terribly good at games. Later on he became British chess champion 10 times. When we were young, it wasn't just chess; it was all games. We developed a sort of competitive attitude. But I didn't play chess much. My younger brother was too good already and so was my older brother, so I opted out.

Did you compete in mathematics with them?

I don't think compete is really the right word. I learned. My younger brother was never mathematical. He was good at chess but not particularly good at mathematics.

How much older is your older brother?

Two years and two months.

Did you learn calculus before high school?

Let me explain. I was first of all in the UK until 1939. I was born in 1931. In 1939, just before the war, my family moved over to the United States. We were there for a few months in Philadelphia. Then we moved to Canada, where my father had a job as director of psychiatric research at the Ontario Hospital. It was in London, Ontario. And so we lived the war years in Ontario. I had just moved over to high school for the last year I was in Canada. Then we came back to England, just after the war. My father had obtained a position as a professor of human genetics, or eugenics, as it was called until my father had the name changed.

Where was that?

This was University College, London. In England one would learn calculus at what I suppose you call high school. My father was so keen to get the thrill of telling me about calculus that he taught me just before I learned it at school.

So it might have been at age 15, or something?

It would be something like that. I'm sure my older brother Oliver knew about calculus a lot earlier, because he was racing ahead at great speed. I just learned it more or less at the time everybody else did.

Did you ever read anything about cosmology at this age?

Reading wasn't quite the thing I did. I never read much. Oliver read a lot. I didn't. I did do things like make models — polyhedral, and so on. I was very interested in doing things on my own.

Geometrical models?

They were usually geometrical models.

Were they three dimensional?

Yes. I made them out of cardboard. That was just one of the things I did. There were lots of things I made.

Did you talk about cosmology with your brother?

My father would have known more, but I'm not sure how much they knew on the subject, although I think they knew the prevailing views. I'll tell you what did influence me though. That was Fred Hoyle's radio talks. It was probably around 1950. Those were absolutely fascinating.

So you were 18 or 19 then?

I think it was when I was at university. It was when I was doing a math degree at university.

Those radio talks made an impression on you?

Yes, very much so. I was interested in astronomy. My father was interested in astronomy. You mentioned cosmology before, but my father was certainly interested in astronomy. He knew quite a lot about the stars and constellations, and the nature of some stars.

And he talked to you about that a little bit?

Yes, and also he had a telescope. He had a nice brass telescope, a refractor. In fact, he and my mother went to see an eclipse once. They climbed up a mountain and saw the eclipse through this telescope, which had a special attachment for viewing the sun.

How old were you?

That was before I was born. But the fact they had this interest in astronomy is certainly true — my father especially.

When you heard Fred Hoyle's talks, did you have a particular preference for any type of cosmological model? I imagine that he was emphasizing the steady state model.^{[1]}

Yes. I didn't know anything about that at the time. It was a series of radio talks. I can't remember how many exactly.

Some of them were just on astronomy.

They started off, I think, at the local level. First planets and then stars. Then he talked about cosmology and the steady state. And he definitely talked about this, because I remember his saying that galaxies, when they reach the speed of light, disappear off the edge of the visible universe. In the steady state model, as he was describing it, the galaxies would disappear from view. The idea was that when they exceeded the speed of light, the light wouldn't get here. I remember being very puzzled by this and drawing various pictures and so on. I went up to Cambridge because my father had this idea that somehow if you wanted to be a mathematician you had to go to Cambridge. I was at University College because that was sort of easy, that was where he was a professor and so on, and we didn't have to pay fees. I'm not sure if this was the occasion, but I actually tried to take a Cambridge scholarship exam, which I didn't get. I thought the whole thing was pretty ridiculous because I really wanted to stay in London. But it was during one of those visits that I first met Dennis Sciama. My brother Oliver was a good friend of Dennis's, and both were research students at Cambridge at the time. They used to eat lunch at a restaurant called the Kingswood, which is where I met Dennis.

This was before you became a graduate student at Cambridge?

Yes. I was an undergraduate. Oliver told me that Dennis was a cosmologist. So I said to Dennis, "Well, you're a cosmologist aren't you. Can you explain this to me about galaxies disappearing off the edge? It doesn't make any sense." I drew a picture and said, "Surely, you can always see the galaxy. You draw back your light cones, and any galaxy crosses the cone." Apparently he hadn't thought about this before and was very impressed.

This was in the context of steady state.

Yes. Fred Hoyle had given the impression that when the galaxy reaches the speed of light, then it would suddenly disappear from view in the steady state model. This doesn't work. So I complained about this to Dennis, and I think he asked Fred. Fred thought about it, and it turned out that what I was saying was right.

You thought about it in terms of light cones?

Yes. I don't think they (Hoyle and Sciama) were used to drawing pictures; whereas, that was the sort of thing I did naturally, being geometrically minded. That experience was quite important to me, especially because I made friends with Dennis. When I did go to Cambridge as a research student, in quite a different area (in pure mathematics), I think Dennis felt it was his duty to look after me. I don't know. You can ask him. But he somehow took me under his wing and had lots of discussions about physics, cosmology, and all sorts of things. Although I was officially a pure mathematics research student at Cambridge, I learned an awful lot from Dennis. Certainly not just cosmology, but physics generally, and a kind of excitement and enthusiasm for the subject that was very important to me. I think he was a tremendous influence, and I suspect this meeting in the Kingswood restaurant was important because it meant he paid attention to me. We just had lunch there or perhaps it was dinner. I can't remember now. A number of Oliver's friends were there. I think Felix Pirani was there.

I read in an interview^{[2]} that Dennis Sciama did with Spencer Weart in the late 1970s that you were sympathetic to the steady state model.

Oh yes.

Can you tell me why you felt sympathy to the steady state model, if you can remember back?

It's very hard to disentangle these things, but it may partly have been Dennis himself. He was very enthusiastic about the merits of steady state, and that is probably what got me involved. But I think I also liked the idea of steady state for itself: a universe that always was there. I think this has an aesthetic appeal of its own. There was always a conflict in me because I also felt very strongly that general relativity was right, obviously a beautiful theory. Again, it was aesthetics to some extent. I had strong feelings for general relativity, and I also recognized the conflict between these two sets of ideas, (the steady state model versus general relativity). This conflict was influential in the way that I thought about cosmology. In trying to resolve the conflict, I was finally able to see that it was irresolvable, in a geometrical way. This ultimately led to singularity theorems — by thinking about light cones and how they focused. At an earlier stage, I had had a rather wild idea about how you might be able to make steady state and general relativity consistent with each other.

So you were aesthetically attracted to steady state. I'm just re-phrasing.

Yes.

But you appreciated the rigor and mathematical foundation of general relativity.

Yes, but again its aesthetics also. But general relativity has much more of a detailed mathematical structure and is, as you say, more rigorous.

And you can't derive steady state from first principles.

It was more that the two were just incompatible. I was using the idea that local energy must be positive. (I would need to reconstruct my line of thought.) If you assume a universe that is exactly uniform, it is hard to see how you could have steady state without negative energies. I was trying to see how you might have irregularities and how the presence of irregularities might allow for a steady state within general relativity with non-negative local energy. But then I started thinking about the focusing of light cones and so on, and I could see that the focusing effect was relentless and would forbid a steady state if local energy was positive.

In a few minutes I want to ask you about the singularity theorems. Let me just go back for a moment. This is a difficult question. When you say that steady state had an aesthetic appeal, could you elaborate slightly on that? I know the word aesthetic almost repels elaboration.

I suppose it's partly the big bang, which was unpleasant in some ways. The idea that you have this singular origin, which seems to go against physics. It's where your view of physics goes wrong. If you have to have this singular state in the beginning, that's ugly. It's ugly because you don't understand it. Aesthetics has a lot to do with understanding. It's a difficult issue.

And the problem with a singularity is that you don't have any physics before then? So you have incomplete understanding?

Yes. It's somehow just magic. That magic might be beautiful, if you like, but if one is trying to be scientific, it is understanding that appeals. And here, at the singularity, you just have to give up, whereas steady state gets around that problem and gives you a kind of picture in which you believe that you can comprehend things.

In the early or middle 1950s, did you have a preference for a particular big bang model? Let's say open vs. closed or flat.

Not at that time. I don't think. Although I suspect I rather like the closed models. I'm not quite sure what I really thought. But I suspect that I rather liked them closed just for the similar reason that you feel you can comprehend the universe in its totality. It seems more pleasurable. If the thing is finite, then somehow it's more comprehensible.

You were about to say that later on you had another preference?

Yes, I think if you are asking my preferences now, I would say — for reasons which are indirect and not really scientific perhaps — I think I'd prefer the open models. So I think I would put my money on the k equals minus 1 (open) model. Not much money. The odds aren't very different from 50/50, but my reason is one that has to do with my feelings about physics.

Could you tell me briefly about that?

Yes. It starts with a long story, and I've written^{[3]} about this so there's no point in my talking too much about it here. The story is about the second law of thermodynamics. The second law implies that the big bang has to have a very special structure. It can't be just generic. This structure presumably has to come out of quantum gravity. Quantum gravity is the thing that decrees what this special initial state shall be. But the special structure doesn't apply to the final state. The final state has to have high entropy and be very complicated. So the question is what kind of simple initial state does one expect? Well, I have put forward this idea^{[4]} that, as you work your way back in time, the Weyl curvature should go to zero in the initial state, which means that the initial state is conformably flat. Now, if you assume that it is exactly conformably flat as you go right to the beginning, then you have a symmetry group which applies to the initial state. This group, depending upon which model you take, is either the orthogonal group of the rotations of the 3-sphere, or the group of Euclidean motions of 3-space, or it's the group of motions of three-dimensional hyperbolic space. Now the group of hyperbolic motions is the Lorentz group. The Lorentz group is a complex group, whereas the other two groups are not complex groups. It's spinors and so on, which have a natural association with the Lorentz group. I have this belief that complex analytic structure (holomorphic structure) is a very basic part of what's going on in physics. So if you want to have something which has a simple description in complex terms — I don't mean just simple, I mean a description which is holomorphic or complex analytic — you are led to expect this kind of holomorphic group as the group of the initial state. This requires that k = -1. In fact, when I've described this thermodynamic argument (that the universe had to begin in a special state of unusually low entropy), I've often used the closed universe, just for illustration. But now I am bringing in something else, which refers to the depth and fundamental nature of holomorphic (complex analytic) structure. The only one of the three groups that has this holomorphic structure is that for k = -1. So that would be my reason for taking an open universe — in the absence of some other reason for believing in something else.

So that theoretical reason compels you more than the observations in your belief in an open universe?

Well the observations would have done at one stage. When observers were saying that k = -1, I would sit back and say, "That's nice." But now, since we know there is dark matter and we don't really know how much there is, it is quite possible that there's enough to close the universe. In fact, I imagine that, for various reasons, the observations may well not be able to distinguish k = 0 from k = -1 or k = +1 for a long while to come. If you think of a closed universe, there is a time scale. It's the time from big bang to collapse. In an open universe, there is, in effect, an imaginary time, which is i times the time scale, which is also a fundamental time. So the question is what one believes from playing around with numerical constants. If the thing that closes the universe is neutrinos, massive neutrinos, then their mass is presumably less than the electron mass. It is this characteristic smallest mass, which would give you a pure number in terms of Planck masses.

Of course, you would have to multiply the Planck mass by a very small number still, e to some negative power or something.

Yes, that's the sort of thing. This is all high speculation. But it could be that one's really wanting to look at a universe in which the time scale is, say, the cube of the ratio of the Planck mass to the neutrino mass, as measured in Planck times. I think I rather hope it's not like that!

You could apply that argument either to an open or closed model. You also could get the time scale for the closed universe out of the Plank time, with the appropriate multiple.

Yes, absolutely. So that doesn't distinguish between the two models. One could attempt to distinguish by using the ideological argument about the holomorphic structure of the group. But these are not things I would attach a great deal of weight to. We must look to see what observations have to tell us. Except that the observations may not be able to distinguish the models either. I suppose that what these arguments imply is that if people find that omega is very close to one, we can't go and say that shows that k is zero, i.e. the universe is fiat. It could well be that we're way off in finding a distinction between the different models. On the other hand, we might turn out to be just incredibly lucky that our observational techniques are good enough, if not right now, then perhaps in a few years’ time, to be able to make that distinction.

I have a couple of questions I want to ask you about that later. Let me just go back briefly to your period of graduate education at Cambridge. You said that you were working at pure mathematics. Was it with Hodge?

I was a student of Hodge's, yes. For one year, because he threw me out.

I didn't know about that. Why did he throw you out?

Perhaps that gives slightly the wrong impression. He did actually have a tendency not to want to hold on to his research students if he didn't think they were working in the sort of line that was interesting to him. Although I was working on a problem that he originally suggested, it took me off in directions that were not quite his interests.

But it was still pure mathematics?

It was pure mathematics. It was playing around with tensors, rather algebraic. The second two years I worked under John Todd. I think part of the reason Hodge threw me out, if I can use that term, was that I decided that the problem he suggested had no solution. He was very polite all the time, and I tried to explain to him that the problem he suggested had no solution. He was always terribly polite, although he didn't believe me. He never quite said so in so many words, you see. He just said "Oh that sounds interesting. Would you like to explain it to me?" And I tried to show him this funny notation I had developed. I was writing tensors in terms of blobs, with arms and legs. I developed this notation partly because Hodge's own lectures were so chaotic. I just tried to make sense of all the indices he wrote on the board. And then I developed this diagrammatic quotation. I said "Well I could use this to develop a tensor notation."

Drawing pictures?

Drawing pictures, yes.

Was that related at all to your space-time diagrams?

No. If you want to see it, you can look in the back of Spinors and Space Time, Volume I.^{[5]} It was just a way I had of writing complicated algebraic expressions, tensorial expressions, in a way which I could visualize. Actually it is a very handy way of writing tensors. I developed that in my first year of research. So I wrote a great screed, a fat manuscript on all of this. As a young and green research student, one always had the feeling that the great professor would understand anything one could say. I wrote this all out, and it was totally incomprehensible to him. I think he thought I was a bit mad or something. I'm not quite sure what, but he thought I wasn't doing the sort of thing that he could understand.

How did he feel about your talking to the astronomers and the cosmologists?

I don't think he knew actually. That was just a side activity I developed. I don't know if he knew, but he might have known. I don't think he minded. He was, after all, professor of astronomy and geometry, although he didn't actually do astronomy.

At what point did you decide that you might want to go into physics as opposed to pure mathematics? Did you ever make that choice?

It wasn't entirely a choice I'd made even then. While I was an undergraduate, when I was specializing in geometry in my final year, I did go to lectures on quantum theory. This was just for fun. So I certainly kept that up to some extent. When I was at Cambridge as a research student, I think in the back of my mind I was still interested in physics. And Dennis Sciama very much stimulated my interests. He kept trying to persuade me, like saying, "What are you doing with this pure mathematics nonsense. Come and work on physics and cosmology." He was always going on like that. And I think to some extent I was persuaded, even though I wanted to finish what I was doing. In addition to all this, I should mention that I went to lectures in Cambridge in physics, not just pure mathematics. The two lecture courses I found most inspiring were not on pure mathematics. One was by Hermann Bondi on general relativity. And the other was by Paul Dirac on the quantum theory. In different ways, they were absolutely beautiful. They influenced me more than any pure course.

Let me move now to your work in the mid and late 1960s on singularity theorems,^{[6]} which we mentioned earlier. Do you remember what motivated you? Actually, you were beginning to describe your motivation from an early time, when you first drew the space-time diagram showing that Hoyle's statement might be wrong. Can you try to reconstruct the motivation for working on the singularity theorems?

One of the things that Dennis Sciama did when I was a research student at Cambridge was to persuade me to come down to London and hear a lecture being given by David Finkelstein. I think it was at King's College.

On the Finkelstein coordinate system?

Yes. It was, in fact, on the Schwarzschild solution, extending through the horizon. Not only that, but he mentioned the piecing together to form the whole Kruskal picture, which I think he had just recently learned.

Were the Kruskal coordinates known?

I can't remember whether he gave the Kruskal coordinates, but he had his Finkelstein coordinates. He had two different coordinate systems, and he stuck together the exterior regions, and then pieced the two interiors to the interiors of two other coordinate systems. There are four regions altogether. This gives you the whole thing except the central, point. You do need another coordinate system to get that. He had all that piecing together. I can't remember whether he gave the Kruskal coordinates for the whole thing or not. He certainly produced the picture. And that I found fascinating. I remember being struck by the fact that although you remove the apparent singularity at r = 2m (the horizon of the black hole), you still had another singularity at r = 0. It seemed that you just pushed the problem somewhere else. When I got back to Cambridge, knowing very little about general relativity, I started to try and prove that singularities were inevitable. It seemed to me that maybe this was a general feature — that you couldn't get rid of the singularity. I really had no means of proving this at the time. But the only thing I did have was [the mathematics] I'd been playing around with before. I should say this was not while I was a research student. This was while I was a research fellow. I'm not sure I've got my years straight. After being a research student at Cambridge, I went back to London. I had a job there. To finish my thesis, I actually took a fourth year, half of which was in Cambridge and half of which I spent with a job at the NRDC (the National Research Development Corporation) in London. This was working with Christopher Strachey on mathematical problems connected with computer systems. He became a professor here in Oxford, somewhat later. He died some years ago. He was very influential in developing computers in this country at the time. So I had a job with him doing mathematical work on computers, for about six months. I thought I was going to write up my thesis part time. I found that wasn't possible. I could finish it only when I gave up the job in London. After that, I had my first academic job, which was lecturing at Bedford College in London. Then I went back to Cambridge again on a research fellowship. So the Finkelstein lecture was after I'd gone back to Cambridge. At Cambridge, I had decided I was going to start learning some physics, in a serious way. I had had to learn it myself because I'd never taken a proper physics course (as opposed to just applied mathematics). I'd learned a lot from Dennis.

So you gave yourself this problem to learn during that time?

I got myself started out of historic order. When I was a research student at Cambridge, I had started thinking about spinors. I'd been mystified by spinors.

Did you learn about spinors from Pirani?

No. I finally learned them properly from Dirac. But I must have already been interested in spinors when I talked earlier with Dennis Sciama. While I was a research student, I'd been intrigued by spinors, because my thesis work had been on tensor systems. Then I heard about these funny things, which were square roots of vectors. How do you take the square root of a vector? That was very intriguing to me. The fact that spinors were important to physics was an additional intriguing aspect. But I had never been able to think about them seriously, although I know Dennis Sciama had tried to make me read Corson's book.^{[7]} I think I had asked him how I learn about spinors. He said to look at Corson's book, which was totally unreadable. I remember taking it to Switzerland on holiday once and reading about two pages. And then Dirac, in his second advanced quantum theory course, gave a little series of a few lectures on two component spinors, which were beautiful. You think of Dirac's spinors as having four components, "the Dirac spinors," and so on. But he had quite an appreciation of the importance of two-component spinors. In fact he wrote an important paper^{[8]} on the higher-spin wave equations, which most people didn't look at because it was written in only two component spinors. People later did it all over again with four component spinors. Dirac's lectures on spinors made the subject come clear and alive to me. And that was really the thing that made me feel I understood two component spinors. But I didn't work with them until afterwards, when I came back to Cambridge as a research fellow.

This was before your work on the singularity theorems?

Yes, this was still before the David Finkelstein lecture. When I went back to Cambridge as a research fellow, after my job at Bedford College, I started thinking seriously about physics, and I thought I didn't understand electrons. I wanted to understand quantum electrodynamics. Actually, I failed to understand quantum electrodynamics. I thought in order to understand that, I really had to understand the Dirac equation, and I wanted to understand spinors properly. I think one thing concerning spinors that I did get from Pirani is that he put me onto this paper^{[9]} by Bade and Jehle, which was a review paper on two-component spinors and very helpful to me. I worked out the geometric description of spinors, and the celestial sphere, and the fact that the Lorentz transformation of the sky is a circle-preserving transformation. And I began to appreciate the complex analytic character of spinors. So I had started to appreciate the importance of complex analytic structure. (And this again gets back to what I was saying about k = -1. It's the same group.) Pirani was interested in the circle-preserving property of Lorentz transformations. He wrote to J .L. Synge mentioning this funny idea about spheres always looking spherical. Synge then wrote back to Pirani a letter, which Pirani sent me a copy of. Synge's letter started by saying "This idea of Penrose's that you mentioned, surely that can't be right." But then he wrote, "But wait a minute ...” and produced a proof right there on the spot. A beautiful geometrical argument. It was really nice.

That's wonderful. If I understand you, you started working on the singularity theorems because you were stimulated originally from this lecture by Finkelstein.

That's right.

Did you think originally that there would be any cosmological application, or were you just thinking of black holes at first?

I think both. I'm sure I had the cosmological singularity in my mind at the same time. I was aiming more at the time at the black hole singularity, the Schwarzschild singularity. When I got back, this digression about spinors was relevant because it was the only thing I knew, or thought that I knew, that perhaps would provide a different angle. You talk about psychological aspects. It was important for me always, if I wanted to work on a problem, to think I had a different angle on it from other people. Because I wasn't good at following where everybody else went. I wasn't the kind of person who could pick up the prevalent arguments and knowledge of the time. Other people were good at that. They could suck it all out and put it together and make advances. I was the kind of person who'd have some kind of quirky way of looking at something on my own, which I would hide away and work at. So it meant that I had to have some way of looking at a problem that was my own. And since I was thinking about spinors very much, and I felt I got hold of the geometry of spinors, I thought this has got to be my way of looking at general relativity. Perhaps I could get some insights into this singularity problem by looking at it in a spinor way. That was the motivation for me to think about the spinor way of looking at general relativity seriously. It was the singularity problem. In fact, I never got anywhere with the singularity problem at the time. I just started developing the spinor method and writing everything I could think of in terms of spinors. I was rather startled by how magnificently it all worked. The Riemann curvature suddenly made so much more sense when you looked at it this way. That really is what started me thinking about general relativity as a serious activity and something I could really work on, on my own.

Let me ask you a question about the singularity theorems. After you and Hawking and Ellis applied them to cosmology,^{[10]} did you feel then, and do you feel now, that the observations of our universe sufficiently satisfy the conditions of your theorems that you believe there had to have been a singularity? At least in the classical domain.

I always felt that the required conditions were probably satisfied, without knowing much about the observations. The words one uses here sometimes shift a little. When I first looked at the problem, I didn't like to use the word singularity or think of the universe as singular, because I always believed it just meant that classical general relativity has to be replaced by some other theory. We shouldn't be throwing up our hands and saying physics gives up. We should be understanding what physics is. And so I would tend to say that you don't really have a singularity; what you have is some change in our physics.

Well, let's use that definition of the word singularity. If we can go back to the Planck time, we will call that a singularity.

Yes, okay. But then I think I did use the word singularity, because Stephen Hawking tended to use it that way to say that the universe had a singularity. I'm not sure that his viewpoint was the same as mine, but I got round to thinking that way. I tended to say it's some new physics, maybe its quantum theory. Quantum gravity is what I think I really said. We tended to talk more about singular states, just as a way of talking. But I did think that we knew enough to believe that is what the universe did.

Do you believe that now?

Yes.

You really think that there had to have been this quantum era, based on your theorems and the observations?

Yes. As you mentioned before, I was sympathetic to steady state for a long while. Then I began thinking about how it was really incompatible with general relativity in terms of focusing arguments, and so on. And then Dennis Sciama, who had been a strong supporter of the steady state in the early stages, swung around. He had been swung around by observations. I was impressed by that fact that he was swung around, having been such a strong supporter of the steady state before. The observations must have produced a strong counterargument. So I think it was indirect. My belief that the observations were powerful in support of the singular initial state (thus favoring the big bang model over the steady state model) was rather vicarious, through Dennis, and the fact that he had swung around himself.

I asked you this question because the vast majority of people who work in cosmology assume a singularity, but may not really question how much that depends upon certain idealized assumptions. You are one of the few people who could think about it in more general terms.

It was a fair question. Stephen Hawking was more cognizant of the direct observations and what the presence of the microwave background meant. We wrote this paper^{[11]} together, with an appendix that addressed this very issue. Stephen had already addressed this issue in an earlier paper.^{[12]} It's just that since we had a more powerful theorem, we could weaken the physical assumptions. But I think that exactly to what extent the presence of the microwave background could satisfy the required conditions was something more Hawking's than mine. I was prepared to believe that the universe was very close to a singular state at one time, from observations. One might then have believed in a previously collapsing phase, such as in an oscillating universe model. But the singularity theorems really showed that that couldn't happen unless one had a gross violation of energy conditions.

Let me ask you a question about the twistor theory,^{[13]} which you touched on briefly. How have you felt about being the leader of such a nonconventional approach to space time-physics for over twenty years now? How have you felt about the unconventionality of it?

I suppose I don't care about that. It's a complicated question you're asking. It's unconventional and it's not unconventional. It's unconventional because people don't know about it. A lot of it is mathematical techniques. I like somehow to make the analogy more with, say, Langrangian theory or Hamiltonian theory. See, there was Newtonian physics, and that had become conventional physics. Then a body of mathematics developed for how to treat Newtonian physics. It may be that Hamiltonian theory was not the best way to do Newtonian mechanics. But it was just nice, because it related to mathematical ideas that were interesting for their own sake. But it didn't actually change the physics. Twistor theory is to a large extent like that. It's a way of formulating physics which is unconventional — people don't know about it much — but it doesn't change the physics. One can use twistors in standard physics, like Maxwell theory or Yang-Mills theory. On the other hand, a lot of the motivation behind twistor theory is unconventional.

Could you describe that?

Somehow it's the idea that space-time isn't the primary concept, that it is a secondary concept, arising from some other, more primitive ideas (twistors and complex analyticity). So it leads one in unconventional directions.

I've read in one of your papers^{[14]} that you thought the primary ideas were just numbers, finite quantities.

Yes. There are different motivations, which to some extent are at odds with each other. One of them is the finite discrete, which I was certainly very strong on in my younger years. And the other is the complex analytic. So I had these two things that have appealed to me from very early on. I think even when I was an undergraduate; the two mathematical fields which I liked the best were, first, those dealing with integers, mod two, just "on" or "off” states. And the second is the complex analytic, complex numbers, which have such a tremendous power and beauty on their own. At first sight, that second set of ideas was quite different from the first. I've either been drawn by one or the other. Early on, I was much more drawn by the combinatory. Later on, the power and beauty of complex analysis has been much more influential. What I really like to think is that these two things come together and that they are kind of dual to each other, in a way which I can't explain. But some things that impressed me were how complex analysis could be used in number theory, in tremendously powerful arguments.

Would you feel that's primary to understanding physics?

Yes. It's mathematics first of all. But I think that you cannot separate physics and mathematics. One of the things that have always impressed me tremendously is how complex numbers are so fundamental to quantum theory. Complex numbers forced themselves into mathematics, even against the mathematicians' wills. Numerous mathematicians kept imagining that they didn't exist, but complex numbers kept coming back, and they became a powerful way of looking at mathematics. Obviously, one now accepts complex numbers as a very fundamental ingredient in mathematics, but they forced their way into mathematics for purely mathematical reasons, and then they were accepted. The fundamental theorem of algebra is an example — the relation between complex exponentials and sines and cosines. But that was just mathematical trickery at the time. Complex numbers have a kind of mathematical reality, which is very powerful. Here we see they actually have a fundamental role to play in physics. Whereas, up until that point, one always thought that physics dealt with real numbers. Complex numbers were kind of funny and auxiliary. But here they were, sitting there at a fundamental level in physics. So I've always been impressed by this interrelationship between mathematics and physics, which is no accident, I'm sure. The way that physical theories are so beautifully accurate — not just that the mathematics is so accurate in physics, but it's that mathematics which works well in physics which is also very fruitful within mathematics. Calculus, for example, is a tremendously powerful idea.

So you are saying that ideas that are needed for physics also are particularly powerful in mathematics.

Such ideas seem to be underlying physics. It's not simply that we have a powerful body of mathematics that's used in physics, but somehow it's already sitting there. One sees this in Newtonian mechanics, in Maxwell theory, differential forms, fields, and quantum theory, complex numbers, quantum field theory — where one has these infinite dimensional ideas, which are now becoming tremendously important and feeding back into mathematics and opening up whole new ideas of mathematics. This seems to me to be no accident.

Let me change the subject. One of the things that we're interested in is to what extent scientists use metaphor, visualization, and pictures in their work. This seems to have been a strong characteristic of your work, your space-time diagrams^{[15]} and your geometrical tilings,^{[16]} and other things. Could you tell me a little bit about the role that pictures play for you?

They've always been extremely important in my way of thinking. Mathematicians certainly don't think the same as each other; some are much more geometrical, some much more analytical. This impressed me very much when I first started as a mathematical undergraduate student. I thought they would be more uniform in their thinking. I found myself to be much more geometrical than my mathematical colleagues. I tended to think about problems that other people thought were not geometrical in a geometrical way. So it's always been very important in my thinking. On the whole, I think mathematicians are not always that geometrical. Sometimes they find geometrical arguments quite hard. In physics, people are a bit more geometrical. It's curious that often if one is going to write a popular work, the editors say that one must have lots of pictures, and that's to explain it to people who are not mathematicians. Whereas, mathematicians often find it difficult to think that way.

Do you usually form a picture in your head when you are thinking about a problem?

Yes, very much so.

Do you ever try to picture the big bang?

Oh, yes. [Penrose laughs.]

What picture do you form?

One uses many pictures. Certainly if I wanted to visualize something four dimensional — and I often have to try and do that — I might use lots of different descriptions, each of which would be only a partial description. So, if you asked me about the big bang, I tend to think of lots of different things. I might try to think of it rather physically. That would be imagining what it was like to have everything all scrunched up together. You asked me about reading, and one thing I did read was Mr. Tompkins.^{[17]}

There were two of them — one was on quantum theory and one was on relativity.

That's right. In one of those I think he did make some attempt to picture what the universe was like at the big crunch. So, I have tried to imagine myself in a situation in which the universe is collapsing. How helpful that is to me, I'm not sure. Or I would tend to think of space-time pictures. All these things together are helpful. It's useful to try to get a physical feel for things. Or you try to imagine yourself near a black hole and watch it evaporate.

Do you picture yourself as being there on the scene?

Oh, yes. [Penrose laughs.] But that has to be coupled with a much more rigorous mathematical way. But it's still often visual.

Would you picture your space-time diagrams in your head?

Oh yes, but one might throw a few dimensions away. And one obviously uses this conformal map to stretch things out. But I know in the early days of worrying about the collapse to a singularity in the Schwarzschild solution, and what irregularities would do, I was very much trying to visualize it. I was trying to see what kinds of instabilities there might be. That's a tough problem, because what you're trying to show is that the singularity in the Schwarzschild solution is stable — if you perturb it, it's still there. Whereas that's not quite true. It really is unstable. If you perturb the Schwarzschild singularity to get that of the Kerr solution, it's still there, but it is a long way off, so there is instability in that. One says the singularity is stable, in a sense, but that's not right. The presence of a singularity is a stable property, but the actual singularity of the solution is probably unstable, and where the perturbed singularity resides is somewhere else. But I was very much trying to picture what's going on.

I wanted to ask you about your reactions to some recent developments. Do you remember when you first heard about the horizon problem in cosmology?

I remember the first time I looked at this. Well, it depends on what you mean by the horizon problem.

The causality puzzle.

I knew about horizons from Rindler's article.^{[18]}

Yes, but I mean the problem of communication that we see in the microwave radiation for example.

I suppose I had vaguely thought about that for a long time. Whether I worried about it is another question.

Do you remember at all what you thought about it in the early days?

The first time that I began thinking about this in connection with the sort of thing that people now worry about — in terms of inflation and so on — was when George Sparling was talking to me about symmetry breaking. The question was: if one believes the standard models, then how it is that if you look at two distant quasars in opposite directions, they both seem to have broken their symmetries in the same way. My reaction, being unconventional in other ways, was completely the opposite from other people's reactions. It's a problem. If you take the standard view, you have to believe that those two quasars were causally connected in some way.

At some earlier stage.

Yes, and therefore the symmetry could be broken in the same way. Or else you have some domain wall problem.

When you say the quasars needed to have their symmetry broken in the same way, do you mean so that they would have the same properties?

I mean with the symmetry broken in the same "direction." I'm a bit hazy about which thing happened when. If you think about, say the weak interactions, and then you have to worry about whether what you are looking at is a W boson or a photon. If it starts off as a photon and it's broken the wrong way, and it's really a W boson, then how can it be that what you see are photons in all directions? The galaxies in those different places apparently had to have chosen the same direction of symmetry breaking.

I see. You are talking about now a different version of the horizon problem than I usually think of. I usually think of the horizon problem in terms of uniformity of cosmic radiation or temperature, but this is basically the same problem.

It's the same problem. There are two separate problems if you like. They are different in that one of them is the problem of the symmetry breaking mechanism, and the other is the problem that we have anyway, because we see the uniformity of the early universe.

Let me talk about the uniformity problem first. When you first knew about that, or thought about it on your own, did you regard it as a serious problem with the big bang model?

I can't quite remember. I remember other people worrying about it. My reaction was not being quite sure about what they were worrying about. I suppose it's partly that we don't understand what the big bang was anyway. I know what I think now, but you are asking me a historical question and I'm not quite sure. I think I probably wasn't worried enough about it. I remember Dennis Sciama talking to me about this.

Did you worry enough about it to try to imagine how it might be solved?

That's what I am not quite sure of, because I might be projecting my present views back to then. I know what I think now.

What do you think now?

I think it's all coupled to the second law of thermodynamics problem. The horizon problem is a minor part of that problem. It's not the big problem and, in a sense, that's what I always thought. Now I think I have a more coherent picture of why it is not the big problem. Why I didn't think it was the big problem then, I'm not sure I can answer.

And you think that when we understand why the universe began with a relatively low state of entropy, we will understand the horizon problem?

That's right. It's one part of the initial singularity and why the initial singularity is so special. People nowadays, over the last several years, have tried to make inflation solve the problem, which I just don't think works.

Let me ask you about inflation. Before you tell me why you don't think it works, let me ask a sociological question. Why do you think that the inflation model^{[19]} has caught on so widely?

I think it's like so many things that are fashionable. I suspect there are many factors, and I wouldn't know if I could get the balance right.

Don't worry about the balance. I don't think anyone can get that right.

I first heard about it when I was at Princeton. I heard that Alan Guth was giving a lecture, and I didn't go to it for some reason. I can't remember why I didn't go, but people described the idea and I thought "gosh, that is a horrible idea." My initial reaction was I didn't like it at all, whereas many people thought it was a wonderful, beautiful idea.

Why didn't you like it?

It was partly because I suspect one has got one's idea of what the universe is like, and this was disruptive to my ideas. I don't know to what extent my reaction was due to that. I suppose it's partly because I didn't think it was really solving the problems of physics. It's partly another thing which I didn't like. I had already heard about false vacuums and things like this. I remember going to a lecture by Sidney Coleman, and he was talking about false vacuums. This was somewhat earlier. I just thought the whole thing was so ugly, I felt ill. This is a very personal reaction. Why did I feel that way? I don't know. I suppose it's partly the feeling that one's got a picture of what the vacuum is like, and this was a very different view. I just thought this was way out. Partly, it was because it was very much dependent on this current idea of symmetry breaking. So much of modern physics is dependent upon symmetry breaking. I never liked the idea right from the start. We know that symmetry breaking takes place in ferromagnetism and so on, and that's part of physics. But whether it's part of physics at this fundamental level… I'd always felt that this is an ugly idea because one is looking at a big symmetry group. This is really the whole idea. You look at a big symmetry group, which is supposed to be beautiful because symmetry is beautiful. This big symmetry group is supposedly more beautiful because it has more symmetry. Then you reduce it to a small symmetry group, and you do this through some symmetry-breaking mechanism. But to me, a big symmetry group isn't a beautiful thing because you have a big group, and a big group is a complicated thing, like a big number. Why is that beautiful? I always thought that symmetry comes in physics often because you ignore certain features. It's useful. Symmetry itself is a very beautiful idea, I admit, but to say that physics has that symmetry on a fundamental level, and that the symmetry is some big group — why should it be that big group and not another one?

You would rather it is a smaller one?

I would rather the symmetry group is smaller. The Lorentz group is so beautiful because it's a very simple one, and it even comes from lack of symmetry. You take an arbitrary complex manifold with the topology of a sphere — that's the simplest possible complex manifold — and it automatically has symmetry. It's something that is produced out of something without symmetry. Yet these other ideas are imposing symmetry from the start. I don't see the point of that. I've held my own type of view from way back, much before people were worrying about symmetry breaking. So when the symmetry breaking idea came along, it didn't strike me as a beautiful idea. It struck me as an ugly idea. This has colored my thinking so much right through. When I heard about false vacuum, the main reason I didn't like it was because of symmetry breaking. Then Guth's idea for inflation was just using symmetry breaking again. So again, I didn't think it was beautiful right from the start.

Let me go back to the sociological question. Why do you think that so many other people like the inflationary model?

I think other people haven't felt this way about symmetry. They think symmetry is a beautiful thing.

But you said that you yourself like symmetry.

I like symmetry as a mathematical thing that you use. The question is whether symmetry should be there as a fundamental postulate of physics. Symmetry is certainly a beautiful mathematical idea; there is no doubt about that at all.

But it bothers you that you have to postulate it?

Yes. And also that in GUT models you postulate a big symmetry, not the simplest. This goes back to when I said I like the field of two elements, zero and one. But a group of 792 elements, why is that beautiful?

I can see that other people's liking symmetry would have prevented them from rejecting the inflationary universe, but why do you think they were so attracted to it?

I think they thought it was solving these problems. They also saw it solving what I call the self-made problems. In a certain sense, it does solve those problems — the monopole problem, and so on — which are produced by some of the grand unified theories. That is legitimate. I'm not saying it's not legitimate; it's just to what extent one is attracted by it. Then there is also a sociological factor that I believe is important, which I find disturbing. It's a question of which kinds of people are working on what subjects.

You don't have to mention any names.

No, I'm not going to mention names, just generalities. See, I've spent my life working with general relativity, but a lot of other people have spent their lives doing other things, like particle physics. In a sense, I did the same sort of switch in the beginning. It was only later that I came to feel that general relativity was so fundamental. Then people come in from outside, not being experts on general relativity or cosmology particularly, knowing about particle physics, symmetry breaking ideas and so on, and bringing this expertise into the subject. I think there are very many more of those people than relativists. Locusts would perhaps be the wrong analogy, but there are huge numbers of people and they see an opening into this subject, and they come in and almost take it over. I felt this a bit with super-symmetry. In general relativity, I felt this again with a lot of the ways people tried to quantize it. This is a bit personal, and I'm not sure I'm being fair, but certainly people often use types of arguments which are very foreign to my way of thinking — very un-geometrical. Bringing ideas in from other subjects is fine. The union of particle physics and general relativity is a magnificent thing, and what one can learn about the early universe from both ways around is fascinating. That's fine as long as the particle physicists appreciate the problems of general relativity. I think often they don't. This is the thing, particularly if you are trying to quantize general relativity. People come in without being aware of the very fundamental problems we have argued over endlessly among general relativists. There are very fundamental difficulties that one has in trying to quantize, and these people just try to sweep them away. I think here in cosmology one has something similar, people coming in from another area, from particle physics, thinking they can sort of take over. There is a lot of input they can produce, but that's different.

I understand that. Let me ask you about another problem with the big bang model, and that is the flatness problem. Do you remember when you first heard about that or conceived of it yourself?

Again, it's a thing where I felt I never worried enough about it. I remember Jim Hartle trying to persuade me this was a much bigger problem than he thought I thought it was.

When was that?

This was not all that long ago, it was probably at one of our quantum gravity gatherings.

In the last ten years?

Yes, I suspect it might have been about ten years ago.

Was that the first time you began thinking about the flatness problem, or had you thought about it before then?

I think he persuaded me that the problem was a bit more serious than I thought it was.

Do you remember why you first thought it was not a serious problem?

I've always lumped all these problems into the same thing; they're all part of quantum gravity. You have to understand what it is that made the big bang singularity what it was, and that is something that we just have very little conception of. We have little conception of why the singularity was produced as extremely uniform. There are two problems here: one is the uniformity, and the other is the flatness. Jim Hartle correctly persuaded me; he certainly worried me a bit more than I had been. I thought you can get the uniformity out of some quantum gravity — the state must be simple, maybe it comes out of twistor space, maybe it is conformably flat and can be described very nicely. So I felt that was something one could come to terms with. But the flatness problem is why a certain parameter is very close to zero. That is something one has to face up to. But, again, it's the large number problem. Why is the time that the universe will take to collapse, whether it is real or imaginary, so many Planck times? It's a large number problem. I don't worry about it quite in the way that other people do. I think of the problem as the uniformity problem together with the large number problem. But the large number problem is a problem. We see it all over the place. We see it in the gravitational constant. Why is the Planck mass so much larger than the electron mass?

And you imagine that might be resolved by a fundamental theory of the singularity?

The flatness problem, yes. What underlies that is a problem in physics. The way I look at it, even in physics today — don't worry about the big bang — we have the large numbers. Dirac, as you know, pointed this out.^{[20]} We’ve got these big pure numbers. Are they something that one should expect to get out of a fundamental theory? I believe that. I think the numbers should come out of a fundamental theory, rather than Dirac's view, which is that they are evolving numbers.

There is also Dicke's view,^{[21]} the anthropic principle. What do you think about that argument?

That's a tricky one.

That is another way to explain things.

Yes, it's a possibility.

Do you think there is some legitimacy to that argument?

Yes I do, but I prefer it wasn't right.

Why do you prefer it wasn't right?

One thing is the numbers have to be constant. Somehow they have to be fixed numbers. The anthropic principle doesn't do as much for you as you'd like. It doesn't explain the second law of thermodynamics — why the universe was created in a state of such low entropy. On the whole, one finds that the anthropic principle is something you bring in when you haven't got a good theory. People say, "We've got to fix these constants, and the anthropic principle does it for us." It's a way of stopping and not worrying any further.

Of course, Dicke's use of the anthropic principle was rather weak — explaining certain numbers in terms of the lifetime of a main sequence star, and so forth.

I think that's all right. It's a partial argument, because why do we have to live near such a star? But it's all right as far as it goes. The weak anthropic principle I can accept, but the strong one is different.

Let me ask you about your reaction to an observational development, a recent one. Are you familiar with the recent results of Geller, Huchra, and de Lapparent^{[22]} on these hubbies, the large-scale structures in the Center for Astrophysics redshift survey, the finding that galaxies seem to be distributed on the surfaces of shells?

I know what you mean, yes.

Does that observation have any effect on your thinking about some of the assumptions of the big bang model, like the assumption of homogeneity?

Yes, it's intriguing. I wouldn't like to put my money one way or the other. Since it's an observational question, to some extent I don't know. I would like to wait and see how impressive those observations are. I have sympathies with both sides. I suppose that if something lies on shells, maybe some explosion or something did it, some wave that's been able to compress everything. We know that galactic centers explode. And there might be some rather intriguing general relativity going on which is responsible for this. On the other hand, if you think it's a false vacuum or some phase change, then that I find disturbing. I would be very troubled if that turned out to be the explanation.

For reasons you mentioned earlier?

Yes. I would rather that not be right. For the same reason, cosmic strings would disturb me. Although studying these things is fun. (I don't think I believe in them!)

Have any of these observations changed your opinion about the basic assumptions of the big bang model?

I don't think they have, no. Obviously one needs some kind of irregularities in the distribution of matter. We can see irregularities. You see, we're pushed back into physics we just don't know. I think the symmetry breaking and bubbles and all of that stuff is premature. We even have ideas in current particle physics which I personally wouldn't want to survive. That's just an opinion. My guess is that they (broken symmetries, cosmic strings, etc.) are going to turn out not to be the right explanations of the observed inhomogeneities. But on the other hand, there might be something quite extraordinary happening in the early stages of the universe which we've seen with quasars, explosions and so on, and which might involve physics which we simply don't understand. I don't think that our picture of particle physics is satisfactory at the moment. I believe that there will be some fundamental changes. Of course, the standard models look quite good — QCD and the electroweak theory — and I suspect that any really fundamental picture of particle physics will have things like that. These current theories are indications, but they don't look to be fundamental. If you are looking at huge energies, there may be some quite different things from what we understand at the moment. In the big bang, there may have been enormous particles. I don't know. I think we should be prepared to accept big surprises. So if one sees these spherical shells, that don’t repel me at all. I just have to wait to see what experiments and observations show in the future.

I just have a couple more questions. Could you tell me briefly what you think are the major problems in cosmology right now?

Well, clearly, the actual structure of the big bang, and the physics which must have been going on in the early stages. That perhaps isn't a cosmology problem. It's physics, but cosmology depends on it. Galaxy formation, obviously, is a big problem. I just don't know what other people say on that problem now. I haven't studied it myself. It may be that the problem of galaxy formation will be resolved in terms of this other problem. Another problem is what the missing mass is. And there is missing mass. We'd like to know whether it's enough to close the universe, which I hope it doesn't, or leave the universe substantially open, which I hope it does. Obviously, that's a big problem. What is that stuff? Again, that depends on current views in particle physics. It could be something that we just don't know — some other particles running around which we just don't know of. It could be that the missing mass is neutrinos, massive neutrinos. I am no expert in this area.

As far as I know, there are no believable measurements of a nonzero mass for the neutrino.

Yes, so that's up in the air. But it's a possibility. And the neutrinos from the sun.

That may or may not be connected to the other problem.

Yes, that's right - it could be. There could be neutrino oscillations. Maybe there is some asymmetry between neutrinos and antineutrinos. That could be related.

Let me end with a couple of philosophical questions. For the first one, I may have to ask you to put your natural, scientific caution aside. If you could design the universe any way that you wanted to, how would you do it?

Oh, I'd take the one we've got. [Penrose laughs.] I can't deny that. There are so many things that make it incredible. I couldn't design a universe that could compete with the one we see. Let's put it like this: The sort of things I hope would underlie the actual universe, as I have said before, would be the complex numbers. Complex numbers constitute just one aspect of this unity with mathematics. I believe in a deep unity between mathematics and physics. Whether they are in a sense the same thing is an intriguing question.

You are designing the universe, so you could make them the same.

My universe would have to be mathematical. Yes. When you put a question that way, I turn the question around myself. This is perhaps more the way Einstein did when he's talking about God and he says "how would God have done it?" So he tries to put himself in God's place. It certainly would be a mathematical universe. It's the sort of mathematics which is profound as mathematics, fruitful as mathematics — as opposed to certain things that are dead-ends as regards mathematics. Fruitful mathematics tends to be fruitful to physics also. You mentioned the anthropic principle, and there is also the many-worlds point of view towards quantum mechanics. I feel uncomfortable with both of those ideas, particularly the many-worlds interpretation of quantum mechanics. I mention them because they relate to the question of determinism. If we think that there is just one universe, and if it is a determined, mathematical universe, then that universe is not just deterministic in the ordinary sense, but it is completely fixed, with no freedom of initial conditions. I'm not sure what Einstein meant when he asked whether God had any choice in the creation of the world, but perhaps it was this. You think about determinism in the ordinary sense meaning that if you know the state of the universe at one time, then in principle you know it at any other time. But that isn't too much help, because you don't know what the state of the universe is at any one time. So although the idea of a deterministic universe may tell you something about the form of the equations of physics, it doesn't give you much real predictive power.

I saw an article of yours^{[23]} discussing whether a fully deterministic universe could be consistent with special relativity. You considered a universe with world-line branches going out, and you tried to do that in a self-consistent space-time.

That's in a way related to the anthropic and the many worlds view. You might have a universe that was branching all the time. The totality of all these universes is the whole universe, in some sort of an ensemble.

Is this part of your ideal universe — the one that you would design?

This is a point of view. It could be like this. Then which branch one finds one's self on depends on the strong anthropic principle. So if you have that branching picture, together with the anthropic view, that gives you a picture of the world. Then you can take the view that this whole totality, this omnium, has a nice, precise, neat mathematical description — although the universe we see around us may not have this nice, precise description. So that would be a way of making the unity of mathematics and physics complete. That is a point of view I have sometimes toyed with, flirted with. I suppose in the early time of my life I sort of believed it. When I say early time, it was actually before Everett.^{[24]} I think my present view would be a much happier view, in which the unity of physics and mathematics is still complete, but complete in the universe we actually see about us. So there is only one branch. The description of that universe forms such mathematical subtlety that we can't appreciate it yet. The best example I know of this sort of thing in mathematics is the Mandelbrot set. One has a structure that has a beautifully simple mathematical description — it's just the complex plane with a simple iterative rule — and you have this extraordinary structure with this fantastic complication. But I don't believe the universe is a Mandelbrot set!

But it's an example of getting a lot of complexity from a very simple law.

That's right. You get this extraordinary complication with a very simple rule. Seeing things like that tended to give me certain optimism that the universe as we know it might have a very simple underlying principle. So I think that is the way I would try to design the universe.

Let me ask you one final question. There is a place in Steve Weinberg's book The First Three Minutes where he says "the more the universe seems comprehensible, the more it also seems pointless."^{[25]} Have you ever thought about that issue?

I don't agree with that sentiment at all. Is that his viewpoint? I remember the statement. Of course, he may be sowing seeds. I don't know whether he believes that.

What are you’re feeling about that?

I suppose my reaction is the opposite of the sentiment which seems to be expressed there, namely, that our comprehension does give the universe a point. It's part of how I look at mathematics. The understanding of something in terms of mathematics doesn't eliminate a problem; it gives it a deeper character. Suppose you have something in nature that you are trying to understand, and finally you can understand its mathematical implications and appreciate it. Yet there is always some deeper significance there. I don't know how to explain it.

It is a very difficult question.

I don't think our understanding removes the point. You see, in a sense, understanding nature is making it more mathematical. That's what we are doing all the time. Mathematics is logical structure, a disembodied logical structure, and you might think that when you put your physical problem into that disembodied mathematical structure, you have removed its point. Maybe that is the sort of thing Weinberg was saying. And many people might think that. But my view is: once you have put more and more of your physical world into a mathematical structure, you realize how profound and mysterious this mathematical structure is. How you can get all these things out of it is very mysterious, and, in a sense, gives the universe more of a point.

For you, the question of a point is intertwined with this mystery somehow?

I think that's true. I suppose the point has to do with one's own existence. When it comes down to it, the question has to do with conscious perception of one's own existence in the world. A world which has no people in it is pointless. A universe that is just chugging away by itself with nobody in it is, in a sense, pointless. I may be diverting the question.

No, you're not diverting it. That is a very good answer. If I may paraphrase — what you said: "Having people here to appreciate the mystery of the mathematical foundation of the universe gives the universe a point."

I think that's true, but I'm saying that the whole thing is more interconnected than that. We are all part of the world, and we are conscious beings, so if the world is itself describable mathematically, then this whole idea of conscious perception must be describable mathematically. Or if it is simply a Platonic world of mathematics, how do you find yourself in it? That world seems at first dry. Most people's picture of mathematics as just adding numbers, and so on, is the last place they would want to be. I always think a good analogy is cohomology. Twistor theory is a bit like it. You have an area which is mathematically well-definable, and you try to explain it to someone. If they say "what is such and such," you say "well, it is so and so," and you give them a definition. And then they say "I think I understand that," and then you say "wait a minute, it is also the following," and you give them a completely different definition which doesn't look anything like the first one. Then, by some complicated theorem, you prove these two things are the same. And they say "I understood that. I think I understood your proof." In a way, they less understand what you said. Then you say, "But wait, I haven't finished. There's a third thing." Then you bring that along. Then you prove that that these things are all equivalent. Somehow, it's none of them. It is not the first, the second, nor the third thing. Because these things are equivalent, there is some more mysterious further thing that these three things are all manifestations of — and by learning them all, you get more of a grasp of what this more mysterious thing is. I think there is a lot in mathematics that is like that. A definition takes the soul out of something in a way. It's where you see these connections and relate all these things together. That's what gives it soul. Using a term like that is suggestive. Ultimately, we may understand these very mysterious things that give the universe its point in terms of ideas that are now very remote from us.

That's a good place to end.

^{[1]} H. Bondi and T. Gold, Monthly Notices of the Royal Astronomical Society, vol. 108, pg. 252 (1948); F. Hoyle Monthly Notices of the Royal Astronomical Society, vol. 109, pg. 372 (1948)

^{[2]} Interview of Dennis Sciama by Spencer Weart, April 14, 1978, Sources for History of Modern Astrophysics, American Institute of Physics, New York

^{[3]} R. Penrose in Confrontation of Cosmological Theories with Observational Data (IAU Symposium 63), ed. M.S. Longair (Boston: Reidel, 1974); R. Penrose in Proceedings of the First Marcel Grossman Meeting on General relativity, TCTP Trieste, ed. R. Ruffini (Amsterdam: North Holland, 1977); R,. Penrose in Physics and Contemporary Needs, ed. Riazuddin (New York: Plenum, 1977); R. Penrose in Theoretical Principles in Astrophysics and Relativity, ed. N.R. Lebovitz, W.H. Reid, and P.O. VanderVoort (Chicago: University of Chicago, 1978); R. Penrose in General Relativity: An Einstein Centenary survey, ed. S.W. Hawking and W. Israel (Cambridge University Press, 1979)

^{[4]} R. Penrose, “Conformal Treatments of Infinity,” in Relativity, Groups, and Topology, ed. C. DeWitt and B.S. DeWitt (New York: Gordon and Breach, 1964)

^{[5]} R. Penrose, Spinors and Space Time, Vol. 1 (Cambridge: Cambridge University Press, 1984)

^{[6]} R. Penrose, “Gravitational Collapse and Space-Time Singularities,” Physical Review Letters, vol. 14, pg. 57 (1965); S.W. Hawking and R. Penrose, “The Singularities of Gravitational Collapse and Cosmology,” Proceedings of the Royal Society of London A, vol. 314, pg. 529 (1969)

^{[7]} E.M. Corson, Introduction to Tensors, Spinors, and Relativistic Wave equations (New York: Hafner,1953)

^{[8]} P.A.M. Dirac, “Relativistic Wave Equations,” Proceedings of the Royal Society of London A, vol. 155, pg. 447 (1936)

^{[9]} W.L. Bade and H. Jehle, “An Introduction to Spinors,” Reviews of Modern Physics, vol. 25, pg. 714 (1953)

^{[10]} S.W. Hawking and G.F.R. Ellis, “The Cosmic Blackbody Radiation and the Existence of Singularities in Our Universe,” The Astrophysical Journal, vol. 152, pg. 25 (1968); S.W. Hawking and R. Penrose, “The Singularities of Gravitational Collapse and Cosmology,” Proceedings of the Royal Society of London A, vol. 314, pg. 529 (1969)

^{[11]} See Ref. 10, Hawking and Penrose

^{[12]} See Ref. 10, Hawking and Ellis

^{[13]} R. Penrose, “Twistor Quantization and the Curvature of Spacetime,” International Journal of Theoretical Physics, vol. 1, pg. 61 (1968); R. Penrose and M.A.H. MacCullum, “Twistor Theory: An Approach to the Quantization of Fields and Spacetime,” Physics Reports vol. 6C, no.4, pg. 2421 (1973); R. Penrose and W. Rindler, Spinors and SpaceTime, Vol. 2 (Cambridge: Cambridge University Press, 1985)

^{[14]} R. Penrose in Magic Without Magic: John Archibald Wheeler, ed. J. Klauder (San Francisco: W.H. Freeman, 1972)

^{[15]} See Ref. 4

^{[16]} R. Penrose. “Pentaplexity: A Class of Nonperiodic Tilings of the Plane,” Mathematical Intelligencer, vol. 2, pg. 32 (1979); B. Grunbaum and G.C. Shephard, Tilings and Patterns (New York: W.H. Freeman and Company a, 1986); M. Gardner, Penrose Tiles to Trapdoor Ciphers (New York: W.H. Freeman and Company, 1998)

^{[17]} George Gamow, Mr. Tompkins in Wonderland (Cambridge: Cambridge University Press, 1940); Mr. Tompkins Explores the Atom (Cambridge: Cambridge University Press, 1944)

^{[18]} W. Rindler, Monthly Notices of the Royal Astronomical Society, vol. 116, pg. 668 (1956)

^{[19]} A. Guth, “Inflationary Universe: A possible solution to the horizon and flatness problems,” Physical Review D, vol. 23, pg. 347 (1981)

^{[20]} P.A.M. Dirac, Proceedings of the Royal Society of London A, vol. 165, pg. 199 (1938)

^{[21]} R.H. Dicke, “Dirac’s Cosmology and Mach’s Principle, Nature, vol. 192, pg. 440 (1961)

^{[22]} V. de Lapparent, M.J. Geller, ad J.P. Huchra, “A Slice of the Universe,” Astrophysical Journal Letters, vol. 302, pg. L1 (1986)

^{[23]} R. Penrose, “Singularities and Time-Asymmetry,” in General Relativity: An Einstein Centenary Survey, ed. S.W. Hawking and W. Israel (Cambridge University Press, 1979)

^{[24]} H. Everett III, Reviews of Modern Physics, vol. 29, pg. 454 (1957); J.A. Wheeler, Reviews of Modern Physics, vol. 29, pg. 463 (1957)

^{[25]} S. Weinberg, The First Three Minutes (Basic Books: New York, 1977) pg. 154