# 2023-12-04 - IAI - The Edge of the Universe [YouTube](https://www.youtube.com/watch?v=uYYoZupmWXc) Duration: 00:49:07 ## Transcript ### Robert Lawrence Kuhn 00:00:00 - 00:02:59 Welcome to IAI's debate, the edge of the universe. Is infinity the greatest crisis facing physics? I'm Robert Lawrence Kuhn, host of Closest to Truth. Closest to Truth is pleased to partner with IAI. The idea of infinity makes us feel a little giddy, yet we refer to it often, as if it is understood. Mathematicians Georg Cantor, who himself went mad, discovered different degrees or levels of infinities, in fact, an infinite number of them. We have countable and uncountable infinities, potential and actual infinities. Infinities show up in physics. In quantum mechanics, clever tricks like renormalization try to make nonsense infinities go away. And the many-worlds interpretation would reach for a quasi-infinite number of splitting off worlds. In cosmology, we find claims of infinite spaces, infinite time, forward and backward, infinite dimensions, infinite density of black holes in the Big Bang, infinite size of our universe, and infinite numbers of universes, the multiverse. Would an infinite universe, a multiverse, undermine all prediction in cosmology? But are all these infinities perhaps a mistake, some say? Can infinity be applied to the natural world? Is infinity a profound problem or a key insight? We can't forget infinities in metaphysics. Some Eastern traditions have infinite consciousness, and Abrahamic traditions have an infinitely powerful, infinitely knowledgeable deity. The Argentine writer Jorge Luis Borges argued infinity is a concept that corrupts and upsets all others. We have a superb panel. David Deutsch is a trailblazing physicist who is a visiting professor at the University of Oxford. He wrote two acclaimed innovative books, The Fabric of Reality, which inspired me, and The Beginning of Infinity, which offers a new take on infinity. Sara Walker is a theoretical physicist and leading astrobiologist, associate professor at Arizona State University. Sara's insights into the origins of life, astrobiology, and the physics of life have been featured on TED and the Lex Fridman podcast. George Ellis is one of the world's leading theorists in cosmology and an old friend. Distinguished professor at the University of Cape Town in South Africa, he is co-author with Stephen Hawking of The Large Scale Structure of Spacetime. I love George's strong views on the physics of infinity and possibility spaces, which we'll discuss. Now we're going to start with the question, does infinity cause insurmountable problems in modern physics? ### David Deutsch 00:02:59 - 00:05:47 Each speaker has three minutes and will go alphabetically. David Deutsch. Well, there have been infinities in fundamental physics since antiquity, mostly harmless. I'm not actually sure which ones are most feared today, but never mind that for the moment. Suppose we were discussing zero instead, like medieval sages who didn't have Zoom, but might have gathered in their cloisters gravely debating the propriety of zero and deciding no, it's improper because zero by definition can't refer to anything in the real world. It's nothing. No one can imagine nothing. It's incomprehensible. To dare to apply it to nature is sheer hubris. Nature abhors a vacuum. We may laugh at that, but this archaic fear of zero is inseparable from the fear of infinity. Take Zeno's paradox. To walk across the room, you first have to get halfway and before that a quarter way and so on. So to move at all, you have to perform an infinite number of actions, physical actions. Zeno was wrong because a philosophy of mathematics can't determine whether walking across a room makes sense or not. If a good scientific explanation says it does, then it does. And the classical kinematics does indeed explain that a vacuum, empty space, is not the same as nothing. Nothing wouldn't have a geometry, for instance. Today, both of our two deepest theories, relativity and quantum theory, give empty space much more structure than that. And again, that very structure leads to various infinities, infinite physical quantities. But this time, neither theory says that you can walk right across the infinity. So what's lacking in these theories is not that they invoke infinity, it's that they invoke something they don't explain, what these infinite things do. And that's a problem, but not a crisis. There are always problems at the cutting edge of physics because a deeper explanation always reveals a still deeper problem. If that were ever not so, then there would be a crisis. ### Robert Lawrence Kuhn 00:05:47 - 00:05:50 George, George Ellis, please. ### George Ellis 00:05:50 - 00:08:10 Okay, well I support David Hilbert, who said many years ago, one of the greatest mathematicians of last century, and he said, infinity is needed to complete mathematics, but nowhere occurs in the physical world, and that's correct. And I agree with what David has just said, a corollary of that is zero doesn't occur anywhere either. And let's start with the second. There are all of these problems at the foundation of quantum mechanics about infinities and so on, and quantum cosmology. But from my viewpoint, I think it's very, very plausible that everything quantum theory says everything is quantized at a very broad level. And space time, I think it's very plausible that space time itself would be quantized in some sense, it would emerge out of a discrete kind of structure. If you could look at it at a high enough level of detail. And that, if it does emerge, it will get rid of quite a few of those infinities in quantum physics, because you will no longer be summing over an infinite set of points, you'll be summing over a discrete set of points. In cosmology, all of this stuff about infinite domains and all the rest of it. The fact of the matter is that in the universe, what we can see is restricted by our visual horizon; we can't see beyond the visual horizon. So you can claim there's infinity, I can claim there's not. We're talking metaphysics, we're not talking physics, because there's no way whatever, that either of us can be proved true, because I take the strong viewpoint that science is determined by what can be experimentally or observationally determined. And any claim that there's an infinite number of multiverse domains, no matter what kind of view you have on it, it cannot be verified. And so therefore it's a metaphysical statement, which does not actually relate in some serious sense to the real world before us. If you take the strong view, which I do, that science relates to what is observationally or experimentally determined. ### Sara Walker 00:08:10 - 00:10:01 Sara Walker. So, most of the ways I think about what we're doing in modern physics are from the perspective of thinking about ourselves as physical systems because I'm interested in the life problem. So a lot of the paradoxes that emerge in modern physics can be thought about from a different perspective if we think about math itself as a physical system. So what is it for physical systems to emerge that then write down laws of physics and describe those by mathematical statements. And from that perspective, mathematics itself looks like a kind of information. It can be copied between different things. So this is a very sort of concept that I think David has talked about quite a bit, that it can be copied between different physical systems. And then you can ask questions about why the concept of infinity would exist from that perspective. And I think infinity as a physical concept is not, it's a metaphor in the sense that George was talking about, that it doesn't describe something real physical out there. There is no such thing as a physical infinity, but infinity exists as a kind of information that we can use to manipulate matter and to understand properties of the universe as it exists here and now as sort of a counterfactual property or a little bit of allowing us to see where we're missing things in our measurements or where there needs to be a new creative solution. So I think it has a kind of physicality in the sense that obviously it exists in our minds, but it doesn't have the kind of physicality of what it in our minds represents in the world. It's something else. And if we could figure out why infinity is a useful concept in terms of what it's doing when we write down laws of physics and maybe how it enables more creativity for identifying where our laws of physics are breaking down or thinking about counterfactual possibilities, then we might be able to rein in infinity and understand a little bit more about why and how it behaves and what it's doing. ### Robert Lawrence Kuhn 00:10:01 - 00:12:13 Thank you, everyone. I see the tension already building up. Now we have three themes to explore each of these in some depth. Theme one is what about all those claimed infinities and cosmology? We're talking generally. What about space, time, dimensions, density of black holes in the Big Bang, the size of our universe and the number of universes? Is this reality or is it metaphor or we just can't know and therefore it's metaphysics as George says? Sara, I want to begin with you and focus on your work. You search for life in the cosmos. But as you do that, how real or how relevant are all these potential infinities such as the potential infinite size of our universe even beyond the light horizon? Yeah, so I don't believe the infinities are real. I think they're short-term problems in the way that we construct laws of physics and what we're trying to describe. So I think, as I said before, the multiverse to me is real in the sense that it's a counterfactual property that exists in human minds that we can reason about and then use it to try to inform models for our own universe. But it's not real in the sense that I think multiverses exist. And I think this is really important because when we're talking about the nature of alien life and the kind of physics that they discover, we have to ask questions about whether they'll discover the same physics, the same kinds of infinities or the same kinds of laws of nature. And I've really sort of, you know, I'm happy to be proven wrong at some point, but I've adopted a stance that pretty much everything on this planet is a product of evolution, including the way that we've architected our theories of physics. And they do correspond to a real physical world, but the mathematics and things that we describe might be a feature of where we sit locally and is not a global feature. So we can't talk about things that we don't actually have direct interaction with. It's important to have that kind of locality when you're talking about evolving systems. David, your book, The Beginning of Infinity, is about possibilities and explanations. But from that perspective, are there possibilities for actual infinities in cosmology, which your two esteemed other panelists deny either in reality or in knowability? ### David Deutsch 00:12:13 - 00:13:07 Yes, I think there is that possibility. And I think that we shouldn't a priori reject any kind of mathematical description of the physical world. We should judge these descriptions rather by whether they are good explanations or not. And that includes testing them by experiment. But the positivistic idea that philosophical issues should not be expressed in physics just falls by its own criterion. That is itself a philosophical criterion introduced into physics. I don't think there should be any a priori restrictions on what physicists can postulate about the world. They should always be a posteriori critical. ### Robert Lawrence Kuhn 00:13:07 - 00:13:21 George, I think that hits you directly. You don't like infinities. I liked your 2018 paper, The Physics of Infinity, in which you argue that the supposed existence of actual infinities in nature is questionable. ### George Ellis 00:13:21 - 00:15:31 Yeah, so I want to double down on this because what one mustn't forget is the following. Infinity is not a big number. And I think physicists make the error of thinking infinity is a very big number. It is not a very big number. It doesn't matter how far you go, how long you travel. You haven't made the beginning of the start of getting towards infinity. As we as relativists, we like to draw these diagrams where you see infinity, but that's highly misleading. It doesn't. If you take infinity and multiply it by 10, you've got exactly the same thing. That is not the way any real number behaves. You can add 700 million million. It is still exactly the same number. Infinity is not a number. And I think physicists confuse infinity with a really big number. That is the reason why it is a mistake to have it in any theory. And so David said that you should reject models if they don't provide a good explanation. I take a solid view here that if infinity occurs, your theory is broken down. Your theory is not good enough and you need a better theory. So that is my view on it. If infinity occurs in your theory, that means your model has gone beyond its domain of validity and something else is needed. And so, for instance, when I wrote the book with Stephen Hawking, we explicitly said that the existence theorems for singularities that we were talking about, we explicitly said they don't actually mean there's infinity. They mean you need a theory, namely quantum gravity. And of course, we don't know what the theory of quantum gravity is. I stick by my position now that through quantum gravity, if we get to the real foundations, we will find it as a discrete theory in which there are no infinities when you hit the conditions at the start of the universe or at the end of black holes. I'm doubling down on my position. ### Robert Lawrence Kuhn 00:15:31 - 00:16:12 Okay, so let me, you do not, though, require that the universe be closed because the standard model of cosmology would say if the universe is closed, then it's finite. But if it's open or it's flat, it will go on forever. And it does not seem that the density of our universe can give you that. Even a flat universe with the critical density would take an infinite amount of time to halt its expansion. So a flat or a hyperbolic universe is said to reach infinity. But your claim is that even if that were the case, that it would still not be infinity? ### George Ellis 00:16:12 - 00:16:27 My claim is when you say that to reach infinity, that's a fundamental mistake. It will always be expanding towards infinity, but it will never reach infinity. That's the fundamental distinction. You won't ever get there. ### Robert Lawrence Kuhn 00:16:27 - 00:16:36 David, respond to that. That you are bigger and bigger forever, but you never reach infinity because it's not a number. ### David Deutsch 00:16:36 - 00:17:36 This is just a prejudice. And now it's become a prejudice about what words we use. The space-time manifold either is infinite in time or it's finite in time. Right. And if it's finite in time, then it comes to an end after a finite time. And that means if you were rejecting the infinite ones, you'd be rejecting a whole class of cosmological theories for no physical reason. I'm not. I'm not rejecting models which expand forever. What I'm saying is that they never reach that final state. It is always in the future. The space-time doesn't reach things. It is there. It's a four-dimensional thing. Yeah. So this is a deep underlying dispute. I don't believe in a block universe. I believe in a growing block universe. That space-time is expanding. It has a future boundary. ### George Ellis 00:17:36 - 00:18:09 Now it turns out that NASA agree with me. They think the universe is 13.7 billion years old. If it is 13.7 billion years old, it doesn't stretch to infinity. That is its age. It will be bigger later. It will be bigger later in the future, but it will never be infinity as old. It will be getting older and older and older, but it will never actually be infinity. So I believe in a evolving block universe, not a block universe which has already reached infinity. That doesn't make physical sense. ### Robert Lawrence Kuhn 00:18:09 - 00:18:43 George, I want to press you on this for a second. On the one hand, you're saying the nature of science is we can never subject it to experimental observation because of the light cone, etc. So it's outside of science. We shouldn't discuss it. On the other, you're making an affirmative claim that infinity does not exist. That seems to be two different views to the same question. ### George Ellis 00:18:43 - 00:20:11 I don't think so. So, right, let me put a challenge to David. What experiment would prove to you that infinity exists? What's the experiment? Well, as I said, we don't judge scientific theories by what we can prove. An experiment can never prove anything. It can only disprove at best, and it can explain. And those are the two criteria by which we need to judge science, not by whether we believe it's a block universe or not a block universe. It's whether the theory of its being a block universe is a good explanation of what we want to explain in the world. There's a distinction here between some mathematical physicists who have theories of space-time, which produce a block universe. For real working cosmologists, the people who analyze the Planck data do not believe that. They assign an age to the universe, which says it does not stretch to infinity. It is moving towards it, but they assign an age to the universe at the present time, which is in contradiction to the people who say we live in the block universe. OK, so the question of the block universe, whether it's growing or static, is a fundamental metaphysical question. I don't think we can ever subject that to some sort of observational experimental confirmation. ### Robert Lawrence Kuhn 00:20:11 - 00:20:25 By assigning an age to the universe, it has existed for a certain time. That is a definite statement which comes out of standard cosmological theory. Sara? ### Sara Walker 00:20:25 - 00:21:03 No, I was just going to say it's always perplexing to me because I think infinity and time get convoluted all the time. So if you think time is finite and time is a real physical thing and time has to progress in order for certain things to be computable, like you have to have enough memory in the universe or enough computational power, then it might be the case that we can approximate infinity or we can get closer to it. But by what George is saying, it's never actually going to get to infinity. So it's almost like infinity is a placeholder for some future trajectory set by the theories we're building. But it's not a real physical thing in the universe because the universe hasn't had enough time for it to be realized and it will never have enough memory to be realized. ### Robert Lawrence Kuhn 00:21:03 - 00:23:15 Sara, your approach to this, again, I hear two different kinds of approaches to the same. I think I'm halfway between George and David. I'm not really sure where I stand on this argument. Well, you do make an affirmative claim that our biological evolution and the structure of our brains has sort of caused us to have our mathematical thinking and ways of explaining it. And so that is an additional barrier, if you will. George is talking about the barrier of scientific observation or experimentation. You're talking about an even earlier barrier in terms of the structure of our brains and enabling us to understand. Yes. But I think those things are related, right, because our ability to understand things is always limited by the horizon of what we can measure and how precisely we can measure it. So it's bounded by our technology. But you're making an additional claim that your feeling is that there are no real infinities, even though your affirmative claim is that it's beyond our structural neurology or our observational capacity to determine it one way or the other. Yes, because that's a useful thought exercise for thinking about what we're doing with infinities is to assume that they're an object that we're manipulating in our minds and using for specific purposes. OK, we're going to go on now. I think our panel skews to if real or imagined infinities are determined by democratic vote, which it of course is not. But if it were, we would skew to no real infinities. But I'd have to say that most cosmologists that I've met probably lean a little bit the other way towards David's side. Clearly, this is a fundamental problem that remains unsolved and should continue to be addressed. Our theme, too, is what to do with the nonsense infinities and the measurement problem in quantum mechanics and the disrupting infinities and the measure problem in cosmology. I want to start with David. David, you are a long term supporter of the many-worlds interpretation of the Schrödinger equation that governs the wave function of quantum mechanical systems. ### David Deutsch 00:23:15 - 00:25:39 Are these splitting off worlds infinite in number? We'll start simple. Oh, it's not known, and it doesn't make much difference to the theory whether they are or not. But I think the infinities you were talking about, the cosmological ones, they're different from the quantum mechanical universe. The universes that are invoked in cosmology are universes with different laws of physics and so on. And there is a genuine problem. If there's an infinite number of them, there's a genuine problem that means that no predictions can be made about them. Because if they're all equally likely, that doesn't mean anything because then each of their probabilities is zero, strictly zero. So that's completely different from the quantum mechanical universes proposed by Hugh Everett and by Schrödinger, by the way, which I think there is incontrovertible evidence for and which precisely do not have this measure problem. So if the cosmological infinity is to be tamed, some way of solving the measure problem for them has to be appended to the existing theories. Of course, if there's a finite number of them, like 10 to the 500, then there's still an issue. There's still an issue of whether the number should be regarded as a probability. I think not. Okay, there are many questions that one has. I do understand the many-worlds interpretation is theoretically the only interpretation that doesn't interpret the Schrödinger equation. It just takes it at face value and doesn't have to go through any type of decoherence or anything else. We'll get to that. But I still want to press you on this. Is the number infinite or is it just so many gazillions that you don't know? Can you give an answer to that? Well, I think it's not known. It could be. It could easily be infinite. The number of distinct distinguishable universes is more likely to be discrete, but it could still be a discrete infinity like the infinity of the integers rather than continuous infinity. But we don't know in either. ### Robert Lawrence Kuhn 00:25:39 - 00:26:48 Would that be an infinity in both temporal directions, backward and forward? I'm asking maybe naive questions, but these come to mind when I hear about them. No, it's not naive. Now you're raising the question to involve quantum gravity as well, because different times are special cases of different universes. But what that means in a specific theory, we just don't know. That would require a theory which unified general relativity and quantum mechanics, and we don't have a satisfactory one of those. So, you know, I don't know. I would tell you if I did. No, I know. I know you do. You're not bashful with your ideas. That we know for sure. George, when I first heard the many-worlds many years ago, I mean, the first reaction is, you know, is that a joke? Who can believe that? But as I began to understand it, the measurement problem is real in quantum mechanics and the many-worlds interpretation claims to be and may seem to be the only literal approach to the probabilities in the Schrödinger equation. But, you know, is this cure worse than the disease? ### George Ellis 00:26:49 - 00:28:20 Well, you see, I start off with a very different position from David. On this, he said there's indisputable evidence for the Everett multiverse. I don't even begin to believe that. So we start off in such a different place. And I really have to raise the problem here. The Schrödinger equation is a linear equation. And the idea appears to be, and I've been having a debate about this, that there is one function, the wave function of the universe, which obeys this equation and that determines everything. Is that right? Yeah. So if that is true, then how do mice and atoms and sheep and cows and birds exist? I simply cannot believe that one function, linearly evolving, results in all of that outcome. So we start off in a completely different position. And to explain that a little bit, I believe there are local wave functions everywhere in the universe, but no global wave function. And that puts the whole thing in a completely different perspective. So I just don't have this problem with all these branching functions and all the rest of it. Because on my view, there are local wave functions, wave function collapse takes place locally, depending on local conditions. And I never even get near the problems that David is approaching because I don't have this infinite branching wave function. ### Robert Lawrence Kuhn 00:28:20 - 00:28:57 Sara, from your perspective, looking at both the measurement problem in quantum mechanics and the measure problem in a potentially infinite universe in cosmology, how do you dig below that to get to some fundamental questions? So I don't take either of the theories seriously when you get to that stage. So I think there's a tendency to take theories and they have sort of an explanatory domain. And then we always want to push their explanations beyond what they're actually designed to explain and what they actually tell us. ### Sara Walker 00:28:57 - 00:30:17 And I think that can be useful, but I think in some ways that extrapolation is quite poor. So for example, the idea in the multiverse that there's an infinity of copies of me doing this somewhere is to me a sort of gross overstep of what the theory actually says. Because quantum mechanics was designed for a certain type of phenomena at a certain scale of reality. And one of the reasons the measurement problem is difficult because it has to deal with observers and how is it that physical systems that are information processing systems or acquire information from another physical system? You know, what is the physics of those systems? And I see that as being a completely different domain of physics than what quantum mechanics describes. And that has to do with the physics of evolving systems, systems that generate information and use that information to actually make new possibilities occur. And there's traces of what that looks like in quantum mechanics, but I don't think quantum mechanics describes it. So I think this idea of the multiverse is actually taking uncertainty that we have about micro scale phenomena, assuming the entire universe is that when we actually live in a completely different space that's constructed along specific trajectories where information has been building up over four billion years on this planet to make specific features that are us that are very local structures and probably don't exist anywhere else. ### Robert Lawrence Kuhn 00:30:17 - 00:31:43 I think we need to tease apart the different ways that we're approaching this question. We're using the word infinity in many different respects, the infinity in quantum mechanics is a radically different kind than the infinities in cosmology. I think we recognize that. I think there are then questions in epistemology, both from an experimental and observational point of view as George stresses and from perhaps a cognitive capacity, obviously escaping from hyenas and jaguars on the African plains did not by force give us the capacity to understand quantum mechanics. So our brains evolved for a certain thing. I think that's Sara's point. So I think rather than blur all these different ways of thinking together, we need to tease them apart and address them separately. So Sara, one more question for you is that in the measure problem in cosmology, because of all these infinities, and you know, and because of the nature of infinity, if something is even remotely possible, if it's not like putting it in reverse, it's not impossible in infinity to occur, then that almost impossible thing but not impossible will also occur an infinite number of times, maybe with the same level of infinity. So it becomes very complicated. If there were an infinite universe, which you don't believe, but let me put the counterfactual to you, how would, ... ### Sara Walker 00:31:43 - 00:32:40 How would, would there be implications for identifying intelligent aliens, which is your life calling? Yeah. So the interesting thing to me is that question doesn't actually make sense in an infinite universe. And what I mean by that is I think the fact that we live in a finite universe and there's some locality to where we are, is in part how I can explain our existence because the information is built up over time. Whereas I think if you think, you know, there's no new physics needed in life and biological things can just fluctuate into existence anywhere, it suggests to me that the information necessary to generate a living thing, a complex thing, exists at every point in space, time everywhere. And therefore there's nothing special about it and there's no evolution or knowledge to be gained by the actual physics in the system. And I don't subscribe to a philosophy where everywhere there exists the design of complex things. ### Robert Lawrence Kuhn 00:32:40 - 00:33:24 David, I want to ask you this question because as I've been dealing with the many-worlds interpretation and thinking it absurd at first, I've experienced the fact that over these couple, three decades, the many-worlds interpretation has become more and more accepted by more and more quantum physicists. So why were you right in The Fabric of Reality? You talked about this as one of your four big ideas. Why were you right and I was wrong? Why are there more people now committed to the many-worlds interpretation? What's been happening? ### David Deutsch 00:33:24 - 00:34:55 Well, one thing is that people are trying to build quantum computers. And if you want to ask how a quantum algorithm works, then there's really no choice but to work out what it does in each of the separate branches of the Schrödinger equation or what. And one of the problems with, a major problem with trying to confine theories or ideas to the realm for which they were invented, like the tigers and whatever it was in our ancestral, no lions it would be, wouldn't it? In our ancestral environment, then the trouble is you would be forced to say that the theory of quantum computers for which the theory, the quantum theory was not invented, should be abandoned, shouldn't be explored and they won't work. And even to this day, the universal quantum computer hasn't been built. So no such computer has ever been experienced by anyone and the theory was not designed just to describe it. So why should we take it seriously? Well, I think that such things have to be taken seriously. And one day it will, if we don't, then one day one of these things that we haven't experienced that our theories weren't designed for will step up and bite us. ### Robert Lawrence Kuhn 00:34:56 - 00:35:41 I'm going to give David the last word on that for theme two, that nonsense or disrupting infinities can be dealt with in various ways and perhaps dismissed. But I personally feel a lingering anxiety that we're all missing something big here. So theme three goes back to Borges and his idea, is this correct to argue that infinity is a concept that corrupts and upsets all others? And I want to begin with George, because George, you pose the idea of possibility spaces, which absolutely entranced me by the innumerable ways that the total state of affairs of everything could possibly be. So what are possibility spaces and are they infinite? ### George Ellis 00:35:41 - 00:38:57 Okay, well, there are physics possibility spaces and mental ones. The physics possibility spaces, for instance, in quantum, in classical physics, they're phase spaces, in quantum physics, they're Hilbert spaces. And that describes the possibilities that can happen. But the ones which really fascinate me have been described by Andreas Wagner in his book Arrival of the Fittest. He talks about possibility spaces for proteins and for genotype-phenotype maps. And it's an absolutely wonderful book. There's a whole set of possible proteins and any protein which actually exists is chosen from the set which is possible. And evolution on Earth has explored a subset of the space of possible proteins and there's still possible ones out there which haven't yet existed. You can continue this, there are possibility spaces for beings, living beings and for brains. And that leads on to the thing for mathematics. And most working mathematicians are mathematical platonists. They think they're not inventing mathematics, they're discovering it. And so there's a space of possible mathematical results. Like for instance, if you calculate the number pi, what is that ratio? You better get the number you all agree with. That's the only possible value. There are all the possible symmetries which underlie present day physics and so on. So there's a space of possible mathematical results. And the thing you have to do is you have to distinguish the space itself from what humans know about it. And those are two different things. Once you realize that then you realize that criticism is saying, but why does mathematics change with time? Mathematics doesn't change with time. It is our understanding which changes. And it's the fact that we believe that every intelligent person everywhere in the universe understands this, which is why we have sent mathematical numbers attached to the spacecraft we sent out so that people would realize this. Now, the fascinating one where Borges comes in is his library. The wonderful library of Babel is the space of all possible books and therefore it's a space of all possible thoughts. And I think that is a most wonderful idea and it is finite. If you look at it, Borges had a finite library and the reason is quite simple. To describe a thought needs a sentence. And by the time you hit the end of the sentence, you must remember the beginning of the sentence, which means the sentence must be finite. And that means that a computer can actually print out all possible thoughts and we can be generous about this. We can say, let's say that a possible thought might be written in a sentence. We'll be generous. A hundred thousand words if you want to say that. You can print out all possible sentences of a hundred thousand words that is finite. And so I think Borges' library is a wonderful example of the fact that the number of possible thoughts that we can have is finite. ### George Ellis 00:38:57 - 00:39:01 I know that's contentious, but I'm going to defend that. ### Robert Lawrence Kuhn 00:39:01 - 00:39:16 Yeah, but isn't that the same argument in philosophy? Is this the best possible world? Because you could always add one more. So if you have those sentences, you could append one more idea to that. So it's a compound sentence and you could do that, add in for an item. ### George Ellis 00:39:16 - 00:39:54 Yeah, well, that is we have this problem of people who forget what language is about. Language is conveying meaning from me to you. Adding all of those sentences. You can remember the end of the sentence from the beginning, by the time you reach the end, only if it has a finite value. And that is simply a linguistic statement. It's a characteristic of some theoretical linguistics people. They've divorced language from what language is about. Language is about conveying meaning, it's about conveying understanding. And that theory divorces linguistics from what linguistics is supposed to represent. ### Robert Lawrence Kuhn 00:39:54 - 00:40:21 David, what George is saying sounds like it contradicts the title of your book, The Beginning of Infinity, which deals with possibilities and explanations. Are you using the term infinity as a sort of a nice adjective to mean a really big number of things that we can do? Or do you mean it in the literal sense that the number of explanations and thoughts can go on and is properly represented by a real infinity? ### David Deutsch 00:40:21 - 00:42:57 Well, this thought experiment is not realistic. So George should be ruling it out from the beginning. But the important thing is not what the space of all possible thoughts is. The question is whether the thoughts are bounded or unbounded. And the same with the computers. If I say this computer can compute all Turing computable functions, that's an infinite set, but it's never going to compute even a tiny proportion of that set. I mean, this particular computer can only compute a finite number because it's only got a finite memory and so on. So is it possible to add memory to a computer ad infinitum? Well, again, the question should be, is there a fundamental bound imposed by physics on how much memory we can add to a computer? And in terms of the algorithms, and this is answering what Sara said also, the algorithms that are available to solve problems are unlimited in the same sense that the set of mathematical functions that we can compute are unlimited. A particular instantiation of them will always be finite, it is finite, but unbounded, we can add to it. And then when we reach the limit of 10 to the power of 10 to the 100 or whatever it is, if the universe were to last forever, so that all possible thoughts had already been thought, then even then, the meanings of those, by that time, we'll have brains, you know, 10 to the 100 in size. A particular representation in words may mean something else like today, if we speak of things like concepts like freedom, or science, we mean something different from what they meant 200 years ago. And that difference is not expressed in the words, it's expressed in the content in the context of the problem in which the words are spoken. And there's no boundary. And there's no finite. ### Robert Lawrence Kuhn 00:42:57 - 00:43:03 Sara, do you impose a boundary because of this cognitive capacity argument. ### Sara Walker 00:43:03 - 00:45:14 Yeah, so I think, I think there's a huge confusion that we think because in our minds, we can imagine all possible, like iterating over the space of all possible things, based on some mathematical construction, that that's a reflection of what's out there objectively. And I find that deeply problematic because we don't live in a universe where all things exist. We live in a universe where certain things were selected to coexist. And the selection process actually requires certain physical contingencies. So for example, if you want to think about, you know, if I have a graph of a molecule, like an abstract representation of a molecule, of course I can write all kinds of different algorithms to produce that graph as an output. But the way the actual physical universe makes a molecule is bound by physical law, it has to make bonds. So there's a very restricted subclass of algorithms that are physically implementable. And that's never going to change because chemistry is actually built out of those algorithms. Now, as you progress and you build increased layers of abstraction and biological evolution, they become physical systems that are more decoupled and therefore maybe have a broader class of algorithms that maybe can implement those things and describe them. But I think so maybe I'm agreeing with David. I'm not sure if I'm agreeing or disagreeing. I think that doesn't matter. I think the main point I want to make is that what exists is historically contingent. It depends on what existed before, right? So our technology right now on this planet is a restricted class of all possible technologies based on the historical contingency we have of inventing the transistor, right? So it's not like all things we can imagine can exist. There are certain things that we can build based on what exists now. We can build universal things like computers and the brain has got a universal computer built in. And so in that sense. But what is it? But it's not universal over physical things. It's universal over abstractions we might build a certain subclass. Yes. It's universal over abstractions. Those abstractions in our minds have their own reality. That's a whole other conversation. Right. And then there's always things that exist outside of the universality of the machines that we might describe. So I think. Yeah, I don't know. ### Robert Lawrence Kuhn 00:45:14 - 00:45:48 I have a quick question. I want to begin with George. If actual infinities are not realizable in physical nature, could there be infinities and could they make sense in a non physical realm if such exists? I just want to pose like the Abrahamic God is supposedly infinitely powerful, infinitely knowledgeable and infinitely present. Whether it's true or nonexistent, that's not what we're talking about, obviously. But would that make conceptual sense outside of the physical realm? ### George Ellis 00:45:48 - 00:46:56 I think it's a metaphorical way of speaking, which leads to all sorts of problems. The word infinite applies within. It relates to the physical universe, even though it doesn't exist. At least I know that within the physical universe, it's bigger than anything I can ever conceive. I don't even know what it means when you're outside the universe. But you end up with this kind of statement. What happens when an irresistible force meets an immovable object? You just have to shrug your shoulders and say this question kind of has no meaning. And if God is infinite, is this in space, in time and thought and power or something? It's just a metaphor. It's a metaphor, and I think it's making exactly the same mistake as physicists do when they confuse infinity with a really, really, really big number. It's a way of thinking. ### Robert Lawrence Kuhn 00:46:56 - 00:47:38 Okay, I don't think it gets you anywhere. I think we all agree that infinity is a deep conceptual probe. One of the theses of this debate is that should we get rid of infinity and therefore we would be able to understand and make more progress just getting rid of it? Do any of you agree with that? I know David doesn't. And, you know, frankly, to be fully honest, I don't either. I think infinity is a powerful probe of reality. But I want to ask George and Sara, should we just get rid of it? Would that make our lives better and more accurate assessment of reality? ### George Ellis 00:47:38 - 00:48:04 I think we should keep it as a theoretical concept with the statement. It doesn't occur in reality. So I think it's useful as a theoretical concept with this additional statement. It's not going to occur in reality, as I said, because no matter what you've done, how far you've gone, you have never even made the first step on the road to infinity. Sara, you want to get rid of infinity? ### Sara Walker 00:48:04 - 00:48:50 Not everywhere, but I think there's certain places that it makes a lot of sense. And I guess the one that I'm always most interested in right now is the connection between infinity and time. I think if you accept infinity as real, then you can kind of accept timelessness also as real. Like when you're talking about this other world of infinite beings, infinity takes time to calculate. So it seems that there's a deep connection between infinity and time. And I think I'd like to just remove it and say that actually anything that you want to produce takes time to make it. And you can't just put like a placeholder that this just exists and it's infinite and it doesn't. It's useful as a concept in mathematics. Every undergraduate learns to sum infinite series. And that's actually really interesting. So it's useful in that context. ### George Ellis 00:48:50 - 00:49:00 Yeah, but with the big asterisk, George, and then 60-point type, this doesn't occur in reality. Use with caution. ### Robert Lawrence Kuhn 00:49:00 - 00:49:04 Okay, everyone, well, this has been terrific. I thoroughly enjoyed it.