Quantum vs Classical Statistics — Metaphysics of Individuality

0:00 Thank you very much everyone and I'm very grateful to Michelle for the invitation to take over from Catherine for her unfortunate ending of the series. So, many thanks. So I want to talk about quantum and classical statistics and I want to do so from the point of view of a certain project. So I'll start off briefly with that because it motivates the problem. So as I see it, there's still something quite useful to be done with a formal linguistic framework in geostatic science. I'm surprised. It's common with a lot of people. Lessing from the theory of mathematical empiricism, formal logic doesn't do a lot for us, it seems to me that what it can do is not to reconstruct physical theories in a first order formalism, not to axiomatize in the way that Hilbert thought we should, in the way that Russell and Carnett thought we should, but to be clearer on how to talk. And the reason why formal logical methods might be useful, something we do all the time of course, is because the criterion for what is an object is very clear and it does seem to me it's a great achievement.

2:30 To see how bound quantification really is the key to ontology. Now the further ingredient that really drives the program along is when one supposes that the equality sign is to be defined. And the reason that drives the program is because if defined, in terms of predicates, the first question in determining the ontology of a theory What are the acceptable predicates? And once those are fixed, the criterion of object which is fixed, whatever is discerned using those predicates, is then fit for bound qualification. And furthermore, there's a very natural criterion for what should be the predicates in talking about the world in the light of the physical theory, and namely they should be invariant under all the symmetries of the theory. So that, I think, is a necessary condition, if it's not a sufficient one. I'm not so much interested in a sufficient one, but I think necessity is good, and if one then, as it were, runs this program, consider a physical theory, look at the invariant quantities, define predicates accordingly, which are going to be the invariant predicates, determine on the basis of those what What one's going to quantify over, and thereby the domain of objects. And one finds one has good results in each case, so I found. So I've rather been trundling my way through physical theories. Newtonian gravity, special relativity, general relativity, classical system mechanics, quantum system mechanics, quantum theory, quantum field theory. So I haven't looked at quantum gravity particularly, although I think it applies just as well to

5:00 Quantum loop, quantum gravity program, string theory. So it seems that this program works quite well, and today in particular I want to talk about one bit of it, which is the quantum statistics, mathematics, statistics, physics... Okay, well the first thing to say, well James has already said it before, is that there's no real problem with fermions at any rate. This enterprise applied to indistinguishable particles will result in taking the permutation group for n articles as the relevant symmetry group, and one then wants predicates, so if I can put superscript n to mean it's an n-place predicate. As long as this thing is invariant under permutations, any permutation in this permutation group of elements, then can one use such predicates in order to discern fermions? Yes, is the answer, and James said it, one discerns them in terms of having opposite value of the component of spin. Now, with bosons, there's a question mark, and in fact, I think there is a case that elementary bosons should not be counted as objects, but I don't think that applies to bosons which are composites, permanent composites, which, strictly speaking, can't be the same quantum state. Usually in physics text one describes them as all in the same state, you know, helium or whatever, but that's just an approximation. Two helium atoms can't be precisely the same quantum state, because if they were, their fermion constituents would be two.

7:30 And so, I think in the case of bosons, these are going to be objects just as fermions are. The indication of that is that there's something wrong with them, that the status of those individuals is going to do any explanation to the explanatory work in statistics. If both bosons and fermions are perfectly kosher as values of bound variables, they're objects in the conservative traditional sense, it's a very thin notion of an object. If that's so, then maybe they're not individuals, if they're individuals, for example. Let's see how that could explain the difference between the two. And I can't see that it can. There's really, I suppose, five categories, and let me write them down. So, James has already gone through this a bit, so absolutely discernible objects are those to which, if one is considered as a collection of them, there are predicates and one free variable will apply to one and to no other member of the collection. There is discernibility where the relative product of these two objects is irreflexible. But then let me introduce a category of formal discernibility where you're essentially using the negation of the equality sign as an irreflexible program. And I think, thank you, I hope you've seen discern objects. There's nothing in the framework that the object in the diamond circle which says equality has to be defined.

10:00 I think my program of using definition of defined equality to give us leverage over ontology I think is quite a good one, but if you were to ask, metaphysically, what's the problem with objects discerned only by the negation of the equality sign, they're not using anything suspect, like names, then I don't really know what to do with that. And now let me add, though, what if one further introduces names, and let me put in unnamed, okay, so we've got really actually six classifications of coming out of straight logic for objecthood, and I think what we find is, let me put that here, fermions, composite bosons, and then elementary bosons, draw a little table, So fermions are weakly discernible invariably, composite bosons weakly discernible, elementary bosons formally discernible, but now if you use names the weak and relative categories collapse into the absolute, so stick in names for all but one of these So a k minus 1, a k plus 1, a n, and here I have an x, a variable, so this will be a predicate and one free variable that will apply to a k and a n, right so. This has been a cartoon of absolute dissonant ability. So you can introduce names and then you get everything as being absolutely dissonant all. Fermions and composite bosons and a bunch of other things. And you may well think that the use of names is now going to destroy whatever meaning one might give to permutation symmetry.

12:30 Actually, I'm more in the view that it makes no difference whether you use names or not from the point of view of quantum statistics and the difference between classical and quantum statistics. But perhaps we can come back to that at the end. There's just nothing you can say using names that distinguishes one arrangement of parts and a permutation. So one can't express the fact that these are distinct states and terms, not even using names. Well, that might be a point to come back to for discussion, because it may be a bit surprising. Okay, but then, coming to quantum statistics, if there's some notion that quantum mechanics don't have individuals, Well, in all of these cases, I think we have objects. So the issue is what more does an individual have to be over and above being an object? So does it have to be absolute design? Does that state your case? And there just isn't a way to put it that separates what you can classify. There's just no category you can hit on that will cleanly separate what you can classify. Well, what you can do is you can go for permutation symmetry. You can say what permutation symmetry is. But even that doesn't work, because you can apply permutation symmetry classically, and you still have maximal possible statistics. Okay, so, to put it to you, there's a problem. I mean, what, then, distinguishes one from the other? It's not imposing permutation symmetry. It's not demanding absolute descent validity, or relatively weak descent validity. It's not using names, or not using names. So it's none of these things. And this is what makes me think that the question of individual, whether the object is individual, is a common question. It's just irrelevant. It's not too strong, but it's not on its own very helpful. So, what is responsible for the difference? And that's what the rest of my talk will be about.

15:00 The first point to make is that certainly... Oh, and if I can just put all my cards on the table. Permutation symmetry must have something to do with it, because if you look at quantum mechanics, if you don't have permutation symmetry, then you get Maxwell-Bosman statistics. So that would seem to say, oh, well then, yes, permutation symmetry does explain the difference within quantum mechanics. Well, okay, but still I want to know what is the difference between classical particle statistics, right? Not even. Now, the first point to make here is that permutation symmetry was only, did not come naturally. Boltzmann didn't require it. Nor did any of the other family fathers of system mechanics. But it first came to the fore with Coyle, and this is his clear statement on what he was doing in 1900. The distribution of energy, remember he was dealing with black-body radiation. He thought he was using Boltzmann's combinatoric methods to define the entropy, because he had run into a problem with how to define entropy. He hit on the right entropy function. By interpolating between two quantities extracted from empirical, it was real curve-fitting stuff, you know, it was kind of nice. Anyway, so he hit on the correct expression of the entropy because it gave the client distribution and then further tests showed that that was doing very well. He wanted to establish that this is indeed the correct entropy function. He turned to Boltzmann's methods and probably didn't understand the problem. And this is what he did. Distribution of energy over each type of resonator, these are in the walls of a black body cavity, must now be considered. First, the distribution of the energy E over the N resonators with frequency V. If E is regarded as infinitely divisible, an infinite number of different distributions is possible. We, however, consider this is the essential point E to be composed of a determinate number of equal finite parts and employing their determination in the natural constant.

17:30 It seems clear that Planck didn't really understand that. What he thought he was doing, by dividing the energy e over the n resonators, or dividing e into discretizing e, he thought he was doing more Boltzmann than me. When Boltzmann divided up phase space into cells, that's what Planck thought he was doing. So the whole subject began in confusion. And it took a very long time to disentangle all of the threads in it. I think it was only with, not even Bose's paper of 1924, but Einstein's paper, it was really all of the pieces that were in place. And then and only then had we been fully able to understand all of the peculiarities of the various derivations that had been given of the background of the public distribution. And of course the year 1924 was not a good year in which to find the last piece of a jigsaw puzzle. Absolutely the wrong year to do that, because the whole thing got, as it were, swamped by the developments of next year. And I don't really think the physics community ever recovered. If you look at what physics textbooks say about the difference between... What they were saying in the late 20s on is it's to do with indistinguishability. So this is a failure to understand required limitations, and you still get that, I suppose. Failure was corrected, as I think it was, come the 50s and 60s, and you start to see some rather clear and fairly visible papers that say, look, there's nothing wrong with treating the classical part of the business as well. Then the textbooks start to talk instead about quantum non-locality as maybe gen-identity. And I haven't really seen any clearer statement than that. Now, in the back of everybody's mind, I hope there will be a memory of the fact that certain quantities evaluated using integrals, continuous integrals, give you the maximum Boltzmann statistics, the Vim distribution, for example, whereas if you replace the integral by a summation sign, you discretize, you get the front distribution. That was clear, I think, already in 1906.

20:00 That point is actually the key point in this conclusion of what's to do with discretization. And I think physicists understand that discretization has got something to do with it. The issue is precisely where it stands. And that's what I just want to go through. Just exactly how does this work? It works like this. Boltzmann considered the problem of Given a set of numbers, n1, n2, nk, where these correspond to energy levels or an interval in some cells, some interval in energy, ek to ek plus delta. And he's asking a certain number, nk, of particles in this energy interval and so forth for all the others. And then he considered how many different ways to distribute a total of n particles. And then he was led to a combinatorial factor like this and then you have to consider that's the number of ways of breaking down these n particles into, so that's a total combinatorial factor, but then you've got the question, well given that you've got n2 particles in this energy interval, how many different ways can you distribute them? Well the answer is infinity of course. So this is where he introduced the idea of a cell to discretize these energy intervals, so we looked at the one particle phase space, some energy, some cell, tall, arbitrary, and then the issue is how many ways can you, and suppose there's ck of them, so c2, ck such cells in each of these intervals, so you could end two particles over the c2 cells.

22:30 How many different ways of distributing them now are there? It's going to be ck to the nk. First particle, I can put it in any one of the ck cells. Second one, I can do the same. Third one, I can do the same. Just multiply ck by itself, nk times. And, of course, I've got to multiply. That's just for the nk particle, so it's ck. Well, very often, CK is just the one, actually. Sometimes that happens. All you're left with is this. And then it gets extremely confusing. Anyway, what Plouffe did, he came along, and instead of writing down this expression, he wrote down this expression. And it wasn't clear just how he arrived at it, either. Aaron Test showed how to get this a few years later. It was really not clear, and as I say, I don't think it was possible to propose Einstein's work. One of the reasons why it's a bit baffling is because you might think, well look, if I'm going to impose permutation symmetry classically, surely I'll just divide through line factorial. That's right. That's exactly what you ought to do. But there's two things that are wrong with that. One is some philosophical quibble, perhaps. It didn't occur to me to do that. They saw indeed that they ought to divide through line factorial. In order to get an extensive entry. And that bothered people. Okay, and this is... how much time do I have? Not that much actually. I can go into it a little later if anybody wants.

25:00 But if you use this thing, what you get, you get a term k n log v in the entry function. k is the Boltzmann's constant. It's plus other stuff. But this thing does not behave in the right way when you double the volume and double the number of particles. So if people realize that what you need to do is subtract off k n log n, that will do it, because then you'll divide v by n, that will just do that, and then you double n and you double v, that thing will stay constant, and then you just keep on doubling, so that's the right thing to do. But they didn't immediately say, oh, well, but anyway, of course, we all divide by a new factorial, because of course these are all distinguished by their capacity. Why didn't they say that? That's a very interesting question. Why didn't they say that? Somehow I'll come up on that it must make a difference to commute the particles, and that's what I was thinking. Why don't they just divide through by a factorial? The other reason you may not have to divide it through is because when you do, you don't get, this is the number of waves, it's an integer. You divide by a factorial, you don't have an integer. So this can't be understood as the number of ways of doing anything. Now I don't know if that's the reason, I don't know what is the reason actually, it's a bit of a psychological investigation as it were, why didn't people naturally just divide by a factor? What was needed to get over the integer problem is the recognition that the real thing that counts is not the number of ways of doing anything, it's the volume of the phase space. That's what matters. And once you have that understanding, then you see that indeed you just divide by a factor, and it doesn't matter that you didn't have an answer to it, there's no reason to have an answer to it, you've got a phase space logarithm and a phase space logarithm. Now, so this is sometimes called the reduced phase space that we indicated by the equation space of the n-particle phase space over the permutation group, where you identify points of the phase space that are related by permutation.

27:30 Let me look, not at the whole grand thing, let me just look at the k-th energy interval, okay, to see more clearly how this works. The phase space volume, the reduced phase space volume, quotienting out the permutations, is just then, I won't even bother with the subscript k. I've got a volume c to the n if it's unsymmetrized. If I, in fact, if I quotient out the permutations, that's the phase space volume. Look at what this means, let's take a simple case. We'll take that C is equal to 3 and N 2, so two particles, represent their phase spaces by one dimensional line, and I've got three cells, so here they are, so here's particle A, here's particle B, and the phase space volume would just be C to the N, so the few squared is 9, and there they are, those are their own cells. Now divide by n factorial to get the reduced space volume, and what is the volume? Four and a half, so not four and a half ways of doing anything, but volume four and a half, but now look further, the volume of these is less than the volume of these, so what happens is you can still talk about the number of, the distributions of the particles, how many have you got in the various cells. But the cells have got unequal volumes. That's the difference between quantum and classical. But classically, sorry, quantum mechanically, what you're doing, because count the number of possible distributions now for six, okay, which is what this formula gives you.

30:00 So, two factorial, sorry, two plus three minus one factorial over factorial two factorial. Four factorial over, so four times three divided by two, six. One, two, three, four, five, six. There are six possibilities here, but the volumes of them are unequal. Now, what do you do quantum mechanically? Quantum mechanically, you do exactly the same thing, but you don't talk about volumes, because you're in Hilbert space. What you do instead is, I can represent the possible states just as dots. I've got a dot here, a dot here. I've got the nine dots. Let's do the unsymmetrized case. I've got nine possible orthogonal states, two orthogonal states. And the count of the number of dots gives me approximately, well, exactly the same actually, as the volume, nine. But when I symmetrize, I lose these guys. Now I've got six dots. No longer a good approximation to the volume. And that's it. That's the difference between quantum mathematics and statistics. It's all down to that. Well, and further, in the fermionic case, of course, the very possibility of being able to make the constraint that no more than one particle can be in a state is actually a matter of privileging the size this tall. I'm sorry, I need to say a little bit more. What if I take in a finer grainy? If I do that, then the count of the possible distributions here does approximately equal the volume. And what that's to say is that the distinction, the contrast between the count of the possible states and the volume, the contrast, is tor-dependent. And it's by having tor as an absolute quality that it's possible for this to alternate.

32:30 And, I mean, by the same process you can see why in the dilute limit you can't make a difference, because the dilute limit is where you've got the same cell along the diagonal, that's exactly along the diagonal. The topology of reduced phase space is quite complex, and you can visualize it quite hard. But basically the issue is that the cells close to the diagonals are the ones where the volume assigned to them goes down classically, but doesn't go down quantum mechanically, where the representative point of quantum mechanics is just the state, counting states is what we do in mechanics. Okay, so I think I'd just like to end with a comment about getting back to more metaphysical questions. This business about classical symmetrization does not make a difference to the statistics. All it does is it gives you an extensive entropy. It was made by various physicists in various papers. But Mick Huggit made it in the J. Phil in 1999. I think David Elbert must have read that because David took it up in his book, On Time and Chance, and very much endorsed Mick Huggit's point. Here's what he says. There's a certain fairly trivial sense in which ought to have been obvious from the outset, if we had stopped to think about it, that the facts of thermodynamics cannot possibly shed any light on the truth or falsity of the doctrine of hexadecimalism, that's spelling. The question of the truth or falsity of the second law of thermodynamics is, after all, a straightforwardly empirical one, and the question of hexadecimalism, the question, that is, of whether or not certain observationally identical situations are identical in simplicity, manifestly is not. Now, he goes on to say, nevertheless, it might have turned out that the hexeocistic description is somehow simpler, or more natural, more compelling, or more of an explanatory success than the non-hexeocistic one.

35:00 And the thing we've just learned, which seems to be substantive and non-trivial and impossible to have anticipated without doing the work, is that that is not the case. Well, okay, then the stronger claim to make is that, on the contrary, it's the non-hexeocistic way of doing it which... Which is simpler, and more natural, and more compelling, and gives you things like extensive entropy, but okay, he discounts, and Nick Huggick discounts extensibility of entropy, a straightforward piece of empirical fact, and that's a real can of worms, we talk about it when anybody wants to, but the point I want to make is that this just can't be true, because the point about identifying or not identifying beautifully distinct states of affairs is exactly whether or not to impose permutation symmetry, the permutation rule. And the point here is that in classical phase space it doesn't make a direct and direct difference. The reason is stated. You're looking at volume measures and whether or not you divide by an overall n factorial doesn't impact how you make this clear. But certainly when you go to a discrete phase structure, you don't even have to call this Hilbert space, just imagine a measure which is concentrated on a point. You do that and it makes a direct and direct difference. You can probably think about them in connection with this talk. So, just a very primitive lesson that I take from the various telemodels which are developed, is that if you focus on combinatoric issues, then you can imagine a telemodel which behaves like The best quantum system you have. So you can model with two models which are not quantum models, just you know some sort of games. You can model the combinatorics of quantum mechanics.

37:30 You can do whatever you like with this hexase, this pronunciation, but then the point is that it is not quantum mechanics, and there is more to quantum mechanics than the counter-gentauric issues. So, well, I don't know if you've been thinking about that, but does this somehow elicit... What is the necessity to look at more things than humanitarians, etc., in connection with the argument? Well, not particularly, to be honest, because the point about this, and look, here's the line that someone like David Elwood likes to come back and say, look, it's not anti-hexatism that is doing the work, it's entanglement. Because, you know, okay, you can call it anti-hexatism, quotient out the interpretations, but what that means in quantum space is that it's not extensive positions. And whoever said that that might not have empirical facts. But I think the point to make to that is that you don't need quantum mechanics to get out Einstein and Bose statistics. You don't need quantum mechanics to get Bose compensation. You can do it all working in classical closed space with a discrete measure. You need times constant, but you don't need the algorithms to do those kinds of statistics. And really the only basis now, okay you've got a discrete measure and you can say, That's going to have empirical consequences, fine. But then, use a discrete measure, hexatide, be a hexatist, you get mental dots and statistics. Don't be a hexatist, you get mental dots and statistics. Isn't it something like continuous space? Continuous space washes out the distinction, that's the point. But the issue about whether hexatism has empirical consequences is that it doesn't on some phase structures, it does on others. So clearly it can have empirical consequences. And the point about the Speckens thing, yes, I agree that I would be critical of some of what Speckens has done, because he's not working on mechanics, but I think he did. You don't need quantum mechanics to get those enzymes, because those enzymes don't have them. Oh yeah, sure, but he had something like, you know, something like continuous models, etc. But then this may suggest actually that you can do a toy model, which will have nothing to do with either classical or quantum mechanics or any mechanics. That's what will reproduce those Einstein kind of things.

40:00 Well, I think this is a toy model. I mean, to just take a discrete measure and have it replaced by this is a kind of a toy model. You had four more questions. A question about discernibility of fermions and bosons. In the singlet case, singlet state case, you can do weak discernibility if you've got the two particles localised opposite ends of the experimental set-up, and that you could do both with fermions and with protons, experiment with polymerised protons. So there I don't see a difference. And I don't actually see that you can say that fermions, in the same way, have opposite spins, because the spin state is just an optimal mixed state. They don't have any spins. Opposite components of spin. They do in the common sense, but the relative quantity is, so I'm taking pi plus and minus to be the case. So this thing, the expectation of this in the EGR state is expressing the anti-correlation of this. Do any of you want to comment? Well, I have essentially the same question as Guido, but let me put it differently in terms of the way you've explained it on the board below. When you were talking about fermions, you were talking about a completely What is a complete predicate that has all of its permutations, right? A symmetrized predicate. Well, that's logically equivalent to any of its own permutations. So, it's not asymmetric. So, the general explanation of the completely symmetrized predicate where you have f and pi f and pi pi f, right? So, that's a symmetric predicate.

42:30 So, weak discernibility can consist in them satisfying an asymmetric radical. No, no, it's not. That's relative. Weak discernibility is irreflexive. Irreflexive? Irreflexive. So, you're saying, okay, not reality, but, okay, I'm interested in weak discernibility. Could you say a bit more about how the use of names still doesn't alleviate it? Well, yeah, right. I mean, I was pretty mumbled about this for a long time, I must say. I think the fact of it is, and this points to a lack of clarity in my project, I think that's also my problem, but is one allowed, especially describing it in particles, are you allowed a one-place predicate? F1 light particle is in Under Lake Geneva, under the lake, away, you know, in one corner of the apparatus. So that's a predicate, one place. Well it's one place, so the symmetrization condition doesn't... Now, if I've got a name, a particle name, and I stick it in there, a k-particle, that's which particle it is, x, and that's the k-particle. Well, now I've expressed something which, truthfully, changes on the permutation, so I've broken the permutation symmetry. But in particular, what I was, this is what I was wondering about, am I really entitled to a predicate like that is located in, should I not write down this predicate? I think this is really what I want to be doing. I think this is the right sort of predicate to restrict myself to, if your requirement is that predicates are variantized permutations. And the point being, because f , if I'm mute, I realize that you're thinking I'm using variables, but I think that's not...

45:00 Really the issue, I mean I can, I'm just using distinct variables and the question of whether or not they take values on distinct objects is a question of the analysis of the sensibility of the machine. But I think, I think this is what I should be driven to. If you use parameters like this, then you, even with proper names, you just can't say anything that breaks the symmetry. You can't express the fact that you've got two distinct pattern particles, even using proper names. There's something right about that because the work really is in the symmetry condition, you know, the use of what the names are not. I mean, philosophers often worry about, you know, what do names do, what do they mean, and sometimes give them substance. They're suspect. So in that respect, I'm happy to stay away from them. But I think actually, at least in this situation, they're innocuous. And I agree that the indistinguibility by itself is now no explanation for these statistics, and I think it was badly put into the manuals and so on. Now, we can come back to the way some of the founders of these statistics have explained them. You say that it could be crucial also. And if you see, for instance, for the Bose-Einstein statistics, Bose didn't really explain that. Einstein invoked some mysterious interaction at the distance between the various bosons present in the gas. I don't think he invoked that exactly. He remarked on it as maybe this is an explanation. It wasn't what drove his derivations to existence. And finally, I think Schrödinger had an explanation, a very serious one. He just said that the reason why there is no point in permuting these entities is that there are no entities at all. There are just modes of excitation of them. So he invented, so to speak, the theoretical formula. So this is also an explanation.

47:30 Oh, absolutely. And when it does come to elementary problems, and because I don't actually like formal discernibility, and I don't like names, and I'm going to stick to my Aquinian framework, and I'm going to give you a definition of what I'm saying, I want you to understand that the point is that... Because most gases are boson gases, they are objects, so I've got a problem. ... going to be an inquiry into the meaning and truth of structural realism, and it is all centered about this question. Do structural realists need a new theory of...

50:00 I'll present you now with a deductive argument, which gives an answer in the affirmative. An argument which is of course based on premises. The first is an assumption. If structural realism is a tenable view of scientific knowledge and the change of scientific knowledge, blah blah blah, then structural realism possesses fitting concepts of structure and object and properties. And by fitting I mean that perform the required epistemic and ontological labor. So epistemic in the case of the epistemic version and ontological work in the ontological version. Next premise is a thesis. If category theory provides fitting concepts of structure and object, then so does set theory with ur-elements. The third premise is a claim. Set theory does not provide a fitting concept of structure. This is what many structuralists, certainly those in mathematics, believe. Well, then we can perform a modus pollens or tollens with these pieces, and then we then add the following truth. Category theory and set theory are the only two available theories that provide a concept of structure. All possibilities of this argument can be no other than that structural realism meets two theory of structure, if it is to be achieved, and one witness for this conclusion, or James Layman, and other quotations can be given. Now, what is the status of these premises? Well, the ones which are in green, they are safe. They are not open for dispute, right? Which you never want to lose. This is a simple truth, so it boils down to the claim and the thesis. The thesis is in red, so I'm going to devote some time to arguing for this thesis.

52:30 That is what I'll do now. Right, here is the thesis again. If category theory provides fitting concepts of structure and object, then so does set theory with four elements. So I assume that you are somewhat familiar with what category theory is, at least on the elementary level, set theory, and well, all of us are. Alright, well, we have to argue for this, so we may assume here the antecedents. So assume that category theory provides fitting concepts of structure and object. The overwhelming majority, but not all, of category theorists, they have a language which is two-sorted with object variables, arrow variables. There are two dyadic predicates. A is the domain of F and B is the codomain of F. And there is a triadic predicate which says that H is the composition of the arrows F and G. So this is the usual. Now, the category consists of objects and all arrows between them, with the objects, both on their composition, which is always assumed associative, and it includes all identity arrows. So this is how it all starts. You really need to know. No more. Now I'm going to make a few remarks. I only have to assume it in my argument. I nonetheless want to add that even as an assumption, it is endorsed by many. My first remark consists of two questions. Is a category an object or an arrow? There are only objects and arrows. Anyone knows him? Yes?

55:00 The next question is, how to express in the language of category theory a category consists of? Now that is going to be more difficult. If I want to say this particular object belongs to this category theory, well, as a set theoretically trained mind, I would use the membership relation, but I don't have that here. So there are certain questions I cannot answer in the language of category theory. So it is not entirely appropriate in that it cannot formalize everything that category theory is set. So there is always in the background this membership relation, which is of course never formally admitted. This is a sort of criticism that has been leveled by various people. About objects, next to nothing is assumed. They can be sent to each other by arrows. And they can even be arrows. They have no nature. They are just objects about which nothing is assumed. So you see that is good for structural realism, particularly the ontic version. Objects are categories. Can I just interject? Does that mean that you've got to have arrow, arrow, arrow? You've got all the identity arrows between objects, so an object can be arrow. Now structures are categories. So all structures in the language of category theory, they are categories of some sort.

57:30 So groups and partial orderings and whatnot, they're all to be conceived as particular kind of categories. It's fitting for algebraic topology, universal algebra, and other rather abstract branches of mathematics. And it is exactly from these branches of mathematics where category theory is fitting the structured concept for structural realism. That is the question. There's so much about category theory. So it seems to work alright, although there are some critical remarks to be made. Consider now some axiomatic set theory. Well, canonical choice is set at z, but now with primordial elements, which is usually just moving elements, but let us call them objects, right, what's in the world. We then have set variables, we have object variables, we have a dyadic predicate, and a is a member of x. Or Y is a member of X. And this is the usual notation. And then we can have set theory with objects. So that is set of C reformulated in this language, where also these object variables occur. And we have some axioms say U, Kappa. Kappa asserting there exists a set of cardinality K which consists only of Now a side remark, this theory is conservative over set theory. You can add all this and nothing goes wrong. Now suppose we want to do category theory in set theory. It's almost obvious how to do this. An arrow is an ordered pair of a function, so f is now a function, and a codomain.

1:00:00 And now the category is simply a Burbakian species of structure. After all, we are here in Burbakistan, are we not? And this is simply a species of structure. So we can put it like this. We have here a set of objects. We have here an associated arrow algebra. And this is the algebraic operation for its composition. And we have here the set of identities. Functions, grid theory, sets of objects. I'll call this a few remarks. Now here everything is well defined in this language. We know exactly what the category theory, the category is. This is an ordered pair. And we can express what it is. We can express sentences like this object A belongs to this category theory. So it is simply a repeated application of the membership theory. We can say the same, we can say the same for arrows, for some kind of, say, n times repeated membership relation. We can say the same thing about objects as we can in category theory here. About objects, next to nothing is assumed. They can be collected in sets. That's all. They have no nature. Nothing is assumed about them. So that's again good for structural realism. Now, here we have to converse. We have here that categories are structures. Besides category structures, there are lots of other structures that you do not naturally write as category structures in the language of set theory. Now, just as in set theory, arithmetic proceeds as usual after we have identified, say, the set of natural numbers with the set of all finite ordinals. Zero at the end of the set, and the successor operation with von Neumann's successor operation.

1:02:30 And real number theory proceeds as usual after we have defined the real numbers as the set of data game cuts. And group theory proceeds as usual after we have defined what a group structure is, etc. And so category theory can proceed as usual after arrow, category, third natural transformation, etc. have been defined in the language of set theory. I am mentioning this in order not to get confused between two things. And the first thing is mathematical thought and the other one is the language in which we describe it. Category theory emerged at a certain point in time in the history of mathematics as a language that was first wholly defined in set theory, in von Neumann-Bernays set theory, but later the set theoretical basis was sort of deleted and the language stood on its own and that was because in proofs Topologists and other types of mathematicians, they didn't want to be bothered with all these set theoretical details. They preferred the language of category theory. But it doesn't mean, it doesn't speak at all against the set theoretical basis of category theory. As soon as you know the definitions, you can do in mathematical thought whatever you want to do with the concept of category. Just as most mathematicians do not think of functions as sets of ordered pairs, but they simply think, okay, there's a rule that maps one thing to another. That is an element of mathematical thought. But no mathematician objects against if pressed to give the definition in set theory. It's a particular set of ordered pairs. Now, the kind of structures in science that say phenomena are much more easily characterized in the language of set theory than in those of category theory.

1:05:00 And why is that? Because they are comparatively mundane. If you think what kind of structures does the most mathematical branch of natural science use, We need some integrals, we need Hilbert space, and a few other things. And that covers it, really. So mathematical physics is much and much more advanced and developed, but they are not in the business of saving the phenomena. They have the difference. There's a difference there. So I conclude, certainly, if category is good enough for structural realism, then set theory with four elements. That was the most controversial premise in my argument. Another premise was this claim, set theory does not provide a fitting concept of structure. Now is this claim true? Well, I have a few remarks. The first is this, set theory with The claim of the structural realists of the ontological irrelevancy of objects, and it makes it good by means of these two theorems. Let D be a domain of objects of some structure, then every domain equinumerous to it, replacing D in this structure, yields, of course, a structured isomorphic to it. So it doesn't matter which objects are true. It's all completely arbitrary. Every object A can play the role played by any domain member in any structure.

1:07:30 So any object can play the role that you want it to play in any structure. Now the usual complaint leading up to the claim is that the claim is actually false. That is the following complaint. In set theory structures are sets and sets are abstract objects even if they contain only objects. Where structures that structural realism posits are somehow concrete, they somehow live in physical reality and not in some platonic realm of existence. So let us clarify this objection a little bit more by taking an example for a change, right? We're talking about science after all. Consider, say, a helium atom in a magnetic field. Well, so the standard quantum mechanical treatment by means of what is a candidate for a physical structure looks like this. So, suppose we have here an object, we have Hilbert space, a Hamiltonian, a state, and we have a probability measure. Now, when ontological structural realism claims that this structure exists, it seems metaphysically thicker The aim of structural realism was to be more modest, or somehow less metaphysical than object structural realism. If we take this seriously, and this whole thing is supposed to exist out there in physical reality, well then we have of course a set. What can we do in the face of this? Well, one thing is this. We can stipulate that only some ontological substructure of this structure corresponds to something in physical reality.

1:10:00 So just as we can speak about empirical substructures, we can also speak of ontological substructures. This is a suggestion for example. So suppose it is this, the Hamiltonian, the state, and the probability measure. Now we are speaking quantum mechanics, so issues of interpretation are always just around the corner, and one may interpret this probability measure as a set of potentialities or propensities. I noticed that this is your ontological substructure, that this is something that we already have in pure set theory. So now we have eliminated the object. So you might say that if you stay, if you do your structure or your reconstructions of physical theories in pure set theory, already you have eliminated objects. So perhaps this already was the paradise of the structuralists to begin with. Now the question is, does category theory perform any better? So if we would write down some sort of category structure here. Would we then be out of trouble? We would then be committed to say that all kinds of constructions of arrows and objects exist, and these arrows are, of course, no less abstract objects than sets are. It doesn't seem to pair any better. A bit about the truth of structural realism. We're going to set one step back. So to say precisely what it is what structural realism claims, the ontological version at least, it is simply not clear to me what exactly it claims. So this makes us wonder, why do we want structural realism in the first place?

1:12:30 Well, we can interpret this question in two ways, depending on the contrast class that we use. Well, why do we want it rather than object realism? Well, because structural realism is a less easy prey for pessimistic meta-induction of the history of science. And because it is a less easy prey for attacks launched from underdetermination, from the underdetermination pieces. Let me give a brief interlude here. Look upon structural realism as a philosophical research program. Now, the motivation of, for instance, Leitman and French to say that structural realism Rather than object realism, namely that the concept of an object or an individual is problematic in quantum mechanics, I don't see that as a problem at all. What I see as a much more almost devastating problem, that is of course the theorem of melamines. There are no relativistic quantum particles. David Lewis once said, there are no knock-down arguments in philosophy, but there are knock-down arguments in philosophy of physics. I mean, this theorem of Melman, that is a knock-down argument. There is no way you can speak of particles, so objects, in quantum field. That is simply the conclusion of a theorem. Unless you are prepared to change the meaning of particle in such a way... If you admit you are going to speak of something different, there is of course always a possibility to change the rules. And that is a motivation from the Lakatosian belt of the philosophical research program, of which the core is of course the structural fields.

1:15:00 So you see, the belt is rich enough to withstand a text. There is no need to give it up yet. Now, why structural realism in the first place rather than empiricism and instrumentalism? Well, because structural realism provides an explanation of the success of science, which is a striking phenomenon indeed. So, some version of the miracle. And I have now specifically prepared a transparency for the audience members here, because I want to convince you that empiricism, not the version of Wassmann Phrasen, Does not have an answer to the miracle challenge. I read a few things of our beloved boss about the miracle challenge and I keep thinking that he is confusing two things, see. So I hope that at the end of my talk he is going to convince me that I am a confused mind or he is going to admit that he was the confused mind. So here comes my analysis. Farfrazen confuses two questions. Why do we have accepted, say, quantum mechanical models of atoms and molecules and rejected classical models? Well, the answer is the quantum mechanical ones saved the phenomenon and the classical ones did not. An empirical inadequacy is sufficient for rejection. The game of science, the aim of the game is empirical adequacy. So if a particular theory keeps producing empirically inadequate models, it must go. And that explains why we have rejected them. But that is not the question of the realist when he gives his miracle argument. An answer to a question about acceptance and rejection, it is the following question.

1:17:30 Why are quantum mechanical models of atoms and molecules empirically adequate and classical ones are empirically inadequate? Well, the real is desired answer is the quantum mechanical models got something right about atoms and molecules, while the classical models got wrong. That is the direction in which the realist seeks an answer. Let me give you a comparison. Suppose I want to buy the fastest car and I can choose between a Volvo and a Porsche. I'm going to test these vehicles. I'm going to make them speed for 10 kilometers. The least amount of time is the one I'm going to buy. Now, someone comes in and he says, wow, I see you have a Porsche. Why is a Porsche the fastest car? Now, Van Fraassen says, well, he decided in advance that he would buy the fastest car. But that is, of course, not an answer to the question. If someone asks why is the Porsche the fastest car, he doesn't inquire about my behavior, but he wants to hear a story about the aerodynamics of the car, about the quality of the engine, etc. He wants me to tell something about the structure of the car. And so the same for this question. You want to hear something about the nature of this theory. But this is still only pointing in the direction of an answer, yes? Because here, so I'm now going on a bit with this dialectic with the empiricists, is of course going to ask here, what is it then that these models got right? Well, the structural realist is going to say the structure of micro-physical reality. And the empiricist says, yes, but what does that mean?

1:20:00 And I put this question also, I endorse this question, what does that mean? To say this, a bit of a Sesame Street picture now. Here I have the domain of discourse of whatever theory or structure I have. So here are the quantum mechanical models of atoms and molecules. I look at one with magnifying glass and I see the structure here and here I have physical reality. I have the observable part with its phenomena, data structures, and these are supposed to be embedded in the empirical substructure. Now there must be then some sort of ontological structure, unobservable, which I can embed in the ontological substructure here. Now, we have atmospheric access to observable physical reality by means of sense experiences and the language of quantum mechanics, but independent of the truth claims of quantum mechanics, we can test what quantum mechanics test about the observable part of physical reality, but we lack at least this kind of atmospheric access to the unobservable part of physical reality and must postulate this ontological substructure wholly dependent On what quantum mechanics claims about unobservable physical reality. That is why we cannot test this. That is empiricism's challenge. So you might still say again, why do it? Well, because to repeat, we are in the process of giving an answer to the question, why are these scientific theories empirically successful and those others are not? How does this explanation work, eh? Some miracle argument. Well, I've drawn the same picture. Now I submit to answer this question, and this is one, so remember, this is one of the most,

1:22:30 perhaps the most philosophical basis to be a structural realist at all. If we don't get an answer to this, the whole program is ill-motivated. But to answer this, and question two, we need to know, well, what an explanation is, what this structure in physical reality is, and what this structure is, right? Well, when we break up the first question, as everybody does, and be semantic realists, so that we can identify the... Ontological substructure simply as the thing that the theory says it is. So we take our theory literally. We only have to know C. We only have to know C. Well, to answer C, so what this structure is and therefore what the structure of physical reality is, structural realism needs a new theory of structure. So along a different path, I have reached the same conclusion. I think Olson has two readings. One is a perfectly legitimate and sensible scientific question that the scientists would be willing to answer because every theory will explain, on the basis of that theory, the successes and failures of the past ones. So Newtonian theory would explain the successes and failures of the Aristotelian, quantum mechanics would explain the Zootopia. These are scientific answers to scientific questions. Makes sense. Now, it seems that these realist philosophers, they read the question Q2 in a different way, so that the scientific answers are not sufficient for them. And I think on that reading, it's an unencouraged request.

1:25:00 Yes, because what they are asking for could only be answered by something purely metaphysical, because they have already rejected the adequacy of the scientific answers. So it's not good enough for them to say, here's an accepted scientific theory, and using that theory we have an explanation of the failure of the past home. So they want an extra-scientific answer, if any answer at all. Just about your controversial point, of course I had no time to just add it to the contrary, but just one point, I think it's sort of pre-judgment that many people believe that it's anything easier to make things in set theory in the way you suggested, but just on the opposite, if you just look more by key in this fourth chapter of set theory where the concept of structure is. The concept of topology is defined formally. It's sort of this greased categorical concept which simplifies normally when they are defined in categorical level. And then if you're trying to do anything like topos theory in that way you suggest it in set theoretic. You cannot prove anything in CFS or whatever. You just get paralyzed absolutely. You cannot prove anything in CFS, what people prove in topo-steering, things like that. And actually, of course, the question is very interesting. The principal result is about relationship between category and set theory, which is a rather weaker result. It just says if you have a category and then you have a functor to category of sets. Then, whatever other functor you have from the first category to the third category, then between those two functors you have natural isomorphism. That's not a very strong result, but it's a better way to say it. But just to conclude, I would say I agree. Actually, I think that category theory actually changes the concept of structure because normally people often say that category theory is sort of manifestation of structuralism, the same structuralism that we have to say, but just better working. I think that's wrong. It's really category theory will change the very concept of structure. I agree with you that it needs to work.

1:27:30 The idea is that from the standpoint of a further theory, you can say why a previous theory is a failure. You can also say from the point of view of a further theory, why a theory was adequate to its limited domain of validity. So I think this is exactly the model of the thing that the realist is looking for. He's looking for something that would explain retrospectively why... His present theory is adequate at least to its domain of validity, but of course this means projecting oneself to a completely unknown future of science, inside science, but always maybe to this sort of asymptotic truth that realists are looking for. Doing more than that is obviously, as you say, completely meaningless, and Schrodinger also had a very good way of putting that. He said, okay, for instance, he showed that the only way of characterizing the structure in the liminal, H, is to say that it's completely isomorphic to the structure, S, that you are postulating in your theory. It's the only way to do that. And Schrodinger said, well, if this is the case, then your attempt at explaining the success of structure S by its isomorphism to structure H is just a way of being redundant, of reduplicating the question without really answering anything. Yes, well, to a certain extent that is true. Of course this is a question, as I indicated, the miracle argument is not supposed to be something that is inquired into by science. Of course it's a philosophical question. And if I write down a quantum mechanical model on a magnetic field, and I write down, say, a Lorentzian model of exactly the same situation, well, the realist asks, why is it, to put it like this, why can I embed the phenomenon in this model and not in that model?

1:30:00 Okay, so now the answer is, well, this is an unintelligible question. Well, that depends, of course. I mean, it is not inconceivable that there is an answer to this question, that I have a theory of structure and I know what the physical structure is, which is implicitly defined by my new theory of structure, and then I say, well, this theory works with this structure and this with this, and the only explanation I can think of is that reality is really... This structure latched on to reality and the other one did not. This is more or less an adequacy condition on what it means to give a correct explanation. So you might not be convinced what to say is unintelligible. Okay, if it is unintelligible then it needs not an answer. We have ended the discussion about the miracle argument. I think that this miracle argument is one of the... Well, what other arguments are there to prefer realism over empiricism? So if this is out of court, then what is it? So rise up you realists and tell me. What other arguments are there then to be a realist? Another question, and then we are losing the pause, if you are happy, we can skip over the pause, but if we want to stay on time, then there will be no time for the pause now. Am I taking up the pause? It's not your question. My question is on this point, actually. It seems to me that the right thing to say here is that the realist view is not that the structure matches on to reality.

1:32:30 I think that is rather unintelligible. I think the simple thing is one states the theory, and one says there are such and such objects undergoing such and such interactions. And that is what was intended to support the success of the theory. Now I don't think there's anything unintelligible about that. One may say, well, I'm not concerned about whether or not we're getting that right. Okay, but it's not unintelligible to say, well, this is why the theory is successful, because there are molecules that are in the ways that the theory says. That's why the theory is successful. I resist this view that one has to talk about latching onto reality in a mysterious way. But you've just stated Arthur Frank's position. Well, indeed. But Arthur Frank, I never understood why he began that paper with realism is dead. You know, it's more, this is how realism lives. Yes, the short course and then the...