Oxford Mathematical Institute — John Horton Conway

0:00 Hi, I'm John Wilson-Sugway. I'm a common educator in the UK. At Cambridge, I'm afraid to say this is where it started at Education and stayed there for a while. I had the privilege of actually being taught by him when he was there. And if I could add one personal thing, I remember very vividly how at Trinity and Haybridge we had a room called I-4 in Grecourt. I don't remember. But I-4, for some peculiar reason, always ended up having a mathematician in it. And the mathematicians, a group of them, used to gather for lunch and bread and cheese once a week. he used to come to this, which I think if you think about it, it's an amazing commitment for the university. Anyway, at some point he got fired at Cambridge, and he moved on to Princeton in 1986, where he still is, and he's currently the John von Neumann distinguished Professor of Mathematics. He's known as well, I guess, as well as everybody here, for many different parts of Appalach, from algebra and combinatorics, the sphere hacking, the geometry, and it's been highly influential in all the world's good. He's very well known for his exploratory articles and his willingness to engage those like myself at that time who were new to Appalach, and for his games like Go, which won too many prizes to mention them all. So I'll just give you John Conway, who is about to talk to us about free will, quantum logic, parents. This is an unusual sort of subject for me to be talking about, but I had a prior talk about it last week. At this time, except this is part of our difference, I spoke on

2:30 this topic to a slightly different title, to the Philosophers, Philosophy Department of Princeton, which included a bunch of well-known physicists and mathematicians as well, which I think was a little bit more of an ordeal. But anyway, it's a bit peculiar to be talking about free will, but I think it's appropriate because I should mention that everything I'm doing, that everything I'm doing is joint with Sergeant Koshin. He has been interested in quantum logic for a long time. He had an education as a physicist, but as a mathematician, he's really on the logic side, and he did a number of things, usually with other people, with Ernst Specker from Missouri, he found this Cochin-Specker paradox, which is a good part of what I'm talking about today, some 40 years ago. with James Axe, he solved a famous conjecture of our team in number theory using a mathematical logic way of doing things. So he's done a little bit of different things. But okay. I've been talking with him about quantum mechanics and quantum logic. for a few years of coffee table time, coffee times. You know, we get together, and he mentioned some problems of the pure mathematical kind that he and Specker had come up with at the background of logic, and we managed to solve them together. And then we were working on refining this Cochin-Specker paradox. And a few months ago, it's about as many We were up in his office, and suddenly we realized that we'd reached a particular degree of precision in this theorem that was new.

5:00 I mean, the precision was this launching thing. The ideas are old. They date really back to Einstein and Elskir Rosen's famous paper of about 1935. But what's wonderful is, over the years, various people have come to this subject, done various things. There's a famous paradox of Bell in 1965, and the Kirin-Specker paradox is actually co-revaled with that slightly earlier. Of course, they found it slightly earlier, but they published it slightly later. And what it implies is, in some sense, I mean, there's some astonishing things about it, but what this type of argument implies is sort of twofold. First, that there is what Einstein called spooky action of the distance in physics, which is really rather surprising. But second, there is a certain indeterminacy or unpredictability. And normally, that unpredictability is stated by saying it's all predicated on a theory. It says, perhaps, that quantum mechanics is not a totally predictive theory. It's not complete in some sense. The real fact, however, is a property of the universe, not of any theory. Physicists probably will think everything I'm saying is very old, but it's not. The ideas are old, but the simplicity and the precision and the absence of assumptions is new. So, okay, so let me introduce it by talking of the Cauchin-Specker paradox. So, what happens, there are things, I won't put this on the sound, isn't it? There are things called spin one particles. There you need to know what they are. There is also a thing you can do to them, which is traditionally called measuring the spin of the particle in a given direction.

7:30 That's a very bad name for it, a slightly better name is observing the spin of the particle in a given direction. Let me call it that, but part of what we're proving is that there's no such concept of the spin as the spin of the particle in a given direction. So what you do, well, I don't know. I mean, if we could only manipulate tiny little magnets, it would just sort of bring the magnet near to it and see whether it was attracted or repelled in that direction. In fact, you'd have to do more complicated things. But you measure the spin of the particle in a different direction. But I'm going to say that you can measure the absolute spin. Well, let's say, let's say, let's say the spin of a particle in any given direction is 1 or minus 1. I'm putting that in quotes again because it's sort of strictly a meaningless a statement, it's only defined by doing various experiments, but I want to not talk about the spin, I want to talk about the absolute spin, or actually the squared spin, to the main point. And then, one and minus one become the same thing. And here's an axiom I'm going call the spelaxing. If x, y, and z are the perpendicular directions, the answers will be Well, let me put 101 in some order. I find it rather easier to say 101 or 110 in some order. And I just, in this, use the term answers. But let me say, you can do something if you measure the spin, well, a little bit of the

10:00 other than meaningless words for a few months. If you measure the spin in the x direction and then in the y direction, those measurements don't interfere with each other. They commute technically. So had you measured it in the y direction and then in the x direction, you would have got the same answers. Provided those directions are orthogonal. So if you measure in three different orthogonal directions, one after the other, you'll always get the answers 101 in some order. Now, I want to point out about this axiom that it's meaningful. And a lot of what I say, and a lot of other people say occasionally it's meaningful, is not meaningful. And what do I mean by saying it's meaningful? I mean that it's observationally meaningful or operationally testable. If you don't believe this, and you have suitable equipment to perform the experiment of what that is, then I'll hand you a spin-one particle in some sense, and you can do the operations and you'll get the answer 101. First time you do it, next time you might get 011. Next time 011 again, or later time 1100. You do this experiment a thousand times on a thousand different particles, and you'll gradually come to believe this accident. The axiom is operationally meaningful. There's a sequence of operations you can do that will convince you, if you believe in the principle of induction anyway, that this is all that can happen. Okay, and let me talk about what I'm going to call the old form of the Kochen-Specker paradox, since it's 40 years old, is this, but it's tempting to believe that the answers to these potential experiments exist before you make the experiments, obviously. That's why time used the word oblation, that the answers to these observations exist, or if you're just finding out what they are. Now what does that belief amount to? Well, it amounts to this. Here's the, I think of this as the particle, but it's really the sphere of directions. So, suppose that naive belief were true, then I'm not actually going to do this experiment, I'm just going to think of doing the experiment in this particular direction,

12:30 wonder what the answer will be. If it already exists, oh, I'm harking on the stick. So if I ask it in this direction, it'll come out to be a one. And now I've asked it in this direction, it's also there one, and now it's a zero. And one's definitely to believe that the answer exists in each direction before you actually make the experiment. Now, the first thing I'm going to do is prove that that is the step. Okay? And that just follows from this axiom, Paul Gretchen. Okay. How are we doing it? The reason why it ends so is that there isn't a function defined on the surface of the sphere that takes only the values zero and one, of course it's already a rather funny kind of function for a physicist to believe in, because it's a sort of point set notion really. There's a function defined on the sphere taking the values zero and one with the property that for any field of normal directions, it has the one at one point. So, in other words, the reason why we know that these answers are not determined ahead of time is that there's no consistent set of answers to these questions. And that's really quite an easy theorem to prove. If there were such a function, then a little sort of principle of logic, the compactness Sorry, no, if there weren't such a function, sorry, then this compactor's principle tells us that there will be a finite subset of the sphere on which it's already impossible to define. Well, let's put it the other way around. If one could define such a function restricted to every possible finite subset of the sphere, then one could define such a function on the whole sphere. So there must be a finite set of directions where already you can see a combination. And that finite set has changed over the years. In the original Kochen-Specker paper, they had a certain set of 117 directions. One of the first problems that Simon introduced me to was the question of minimizing, you know, reducing the size of that set. And I reduced it to 31 directions, but the example I'm going to use is not, I think it's not Turkish Speckers.

15:00 It's a very nice example. And I'll tell you how to define it. There's a very nice way to think of it, which isn't this, but this is peculiarly appropriate for my proof. I've drawn a circle in each face of the cube here, and I'm going to mark the directions on the surface of the cube, not on the surface of the sphere. And within each circle, I'm going to describe a square parallel to the face of the cube. as it lies in, and then, oops, all those words, the force was, but what you see, and then the set of directions I'm talking about, I have nine of them on the interior of each face, so the directions go into the midpoints of the edge. When I'm drawing a bar here, I really mean the line joining that blob to the vector of the cube. And by the way, these directions are really sort of bidirectional. The negative of a vector really tells exactly the same as a vector. Let me explain that. And, look, suppose I take these three vectors and then two of them will be 1. On this counter-fessional function, when I decrease the function, two of them will be 1 and 1, 0. But if I look at this one, the one with 0 here, and replace it by its negative, that's also got to be 0, because the negative, together with the other two vectors, is again a non-partum multiple. Well, I could have read that one with the ones, and we basically find it's a negative. So it's easy to see that if there were such a function, it would have to be, as it were, antipotally symmetric. So now, whenever I speak of direction, you'll be pleased to understand it in that antipotally sense. Okay, so these are the 37 directions. There are nine on each face, but there are effectively between faces, so that is 27, sorry, not 37, but 33 reactions.

17:30 And then there are essentially six sections of the cube, so 27 plus 6 is 33. And it's not possible to define such a function on that cube. And let me sort of give you a quick proof of that fact. Oh, by the way, I want to remark, sometimes we don't see the entire triple of orthogonal reactions here. But if I see two orthogonal directions, I want to say that the values assigned to them cannot both be zero. Because then the third member of the orthogonal triple that they're part of, Well, whatever you make it normal 1, it doesn't depend on the 1 and 1. You can't have two 0's in orthogonal directions. And so let me prove that no such function exists. Well, I'm going to draw this picture of the cube. And I think it's most helpful to think of this as a little Picasso-like cube. Now, you're looking at this cube, you can see the left face and the front face and the right face together. So if you want to, look at what's happened when you're looking down into the sink. And this is the back face then, there's the one in the middle. But I think it's rather easier to think of this as the front face. Okay. Well, let's take the vectors that actually go to the sensors and the faces. They have to have the 101 property. And without loss of generality, the one with zero might as well be left hand base. Okay. Now that implies, because I can't have two orthogonal zeros, that implies that all the spots I marked up there, that I'm sort of marking them here, that are orthogonal to, what am I trying to do now? There's a problem with this line. They all have to be marked at once. Okay. Now what I want to do is, well I want to make an observation first. And the better way of defining this in terms of this circle makes something very clear. Namely that the angles are tended by

20:00 these two blocks at the center of the cube is a right angle. How do I know that? Well, it's the same as the angle subtended by these two crosses, because they're both diamonds in the same circle at the center of the cube, and that obviously is a right angle. Okay, that's this vector, this vector, this vector, so let's look at these two points here. I'm looking at this space now. Well, let me continue to look over here. I'm looking at the left-hand face. It's a zero. The vector orthogonal to these two has already been marked with a one. So one of these has to be a zero, and one of them a one. Now, it needn't be the way around that I put it. But if it is the way around that I put it, then I'm going to sort of think of an arrow joining, pointing from the 1 to the 0. And there's got to be such an arrow in each of these two directions. Okay? Okay. But up to similarly, there's only one way of doing that. I mean, so I might as well suppose that this is a 0 and this is a 0. Okay. And then those two points are antithetical to these two points, so those are zeroes too. Now I want you to look again. There's a column of three marked spots here. You can see them perhaps more easily, this kind of thing I'm talking about. And this similar column here, these dots is orthogonal to these three. As you can imagine by seeing them from the top. You obtain this set of three dots from this set, but it's rotated to a dynamic. And we already have, so what I've just said really, includes the assertion that this dot is orthogonal to that zero. It's actually orthogonal to all three of these things. But because this If this is marked with a 0, this must be a 1. If anybody's taking notes, I'm going to put 0 subscript little a here.

22:30 And that implies 1 subscript little b. Otherwise it'll be 2 or 1 from 0's. Fine. Now, this one here, which I think I'll call 1 subscript x. That's the same as this one. This, this, and this are an orthogonal triple. Okay? And therefore, we've already marked two ones, so this is better to be a zero. And I'll call this zero substitute c. So that implies zero substitute c in the presence of one x. Okay? But then this zero is orthogonal to all theories. And so, these have to be 1s as well, and I'll call these 1d and 1p. Okay. But in an exactly similar way, this 0, which is 0 alpha, implies 1 beta, which implies 0 gamma, which implies 1 delta and epsilon. So, let me just drop that over here. Okay. Actually, it might be better if I had to change delta and I'm silent here. And now, one D and one delta implies zero, whatever the next. The letter is, we're going to put F. We have the correspondence between the green and English of Latin alphabet. So here we have 1d in delta equals 0x. equals 0, zeta. Let me use diagram. And now, this is a contradiction.

25:00 So that's the proof. It's very interesting. So let me sort of anthropologise that by saying the particle cannot have made up its mind ahead of time. What answer is it going to give you to any particular question of this time? And that's really the key to the reaction. But that's not new. That's the old statement. The 1965 version of this there. It doesn't sort of show you that something funny is going on. But you see, it might, for instance, have made up its mind one answer it's going to give to any sequence of experience of this type. Maybe it sort of says, oh, well, if you ask this question first and then this question, I'll say zero for the first one and one for the second one. But maybe if I ask them in a different order, I'll do something else. And that's not possible either. By the way, some of these facts are also old, but as I said, the exact precision of the statement is really quite new. If you read the sort of physics papers in this area, you find they make various assumptions of contextuality or non-contextuality. I don't know which is which, actually. And the philosophers are kind of even worse, a man called Redhead is going to talk about this subject, in which he has, I think, 12 different notions of locality in himself. And one of the merits of the thing I'm going to say here, I think it's very simple. The physical assumption, another thing that perhaps I should say, to read the physics papers, they tend to assume the, not always, assume too much quantum mechanics and then it's not at all clear what you assume. So when you finally get a contradiction, you can say, oh well I don't believe all that quantum mechanics, I never did all that with anyone, or something, you know, and thus you don't actually have to believe anything except three particular axioms. So here's the theorem. If there exist experimenters, I've got to tell you what it means, with free will.

27:30 I'm sorry, I seem to be a bit redundant, because I'm thinking it's a damn statement. I'm also guessing that this is probably the hell most of the way I would say that. Then, let's say, elementary particles. As I say, all of them know the names of elementary particles. How can we work? Now, that envoys a concept that you don't normally expect to see in mathematical theorems. But I'm going to give a definition of this in a mathematical. But this is subject to green, I'm going to just call them axioms, and I call these axioms spin, twin, and thing and okay and these axioms are better that really do seem to be typical and moreover two of them are operationally meaningful in the sense that I just proved that this one is, and this other one is one that you've all heard of, and most of you will believe, I hope. But you don't quite know what free will is. Well, for the purposes of this talk, what it will mean is that the behavior of the particle, or the experimental object where everything has free will, is not a function of all the information that was accessible to it. So I feel I have done stuff. I mean, here's this pen, okay? I feel free to either drop it or not drop it. I think that's right. In Princeton, I didn't bother to drop it. Well, I can't prove that I have free will, or that you have free will. And it's irrelevant, anyway, from the point of view of the theory. The theorem has an ethic. It's very significant that this concept appears at both ends of the arrow.

30:00 I mean, I couldn't possibly prove, you know, something like this at this end of an arrow. And I guess I could use any meaningful things not mentioning free will from this one. But let me sort of remark that if we don't have free will, I mean, this has been a very long argument. You know, Aristotle has a piece called The Sea Battle, in which he discusses the proposition as to whether the statement that there will either be a sea battle tomorrow or there won't be a sea battle tomorrow, whether that statement is true today. I think they were discussing some political zones, don't you? And, you know, you see what is not true today, or what was not true today, is that there But also, it's not true today, that there won't be a sea level tomorrow, so what was true was the disjunction of those two things. Of course, when it got to tomorrow, one of them became true. And anyway, this has been, you know, still kept the discussion by philosophers for a long time. But I think, you know, this is a valid theorem. And it's added an asylum to the situation. I mean, it's remarkable that the mathematical level I'm talking at won't compare anything backwards. I want to say something else. If you don't believe in free will, then you're a distinguished company first. I mean, there's a famous piece in Lavasse's work on probabilities involved a similar one in which he says that if an intelligence knew the positions and what we would now say the same momentum of velocities or all the particles in the universe, and if also that intelligence was our intellectually gifted, I forget how he says that, then the intelligence could work out the entire future of the universe. And that clearly is expressing a determinist belief. However, Laplace, like so many other people, was what I call a disconnected determinist. I mean, he believed that physics was determined, okay, but then in his daily life he did that way. And I've had a number of arguments since I've been discussing this theory with people, with people who express a determinist philosophy. I want to just sort of mention a few of them. It's really almost impossible for people to stick consistently to this.

32:30 I was talking to one physicist, and he said, well, you don't need the assumption of free work in your field. He said, instead, you can replace it. You could have the experiment performed by some machine, which is programmed with a pseudo-random number generator, and you can just postulate that the particles aren't listening to the program of that machine. How do you know that you can program your machine to do a certain experiment that you should have done? I don't presume you can do an experiment if you like. That does not get rid of it. Science is impossible if you don't presume you can do an experiment if you like. Another person was a sociologist who said what I still think is so funny. He believed in determinism, and he was saying really astonishing with absolutist determinist position. And he was talking about children who were abused by their parents. And he said, you know, their behavior is now inevitably a consequence, they're sort of out of a program next time or anything. And I said, well, sort of, what use is this way of thinking of things? And he said, well, you might be able to persuade parents to stop abusing their children. I hope you all got the possibility that if indeed this determinist philosophy is true, then they'll either abuse their children or pass, and you won't be able to persuade them, black children will be able to persuade them. Anyway, okay, let me proceed to prove the theorem at the same time explaining the other two axils are. The twin axiom says you can actually produce two particles in what I shall call the twin state. It's often referred to, it's more commonly referred to entanglement, but I want a particular kind of entanglement. And the way you do it is this, you bring these, I mean I'm not going to mention any real You spin one particle together, at least conceptually. And then you measure that total angular momentum,

35:00 or you do the operation called measuring. And every now and then you get zero. There only are many values for this thing, it's a one-size positive particles. So you get zero with a positive probability. If you don't get zero, you just throw them away, and do it with another guy. Whenever you get zero, you now let them float apart, and then if you measure the spin of the particle, at the moment I'm talking not about the absolute or square square, but the actual spin, which is zero or minus one. If you measure the spin of this particle in a given direction, and you measure the spin of this particle in that same direction, then the two answers have to add up to zero. is the negative of the other. But I'm only going to be talking about the absolute or square spin, but I can pretend that the two answers are going to be the same. And indeed, it's really rather important that I don't measure the actual spin. I only measure the absolute spin, which makes the experiment a bit powerful. These experiments haven't any time they've been done, so they're still Gedanken experiments. But that's not anything more than the difficulty, you know, at the moment. Okay, so these particles are now crudely, okay? As far as our kind of experiment is concerned, they have effectively promised to give you the same answers, if you ask them the same questions, okay? You know, believe it. Oh, God, I didn't expect you to read it. I don't want you to read it. I want you to do this experiment at the thousand times. And after, you know, every now and then you choose to measure them in the same direction, you get the same answer. Or you can measure three orthogonal measurements on this one. And the same three orthogonal measurements in the same order on this one. And you'll get the same answers. You'll even get the same answers if you do the same three experiments in a different order. I mean, suppose I did x, y, and z on this particle, put y, and z, x on this particle, then the answer I get to say, the y of x doesn't depend on which two particles it is. So that's the twin-act theory. The experiment we're actually going to do is a little bit more limited. What we're going to do, we're going to have two particles, which I'll call A and B, But I think of them as the away particle and the back home particle.

37:30 And on the away particle, we're going to have our colleague who we sent to Mars or somewhere. It's important that we can go a long way away. We're going to have our colleague measure in the direction of W, the way off the person's direction. And this one, we're going to measure in three directions, which I'll call x, y, z. And both experimenters here are perfectly free to choose exactly which direction or directions they're going to do. They're required to choose them just before they do the experiment. I should have said, this also involves the possibility of separating these particles, spatially, that they're far apart, because we don't want there to be too much, enough time for light to travel from one experimental station to the other, into experiments. difference. And that's this thin axiom. This thin axiom, and I just commented by the way, these two axioms, I want you to understand, this twin axiom is also operationally meaningful. You don't believe it, you do it lots of times. So what actually follows from the twin axiom is that if w happens to be one of x, y, and z, then we'll get the same answer as that. What does the thin axiom say? Well, the simplest way of explaining it says no physical influence can travel faster than the speed of light. No, we're not worried about the speed of light. Faster than some particular band. So, if there's a finite upper band to the speed at which physical influences can travel, that is not operationally possible. There are some people who believe that, say, ESP travels instantaneously, but we don't have to go quite that far. But, I mean, there's no way of testing this accent, but who knows what metaphysical influence is? So this has a rather more dubious character than these two. It's not funny, because these come from quantum mechanics, which has a more dubious character than relativity. But these axioms are quite explicitly testable, whereas there aren't.

40:00 But I'll talk about that later. They're excellent graphs, but they're not quite the same as these. Okay, well now it's easy for us to prove our theorem. But I want to definitely come back and talk about what? Talk about thin again. what I meant by pre-will, the response, so to speak, of the object concerned is not determined by what? The entire history of that part of the universe. So let me draw a little picture So, these two particles, here's the back home particle, now I think I'll make this a bit of a way to the back home particle, and these are supposed to be the light cones, the backward light cones. So, what we're saying is, well, ah yes, one of the things I wanted to say is, it's really It would be rather hard to say some of these things without making counterfactual assumptions. For instance, this is often saying that two operators commute. Or effectively, operators corresponding to two experiments commute. Which means if you do A then B, it gives the same answer as if you have done B then A. Have done, that's subjunctive. If you haven't done, if you've done A then B, you haven't done B then A. And then he talks about what would have happened if you had is in this dubious sense of counterfactuality. One of the really wonderful things about proof is that there's no trace, I'm not quite honest here, there's no trace of counterfactuality anywhere in it. Well, that's not quite true. But the counterfactuality that is in there is the mathematical count. I'm going to make an assumption that, as I shall show you in a moment, the response of these particles is a function of the information available. So, I'm going to call the relevant functions b to a of all the information available to a. Well, what is that available in information? Well, some of it is information that's actually available to A and B. Some of it is information available to A but not B, and some of it is information available to B but not A.

42:30 Well, obviously, whatever particle A is doing must depend on information about it. So A can only depend on that. Okay. Well, now I want to start looking at this again a little bit. Let me give a little bit more room first. What we are going to do with the away particle is just inquire about its spin in the reaction W. So W is part of the information available to A, okay? But it's not available to B, because I'm supposing that B is far away from A, B is on Earth and A's on Mars, and the two experiments have agreed beforehand when they're going to do the experiment and not what they're going to do. By the way, the possible might be listening to it. Or might be listening to Colvin Gray's or something. Might have carefully scrutinized the program somehow. We don't know what this information is. Okay, but W is part of the information that's available to A, at least at the end of the experiment, Well, I'm going to take W out of the information available at V to A, and the rest of the information, I'll talk later about cutting up information in this way, it's quite interesting. The rest of the information is the information other than W that's available at A to B. And when I put a question, and what that's going to be is the assertion, say, that W goes to 0, or maybe 1. So that's the answer to that question, only rather than say, mention W again, and there's a little query against W and say that is, that is I, means that W goes to I. The measurement of W is the answer I, which is a normal one.

45:00 If I put a query there, that's what it means. And now, what happens on the other side of the experiment? Well, the response of particle B is going to depend on information available to A and B, and information available to B only in precisely the same way. And that's going to be what I might think of as a set. x goes to, let's say, i, y goes to j, z goes to k, sorry. But I won't actually use this notation ever. What I'll do first is put x, y, and z there, and move them from this information circle. A slightly different thing now. But then if I put a particular question here, If I have a special mark after one of these three arguments, then I should mean k. I'm only going to ask about the answer to k. And what we have is the following result. And if w is one of x, y, and z, this thing is the appropriate one. of x query, y, z, and all this information, same thing here with y query and z query instead. That's what our . And that's very easy to complete the proof. So what happens here is, take the case, for instance, when w is x, then what we get is that this is theta B of X squared Y, Z, IAB, and I B prime, and K. But, when we look at this, because I should have called X squared, I should have called W, W, Y, Y, Y. When Well, if you look at this, because these are equal, this cannot depend on this information or this information.

47:30 So we can rub those out. And now comes a little rather clever trick, which is missing from most of the... I mean, an analog of this is missing from most of the other discussions. assumptions. They have to make a contextual assumption. But we don't have to. Basically, they have to get rid of this information here. But I remark that there must be circumstances in which this experiment can be performed. Now let's discuss that. What do I mean by of this? Exactly what time am I talking about when I sort of, what time did I put on top of this? Well, here's the answer to that question. Eventually, what happens in these things is actually there's a little spot on the screen that lights up, and it's either okay or it's And it's when that spot on the screen is observed that it's, well, that's not enough. But, you know, Mars is sort of 10, like, minutes away from Earth or something. And the experimenters have agreed to do their experiments. Time is sort of determined up to a few seconds. So what actually happens is this. Let the experiment be done. Okay? And that the times that we're talking about that give these functions be these times at the end of the x-barrow. But I've just shown now that this information here and this information here can't be used in these functions. Okay? So the only information that can actually be used in these functions are these directions W, Y, and Z. Okay? And old information. Information is at least 10 minutes old. Okay? So it didn't matter exactly what time this was. It's infected with the same as if the time was slightly before the experiment. Now slightly before the experiment, the experiment was actually be free to choose which direction he would have done something else. Let's actually think at that time rather than later on.

50:00 Okay, so I just wanted to say that so that we could get there. But now you see we have some nonsense here. I mean, first of all, the free will axiom enables us to say that whichever experiment By the way, I'm only going to require my experiments to use 33 particular directions. So they don't have to worry about getting them exactly, choosing from an infinite number of possibilities. And they don't have to worry about getting those directions exactly right now. So, there is some information that I can plug in, because I can actually do the experiment. So let's choose a particular value of this information. And when we plug that in, we've proved this, but we've also proved by similar arguments these statements here. I should really perhaps use different axis sets, but they have to be the same in different lines. But this thing here, all of these have a common value, which obviously only depends now on W. It doesn't depend on who's doing the experiment. It doesn't depend on, you know, what was, on everything in here, just a particular, some particular value that could happen. There is such a value by the free will axon. And so this is a function which I will call f nought of W, let's say. What have I done? There is a function, F naught, that depends only on W, and has the 101 function. There are many functions, because there are many circumstances. That's a difficulty in some people's proof, because of the corresponding thing. they have to suppose it's independent of this information in some sense which is some kind of textuality that we don't need, or some kind of locality, because all I'm doing, if there are many functions, well we do not prove, if some people have to sort of produce a particular function, I don't exhibit functions that prove that such functions exist, but I proved to you earlier that such functions don't exist, so that's the theorem now.

52:30 As soon as I've come up to time, so unfortunately, I can't say too much about that. We're going on to sort of analyze the thin axiom of it and move its dubiousness. It really comes from...let me just talk about robustness a little bit. The whole of this theory is actually robust. Directions don't have to be exactly or bottom. If they're not exactly orthogonal, then what happens is, you don't always get 101, but you get 101, 99.99% of the time, that's sufficiently close to being orthogonal. And nowhere did we use the fact that you have to get this 100% of the time. It's happened, just if you get it sufficiently often, that's good enough. And the similar robustness elsewhere, we don't need to measure when we're trimming our particles, we don't need to measure in parallel reactions. It's not clear what parallel means, but let's actually close the parallel, which is good enough. And we don't need the speed implicitly mentioned here to be the speed of light, if any finite speed would be good enough. And what else don't we need? There's something from some other . Well, this really comes from the special theory of relativity, and the special theory of relativity is a bit outdated, because it's outdated already in 1915 by the general theory of relativity. But we don't need the special theory of relativity to be exactly true. And if, say, the general theory of relativity were exactly true, then the special theory of relativity would be approximately true, you know, well-awaited in that hole or something like that, or other nasty things, and it's actually more of that than I did. So now, when we talk about the special theory of relativity, what am I trying to say? I mean, if you have the general theory of relativity, then the concept of the same direction doesn't exist, but you have to use directions transported by parallel transport, but we're supposing really the special theory of relativity is the explanation of this. And if the special theory of relativity is exactly true, let's say, I don't really mean it, then the reason why it's true is because if you transmit information instantaneously,

55:00 then seen from another point of view you could transmit information backwards in time and we do sort of know that what is it causes effects can only happen after their causes and not before and that translates by the symmetry principle really into this in action. There's something else about that. What was I just going to say about that? Oh, yes. It's rather difficult to define causality when you have things like relativity, really. And we've submitted this as any kind of penetrating analysis and kind of definition, which is that a sort of physics is causal, or let's say the universe is causal, if it looks causal, from every emotional audience. And then, to say it looks causal is really to say, to refer to the classical notion of causality, rather than the relativistic notion. And that's perfectly possible. And what happens is, how can I say, it's very puzzling what goes on here. Because, you know, I just proved, perhaps you didn't all follow the proof, but I made I did make a supposition, I did make one counterfactual supposition, but I did this as a mathematician. I made the Euclidean counterfactual, where you suppose something in order to prove it's false. What I suppose was that the behavior was a function of the information accessible in the past. That, the only counterfactual assumption I made, that proved it was counterfactual. I still have quite got to the absolute enumerable of this business. But maybe my memory is such that I won't be able to, so I may not have anything. Well, I guess we should go back to that. Before you... No one wants to ask any questions. Or make any comments about yourselves.

57:30 Well, you know, this is one of a number of things that has been around for a few years. exactly how many directions do you need in this theorem or in various generalizations of the sort of class class of what we're going to come down. The thing, how can I say, the more interesting thing is sort of how do we do a reverse accomplish this trick? You see, I believe, I believe I do have to be wrong. And now I can't even say what that means in any operational way, because I'm pretty convinced that that's it. And I think that if you do too, maybe everyone. And therefore I believe that the particles have people, in the very precise sense I believe, that their behaviour is not determined. So these particles are all over the universe letting out to apologise again, taking decisions just like you can make decisions. Okay. Where does our free will come from? Well, I think it's our business. The converse of this thing, which of course I can't prove. But we're made of these particles. They are taking decisions. Somehow in our heads, the particles are exercising their freedom. And that's where our free will come from. So that would be really interesting. Where does our free will come from? But this is a different picture of the universe to the one that Newton's way gives. A very different picture. The universe is actually not determined. These particles are taking decisions, not independently, in a mysteriously related way. It's very strange. It's hard to understand how it can be consistent. But it's very, very easy mathematically to simulate the universe like this. Even in conversation, you can simulate a simple universe that has these properties. And it appears naively as though the particles must be transmitting information to each other faster than the speed of life. We can see that's nonsense. They aren't transmitting information. What actually happens is the answer A is going to give to this question is necessarily equal to the answer B is going to give if you ask the same question.

1:00:00 That is the fact that it's determined ahead of time. What those answers are is what's not determined ahead of time. They're still free. But they usually have half a mind. Okay, because what A gives is what B gives. So, it's all peculiar, but it's consistent. And in other words, consistency means it can logically happen. A universe can work like this. It doesn't appear to work like this. So we should just get into it. Get over it. It doesn't work like this, it's very strange, and it's a spooky thing that Einstein called action resistance. It's there. It's undetermined ahead of time. I believe it's part of our free will. So the reason why I want to call it free will is that we do call it free will when it's humans. This theory really shows it's the same stuff. So whatever time you give it, we should use the same name for that. Is it necessary to have a sub-structure in order to have free will? What do you mean by sub-structure? Well, do you think that you need different components of something that are fighting I'm not qualified to answer that question. I don't believe it is necessary to have such a question. Obviously not. But, I don't mind. I mean, you see the only way I can actually prove to my satisfaction as an mathematician that the particles have it, is to use the spooky action of the distance, you know, which is much more fascinating than free will somehow, how the fact that these answers are mysteriously related. I don't think that's necessary either. It's just necessary to prove. I mean, if only one thing is going on, then we can't tell whether, you know, if there's only one sort of possible thing to investigate, we can't make these sort of things that are analogous to the factuals. But the fact that this is equal to this is what enables us to prove that neither of them But, as I say, I think that's just an archive of proof. I don't think it's necessary. I mean, it would be ridiculous, I think, to suppose the particle has to be twinned with something before it makes the decision. I'm sure they're making these decisions all the time. By the way, I should say, there is a famous experiment, the GHZ experiment, which is the current sort of state-of-the-art in experiments that have actually been performed.

1:02:30 It can also be used as the back-view, instead of this genetic experiment I mentioned here, as a background experiment to support this, it has to be done, not in sufficient strength to prove this theorem. I mean decisions with the things were sort of 50 cent per second, 10 per seconds apart or something. The very time you think that obviously a human could have made money in that short time. But it's the industry standard as I say. But as a combined experiment, it proved over. I mean, Koshin actually came up with an argument like this, but without the precision, in 1975, whereas I think it was the G.H.S. . But, you know, Koshin, of course, was one of the two authors of this theorem. He's very well aware of exactly what he did in 1975 and what G.H.S. did in 1990 or whatever. new crew assumes less and you could, you know, as I say, you could work through the GH dead one and assume less and get the similar result and that's important because it will prove that spin-a-half particles also have the same property that they use spin-a-half particles and probably save too much. Hello. Russell. Well, it's not really a fully formed question before I can't comment. I'm actually rather worried about Finn, even though I'm a relativist. Because I think that the, and the thing that these types of experiments indicate is that there's some notion of entanglement notion which is weaker than the, than actually sending information. And there's the notion of quantum information sometimes called, I prefer to use the term entanglement because it doesn't actually involve, it doesn't, you don't say information. But there's some kind of restrictions on what can happen here and what can happen here, which are consistent with relativity. But they're weaker than saying that one influences the other, or that information is apart from one to the other. And I think the way to understand these things... And there's another example, which is this teleportation example, which I think makes it really rather clear that the thing about quanglement is that it can actually...

1:05:00 You use this argument that if to be consistent with relativity, if A can influence V, then you have to go backwards in time. Well, you face this squarely and you say, yes, Krandall can influence backwards in time. But it's not influence in the strong sense that we understand causality. There's certain restrictions on what it's like. I'd love to talk to you about this in more detail. We actually have some sort of, can in a certain sense simulate this behavior passively, okay? And the way we simulate it is to transmit information faster than speed of light. But there can be two different simulations. We call this the concept of a generous model. In one of them, the decision that A is taken first and transmitted to B. in the other, which you don't expect a difference in the natural brain. And decision B is the judgment at first, and taken there. And each of those models, or each of those simulations, is causal. And the two different things, they're related if you like, I mean they're not invariant under the symmetry of special relativity, neither of those explanations. But the facts they explain are. Now that's really, I think, the explanation for what's not the explanation, You know, we would like, when we try to build models for things, we would like the models to satisfy Curie's demands, actually, that they have the same symmetry as the thing that they are modeling or explaining to you. I'd call that a congruous explanation, which can give incongruous explanations, and there's nothing very wrong with incongruous explanations. And all you can say about an incongruous explanation is not the only explanation. There's nothing very consistent, simply apply that, simply get an explanation in the same place. But an incongruous explanation has this wonderful property that it proves that what's going on is consistent. It's a consistency group. It says this kind of thing that we appear to observe can go on, or should we say could go on. Once you've proved that, that's all you need. I mean, it's babyish to ask how the universe does it magically. you know, how, what kind of screen is pulling the earth to the sun. I mean, he very wisely made no hypotheses.

1:07:30 So, if you just reduce that demand, you don't demand to know how it does its tricks, you can find all sorts of ways in which it could do a trick, but obviously it doesn't. But it does its tricks, so fine, just get out of it. The universe can behave like this. Experiences don't behave like this. Can't open it. I mean, that's my answer. Sorry, that's not very effective. I'll give you one more question. I'm not just letting you exercise free words or something. It's pretty bad exercise, right? Well, yeah. Yeah, so you make actually implicit assumptions by not making them part of the model you're presenting there. But you make an exact mathematical statement. For example, you say there is no extra free variable, for which could be another free will. And you also assume, for example, that... If the particles see the measurements, okay, that's all in the information I'm talking about. I was about to say something else. Because if you include the fact that your measurement device might have free will itself, and that there is an interaction between the measurement device and the particle, which might change the state of the particle, so that observing is not an innocent thing. what you're saying, and what you're really doing is criticizing me for... No, it's not criticism, it's just a mathematical statement. But I don't really mean that the particles have creeper, although I expressed it like that. What I mean is the universe has creeper. I mean, you know, which is expressed in physics, so to speak. Now, it's not the particle necessarily. It's the particle plus the measurement algorithm or whatever. That doesn't matter. The point is that this issue, no matter what it depends on, is not there ahead of time. It's not determined by all the information that's available. That's a big deal. I mean, the theorem does prove this. It gets weird. I mean, you know, when you're starting allowing it to depend on more and more things, you know, some of the proofs went down because we didn't consider that. But this thing we allowed to depend on everything it could depend on, And by the way, this doesn't exactly have to be true, you know, if you sort of say, well, information can travel fast and speak about locally or something, you know, that doesn't matter, because the place we're applying it is just to show the terms of this, the terms of these things, and they're far apart, so, you know, even if there were local weirdnesses going on, it still

1:10:00 doesn't affect the truth about theory. It's not determined by all the information available. I think I know... So let's just back down again. But before, did I just mention that our next micro-incident is another Princeton person, in fact, which often they're trying to use it. I guess the Google call is over there at six months or something. How many people have already been there for you? Anyway, the next time we have the pleasure of Google for Moscow and Princeton. Anyway, so let's thank Mr. Tom Lee very much for the evening. Thank you.