Quantum vs Classical statistics — metaphysics of objecthood (contd.)

0:00 And if one then, as it were, runs this program, consider a physical theory, look at the invariant, the quantum, the quantum, the quantum theory, find the predicates, the calling, determine on the basis of those what the coordinates may be quantified over, thereby determining the matrix. And one finds what has good results is so-and-so. So I'm rather crumpling my way through this conference. So it seems that this program works quite well and today in particular I want to talk about one bit of it, which is optics and statistics. Okay, well the first thing to say, well James has already said it. One more is that there's no real problem with fermions at any rate, or metric articles which are fermions, as objects. This enterprise applied to result in taking permutation group articles as the relevant symmetry group, and one then wants predicates. If this thing is invariant under permutations, if any permutation ends, then can one use such predicates in order to discern the family of T-S, as James said, in terms of having opposite value for the component of spin.

2:30 Now, with bosons, there's a question of, and in fact, I think there is a case that elementary bosons should But I don't think that applies to bosons which are composites and non-composites which, strictly speaking, can't be the same composites. Usually in physics text one describes them as all in the same state, you know, helium or whatever. But that's just a approximation. The two helium atoms can't be precisely the same. Because if they were, they'd only be on the constituents, which would be two. So, I think in the case of bosons, these are going to be objects. And the indication of that is that there's something wrong with the view of objects, the status of particles as objects, in this case, to any explanation of the expansion world. If both bosons and fermions are perfectly kosher values of our variables, they're objects in the conservative, traditional sense, it's a very thin motion. If that's so, then let's see how that fits in, and I can't see that it can, but what I'd get out of this is, let me write it down, so James has already gone through this a bit, so absolutely discernible, it goes to which, which is a collection of them, they're a predicacy, one of the very few, if I have one, if there's no other. Relative discernibility where predicates in two free line cases are applied in only one order.

5:00 Weak discernibility where either relative or absolute, but at least you've got an irreflexibility that applies to the two of them because you're reflexive about the same object. And I think, frankly, I don't really see much wrong with doing this, as to discern objects. There's nothing crazy about the concept of an object, and it can get reached out in a certain way, which says that quality has to be determined. I think my program of using definitions to define 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 negation of equalities, they're not using anything suspect-like of names, then I don't really know what's wrong with doing that. And now let me add, though, what if one further introduces names, so, and let me put in unnamed, okay. So we've got really classification, I think what we find is, let me put it here, fermions, composite bosons, and then elementary bosons, or a little table. Weakly discernible, invariably. Composite bosons, weakly discernible. Elementary bosons, formally discernible.

7:30 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 AK minus one, AK plus one, AN. X variables. So this will be a critical 1-3 variable that will apply to a-k elements. This is a bit of a criterion that I'm going to use in mathematics. You can introduce names and then you get everything as being absolutely seminal. Fermions, and literally. Unnamed, I should say. And you may well think that the use of names is now going to destroy whatever Meaning one might give to permutation symmetry. Actually, there is more and more of a view that it makes no difference whether you use names or not, from the point of view of public statistics and the difference between classical and public statistics. We've actually come back to that. The point is that even using names, you can't defend, as long as you restrict yourself to end-place totality, symmetry, and predicate, there's just nothing you can say using names that distinguishes one arrangement from another permutation. So one can't express the fact that these are distinct states of the term. Well, that might be a good point to come back to in the discussion, because it may be a bit surprising. Okay, but then, coming to quantum statistics, if there's some notion that quantum mechanics, we don't have individuals, 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 an object, so does it have to be absolutely certain? Does that state your case? And there just isn't a way to put it. What you can do is you can go to permutation symmetry, permutation symmetry, and that doesn't work because you can apply permutation symmetry classically and you still have mathematics, doesn't it?

10:00 Okay, so I put it to you that there's a problem. I mean, what then? It's not imposing permutation symmetry. It's not demanding absolute symmetry. It's not using mathematics. So it's none of these things. And this is what makes me think that the question of whether an object is a quantum mechanics is just irrelevant. It's not too strong, but it's not on its own. It's very helpful. So what is responsibility for difference? And that's what the rest of my talk will be about. The first point to make is that certainly, oh I can just put all my cards on the table. Reputation symmetry must have something to do with it because if you look at quantum mechanics If you don't have permutations, then you get Maxwell's. If you do, you get Poisson's. So, that would seem to say, oh, well, there it is. 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? Now, the first point to make here is that permutation symmetry did not come naturally. Boltzmann didn't require it. Nor did any other of the other kind of scholars of statistical mathematics. The first came to the fore with Connes, and this is his clearest statement on what he was doing in 1940.

12:30 The distribution of energy, remember he was dealing with blackbody 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 the entropy. He hit on the right entropy function by interpolating between two points extracted from the empirical. Graphs. It was real curve-fitting stuff, you know. Anyway, so he hit on the correct expression of the entropy because he gave the climate distribution and then further tests showed it very well. He wanted to establish this. He probably didn't understand what he did. The distribution of energy over the type of resonator must now be considered a third-up phase space. So the whole subject began in confusion. It was not a good year in which to find the last piece of the jigsaw puzzle.

15:00 It's been the wrong year to do that because the whole thing got, and I didn't really think, if you look at what physics textbooks say, what they were saying in the late 20s on is it's to do with indistinguishability, so this is a failure to understand to acquire permutation so much of classically and you still get that class of possibility. Now when that failure was corrected, as I think it was, kind of in the 50s and 60s, And you start to see, I mean, some very clear, very visible papers that say, look, there's nothing wrong with treating classical mathematics as interesting. Then the textbooks start to talk instead about quantum non-locality, or maybe gen-identity. And I haven't really seen any clearer statement than that. Now, at the back of everybody's mind, I hope, there will be the memory of the fact that certain quantities evaluated using integrals, continuous integrals, give you the, as if you replace the integral by a summation sign, you discrete it out, you get the function, which is clear in the way that you do physics. But that point is actually the key point in this conclusion, and it has to do with discretization. But then I think physicists understand that discrete taxation has got something to do with this, and the issue is just precisely the way, and that's what I just want to, just exactly how, it works like this, and that's a total common question, well given that you've got, how many ways can you CK such cells, what's it going to multiply, well very often, see, you know what, what Klaus did, he came along, and instead of writing down this expression, and it wasn't clear just how he arrived at the line, but the precise relation between these two things was,

20:00 It was really not clear. The reason why it's a bit baffling is you might think, well, look, if I'd been able to impose permutations, it would have been factorial. But there's two things that are wrong with that. One is some philosophical quibble that happened. It didn't occur to them that it wasn't a natural thing. They saw indeed that there was, they wanted to apply it to them in order to get an extensive knowledge, and that bothered them. And listen, I think Eric has made a good point. If you use this thing, once you get it, you get a term k n log v in the entropy function, in the case of bog. But this thing does not behave in the right way when you double the volume and double the volume of other things. So people realize that what you need to do is subtract off k n log n. That will do it. Because then you divide v by n. That. And then you double n and you double v. That thing will stay constant. But they didn't immediately say, oh, well, but anyway, of course, we ought to do that. Why didn't they say that? But in many of the images it's been somehow hung up on that it must make a difference between the particles, and that's what cuts the scarf.

22:30 Why didn't they just write it through and have it to work? The other reason they may not have been writing it through is because what you do, you don't get. This isn't a number of ways, it's an integer. Divided 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. It's a bit of an investigation, isn't it? Why didn't people naturally divide by a factorial? 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 space. That's why. And once you have that understanding, then you see that, indeed, you just divide by a factor, and it doesn't matter that you don't have any reason to have a factor. You've got a phase space logarithm with a phase space product. So this is sometimes called the reduced phase space, or the equation space of the n-particle phase. Phase space volume, you can still talk about the number of cells that go down in what you're doing, because count the number of possible distributions now.

25:00 So, four factorial over, so four times three divided by two is six, of six. There are six possibilities. But the volumes of them are only one. Now, what do you do quantum mechanically? Quantum mechanically, you do exactly the same thing, but you don't talk about volumes, because you only talk about space. What you do instead is, I can represent a dot here. I've got the nine dots. Let's do the unsymmetrized case. I've got nine possible orthogonal states.

27:30 And the count of the number of dots gives me approximately, well, that is so much, and 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. That's it. That's the difference. It's all down to that. Well, and further, the fermionic case, of course, the very possibility of being able to make the constraint of no more than one particle a complete constraint. This is actually a matter of privileging the size of this tool. I'm sorry, I need to say a little bit more. What if I find a grade, what if I take and find a grade? If I do that, then the count of the possible distributions here does cross. 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 taut-dependent. And it's by setting, it's by having cores. I mean, by the same process, you can see why in the dilute limit, you can't get a difference, the dilute limit, because the dilute limit is where you've got, I mean, any particles along the diagonals. That's along, that's exactly along the diagonals. The topology of reduced phase space is quite, you can visualize it. But basically, the issue is, it's the cells close to the diagonals that are the ones where the volume assigned to them goes down, and it doesn't go down to the state, to counter states and so on.

30:00 Okay, so I think I'd just like to end with this business about classical symmetrization does not make a difference to the statistics. All it does is it gives you an extensive entropy and was made by, in his book, the question of the truth or falsity of the second law of thermodynamics is, after all, the strength of exegetism, whether or not certain is observationally accurate. Now, he goes on to say that that's a real kind of work.

32:30 But the point that I want to make, the point about identifying or not identifying... I think you've probably been thinking about them in connection with this talk, so just a very primitive lesson that I take from various type models which are developed, is that if you focus on combinatoric issues, Then you can imagine a termodule which behaves like... so you can modell with termodules which are not. You can modell the combination of the termodules.

35:00 ...washes out, so clearly it can't. The expectation is this.

37:30 We have essentially the same question as Guido, but let me put it differently to the way you explained it. You were talking about a complete predicate that's all logically equivalent to any of its own connotations. It's not asymmetric.

40:00 This points to a lack of clarity, but is one allowed, suppose you're describing n particles, why do you allow the one place predicate? f1 like particle is n, or whatever, you know, one quarter of the apparatus. So that's a predicate, what is a predicate? Well it's one place, so the symmetrization condition doesn't cause anything. Now if I've got a name, a particle name, and I stick it in there, a k to a k to a k to a k to a k to a k to a k to a k to a k to a k to a k to a k to a k. Well, now I've expressed something which the true pattern changes on the second page, which I've broken. Yeah, but why is this what I've written about? Am I really inquisitive if a predicate like that is located in... Should I not write down this predicate if x1 or x2 or x3? And I think that's truly what I ought to be doing. This is the right sort of predicate to restrict myself to. If you, the requirement is the predicates from that, okay, and the point being, because if x1, the fact that you, I realize that you're thinking of using variables that are different names, but I think that's not really the issue. I mean, I'm just using distinct variables, and the question of whether or not they take values of distinct objects is a question of the analysis of the same language.

42:30 But I think, I think this is what I should be driven by. If you do things like this, then you, even with proper names, you just can't say anything that makes sense. So you can't express, you cannot express the fact that what you're saying happens. Even using proper names. And I actually think that there's something right about that because the work really is in just the symmetry of the picture. I agree that the discernibility by itself is not your explanation, but I think it was You say that it could be crucial also. And if you see, for instance, for the Bose-Einstein statistics, Bose didn't really expect that Stein invoked some mysterious interaction at a distance between the various persons presently present. So then he felt that, exactly, he remarked on it as maybe this is an explanation. It wasn't what drove his derivations to surface, but the thing is. And finally, I'm sure you have an explanation, but very, very small. Just that the reason why there is no pointing from you to these entities is that there are no entities at all, just modes of explanation are going on. So he went, so to speak, between theoretical and mathematical. But this is also a misconnection.

45:00 Oh, absolutely. And when it does come to elementary problems, because I don't actually like formal discernibility, I don't like names, and I'm going to stick to my Quiney framework, and I'm going to do a definition of the physics of equality science, I will conclude the explanation as is and all. But the point is this. Because most gases are gasses, these are also gases, so they are all gasses, so I've got a problem with these. So I assume that there are now some actions. Now is this claim true?

1:05:00 The domain of this course of whatever theory is structured here, and here I have physical reality,

1:17:30 then some which I can have better stemming access to, but independent of...

1:20:00 We are in the process of getting an answer to the question why are these scientific theories successfully successful and those others are not. How does this explanation work and from the same feature? Now I submit to answer to this what this physical reality is and what this structure is and these semantic reals of what this structure is and therefore go along a different path.

1:22:30 It 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 the successes in theory. It seems that these realists feel that the scientific answers are not good, because what they are asking for requires something purely metaphysical, because they have already rejected the adequacy of the scientific answers. So they want an extra science. Just about your controversial point, of course I have 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

1:25:00 in the way you suggested. It's just on the opposite. If you just look more by key this quite formally concept, Define them at a categorical level. And then if you try to do anything like topos theory in that way you suggested in set theoretic CFS or whatever, you just get paralyzed absolutely. You cannot prove anything and things like that. And actually, of course, the question is very interesting. The result, the principal result is Yonaday-Lemba about relationship. Which is a rather weak result. It just says if you have a category and then you have a functor full of sets, then whatever other functor, that's not a very strong result. But just to conclude, I would say I agree, actually. The thing is that category theory actually changes the concept of structure. Because normally, often say that category theory is the same structure, but just better what changed the...

1:27:30 It's inconceivable that there is an answer to this. I know that when I find a new theory of structure, and then I say, well, this theory works with this structure and this with this, the only explanation I can think of is that reality is really, that this structure pledged on to reality and the other one did not. This is more or less an adequacy condition on...

1:30:00 What it means to give a correct explanation. So you might not be can but to say this honey. So if this is court, then what is it?

1:32:30 That's why I think it's interesting. Which I resist this view that one has to talk about natural reality in a mysterious way. You've just stated it over time. Well, indeed. But I'll have to find, I never understood why he began that paper with realism is dead. You know, it's, this is how realism lives. It's dead then, though.