Probability, Determinism, Quantum Mechanics

0:00 Thank you. Never mind. Sit here, I'm going to use this one. Yeah. Oh, he's going to use Will. Might be better. I'll sort of beat myself up. Shall we check if they're all working? She's always interesting to see. She's always interesting to see. Now it's here. It's just down. It's just down. Thank you.

2:30 Shall I go along and see my first place in the room? We do. Is this going to be a lucky day? It's going to be a lucky day. It's pretty. It's just good. I wouldn't mind if this is going to be my fault. Yes, yes. What are we doing tonight? We could. Well, I think he is. Shall I go along with the audiovisual? It will probably be a talk and a long discussion. Thank you. Thank you. Thank you.

5:00 There might be a spare bolt, is there no spare bolts there? Normally they all have spare bolts. Oh, ah! You know John? Oh, he's one of those. He's very good at tech. Thank you. We're getting there. I have to explain the problem. The speaker wanted to have two overhead projectors, a whiteboard or a blackboard. But the audiovisual refused to bring a blank board or a white board on the grounds that for some reason they don't supply that. And then one of the two that he requires has now broken down. We hope they're not causing you any inconvenience. Try and get things started as soon as we can. Thank you.

7:30 Yeah, they're both working, so I think it's in the switch on this one. Yes, I can perhaps a while away at the time by telling you an anecdote of the most disastrous seminar I ever organized, which was given by Mr. Tai from New Zealand, and the talk was about Darwin's visits to New Zealand. And the whole part of the talk was he had all these slides of Darwin from New Zealand, which were a literal part of the talk. And unfortunately, it was the projector to work. So we quite literally spent two quarters of an hour trying to finish the thing. And so by the end, I think the talk was somewhat diminished to quarter of an hour. So I hope it won't be quite as bad. Well, he just doesn't get to work. Okay, I'm just trying to spread this stuff. Could it be the fuse explode? Oh no, there's another time I said. Okay, so I figured that's okay. Oh, dear. Well, I have to begin unusually by apologising to the speaker for the lack of the visual aids, which still I'm sure he'll manage.

10:00 Anyway, it's a very good pleasure to have Simon Saunders here, who's a very old friend of ours, and goes back to the old Chelsea days, and those of happy memory, to those of us who can remember that part of it. And from Chelsea, you went to Harvard next, was it? I went to Oxford, to JRF, to this room, and he pretty much stayed in there. No, I didn't know. Your life was more adventurous than I realised. and now he's back in Oxford again. And so you're involved in this Oxford degree in philosophy and physics, which is quite a rather prestigious Oxford degree now. And so, well, we're delighted to have him here to tell us something about all these mysterious subjects, probability, determinants, and quantum mechanics. Thank you. Thank you. And apologies again for our incontinent. It is a great pleasure to speak in this particular seminar series No, I don't mean that, the descendant plane, the one that I'm going to cut my piece off of the tape on, which was originally set up by Heinz Post, a variable time. Okay, so, oh, let's see, we'll have some more problems. It's the lens, it's the machine, it's the machine, it's the machine. The lens in the machine? Yeah, cool. This has never happened before, Simon, I'm afraid. I could have many worlds in the title because this is also a talk about many worlds, but for you, because after all, the many-world interpretation properly understood is a deterministic theory. Okay, so some coordinates for you, which I was going to leave up on an overhead and I'll go into the other one and so on. I'm afraid because I am only working on one overhead, part of my talk is going to be hard to follow, because I was relying on certain definitions being available that you could

12:30 look at whilst I was writing on a blackboard The latter part of this talk may be a little bit incomprehensible perhaps when I want to re-write it Okay, let me also I'll put this up at the end in case anyone's interested The relevant papers here, they're all available from the Lanol archive Los Alamos archive Essentially version of Many Worlds, the truly correct one, is due to myself and David Wallace. Not that we collaborated exactly, but we just haven't fallen into very similar views. Now the particular letter part of this talk on possibility and the relationship to decision theory is, here I've given a list of relevant titles, they're all available online as well. David Deutsch started it up. Three years ago, Barnum and Caves and others weighed in with a response, more or less dismissing what he had done. David Wallace showed how to do what David meant to do, and I then took out the decision theory. Let's just leave probability. Sorry? It seems you were just left with probability. Well, the probability went already. Oh, you're left with nothing. because he went out here already. So it's quite amazing, actually. Evolution of thought. Okay, now I have... I want to just give you some orientation in... I call it the family tree of many worlds, but I could call it interpretations of quantum mechanics because everything that's important, with the possible exception of quantum logic, is on it. I've got a timeline on the left-hand side, and on the left we've got a fairly clear line-up to set through Du Bois and Bell, Haile, Goldstein in Rutgers University, Peter Holland, David Valentini, Primiter Institute at the moment. I suppose that's alive. I'm not sure if it's going well because it's in contradiction to special relativity and you can't do anything in particle

15:00 physics, not just because of the lack and failure of relativity, but because you don't have any model, not one single model of pair creation and annihilation, for example, the most elementary process in artificial quantum theory. No one's been able to produce that, despite the fact that it's now more than 50 years since birth. By the way, I'm giving you some of this background because in talking about many worlds, it seems to me there's a real credibility problem, and why on earth should we take such a theory seriously? And the reason why it takes it seriously is because the alternatives are Well, not just an R, no, alternatively, it seems to me, if one looks at relativistic one. Okay, now the route through, well, I should perhaps put in a few influences. That's next to me to hear of what's going on. Hang on. Einstein, of course, is there a point? Is there a point? I do. So, Einstein, of course, standing up for realism and as a father of relativity, developed this approach explicitly in order to give an account of the artistic cosmology on campus. I've also got some input from London and Bower. They are on the far right here as the sort of mentalistic strand of thought. Well, very far on the right is Vigna and then you'll put Albert and Lerner, and David too. I don't know if I should put David down here, but let's put somebody in and not get the present Is that a fair position for you David? I'd rather be over in the middle Well you may well be and you see it's see it is my

17:30 fond hope that I can get this arrow in here you see The green arrows mean good contributions. The red arrow means some contribution, but you've got to watch it. So I got Schrodinger in as an input to Everett because, of course, Everett did take the wave function as being a real physical object, but more than that Schrodinger was definitely sympathetic to Everettian ideas as I'll show you in a moment. Now this route through von Neumann, Landon and Bauer and so on, the importance as I understand it for the light of the set that I'm interested in, what I've called many sea worlds, is that they replaced Bohr's classical world with a mental, well, the observer as a point of view or a perspective or a point of particular importance within the physical description. And I think it's clear that for von Neumann and von Bauer, it was the role of observation. It was the idea that whatever happens in empirical science, there has to be a point where there is the sentient being who comes in and makes the observation. Now, the key for Everett, of course, was to replace that abstract point of mental observation, whatever, by something like an automaton itself subject to quantum mechanical laws. So, modeling the observer in quantum mechanics. The observer, thereby, does not become on a par with physics that one is using quantum mechanics to model. the observer remains important because one's key to the empirical epistemological input from the theory the key to it is what is recorded in the brain of the observer, the brain of the automaton the data that that automaton actually records and it's there and as it were only there for everyone that you get something like a classical phenomenology Dida Zay in the early 70s made some important input, as did Brian Davis this is the line

20:00 I've got them, people like Nelson Davis, Perl and Diozzi, these are all people in stochastic state reduction I've got them in the same line as von Neumann because it's heavily mathematical and all of these people So to interrupt, is that our Brian Davis? Yes What he did in his book, Quantum Theory of Urban Systems, published in 1976, I think. That was a seminal book, very important. Now, what people like Zay and Davies and Zurek did with this material, I can't have any mathematical sum of it, is really developed what's now called deconherence theory. And for Zeyn and Zurich, and certainly Van Gogh and Hartle myself, that plays a very fundamental role, because what Decker-Hohen's theory is doing is giving you a kind of preferred basis within an Everettian framework. work. Now, if you look at this tradition on the far right, the mentalistic tradition, the point about this is that there's no physics going into the definition of a preferred basis. It's mentality that picks it out. Why we know not. And that's very clear in Lockwood and Arbottom Love, and pretty clear in the image, actually. And the many minds, then, is a good way of putting that way of thinking about things emphasizing this role of mentality. Now if I can just explore this chain going through DeWitt and Deutsch what they did was try to correct the deficiencies or fill in really blanks that Everett left in the definition how to define the third basis and how to select out how to give the count of probability I'm sorry, I'm using this term for third basis, perhaps I need to explain it do I need to explain it? I do, okay so the point about quantum mechanics is that it's it has the character of a vectorial theory the descriptions that it provides are mathematical vectors in a high dimensional vector space typically an infinite dimensional vector space can decompose that vector with respect to an arbitrary choice basis. Now, which basis you use makes a difference in that various basis are interpreted as values of different

22:30 kinds of physical quantities. Momentum is one choice basis, position is another, energy is another, angular momentum is another, and so on. And because these things can't all, these quantities, physical quantities, can't all have these definite values, as in that uncertainty. It really makes a difference as to what basis you use. Once you've chosen a basis, decompose the vector with respect to that basis, then the length of the vector, the component of the vector with respect to each direction in that basis set, is then interpreted as the probability that the physical quantity has the particular value represented by that direction. okay well I hope that's true enough I mean I can't really go through basics quantum mechanics it won't be terribly important that you understand quantum mechanics the rough picture will do for at least part of what I'll be talking about okay so DeWitt and Deutsch they tried to define the preferred basis they both didn't by new postulates new axioms, and that was quite clear in Deutsch. And with those new axioms, specifying what the basis is, Deutsch also introduced a stochastic process, and so on. So this, as I see it, is disastrous. The only thing going for many worlds is if it is conservative with respect to physics. Otherwise, why on earth should one believe such a thing? There's no point in taking it seriously. If you're going to put in new constituents, new axioms, into of quantum mechanics, then you might as well do it in such a way as to recover a one-world stochastic state production theory. So for me, it's a red line coming from Deutsch. David Wallace, though, well, may I also add, it was a red line because I didn't particularly like talk of worlds at all. I'll come on to a particular perspective on an error that I've been developing over the last ten years or so in a minute. particularly oriented towards worlds. Now, for David Wallace, talk of worlds was quite useful, quite explanatory, and he actually showed me how to take it on board without any quants. So I'm happy to call this line a many worlds theory. I've got a little C in there, and I'll explain that in a minute.

25:00 Just to complete the story, from Bohr through Heisenberg, Zeilinger, Caves, various people of quantum computing, there is a form of neo-co-paganism lying well. It's strongly idealistic. Not in the sense of classical idealism, I suppose, but it's anti-realist. It's the view that physical theory is basically about information, information exchanges, and there's provided a physical systems in quantum theory. Okay, so that's some background. Let me now make clearer how I understand many worlds. Forgive me for focusing on my own interpretation, even if I don't, who will? So, the key idea is that worlds are approximately defined decoherence is only a way of extracting certain kinds of approximations from quantum mechanics the worlds are centred I mean that in the way it's used in standard epistemology so worlds with a distinguished point Now, the right way to think about this, I suggest, is in terms of Minkowski space-time. Minkowski space-time and the matter content within it gives a unified space-to-space-time description of phenomena. but to recover anything like an account of our experience one needs to introduce distinctions between the past, present and future and one does so relative to any point any point event in the cosmic space one then has the light code structure and one has a perfectly invariant definition of what is past in terms of what is past time-like,

27:30 future time-like, and space-like. Given a congruence of roughly approximately parallel world lines, one can define, in a very point, an approximate foliation where one's got a public space, a public present, which is intersubjective with respect to those, that collection of timelike lines, as long as they're approximately parallel to them. As soon as you've got them having no structure at all, then there's no basis on which to pick out one for the issue of another. There's no way of slicing up space-time in an intersubjective way. And that's roughly what's going on, as I understand it, within Everett's approach to quantum theory, that to recover anything like an account of, well, is actual, actual becomes a term rather like the past, present, and future. And one can define it in terms of a relation, just as one does in costly space-time, which is time-like, past, future, time-like, and so on. One has a value definiteness as a relation, it's a transitive relation given certain assumptions from D.P. Haynes' theory, and one's started now. It's not a symmetric relation. Reflexivity isn't the problem, but because it's not symmetric, there's no way you can get an equivalence relation out of it, no more can you in Mikoski space-time. There is no non-trivial relation which is both symmetric and transitive in Mikoski space-time, which is why you can't decompose it into a collection of times in the way that you can determine space-time. And it's the same in Everett, there's no equivalence class of worlds. So you can't decompose the universe's state into a collection of worlds in a natural and invariant way. But just as you can do it in an approximate way in Mikoski space-time, given that you've got some fairly stable sort of structure, you've got reasonably parallel, approximately parallel world lines, so you can, in the Everett framework, I'll try to say a bit more about that, it is going to bear on decoherence theory, and that's the more technical side of this whole approach, so difficult to summarize.

30:00 the other feature of the approach is that probability is basically to do with self-location it's epistemic it's a matter of the probability in the sense of that with which we are ignorant is a matter of where one is located in the universal state and because of the bifurcating structure state, given the basis singled out on different coherence, one is constantly in a situation where one doesn't know where one's located, one doesn't know which branch one's in, but the universal state. So one opens one's eyes and looks around, one listens and observes and so on to find out, and this would be true even given that one knew all that there is to know, perfect knowledge, and so on, you would still need to locate yourself to find out where you are and we're terribly located okay now let's see let me say a bit more about this issue, but I miss this talk is going to be about I've spoken so far about the motion problem so the problem is in the sense of that controlling that of which we are ignorant There is an objective correlate to it, and epistemic probability getting right on that is to try to get right on what the objective correlate is. The objective correlate is the ratios in the modulus square of the branches in the universal state under the unitary evolution, the unitary evolution being the standard on mechanical division that determines everything except what goes on in measurements. so this is supposed to fit in with something like Lewis's framework whereby mechanical probabilities to be given this is now objective probability not credence in terms of categorical properties properties which aren't for themselves somehow in some state of potentiality or indeterminacy or something that is not itself explained this is supposed to be the kind of probability that's entirely transparent

32:30 the branching structure to the universal state is basically determined but it's perfectly categorical as the unitaryly evolving wave function is as real and objective as anything is as it unitaryly propagates develops into a superposition, there is the point of branching, and the ratios in the amplitude squared then give you the relative probabilities of one branch over the other, this being the objective correlate to our step-of-the-art analysis. So when we're trying to get right on the what-to-believe, degrees of belief, what we're doing is trying to get right on what those ratios in amplitude squared are. Part of the point in appealing to Lewis's framework here, and putting it in these terms, is to say that the interpretation is driven by conservative metaphysics, conservative philosophy, as well as by conservative physics. Again, the same reason. There's no point in putting some philosophical principle that is a problematic one, a controversial one, in support of a theory of many worlds, because the theory is unbelievable. So the only way a lot to take it seriously isn't as driven to out by entirely conservative principles. That brings me on to the issue of decoherence. What decoherence really is, and we'll say a bit more now about it, is it's choice of basis in which one's got approximate laws, approximate equations that are informative about the sequence of events as along particular branches with respect to that basis. Informative in all of the ways that one would have a similar view of any other several equations used in physics, or indeed in any in present science. So one really is supposed to be instrumentalist about this. These equations are concise, effective, quick and dirty, if you like, ways of characterising the structure of the universal state. More or less in terms of what happens, if you imagine a tree-like structure, that's the typical liberating picture, there's a point where one has a trunk, which

35:00 what is value-definite relative to the point of initial bifurcation. Bifurcation into branches is the future. This replaces the usual deterministic picture where we are a local event in a single time-like world. And now, if one picks any point in this branching structure and develops a sequential continuous trajectory either forward in time or backwards in time, What sorts of, what is the sequence of events? What equations will tell you that sequence of events? Effective equations. And one of the most important issues here is that for any equations like that to be operating, you better not have interference between branches. And that's the key feature of a decoherence basis. It's one in which you don't have to worry about other components in that basis. evolution of those other components, they are, as it were, non-interfering, well, exactly non-interfering. It's exactly the right term to use, non-interfering components of state, and the basis of such is non-interfering. Now, more than that, these equations need to... there's additional assumptions that are needed in order to have effective equations, locality being achieved on the moment. Once you put in locality, there's every reason to think that this decoherent space is effectively unique. And what locality means is just the equation should take a local form. You want to know what happens. tracing a path of trajectory through this tree-like structure you don't need to have global information in the branch you can work locally if you wish to Okay, let's see. Well, let me say that it's just a fact, proceeding in this way, that what you end up with is quasi-approximately classical equations of motion. And it's an open question whether there are any other equations of motion

37:30 that one can extract that will describe the structure of the universal state. Maybe there are. if they are, and if there's any way of correlating that sort of data with data that we can access to find with respect to the quasi-clantical decoherent basis, then the bet is that this is good, useful physics. We can explore this in the laboratory and let it be Nobel Prizes all around. It's not that somehow thereby the program is upset. one is just learning more structure in the universal state that we can access. Of course, if it turns out that we can't access it, then that would be a shame, but again, it poses no problem as such. I've put up some comments now, this one from Bohr, I have some also from Dirac and Trulia, just to show how this feeds into a certain kind of pragmatism that's been typical of protecting from the tax from the beginning. Here's Bohr insisting in the misconception to believe that the difficulties of the atomic theory are needed by eventually replacing the concepts of classical physics by new conceptual forms. Indeed, it's already emphasized the recognition of the limitation of our forms of deception and by no means implies that we can dispense with our customary ideas and their direct form of expressions and reducing our sense impressions to order. No more is it like the fundamental concepts of the classical physics becomes a description of physical experience now this is Bohr really dogmatically insisting that we use classical concepts where he differs I mean you can read Everett as likewise insisting on the use of the right decoherence basis to extract physical content from the theory but whereas Because Bohr's point is really saying, look, we just can't think in any other way, the Everettian point of view is rather that we just are constituted by physical processes like that. This effective dynamics just is what we are constituted by. And perhaps another way of putting it is that biochemistry itself is our medium. you couldn't see the biochemistry as effective equations of motion operating along particular branches

40:00 and you see how the instrumentalism kicks in. Any kind of system which is composed of sequences of events like that just will be subject to the sorts of laws that we see going on all around us. So one needs to, in a certain sense, oppose that sort of instrumentalism with what? I guess a metaphysics of laws in order to break down that picture, in order to somehow say this picture is incoherent or unintelligible or unsatisfactory or whatever. But you need to oppose it with actually quite a strong, I think, strong metaphysics of what the rules are or could be. Because it's more or less instrumentalism that's giving us back the classical as we see it. I've also got a comment up here from Bohr about relativity, because I think, and Bohr certainly modeled his documentary, what he thought he was talking about. he modeled it on the what he thought was Einstein's methodology in special relativity and what he said here and I'll read from halfway down especially the singular position measuring instruments in the account upon phenomena just discussed it is closely analogous to the well-known assessive relativity theory of a public and ordinary description of all measuring processes including a sharp distinction between space and time what I think Bohr had here, and in common with a lot of other people, including very gifted mathematical physicists like Dirac, is I think a mistaken view of special relativity. They didn't understand that special relativity could be made out as a theory of purely invariant structure, coordinate frame, whereby one only speaks about the American space-time relation between points and never has to introduce commercial coordinate systems and so on. And it's the same, again, within the overarching framework, but one, really, this decahune structure is an invariant structure. It's not basis-dependent in any interesting sense. Roughly the same way that a cloud of gas,

42:30 thermal gas, picks out a preferred basis, one in which the temperature is isotropic, for example. You define the basis in terms of physical structure. It's not something that comes to its next machine. And I think that's something that wasn't appreciated about relativity will be in classical period. OK, well, let me pass on now to talk more in particular about policy. I've already said a bit about it. What is the objective correlate to epistemic probability is branching. There are classical models of this, and let's look at those. I think that our project, as it were, is to either show the framework to be coherent or to show that, on the contrary, it works out as being rather, it has its own intellectual integrity. And classical models, I think, offer a pretty good way to explore the difficulties. The first model I would offer is Parfitian fission. So, we're well used to that as being quite an active field of debate in philosophy, mostly in the field of ethics, but it works very well to understand quantum probability. So, what is the appropriate attitude to have if one is about to undergo a fission? fission, and by all means make it ethically symmetric fission, so it depends on the difference it's operating, what should one expect? So that's a debate I quite welcome. I was going to scribble on the whiteboard, but I'm going to do a little bit of transparency instead. My suggestion is that the following, that there is a trichotomy here which is exhausted, one thought that one can have is that one should anticipate oblivion in the face of arphidium fission so this is, one's going to fission into two perfectly symmetric, and of course it's science fiction but I hope

45:00 that issue has been much discussed in arphidium fission in the context of ethics but I hope you will agree with me in the context of assessing the Everett approach to quantum mechanics the science fiction character of arphidium fission is irrelevant suppose it can be done What should one's ask to be? So one thought is oblivion. The second thought is telepathy, or just something extraordinary. One should expect something extraordinary. One should expect extraordinary experience. and the third is normality we should expect normal experience now it does seem to me that these are exhausted obviously you could be half way between two and three it's not quite normal but it's not so extraordinary either it seems to me these are exhausted and it seems to me that oblivion is unreasonable I mean this is to be we would like to argue for this rather than just a search and it seems reasonable and what seems unreasonable there will certainly be descendants of oneself who talk as though they survived let's put it in those terms by any behaviouristic analysis there is survival by every method one might use to get an ordinary survival in order to conclude without any fission going on that as well one has survived that there has been continuity and so on apply those same tests and you will conclude the same thing there has been a survival everything seems to be ordinary for the same reason telepathy is ruled out which isn't any telepathy going on. There is no extraordinary experience that anybody is reporting. There's no way to get that extraordinary experience. There's no ordinary experience going on. So hence I say normality. But if you do go for normality, then you go for probability. Because the only way to go for normality, if you expect normal experience, you'll have to be divisive two. The only way to understand that is to expect to be one or the other.

47:30 Can I say the case clearly? You're the case where they say the Star Trek beaver makes two copies of me on different positions. I'd rather not do it that way, actually. I'd rather do it that I'm just very rapidly, biologically cut right down the middle and reconstituted so I've got two perfect survivors. so I'd rather do it that way but if we have to go into the science fiction details you prefer the second technology I just wanted to get a face clip because it's not immediately obvious why I expect that I will become two people who have some more experiences it's not immediately obvious outside the original context why probability comes in the kind of expectation I'm talking about is the sort of expectation that we have about whole new lives. What are we expecting now? It's that sort of expectation. And if you expect normality in the memoir, then you can only expect to be one or the other. But let's talk about it at the end. This is one plastic model. The second one is Sleepy Beauty, and you probably, I hope you've all come across this. Let me just draw a little diagram. One's There's going to be a coin toss, and if it's tails, I don't know which way it is, then you're going to be woken up, you'll have whatever, you know, a chat, a meal, whatever, then you go back to sleep again, your memory will be erased. Then you'll be woken up again, same thing, go back to sleep, it doesn't matter if your memory is erased at this point or not. Now if the coin was heads, you were just woken up to once. okay, and so this day this is Monday this is Tuesday and the point is Sunday, this sort of thing so this has been discussed in pages of analysis Alba who brought it up David Lewis responded to it and the question that was posed then is when you're woken up what probability do you sign into it entails and most people break down into their either halfers or their two-thirds as a fair coin you know, I do say it's going to be a half before you're put to sleep back here at this sort of stage for sure you say the coin is going to

50:00 land tails and half but what happens on an occasion of wakening, what do you say then? And the rationale for saying probably this is a third is There are three occasions of weight cleaning, and there's some sort of principal difference operating as to which one this is. Actually, can I just... Does everybody understand the scenario? You only lose your memory, not completely, but just for that section. Yes, that's right. You remember everything that went on here. Oh, yeah. But you just... What was erased is that you were awoken here. That's the point. disawakening, you do not recall that you were woken here. So, on an occasion of waking, everything is indistinguishable. So, the half thing is when you wake up, the first thing is that if you bet on tables every time you wake up, you'll wake up. In fact, you'll win until the odds get 2 to 1 yet. Sure. So surely you should think 10s is 2 to 1. Yeah, from a betting point of view. But the thing is, betting may not be a good guide to the probabilities now. Because this is a kind of a bet where you've got to bet again. If it is 10s, you've got to bet twice. So it's a funny situation in betting. It's betting where you've got to double your stake. Yeah, I wasn't going to want to say that. I mean, can I just have a show of hands? I'm always fascinated by how this goes, and who thinks at heart? So what exactly is this? On an occasion of awakening. What creed is you attached to the coin? You know everything. This is the protocol. You are here at this point. You know exactly what's going to happen. The coin will be tossed. If its tails will be opened twice. On the second occasion you will not recall the first. If it's heads, you'll only work them once. On an occasion of awakening, you will be asked what is the probability that the coin landed tails? I've been thinking about this, I can see that it's important theoretically, and I'll still let go of the answer to that. Well, you don't have to, I'll go for a... If you bet against me, I understand I always bet, of course, it's half up, and I wouldn't

52:30 lose, I wouldn't lose anything more. You'd just lose half away. You wouldn't lose. Certainly you would lose. If you repeated this, you did this every week for a year, you'd lose all your money, for sure. If you bet a half, you'd lose. Because if you bet a half, in this occasion, you have to bet twice and you're losing both times. In this occasion, you only bet once. So in the long run, you will certainly lose money if you bet a half. But, as I say, betting may not be the right guide to the probabilities set up, you're having to double your stake if you're wrong, you double the stake under that situation you have to, as it were, adjust how you bend, you break the usual limit but just a show of hands who would say half who would say a third two thirds and who doesn't know well I don't know I don't know I can just point out Let's make it, not a coin, but there's one in a thousand chunks now that it's heads. We'll make it a one in a hundred. Let's see. Let me make it 99 out of a hundred chunks that you go this way. And you actually die. You lose your house, you know what you're worth. okay so and here you will rework and you just repeat this you know a thousand times a thousand times now by the same logic for those that voted two-thirds you should now think that you've got a 10 to 1 chance that it's tails over heads okay you see where I'm going with this one can as it were create whatever problem is one can increase this problem enormously just by you know constantly being And you can have some terrible scenario here, which will occur at 99 out of 100 per building. Terrible scenario. But if you make this sequence long enough... But on the left hand side, there's no other way to do that. You don't lose anybody. Well, and I think... I haven't talked to you. Well, let's not have that.

55:00 No, but the problem is you've got some extra information, you see, if you've worked it up, you see. You can't say it's a prime problem. No, you don't. You know that you're going to be worked it up. I know, but you're going to be worked it up. Why are you stuck in doing it? For the thing to work, all that's going to happen, you're going to wake up and know nothing else at that moment, and then you're supposed to bet. So why is it relevant for us to know beforehand that if it's a 99% chance that comes up, then after you've worked it up, terrible things are going to happen? I don't quite see relevance to that. Just following through the logic of those that said two-thirds, those that said two-thirds, that logic would also imply that when you have this situation, 99 of a hundred chance of a single occasion awakening, one in a hundred chance of a multiple situation. If you have enough of these, say thousands of times, that will give you a chance of ten to one, you're in this branch, as opposed to this one. So it may be a terrible scenario. You could make a million awakenings. Make it overwhelmingly, on the logic you say two-thirds, overwhelmingly you can take this part. So even if the scenario goes to use your house or whatever, this is a risk you would prefer to take. This is just the classical version of quantum physics. Can I just indicate the obvious connection, I suppose, with Parfittian branching or with Everettian fission. Sorry, I said the wrong way around. Parfittian and Everettian branching, is that what's really going on here, the point about administering an amnesia or whatever, so that you don't perform, from an Everettian point of view, or from a Perfittian point of view, this guy can't communicate with this guy. people. Really, what something like this is, where you're going to be woken up, put back to sleep, and then memory arrays, woken up again, put back to sleep, and then you're awake as normal. What that is, is exactly that situation. The point about the amnesia is that although you're temporarily related here, to the family apart, you're a world line, you can't communicate, likewise here. One can't communicate this here. And if one has then fusion, this would be an exact corollary of the ongoing illness. And one can then repeat this argument in terms of ever actually branching. Okay, so these classical examples are very useful,

57:30 I think, for fighting out philosophical debates as to whether this notion of is really intelligible and that's we can leave it to discussion let me come on to what I really want to talk about I'm actually I not only have I run over time but I thought well look maybe it's just as well because with only you won't be able to recall the definitions that I'm going to use in order to follow the mathematics of the second part of my talk. So I'm cutting this short, the second part of my talk, because, as I say, it won't really be intelligent to you, but I'll go through the basics. The thing about the branching, if one thinks of it, well, think of it classically, if you've got perfectly symmetric fission, then sure, there's a principle of a difference. But in quantum mechanics, we don't have that. We don't have a perfectly symmetric branch. It's the amplitudes that matter. Why? Why the amplitudes? Why the amplitude squared? Why should that be the thing that dictates probabilities? And of course one can say, well, it's just a hostage to the theory. and indeed in my earlier writings on this that was the line i was taking that one never gets an account of what probability is that that explains why something is probability outside of games of chance where you've got high symmetries where you've got the high symmetries then you can do it but even there the dynamics comes in how do you roll the dice how do you shuffle the cards. I mean, that's absolutely crucial to giving any account of probabilities. So I thought that in every case, in every case we know of, you always just have to postulate that something is a measure, an appropriate measure, and you justify it in the same way you justify any scientific hypothesis. But to my surprise, this is the line of argument now coming down through Deutsch and Wallace. It does seem you can do better in quantum It's the amplitude of the matter, and indeed, why it's the modulus square and the ratios of those that matter. Now, let me just start it off. I'm going to give an operational account of this.

1:00:00 Later, I'll say why, from an overarching point of view, you will believe the operational assumptions. On certain other approaches, stochastic state reduction theories, for example, you might question the operational assumptions. There's an explanatory gap of consequence for those approaches. I think for the de Broglie-Gwenk theory, there's no such, not really, an extended gap of the same kind. I mean, that will benefit from this analysis just as much as everyone will, because it is just a certain operational assumption as to what constitutes a measurement process, given which one proves that it's the modulus of the entities that matter. And here's the operational account of the measurement, that there's some physical system measuring device, channels, so D inputs to it capital D possible outcomes each outcome is some physical configuration of the instrument, one outcome that gives you a strawberry and a mug outcome that gives you a candy blossom and a banana so one really can get away from mathematics and numbers here but I do want to use these outcomes to bet to them, so we can think of them as new ones these are new ones so the macroscopic events now the idea is that capital M is the experiment performed when all the channels are open and M subscript K is the deterministic experiment that is performed when only the K-th channel is open, the K-th input channel is open and the idea is that when you do have only the K-th channel open, the others are all closed, literally by shutters just a single channel is open, the instrument functions deterministic always gives you the same outcome. Now let there be identifiable regions of the state preparation device through which the system to be measured must pass, if it is to be subsequently detected at all, call such an experiment a multiple channel experiment. And the way to realize an experiment like this is something like an interferometer where you've got two arms in the from it you've got a beam splitter that's an intimate incident particle beam close off one of the channels with certainty you get one outcome the same outcome every time close off the other channels with certainty you get some other outcome every time now you let both channels open what happens now you don't

1:02:30 get deterministic events okay okay and then So let's suppose that there's an algorithm such that given an initial state psi, given an observable to be measured X, and given omega, some procedure, which boosts up the eigenvalue that one normally thinks of as being measured, to give you one of these U's that I was talking about before, you know, the macroscopic outcome. So the idea is there's going to be some algorithm that is a map from initial state psi, observable to be measured of x with some amplification device omega-yielding macroscopic outcomes. And now that map is such that it equals the sum of probability as pj times the uj, the macroscopic outcomes. I said I wanted them. They don't have to be mathematically meaningful at all, but I do want to bet on them, so let's make them have the sets of new rules. Okay, with the pi sum to 1. no good from an Everettian point of view because who says the probability makes any sense? I'll come back to that in a minute. That's where the decision theory comes in. But if you can just bracket that for the moment, suppose that this is being specified within Copenhagen, in Barrett theory, whatever. So now the issue is, what is this mapping? And what are these P's, these probabilities? Okay, I don't think you'll be able to remember this, so I'll not go on to the little subtlety there which is about degeneracy. I'm going to now give you, I made one assumption already, namely when there is such a mapping, all you need to do is specify the state psi, operator X, amplification omega, and the algorithm functions with that alone. Okay, that's a radian assumption. But here's a second assumption, which is consistency. Here really is just a definition, okay, as to when does an experiment M realize such trickles.

1:05:00 Okay, and I'm saying it does so if and only if, for some region R of the apparatus, the orthogonal state is phi k. Phi k is, I really want to say, the initial state of M k. mk is when all of the other channels are closed and the instrument is performing deterministically so if phi k is the state of mk in that region R I did have a definition two I had a definition one which just gave a deterministic case what happens in a deterministic case so this is really covering you both So, granted that with phi k in region R, with all the other channels closed, we get value lambda k, and we get outcome omega lambda k. Granted that that is the case. And granted further that with all the channels open, the initial state is psi. Then the measuring instrument realizes this triple. Okay, I hope that makes sense. This is very nearly it, so there's not much more to say. Well, I'll just say, consistency condition. This rule, this V, this mapping of these triples, these things, is consistent if and only if it gives the same value on G and G prime whenever G and G prime are realized by the same experiment. So the point about this definition here is that you might have distinct triples that satisfy these conditions because there may be different regions of the apparatus R. I could pick one region of the apparatus R and these will be satisfied for some phi k's and some x, then I can find another region of the apparatus where these same conditions are satisfied, but with different phi k's and for a different x, with different magnum values, lambda, and possibly different O again.

1:07:30 So if I can take one and the same physical apparatus as realizing two distinct trickle-webs, As long as the value rule gives the same value, then the value rule is consistent. Okay, and it is evident, isn't it, that in the deterministic case, all the channels closed except one, that the value of this thing should equal just omega-namb decay. Deterministically, you get that value of lambda-k boosted up to the macrostatic level by omega to give you this outcome and then go down to K. And then, so I think this is self-evident, but then this claim, if it is consistent, then you get formed. This needs a little bit of explanation, because what is omega? What does that mean? Well, it really means that what this is omega and k Okay, that's really a definition of what I mean by omega there. Because omega is a map from eigenvalues to macroscopic quantities, and we can't talk about the expectation value of a macroscopic quantity, it's not defined in our brains as such. This is a really short answer. Okay, so, and I think this is what I, I, perhaps let me just give you some indication of the proof. The first point is, and really this is what the proof works with, these four consequences of consistency. Once you've got these, then the rest is just algebra. And all four of these are obvious consequences of the board rule. If B is the board rule, then certainly all of these things are special.

1:10:00 The first point is this. For an invertible F, and that's really self-evident, isn't it? because by consistency, if some apparatus realises this triple, then that same apparatus can always realise this triple. The state is the same. It acts deterministically in the same way, except that whereas here you get eigenvalue lambda k with payoff outcome, omega lambda k, you're getting f of lambda k, and now the outcome is omega f to the minus 1 acting on f lambda k which just gives you omega lambda k which is the condition that you have here for the apparatus to realise the triple so the apparatus realises one triple and it realises this triple and likewise for the projectors, I mean this is where you need to go back to the definition of consistency or what it is for an apparatus to realise a triple in order to see and on the basis of them perhaps I should give you this is what I was just going to put a couple of equations on a blackboard to show you this, because as it stands it's too much to take in But here we have, given that this is the initial state, we've got some operating X-remeasuring. Now, I'm applying a permutation. I'm afraid I need to think I've really got time to do this. Let me just go through this fourth one, this constants of consistency, which is saying that for any permutation of these spaces, okay, so pi is any permutation of 1 through D, then the value of this triple must equal the value of this, where I've got the permutation acting on the side

1:12:30 and I've got the inverse of the permutation acting on the operator. It's two or three lines to prove this statement, and I hope you can believe it. Basically, if you're flipping say two of the states in which one's going to supposition, you're flipping two of them. But, or just take let's just take the deterministic case so if phi k is one of the confinants here you get x is an eigenstate of x given the eigenvalue lambda k now if I commute that into phi j but I'll still get the outcome omega lambda k if I simultaneously commute the eigen the eigenvalues if I commute those eigenvalues then x will still give the same value on a commuted state that this x gave on the original state and therefore the macroscopic outcome will be the same okay, and now if I'm using that up here to conclude that the value of this triple will equal the value of this. Now I'm permuting the state here and simultaneously taking an inverse permutation of x. And now I'm applying the first consequence of consistency, the one that I did go through, that if I map this thing under some function, I'd simultaneously map this thing under the reciprocal function, and I get back on slightly. So that's what's going to be this here. The state here is the same. and now the function here is actually pi times this and therefore this one here okay now so this is it again I've rewritten it here under the assumption that psi is actually invariant under the permutation with permutation x and the basic condition c k's, then modulus square is the same. Under that assumption, then psi is the same on these two sides, which is this equation. Now, if we

1:15:00 were assuming that this thing is given by some real numbers summing to one times the macroscopic outcomes, so applied to this side I get this expression, on this side I get this, where I've got the permutation acting here. so this holds me any permutation if I interchange j and k then I'll get a sum which is the same on both sides the only difference will be these two terms on the left hand side and these two terms on the right where here I've got say I'm interchanging j and k here I've got omega j, wj, omega lambda j that's this one for k now here is the permutation acting so I pick up omega k omega wj capital omega lambda j And if these two quantities are different, then this forces the probabilities to be the same, the wj equals the wk. And this concerns the case when the degeneracy. Okay, so this is actually a really quick result. It was first obtained by Deutsch under the derivation that was disputed, but it's showing you that you really can't have a probability rule that keys to anything other than these amplitudes. If those amplitudes are the same, the modulus of the square is the same, then the probabilities have to be the same. And you might think, well, surely one could have the probability rule, you know, just make it the biggest value of the observable that you measure is the one that is the most probable. or that the best, the nicest outcome is the one that's in this problem. You know, just any arbitrary rule. And you might think that that could be consistent with these operational assumptions. But they're not. Those sorts of arguments are not consistent with these operational assumptions. So I've really got to be quick, haven't I? So one thing, two final points. The operational assumptions are obviously satisfied in the average interpretation. As long as you've got a black box, as long as it functions in such a way that deterministically gives you a value on some input state, then it will, when you take a superposition of those input states, then it will act to measure, give you a measurement of,

1:17:30 it will act, within the average framework, it will act to cause a branching where the amplitudes of each branch will be the same as the amplitudes with which the initial superposition will spin into the instrument. It must operate that way under the average interpretation. So the operational assumptions are unsatisfied in the average interpretation. Now, the other thing is, well, but in the average interpretation, maybe the probabilities don't mean anything. Maybe one isn't entitled to this notion of probability. So I did assume one could assign probabilities to outcomes. And then the issue is, well, what are determinants given by parts of the born rule? But what if you can't meaningfully assign probabilities to the outcome? If that is disputed, then and only then the decision theory kicks in. Then and only then is there a role for decision theory. I think let me just very quickly illustrate how that. I mean, the kinds of decision-theory assumptions are additivity and zero-sumal. Okay, this is the definition of additivity. It's a payoff. We call it the payoff function. We run around the decision-theory that talks in terms of games and so on. Now, let f be such a function. v is additive if this is satisfied okay it's really the short thing principle they've given two games call one of these setups a black box if given two games it's exactly the same except one of them receives an additional utility over S whatever the outcome is then one should value that game as having an additional utility over S so the value is an additional in this thing if you negate whatever the utilities were, whatever they meant to you beforehand, you

1:20:00 now have some qualifier, you stick a minus sum in front of them, then the value of the gain is negated. The sort of rationale here is if banking is a form of gambling, the difference between acting as the gambler who bets and as the banker who accepts the bet is that whereas the gambler pays a stake in order to play and receives payoffs according to the outcomes, the banker receives the stake in order to play and pays the payoffs according to the outcomes. The zero-sum rule is the statement that the most that one will pay in any utility is the least that one will accept to take this route using it. Okay, now, given those two rules, one can dispense with any assumptions about probabilities for half points to deduce that the value of the game is again given by the problem. That's quite quick. I'll just put it on there. Transparency for you to see. Okay. The permutation pi, before we had over good times the permutation pi, I'm just doing this for the two outcome case, so f of s is adding some quantity s, let this be what one adds, for lambda 1 and lambda 2 of the, likewise the outcome. permutation is now given by this thing and so apply additivity to use this now because omega is additive get the minus sign here and now apply the zero-sum rule to that to get this minus sign here and just take this over to this side you're left with omega s omega versus this thing it's a negative sign in France so you get plus twice this thing equals this so again you get in this simple case now what these actions of decision theory probably do need is the idea that you care about what's the notion of an outcome I think that's why you need these classical models to be convincing that are facing fission classically it makes sense to expect normality. You expect something or other. Maybe probability doesn't make sense, but

1:22:30 in the face of this sort of uncertainty you can then apply these actions of decision theory and that will give you an ordering, a value ordering over the games, and from that into one-known theorem from von Neumann, you can reconstruct a probability function so that you will act, you will order priorities and choices of these games will be as if you assign probably exist in the very same colors as given by the moment ok so thank you very much we're sorry to go over thank you OK, then, so, thank you, everyone, just to get in there. If I wasn't right with the decision theory, the idea is that you make some very obvious assumptions, like an agent will value bets on physical and ethical situations the same if one pays more than the other in the same circumstances you value that category so stuff like that, and then you're going to show that this person will bet as if they had the reasonable leave in line Yes, that's right. Such a rational agent will value games and will wish to play one game rather than another game, as though assigning these probabilities to the outcomes. And that's kind of striking and powerful. Yes, yes. When you do it without the decision theory, what do you establish then? Again, well, without the sort of theory, one's assuming that the outcomes have probabilities. The issue is, what are those probabilities? And then what is established is that they are given by the wrong rule. Right. Assuming the probability... So you must make a assumption that the probabilities

1:25:00 are somehow grounded in the physical reality. Oh, sure. Well, I mean, one's assuming that there's some rule that assigns probabilities to outcomes such that you give me a triple, an observable x and an omega, I will then give you basically the expectation value of the outcomes. So this is like classical probability. Humana, I remember in the matter of chance, observed that if you made a supervenience assumption about probabilities, that the chance of an outcome must have been on the physical situation. So two individual situations must be the same chance of the same outcome. He points out, well then, you could infer that a symmetrical corner must have a few percent chance of coming up in the case of Taylor's because it's two individual situations that are just the same. Well, depending on that's true. But if, I mean, frame might reintroduce asymmetrics, So, one always hoped, in classical probability thinking, that you could get the probabilities out of, you know, here's the thing, it's symmetrical, it's got six sides, they must have the same chops, of putting them to one side a base and it was introduced by the throwing. And then the worry was always, but there's too many ways of cutting things up, that it's not always obvious. that there are six different ways to come down, and perhaps we can come up with eight different ways to come in rather differently. And so what this work does is show that, in fact, one can do better than it seems. from the point of view of the macroscopic classical religion. Yes, we can do that. Can I comment on that? The thing is, what is it about counting six rather than eight or twenty or whatever? It again comes back to the dynamics. How it's thrown, how it lands. And it's only when you analyse that dynamics that it becomes clear that it's the six that matters and not the number of corners in which it's eight. whatever colors that are on the dice

1:27:30 it's when you analyze the dynamics now what you can't do classically is say let the dynamics be exactly the same on each throw dynamics is exactly the same then you can have this coin that buys a man exactly the same way in each throw so how do you how do you get it to be approximately the same and that's when you have to introduce a probability measure on the initial conditions of the dynamics you have to do it now here you don't you can have it exactly the same physical process every time because it's a deterministic branching that is produced exactly the same physical process every time that's the amazing it's an amazing additional way of dealing with the probability of the word that allows you to do better than you can for this reason so you don't need any to probably measure over the dynamics to get the big going Could I make a comment? It's a question, actually, because I don't know a lot about many worlds, but it seems to me that with many worlds, if you are interpreting probabilities in terms of many worlds, and you want an objective interpretation of probability, then I think it's got to be different from the standard ones. it seems to me can I put the argument and see whether you agree I mean in the standard objective interpretations you have a set of repeatable conditions which gives you a set of outcomes over again, this is called God's model so when you repeat these conditions over and over again, and the probability of a particular outcome is the frequency with which it appears in finite frequencies in defining probability well roughly I don't want to go into the details of finite frequencies, but just, I mean, the point I'm trying to make is that it seems that that sort of basic model, never mind the exact details we're going to, isn't going to work, because the trouble is, if I understand the many worlds, I mean, you know, there are a number, we're at this point, we can move into a number of different worlds, we actually move into this one, say. But then the other ones, it's not that they appear at some later stage in the process. Whereas in the Kolmogorov assumption, forget the frequencies, but the key point is if the outcomes are one, two, three, four, five, six, on this repetition you get one, but some of the other possibilities may turn up later.

1:30:00 but I think maybe I'm wrong I don't know anything about many worlds but just reasoning intuitively if we go into this world then all the other possible worlds that they're just not going to appear are repetitions or does even repetition make sense so it seems to me that you may be moving into I mean that may be a good thing you may be moving into some new interpretations maybe I misunderstood The point is that if you take one of these, this branching structure, you trace a line through it, through the tree, I usually go down, and then it's a new, again, you hold the tree, and then you go down, it's a new. Now, you've got a sequence of outcomes, if what's been going on is repeated trials, then we've got a sequence of outcomes. And the relative frequency will depend on which branch you're following down as to what their relative frequency is. You get all branches, you get all relative frequencies. Including the road ones. Including the road ones. If you choose, if the branch, you go down one branch, the other branches can't appear. No, indeed they don't. Or have I misunderstood? I mean, they just... They never join again. They never join again. But this is why I don't understand about what you're saying, Tom, because, look, as you go down one of these branches, you get the repeated trials, you get the unique outcome of the repeated trials. Yeah, but you see, if you're giving some kind of measure over the possible worlds, if you go down one, the others just never reappear. Whereas in the standard model, if you give a measure over these possible outcomes, you get one this time, So there seems to be a difference there. But on repeated trial, you'll get another outcome in the many-months framework. Well, in that case, are you saying you can't actually design probabilities to the worlds, only to certain outcomes? Well, typically, you can if you can design probabilities to the worlds. Well, there'll be a amount of correspondence between worlds and outcomes, and that's the whole point of the story. but all the outcomes will occur within its own world well, hang on, but you get I mean, it's not the case that you've got

1:32:30 I mean, look, you get an outcome but you've got a couple of molecules shifted out of place and those aren't relative to the outcome but it gives you a different world I mean, how coarse, great, how precise is the description of the world? in a realistic experiment you'll get a hundred different states of affairs as the result of the experiment, all of them showing the same result, and they'll differ in ways that will be discernible to the eye. So the issue there of, you know, do you have a long-term correspondence between moral and outcome, this is an idealization, but I'm still not getting it done. Sorry. My point is a terrible simple one. Admittedly, if you're going to say there's a, well, can I just sort of I mean, I'm only making visible my e-pond, actually. But I tell you, can I give you a different... Oh, you can draw on that. No, no, I can draw on that. Yes, I mean, you can, actually. I mean, it's always important. I'm not making. I'm talking about... I mean, just take one of your structures. These are the worlds, right? Is this right? They're all coming down. here, and every time you get a division, you get, so this is one world, this is another one. I used to throw them around and look at it. Oh, well, never mind. So, supposing we have to go down this world, then all these bits here, they're never going to reappear. Sure. So, my point is, if you put your probability distribution over all the worlds, it can't be analogous to the Kolmogorov model. Because in the Kolmogorov model, you have your repeatable and you have outcomes, and if these outcomes are omega-1 and so on, if omega-1 appears this time, well maybe next time one of the others. But in this case, if you put the problem as a solution of the world, then this is one of the possibilities here, but it's never going to appear. From a classical point of view, we'll have one of these outcomes, and we'll go 1, so whatever, you know, at some time t1, then we'll have another outcome, at some later time t2, and then another. So, it's sequential in time. Yes, yes. Now, equally here, you know, I've got one outcome here, say, and I've got the kth outcome here, I've got the jth outcome here.

1:35:00 So as I draw a path through here, I'm picking up a lot of out there, sequence of outcomes. Classically, one would just have a single. We'll use sequence of outcomes as well. So, corresponding to a measure over sequences of outcomes, classically, here you've got a measure over paths through, this structure. And, of course, the measure of those... Oh, no, no, no, you've changed the classical model into sequences of outcomes. But that's indeed what one always does with it. Classically, that's what one's in the future always is. You've got some chance set up, and you get an outcome, and then you repeat the chance set up. No, no, no, this is where we disagree. I don't think you do it over the sequences. You do it, you put your probability distribution over only one, which is over the outcomes. are you talking about a one-shot trial or a repeated trial? A repeated trial is perhaps just a complicated case of a one-shot trial. In a repeated trial, all the outcomes are all the different sequences. If you're talking about a very simple trial, which you just limit 0.1 or toss a guy once, then you'd better cut the picture off after the first note. No, I'm talking about the standard model, which involves the idea of repetition. No, the standard model does not involve that. Look, read Govgorov, read Govgorov, he says, let us take the same repeatable conditions. Donald, you want to stay in the case, please, let me finish the case, where you have a trial where the outcomes are, say, W1 to W2J. Yeah. Okay, and then you assign probability to each of those outcomes. Nothing yet is said about repeated trials or sequences or relative weaknesses. That's specific to the symptom interpretation probability. And if you want to think about that simple case, where you just toss the point once and see what happens, it's not going to help you start thinking about frequencies. If you toss the die a hundred times, then you will observe a sequence of outcomes, which you can think of as a complicated single trial six to the hundredth possible outcomes and each of those will have a probability in fact the probability of those will be imposed

1:37:30 by the probability of those of the trials then the trials are going to be a big trial I don't see where the problem is look I'm just I'm talking about a philosophical interpretation just a standard just a sort of standard which normally occurs among objective probabilities. I mean, we know there are a number of ones, finite frequency, limiting frequency, propensity, and so on. But all these have in common, and it is a way it's formulated by Konkorov in his book, that you do start with this idea of repeatable conditions. it seems to me that you're moving away in this many worlds from that model. This is certainly any kind of free particular I mean, that's really my point. I'm not saying that you should, I'm not dogmatic in saying we must adopt this kind of traditional objective interpretation. So, isn't this right? You can't do the theory of probability with an iterating framework. That's the one thing that does that. Well, I mean, look, it goes classically as well. I mean, you can't define probabilities in terms of finding relative frequencies. They don't be satisfied with a model of that. You know that. Yeah, it's fancy, you've got to fancify it, but I mean, the one thing is that it's a basic point, that if you have the Everettian apparatus, you can't even stop thinking about these things like that, because reality is going to contain all. Now, that was the point. You put the point on trying to make that clear a bit. That's the point that I'm trying to make that clear. What I'm going to say is that some of those frequencies, the branches on which you get road frequencies, are very probable, and all the anyway probable ones will have frequencies probabilities, or just at least all of our numbers. But still, all the frequencies are

1:40:00 there in reality, so you can't start defining probability in terms of the frequencies are there in reality, because we've got too many. Sure, sure. But the reason why I'm not... Oh, well, that's the part I was trying to make. Do you agree with this one time? Yes, I have a qualification. Oh, you have a qualification. Equalification is that if you look at just a single world and you try to define probabilities in terms of relative frequencies, then the answer is, if you tell me there's going to be a hundred trials, can I define probabilities in terms of relative frequencies? The answer is, well, I probably can. And you will never be able to say more than that. You will only be able to say that probably the relative frequencies of the hundred trials will agree with the probability of the check. the same in Everett. You say, probably, you've got a measure over all of these trajectories through this branching thing, and the measure is such that it's enormously concentrated on those trajectories where you get the relative frequencies equal to the relative. So you end up being in exactly the same situation in Everett as you would in one more scenario. In fairness to Donald, every famous figure tried to show that you could take the squared aptitudes to be probabilities because in the long run, the squared aptitudes would correspond to the frequencies. But then he realized, except in the range of launches where they did it, which had very much weird aptitudes, but now it becomes circular. But it wasn't obvious to everyone. This was obviously circular. Well, okay, but look, here's another parallel between the usual frequency type argument. You take n trials, n outcomes, and you look at the relative frequency, and if you look of frequency is n times to infinity, and it equals the probability of nature. Now, in the Ehrlich-State distance, the regression picture, you take n-repeated trials, and you look at the amplitude of all of those branches within trials, and you let n go to infinity, and you find that the amplitude of all of the branches of relative frequency of difference

1:42:30 probability is zero in that moment. So, you know, there's a similar limiting argument applies to the Aberration case as opposed to the classical World War case. So, I mean, there are differences, of course, between the Aberration case and the World War, but there are also these, as it were, structural similarities, the structural similarities. I'm surprised you're reluctant. I mean, surely it's exciting to have a new, exciting interpretation, but you seem reluctant to say that some old classical views are not available. Well, it's not as good as it's not the right account to give I can't believe it. It's clear and there is a new one available which is ratios of these entities square which isn't enough like that in the past. But, oh, okay, why should I do that to me? I suppose because Well, then we'll come in. Going back to this point show that probabilities correspond to frequencies. If you give yourself just Lebesgue measure, then you can define probabilities in terms of frequencies in the many-worlds model. Namely, the probability is the frequency that occurs with measure one taken in all the branches of the trees. So it strikes me... You're using your square apertures too. Because if you try and measure the numbers of different branches in the trees, the rogue branches are going to occur with measure zero. Yeah, but it's not the measurement, it's given by the state. It's given by the one mechanical state. okay okay yeah sure so but i mean the the point is that that you're you're still going to be able to construct a correspondence between frequencies and yeah there's probability the usual link between epistemology epistemology goes to relative frequencies i knew the basis of a lot of frequencies that we can never know anything about for the reason so much to measure. Is that a dispute about how to say this is near?

1:45:00 Well, it depends on what the interpretation is properly. I mean, if you restrict subjectivism... It's not a subjectivist you have at all. This is just what can count... If you're a strict subjectivist, then you don't have to base it on the frequencies at all. You could just say, this is my degree... We're talking about objective problems. We're talking about objective problems. But it's a mistake to suppose that the frequency theory is better placed to account for the fact that we find out that a way to follow the virtues by observing frequencies. The fixed theory is no better placed than any other accounting follow-up in fact, they've all been badly placed. Yeah, yeah, I think that's right. But I just wanted to make a point that the link between the tomology in terms of observing and observing is exactly the same as it is in a traditional framework, the one-world framework. And all you can conclude is that as with high probability, the correct probability is what I've observed as the number of frequencies. And you say that in the one-world cases, you're doing that. But can I try to sound on a conundrum? Because we're now getting into inferring which are the theories from what they observed. What do you think of this argument? So, I get into the box of Australia's cap, or indeed, I spin a coin. And if autodots collapse from the gas is right, there's a 50% chance I can't see heads. But the evidence is right. It's 100%. And they put it neatly. That's how I'll have the survival of people who will see heads. Now I spend the coin, and I see heads when I see tails, and I discover what every creature is 100% that there is an observer who sees heads. So this is strong evidence for effort, now there must be a fallacy in that, but where is the fallacy? I don't think there is a fallacy. It seems to me that if this is what you do, I think that there's a real issue as to...

1:47:30 I mean, the survivors will certainly say, well, look, you know, I didn't expect to survive, or whatever. I mean, make the answer to 1,001, you survived. Are we talking about that? I wasn't talking about the bad situation. Oh, I'm sorry. I thought you said get in the box for sure. Could we make Sir John Bell, where you have a fat or a thin cat, he introduced this to be kind of animals, so both would survive. I wasn't at the moment thinking about the charge that the regions have to attach 100% to the people in their situations. Though in fact the worry I have now was prompted by something people say about that, which is that if you keep on living, then that will be your evidence, sheffield, and sheffield. But then I thought there's a way of running that kind of argument without, forget me, Orthodoxy predicts that there's a 50% chance that my later self will see heads. Everett predicts that with certainty I'll have a later self that sees heads. I spend a coin and I suppose I see heads. So now I can say, look I've discovered something about reality, that Evert said it was 100%, but also Oxygeni said it was 50%. So shouldn't I now take this information to confirm Everett that reality contains this of Michael who sees heads? And that seems too easy. Yes, yes. But where's the fantasy? Should I not say that everyone predicts with 100% that I will have a survivor who sees heads? That seems wrong. You should indeed say that everyone predicts there is a survivor who sees heads. So why should I apply basic conditionalisation? Well, if it also predicts there's a survivor who doesn't see heads. where the fantasy is, but what's to stop me? So, I see here saying, what's to stop me saying,

1:50:00 now I've seen me, and he was predicted with 100% by Everett, but only 50% by orthodoxy, therefore by saying conditionalizations because we up my credence in every to the orthodoxy but that's a too easy a way to but why why choose that prediction where everyone also says the survivor doesn't say I don't know I can make both predictions both of them have 100% and both of them are going to be confirmed well yeah but if you're seeing heads you're confirming one of the predictions and not the other so here's the further thought I had so when you do basic conditionalizations and you're supposed to conditionalise on all the information you have. So when you discover there's a survivor who sees heads, the information isn't just there's a survivor who sees heads in an older reality, which is what you do before, anyway. But you know that, and what's more, you're that survivor! And that's extra information, but that's egocentric information. That's not real information. So, but you said that, and this is where I was trying to come back to, that the domain rating point of view is probably to sort of more egocentric, and I wasn't quite sure. Situational. Yeah, you know what? Situational. Where am I situated? 50% probably I'll be situated there with head and 50% probably I'll be situated there with there. It's more that, as it were, after the branching, I don't know where I am, I don't know in which branch I am. Right, and when you see head you know which branch I am. When I see head then I know which branch I am, but what is the right bet to make, what is the right guess to make? the ratio as to where I am located. Go to the Sleeping Beauty case. Yeah, go to the Sleeping Beauty, yes. Let me get into it, because I'm worrying about this analogy, because in the Sleeping Beauty case, I'm woken up, I know that I'm awake now, I don't know where now is, but I know that I'm awake now. Yeah, that's right. And if one was going with the two-thirds, you'd say, this is good information that you're awake now, and this should make you change your degree of belief in comments coming down heads, even though, say, et cetera, it's information.

1:52:30 But I'm analyzing it, and I'm persuaded. That's not the right way to go. We shouldn't change our freedoms in heads, so we aren't getting any real information from I'm awake now. So, no, being awake now is no new information at all, that's nothing, so you don't know where you are yet. No, I think you do. The difficulty of the sleeping beauty, I think, is that it serves the link between the objective probabilities and the subjective ones. Because on a normal case, if you bet according to the objective probabilities, you'll be okay. But in the sleeping beauty, because the betting set up has been a bit distorted, I think that's the point you made, which is correct. your betting quotient equal to the objective probability, because otherwise, who's ever betting against you could really use you as a money pass. That's unhappy. Just finally, I'm going to kick him out. Because if you do a game in a double unit, that's a step, then you can still own winning money. I don't know how to put it. Strangely, I mean, it's good, but I think you should still, well, there's your betting and there's your belief about rejected probability and what it shows is that they come apart there's a further question about what you should say is your degree of belief I'd be happier keep my degree of belief with rejected probability and explain my betting behaviour diverges my degree of belief because it is my business about doubling the odds yes I think that's a better way of betting it's Frank Gonzini has a nice statement I thought Frank's point was that one isn't learning anything about probability from this scenario because one's impaired, cognitively impaired, isn't that the point? I can't get to the end of Zion, but the beginning of this is it's like a variable stakes gain, where the size of the stake depends on whether you get right or get wrong. And in that case, it can even be advantageous to commit yourself to making bets, which you know beforehand are going to be a bad odds, provided the stakes vary when you're willing to lose your appropriate chance.

1:55:00 Well, I think we have really run over the time. Maybe if anyone who hasn't spoken would like to make a last question. Okay, well, it just remains. Thank you. Thank you.