Incongruent Counterparts

0:00 Transcription by CastingWords There is something coming here, right? I'm not just a person. Okay. Yes, where's that? I think he's just... I think I'll accept it. Why has he already been offered a position? Has he been offered a position at Primiter, one year or something? He has one year. But next year, he has a first doctor. And then after that, he goes. I said that. Um, I don't know. I don't know. I don't know the details. I'm just hoping that it is probable to come back to another movie. Did you show this? Yes, yes. Well, thank you all for coming. I know it's out of term time. Okay, so my subject is Incongruent Counterparts and Mirroring of an Appuori Symmetry. Now, I'll start this off with some elementary background that everybody knows. Forgive me, I put this together partly for a public presentation, and once one's labored over these images. It's a shame not to use them, so I have. So, I'm starting off with

2:30 Leibniz, the foundation of mathematics, the principle of contradiction. But this isn't enough, according to Leibniz. What we also need is the principle of sufficient reason when it comes to natural philosophy, that nothing happens without a reason why it should be so rather than otherwise. And as I understand it, the reason is still alive and well in the foundations of physics, although normally people speak of under-determination or even of indeterminism. This is in relationship to the whole argument and in gauge theory, but I think under-determination is the better way of putting it. So, as we recall, Clark's reply to Leibniz is, well, sure, nothing happens without a sufficient reason, but there need be no other reason than the mere will of God. Okay, so I think this is familiar to us. Now, Leibniz's key argument in the case of space, that if space were a thing in itself, then there could be no reason why the matter should be placed in one part of space rather than another. or by changing east into west. But if space is nothing else but that order or relation and nothing at all without bodies, then we have not succeeded in imagining a different state of affairs. Right, and it's easy enough to understand Leibniz's point of view here. We have a little collection of five bodies, and we have their relations, and then the point is these relations are invariant as these bodies are moved and even taking away this little bit of background space and perhaps we should see Leibniz in saying that space itself, this rectangle is the fiction we can imagine it anyhow and so on so that's all familiar now, the general lesson seems to be that for symmetry transformation which leaves the form of the equations of motion unchanged. If S is a solution, so is the transformation under the symmetry. And now further, and I think this statement was not obvious until relatively recent times, we identified the two solutions. They described the same situation.

5:00 And now the claim is this is quite uniform, as uniformly to all symmetries, all exact symmetries in physics. And I'm adding a claim, only invariant quantities are real. I have actually found that physicists are not entirely happy with this claim. So, let's seem more happy with it. So, we'll be looking specifically at this, and in terms of these relative distances, what hasn't been tied down, of course, is which of these is the case. and make them all individual essences, substances, as Leibniz required, but still, of course, we have the question of which of these do we mean, so relative distances don't pin us down, and indeed if we look at what Leibniz has said about changing east into west, it's not clear whether he actually meant reflections or rotations. Now, we know that Kant did think he meant reflections. There's some textual evidence for that. I think Brigitte Falkenberg unearthed some commentaries of Kant. Anyway, so here's Kant insisting that the difference between left and right must be one which rests upon an inner ground. be made to substitute for the other, no matter how it be twisted and turned. And he also insists that this inner ground cannot depend on relations of the parts. So he claims that everything in this respect is exactly the same. Nevertheless, imagine the first created thing was a human hand. That human hand would have to be either a right hand or a left hand. the action of the creative cause in producing the one would have a necessity to be different from the action of the creative cause producing the counterpart. And I think it is pretty clear that there's a problem for Leibniz and the principle of sufficient reason unless we can say how left and right differ intrinsically in the scenario where the first creative thing is a human hand. Of course, it could be any,

7:30 anisotropic or handed distribution of matter. And then the question is, why handed in one way rather than the other? Now, we know what Kant went on to say. The previous quote was from the Dressions in Space in 1767. This is the inaugural dissertation, 1770. And now the point is that they differ in spite of the fact that in respect of everything which may be expressed by means of characteristic marks intelligible to the mind through speech, they could be substituted for one another. Okay, and now the further claim is that the difference between left and right can only be apprehended by a certain pure intuition. And indeed, that then is generalized to all of geometry, that space does not have more than three dimensions, that between two points there is only one straight line, that from a given point in a plane surface a circle can be described with a given straight line, et cetera. None of these things can be derived from some universal concept of space. They can only be apprehended concretely, so to speak, in the space itself. Okay. But now, let's see. I want to consider that symmetry, sorry, that mirroring is a symmetry of physics, as Kant understood it to be. Now that's not true of the weak interactions, but I think there are good reasons not to appeal to the weak interactions too early on anyway in this game. We should ask, well, what if all of the forces known to physics were mirror symmetric? What should we then say? And in particular, what follows vis-a-vis the thesis that only invariant quantities are real? Invariants under symmetry transformations. Well, then that will tell us that if left is mapped into right by a symmetry, mirroring the symmetry, then the property of being net right-handed is not invariant, and therefore is not a real property of things, likewise of the property of being right-handed. That's how I understand the puzzle, and I think it's not very different from Kant. He didn't have the principle that any invariance from Kant's

10:00 understanding of the problem. Now, here's a crude way to see that under this situation, it may be that being left-handed and right-handed are not intrinsic properties. OK, so imagine a world, FX. We imagine the mirror world, F of GX. And this is precisely the same world, the same physical processes that the mirror image process, down to every last detail, and here we have someone observing something before her, and she says, it's to my right. And in the mirror world, we have the mirror person observing the mirror direction, undergoing the mirror neural processes, making the mirror noises come out from her throat, and of course she also says, it's to my right. Okay, and I think this is a little bit remarkable. It's got to be audio, that's right. She'll write it down and it will look a little bit different, but still she will not write down the word left. And I think what this suggests is that if we really take mirroring to be a symmetry, a global symmetry, then it may be that the world does not differ and would not be perceptibly different from the inside if one takes the mirror image okay, so here's the proposal that just as positions in space goes over to relative distances invariant under-translations so the left-right right, left is a predication of things, right is a predication of things, goes over to congruence and anti-congruence, also invariant under symmetry, under mirroring. Okay, so that's the proposal. Invariant under reflection is right. Now, let's see, do I want to go on? Yeah, I mean, there's various intuition pumps here. I've already used one, which is that the mirror-image person will say exactly the same thing. The merbius strip is another way of getting at the same issue.

12:30 Here's a two-dimensional hand-of-object, and as it goes around the strip, of course, it ends up being not superimposable on the original. So no matter how it's twisted and turned, as long as it stays on the surface, f and mirror f can't be made to coincide. You've got to go around the loop. And now one asks, well, imagine following such a handed shape around a loop in a non-orientable space. So a non-contractable loop. When would it seem to change its shape? And the answer is the answer is that it seems to change its shape. But in some sense, being left-handed, or being right-handed, can't be a legend. And a way to put this, I suppose, is to make it a little more detailed. Suppose Norwegian interaction. Imagine a possible world, in fact, on a non-orientable space-time. And 6 is locally just the same. And then I think it becomes clear that whether something is left or right can't be a matter of simply holding it up to inspection. So actually I suppose this is a very direct challenge to Kant. I haven't thought it would be quite that way before. It's given a common sense, an intuition. You just see that the object is left rather than right, and yet you transport it around this enormous loop in the space-time and come back, and at no point can it look any different, but of course it has become inverted. I don't know what Kantian would say to that, and I don't know what Kantian would say to that, but from my point of view it's grist to my mill, because the claim is that left and right do not in fact differ intrinsically at all. here's another way of getting at the same point let's put it in terms of Galileo's ship now that's a very useful way of thinking of symmetries and how we can learn about symmetries so in the historical case you have various sorts of experiments performed in the laboratory but then you construct an identical laboratory on board the ship you set the ship gently and motionally to form all of the same experiments, can you tell the difference as long as you stay inside the ship? And the answer is you can't. No difference.

15:00 We conclude from that that physical phenomena do not pick out a preferred state of rest. So the analogous case for mirroring would be you have the laboratory, and by all means have an experiment to determine whether a foot is a left foot or a right foot. It consists of a slipper, you see, a machine, and a person puts their foot into the machine, and the machine fits the slipper to the foot, and if it fits, it pronounces left foot, and a red light goes on, and it doesn't fit right foot, green light goes on. So, that's the laboratory, and so forth. Now, instead of setting it all in motion, what have we got to do? We've got to mirror-invert everything. We have to mirror-invert this machine as well. So, of course, it will now be tested with the right slip, whereas before it's tested with the left slip. And you perform every experiment inside this mirror-inverted laboratory, and the same lights go on as before, and so forth. And equally, as with a ship in motion, of course, if you refer the ship to the shore, you say, oh, well, there's a change of relative distance, and that's something physically real. And of course, you mirror-invert everything inside the ship, refer to the shore as a difference. But then the point is, the ship becomes the whole world, a closed system, and there is no shore to refer it to. And thinking that through in the case of, say, boosts, if you boost the entire matter content of the universe, it makes no difference. Another way of putting that is, there's no active interpretation of symmetry as applied. And then the same claim will apply. You take the mirror image of the world, it makes no difference. Alright, so this is the claim. What we understand about orientation. That an object is handed or not handed. That's simple. I can get you to draw a handed two-dimensional shape without any trouble, just by using words without ever using the word left or right, or a word like F or something. And I can get you to draw another one, which is congruent or anti-congruent. I can choose. I can get you to draw it one way, meaning the congruent way or the anti-congruent way. But there it stops. What I can't get you to

17:30 do is to draw the F. Now, what does it mean when we call the shape F? What are we going on? And here's the claim. Certain handed objects, like hands, screws, clocks, cars, are called right, standard, clockwise, right hand drive, and so forth. And we learn how to use these words appropriately. And the reference, so we call a hand right to what exactly? Well, we each perhaps have our own, I think of Wittgenstein's metaphor, beetle in a box. We all actually do have our own beetle in a box. And, oh, I've gone too far now. I refer to the hand that I used to write for. And anything congruent to that I will call right, or I refer to anything anti-congruent to the hand on the side of my heart. Now, all of those referential devices work the same way in the mirror image world. So if I say of an object that is right-handed by these criteria, in the mirror image world, the mirror image object will also be called right-handed. Okay, and the critical point here is that what is happening is being right or being left is being reduced to a non-geometric part. Being on the side nearest to the heart is, for our purposes, a non-geometric part. Okay. Now, um, Weill shared, um, the presumption when he wrote this, which was in 1945, sorry, I think we have to see him as agreeing with everything I've said. According to Leibniz, it would have made no difference if God had created a right hand first rather than a left one. One must follow the world's creation a step further before a difference can appear. Had God, rather than making first a left and then a right hand, started with a right hand and then formed another right hand, he would have changed the plan of the universe, not in the first, but in the second act, in bringing forth a hand which was equally rather than obviously oriented to the first creative specimen. I think Vile was agreeing with everything I've said.

20:00 I think Vile is the forerunner of this view. Perhaps Leibniz, but Leibniz, but didn't. For the commentators, who has said the same? And I welcome again instruction. My understanding is that John Ehrman has rather guardedly hinted at this view that left and right are not intrinsic do not intrinsically differ at all. And I don't think many others writing in Corbett and Counterparts have. So I like to think that it's still somewhat controversial. But Oliver Pooley in his last writing said that it now widely agrees that left and right are not intrinsically differ. I thought that Graham was the only one who ever said the opposite. Oh, that left and right do intrinsically differ. Oh, do you feel that? Oh my gosh. I think the section of articles in the... I forget the... ...seems to be that Kant was right, that it's a difference that has to be shown. oh, oh, oh, no, but the point is not that, oh, no, I should make it see, what I've done, and I take it what Leibniz has done is that Kant is wrong that's what I'm taking this to be saying in agreement with what I am saying Kant is wrong, in other words, there is no further thing to be said there is nothing further to be said is handed, that an object is handed, congruence and incongruence relations among objects, and that certain specific objects are called left-handed. There just is no further understanding to be had. The problem of 3D natural and space, 3D objects, and the problem of whether or not space, 3D is oriented or not, it's a cosmological problem, and for Kant, of course, space was orientable, and so you don't have continuous loops exchanging right-handed, you need a fourth dimension,

22:30 And you have to embed the 3D space in a 4D space. Yes, but look, so Kant is right because it is a problem of isotopy. You have two connected components. Yes, but hang on. Look, the Möbius strip- You have continuous desorienting loops in a 3D space. Yes, yes, yes. But that was only... that's one argument among several. It's this notion of an intuition pump. Namely, something so that you can start to understand what is being thinked. That's the only point of the Madness script. So here was to help you... You use it as an argument? No, it's not an argument. I'm sorry, I misspoke. I did say after the end, oh, this would be an argument against Kant. I hadn't really considered that it was specifically an argument against Kant. I need to illustrate how it could be that being left or being right-handed is not evident in perception. But the claim I'm making is not dependent on that. The claim is that there is nothing more to being left or right-handed than to being congruent to something that is called left-handed. And this being called left-handed does not depend on its geometry. That's complex for that. Indeed. But therefore, take the mirror image world. The mirror image object has the same handiness. And I'm assuming the mirror image of a universe containing a left hand will again be a left hand. It counts as denying that. I mean, what remains to be explained is still rentability, I mean, just in fact that certain many forms are rentable. Well, that's, yes. Well, what remains to be explained is parity violation. That's what remains to be explained. I mean, I've got us to a situation where, assuming mirroring is a symmetry, one can defend the standard view that only invariant quantities are real, in the way that I've suggested, and I see Viola as being the progenitor of this view, and, well, Bass, would you still say that everybody is in the same way? Yeah, because I guess I read Calme differently, that, um, a lot of...

25:00 Okay, but I think that's an enormous region of concepts. I mean, Kant did regard this as intuition giving us key concepts of geometry. No, no, it's more intuition and making a difference there. Non-discursive, they become intuitive concepts. But yes, all right, and then we have this. How do you think that something gets to be called in the writer? What is this calling? I mean, right and left really are being used like proper names. And how do you use a proper name? How do you baptise them? It's a very interesting question as to how children learn left-right. I mean, I think we need to be saying to them... What are the conditions on the rich? Choose any name you like, put a hand on the side of the heart, or that a hand that is typically stronger with any name you like for it, and then anything which is congruent to that gets called by that same name. There's nothing further to be understood because it isn't geometric. If it isn't geometric, then congruent is geometric. That's it. That's the limit to it. Well, that would help me agree, actually. Well, okay, but I'd be very surprised if you could show me any sentence Stanky, isn't it, was the editor of that book, which says this, that the mirror image world is identical to the world. And it's not because I want priority or something like that, it's just that I'd like to know who to cite, you know. Somebody clearly has said this, I'd like to cite them, and Vial is the only person. Would you say that Vial is? I think you meant rotation, because it's not clear why relative distances would solve the problem if you meant mirroring. But I did once discover, but then lost, a fragment of his, which used the same sign as a triangle and its marital image. I think from that actually, Gladys did know that the world and its marital image is the same. The image of the world would be exactly this world, do you mean that from the point of view of any internal relation to one world, nothing would be changed from the world to the mirror?

27:30 I mean that there are different representations of a single reality. And in that case, from the standpoint of these external observers, they are not, I think. Yes, but what relations can that external observer bear to a world, other than his own? I mean, this is exactly what I'm coming on to. So, shall we move on? I do think this claim ought to be counterintuitive. counterintuitive. It ought to be counterintuitive, but still I think it's defensible. Just a comment on mathematics handling symmetries, very similar to the situation in physics, and the claim further, and this is something that mathematicians, I have yet to meet a mathematician who denies it, that left and right do not differ intrinsically. one cannot characterize any structure in mathematics as a left-handed structure as opposed to a right-handed structure. And the other important example is i and minus i. And no matter how sophisticated a mathematical development one has on the basis of a choice, if you measure the unit i or minus i, about twister theory and some complex, whatever. You've always got the complex quantum structure, and it would give you exactly the same mathematical properties. And what about t goes to minus t, time inversions, interpreting it physically? And now the point is that this is not a mathematical symmetry, Unlike i goes to minus i, or the left-right distinction. There's no non-trivial automorphism with real numbers. Now, what I think... Well, I mean, the next comment is that p, c, and t symmetries of relativistic... are mathematically not on the car. There's more work to be done here. What is complex conjug... It's called complex conjugation in quantum mechanics, but what does it really involve? Does it involve i to minus i? It's actually a unitary operator, a charge-maturation operator in quantum field theory.

30:00 It's complex. The issue is a complicated one, but... Just a quick comment on what C is doing. Why is it different in the one-particle theory from quantum field theory? And the answer is that in the one-particle theory, complex conjugation, or P is a better way of putting it, in the Hilbert space representation for a complex field, you typically define the imaginary unit in terms of the decomposition into positive and negative frequency parts. And this is not the same thing as the I that figures in a complex scale of field. Okay? The multiplication by I at the level of the field equation is not the same thing as the multiplication by I at the level of the whole space representation. There's more to say about all that, which I'm not going to, but I hope you get some sense of why this is a subtle question. We can, to some extent, we can put to one side the issue, let's forget about C in the complex conjugation, just look at P and T symmetries. If the point of time inversion is that it maps real numbers representing time into the negatives, then that is not an automorphism of real numbers. So P and T do differ in some fundamental respect, but if you think that the choice of an arrow to time is the choice of an orientation of a one-dimensional manifold, then P and T become exactly the same. And very probably that is the right way to think of what is going on with an arrow to time. if we have a one-dimensional manifold and we have to define a narrow. But if you think that the issue is real numbers going from positive to negative, then that's not really an actual isometric rule. Real number structure properties real numbers as a field of numbers. You could use it in mathematical physics. Directly, bypassing one-dimensional manifolds and the issue of how to define a narrow and one-dimensional manifolds. I have a mathematical failure of symmetry to represent a physical failure of symmetry. I mean, dude, why not?

32:30 And in fact, the equations we actually use are much more appropriate in the other schools in terms of we have a one-dimensional method we use in physics to not exploit the field property of real models. And they don't exploit this sort of T squared or this T. But this is just a natural structure of a reaction, the sense of acting. Art is really clear. Yes, that's right. Let's go on, because let's go on to parity, because this now becomes the interesting question. And I just want to remind you how shocking it was when parity, though I'm reading from rabbi now actually, The complete theoretical structure has been shattered at the base, and we are not sure how the pieces will be put together. And Pauli said, after the first shock is over, I begin to collect myself. Yes, it was dramatic, and so on. And here's Dyson commenting on Lee and Yang's paper, where they first surveyed the evidence of the lack of it, in weak interaction physics, comparity as a symmetry. Twice I said, this is very interesting, but I had not the imagination to say, by golly, if this was true, this is true. It opens up a whole new branch of physics. And I think other physicists, with very few exceptions at that time, were as unimaginative as that. So I think this is evidence that there was a thought that mirroring must be an acrylite symmetry. And I think they were right to think that. Now we have to explain how can it then be violated. So here is an experiment of Crawford eval in 57. Weil wrote his, I think in 55, I think he died in 55. Maybe he wrote it in 53 or 54. We know what Weil would have said. So this is the year of Parade Evaluation 57. We have a pion that comes in. And actually, we've got scattering of the proton, I can show it, we've got a lambda-hyperon produced in the k-on, lambda-hyperon subsequently decays weakly into another pion in the proton. We define one plane, which is the plane containing the incoming pion and the lambda-hyperon, another plane containing the pion, the outgoing pion and the proton. And these two planes are not to the other side.

35:00 And the cross product of the pion momenta and the lambda hyperon product . So that's the violation of the pseudo-scaler. So that's the violation of mirroring. And this is the other way in which it could have gone. And if we saw as many decays like this as like this, then we would have no violation of mirroring. we see preferentially the one, not the other, we have a violation of mirroring. And just to get it totally clear what we're talking about, here I've given you A cross B as the vector C, and the dot product with D being positive is one orientation, and if D were the other direction, it would be the other. So now let's go back to our two possible worlds, mirror world FX and the world FX and the mirror world. Okay, so, that should be the dot, I'm sorry. I've gone across here, that should be the dot. What's going on is that if we imagine this cross product is given, it's ruling out the mirror world. It's saying the mirror world doesn't happen. in which hyperon decays and one orientation is the real world and the mirror world is not allowed. Okay. But then, if we look at how the cross product is defined, and now it's Michel's point. So, here's a hand rule. That's how it's defined. There's no other way to define it. There's no other way to define it. If we imagine this hand as being... in the mirror world, and one uses the hand rule transcendentally to give the orientation in both the world and its mirror image. Then, of course, A and B are vectors. I'm using the hand rule to give me C. I take the mirror image of A and B, and I use the hand rule, and what direction do I get for C? Down. Okay. It's not the mirror image. I have a vector D, and C dot T is positive.

37:30 Then here C A equals C is downwards. The mirror image of D is as shown, and here I'll have it as negative. so this is all with a transcendental use of a hand to tell us what is the orientation what if we abandon the transcendental point of view and I think we have to because what is the sense to this hand, not in the world that is congruent or anti-congruent to this world this is a very special relation we've given no sense to it if we use the hand rule internal Then, here's the hand rule in the world, and here's the mirror image hand. Remember, I'm using the hand that is on the side of my heart. And in the mirror image world, that means it's going to be the mirror image hand. And now, of course, I get that the cross product gives me the same vector, or rather the mirror image of C. And if I have D in both worlds in the mirror image, now the scalar, the study called pseudo-scalar is positive, the pseudo-scalar is positive in the mirror image world as well. So, what this is saying is that pseudo-scalars, in a sense, can't be defined operationally. That's one of the things I might say. Try it with inversions, just if you're thinking there's some hocus-pocus going on. where, unlike mirroring, you invert, it works in odd dimensions only. So here I've inverted every vector. And again, I've got to invert the hand and not just mirror image it. So I could produce the mirror image with a keystroke or two, but I can't rotate them to about 180 degrees, okay? That's impossible. So you see, I've got it there as well. Anyway, and it's the same results. in both cases. Okay. All right. So, now, what that tells us is if we take this transcendental perspective, we ask what will people say in the world, in the mirror image world, well, they'll say exactly the same things on performing parity-violating measurements. They will draw exactly the same conclusions. And yet, from this transcendental perspective, only Only one of these worlds is the real world, the other one is disallowed.

40:00 So there is nothing about the world that tells us it's disallowed, and this, I say, is a violation of the principle of social use. Only one of two worlds is allowed, but there's no evidence available within the world, there's nothing that can be operational defined as, no, we can't give any content, any meaning to what it is about one world rather than the other. All that we're relying on is this transcendental perspective to tell us that one of the worlds is disallowed, or under-determination, and another way of putting it. So, we have two senses of mirror symmetry, one internal to worlds, and as such an apriori symmetry. Whatever is true of one world is true of the mirror image world. Another external to worlds, broken symmetry. Under the apriori symmetry internal to worlds, we should identify the world in its mirror image, but under this external transcendental sense, we should distinguish them. They're both indistinguishable from within, and the suggestion is we give up this transcendental notion. That's the suggestion, or preliminary suggestion, anyway. And then the claim is that the only meaningful use of cross-product and so forth in common institutions within in a way. But... No, I agree with you. No, no, I agree with you. You actually break in. Symmetry breaking is... Tell about it. Ah, oh yes, yes, yes, but that's to be handed. That's saying you have a handed process. And in the mirror image world you also have a handed process. There being a handed process that is... Ah, no, no, but the claim is that the world and the mirroring in this world are identical, different representations of the same reality. The fact that this reality contains a handed process is to say that mirroring is violated. When people ordinarily say mirroring is violated... But if you eliminate the transcendental concept, nevertheless inspired by this, you have the problem that in this world of the principle of a raison suffisante of efficiency written. Well, you see, my claim is that there isn't. Let me carry on. I thought you were going to say something else, actually.

42:30 And you have no reason for the choice of this symmetry breaking or this symmetry breaking. OK, but you mustn't express that in terms of why this world and not the mirror image world. OK, so that's what I'm ruling out. OK, so then we've got to think, well, what do we mean? I think so. In fact, I'm not sure if I should go directly to this. Maybe I ought to say a bit more about this. I think I agree absolutely, even I think there is a mathematical argument for that, just this talk about mirror walls is something not very clear what we are talking about, because if we are already in a kind of manifold space, right there could be different cases, whether it's orientable or not, whether it's like or it's like space, and then it matters, and then the whole talk of mirror world is just not very clear how we can talk about. Well, in a strip it's very clear that mirroring acts as the identity. So the non-orientable space, it's not that there's a lack of clarity about what we mean by mirror image world. I think it's perfectly clear. So whether you've got orientability or not doesn't make it less clear. We've got perfect clarity here of what we mean. I'm not questioning that we can imagine worlds. Okay, but that's a different sort of argument and so forth. That's a very content argument to say we cannot even meaningfully imagine the world. I'm supposing we can imagine worlds, and I'm saying we imagine the world, we imagine the mirror image. as representations of a single reality, that's the claim. Now, I think, let me just wait a moment before getting on to this. This is a comment from Hug It, and it feeds into a debate with Oliver Pooley. I just want to think a little bit more about what's really going on,

45:00 and relating, actually, to Jean's question. Okay, we think that if we stuck with this, but we only imagine these relations within a world, then actually it does feed pretty directly into what Huggins says. So let's look at this now. Okay, so if that's all that we're going to do, then what we've got is orientation to find relative, well, not so much to arbitrary coordinates. He devised a one-dimensional parity-violating quantum mechanics, just to see what he had with the problem. He observed the development of the particles, we could determine the direction of the array to express the orientation, express its direction of orientation, in relational terms, say, by two standard objects and their order orientation. Okay, and that's like taking the decay process and saying it's congruent to my left hand. And he's saying that this isn't acceptable. It's not that there's a descriptive problem or even an epistemological problem, but there's a problem with formulating a theory of the process in suitable relational terms, and a plausible theory should not make fundamental reference to a contingent standard. So there the claim, I think, this is how you have to re-hug it, and this line of thought is, if you do only define left-right internal to worlds by the congruence and anti-congruence relations with named hands and objects, then you're doing, you're setting up equations where the sense of orientation is defined by reference to some particular contingent standard. There's something wrong with that. I think he's right. I mean, I think there is something wrong with that, so I'm tempted to agree with that. And what that makes me think is that we shouldn't be too hasty in, I'm not sure if I've forgotten, this is my next slide, PowerPoint, you never know what the next slide is, once you've practiced for hours beforehand or something. Giving up on this. I think we shouldn't give up on this. I think that we do have an external notion of left and right. They are isomorphic. But I'm allowing that we do have a conception of, say, left-handed coordinates in mathematical platonic heaven. We have a conception of left-handed coordinates isomorphic to the conception of right-handed coordinates. But we can use that mathematical thought, without having any pictures, without having any contingent particulars,

47:30 without holding up our hands. You know, the mathematician is capable. Okay, so I think we should stick to that. You're saying that in mathematics you can make a distinction between Heisman-Worto-Balka. Ah, only relationally. I would characterize the left in itself as distinct from the right, but you can characterize the two structures in relation to one. I can't stand that weak discernibles, actually, are going to be a selection of structuralism. So, or, you know, I don't have to insist on that. I'll grant it, if you want to believe that there is something that a mathematician can grasp in the left, that is distinct from the right, I'll grant it, but I deny that it can be used in some meaningful physical description. Because if it is, I want to see what it is. But my true belief is that you can only grasp them mathematically in relation to them. But let's keep that mathematical concept. Now, what do we have to do if we use that mathematical concept in physics? We have to give an account. What do we have to get right? Ask yourself, for the principle of sufficient reason, how could the world have been different? When Madame Wu did the experiment, and they were all there, crowded around the machines, and they were looking at the simulation screens and so forth, and one laboratory wall is painted blue, and the other laboratory wall is painted red, what did they not know what was going to happen? They didn't know whether the pions were going to go towards the blue wall or towards the red wall. And they had to find out which way was God handed. I think Pauly said that God was weakly left-handed. Well, what if God had been weakly right-handed? What would have been different? Well, you literally, instead of going to the blue wall, you would have got to the red wall. Instead of holding up your left hand and getting the common ones, you would have held up your right hand and got the common ones. Now, this does not, given everything that I've said, this does not amount to taking the world in this mirror image and God chooses one of them. No. It's God chooses a world in which weak interaction process is congruent to, let's not talk about human hands, let's talk about RNA, which is one of the first carbon molecules, congruent to RNA, or God chose a world in which it is not congruent to RNA, anti-congruent to RNA. Those are the choices.

50:00 And this need not involve any violation of the principle of sufficient reasoning. In fact, it may be totally over-determined. And here's another way of seeing it. The weak entraption introduces a shift... How do you look at green? Well, hold on. The weak interruption introduces a shift in the ground state for parallel molecules, depending on whether congruent or anti-congruent. And the lower of the two ground states is the one that would be energetically favored. And indeed, that's the one which gives us the congruent nation that we see. Okay. So that would be a way... I agree with that. Okay, this is a way in which we see that God had no choice as to whether to make the world with RNA molecules congruent to the weak interaction as opposed to incongruent. The ground state energy where you have congruence is slightly lower than the ground state energy where you have anti-congruence. It's a fact. It's a minute, you know, it's one part of 10 to the 12 or something, it's minute, you know, by thermal fluctuations and the rest of it. It's true. No, it's not false. No, it's true. It's true. It could be a false point. Now, there's dispute about whether or not this could... Is this going to force chiral molecules to have a hand-off solution? Is this the explanation for why we see chiral molecules in a hand-off solution? Anyway, so there's a debate about this, but it's just an illustration of how it may be that there was no choice, but had no choice about the matter. The key idea here is that you're using a mathematical concept of left, or right, to describe the world in which the weak interaction is congruent to RNA. And it's clear what you would have to do to write down equations in which you get the anti-congruence, significant difference in the equations. If this story about the weak interaction coupling to these molecules is true, then you've got to grow in physics in order to get out the contrary right-handed versus left-handed. And I think that's an answer to Huggett's point. I'm not defining the equations by reference to, you know, sort of contingent particular.

52:30 I'm setting out equations which describe at least two-handed processes in a definite congruence or anti-congruence relation, that's what I'm doing. Now another sort of rejection, and this is from Oliver Pooley writing about Huggett, But it's a slightly different objection, I think. He's talking about an orientation field. One defines the equations using a contingent standard to define your equations. As long as you build into your equations a description of RNA having the right congruence relation with the weak interaction, Then you've succeeded in giving the correct description to the left. So, um, so Oliver is looking at the proposal of an orientation field. And this is a slightly different way of putting it. And this is the issue of how does, we have one decay process takes place here, another decay process takes place over there. How does the physics, as it were, know how to get the decays both in the same sense? How does it get the decays to be common to one another? How is it locally? What is there locally to guide the physics to tell it which way to decay? And one solution is an orientation field has to be defined. So there's some local field there, which is an orientation field that tells the physics which way to go. And he says, this has the support of Bob Wald, in his book, General Relativity, makes this claim. He says, you need an orientation for it. And he says, look, it appears to involve an unavoidable commitment to the reality of difference that are unobservable in principle. The theory has only electrons congruent to such a three-couple of W bosons, and the theory of coupling to Dublin business must be regarded as distinct even though observations are indistinct. Okay, now it's rather important for Pulley's claim here that this orientation field really is a physical field. I think that's the wrong way to look at it. I mean, here's the point. You see, we have this

55:00 This business, we have the left, is given in platonic heaven, but equally we could have used the right in platonic heaven. Indeed, my suggestion is the ones I'm defining in the nation to know. Will do. Either will do. It's exactly the same as a choice of coordinates. Say, in the Diffomorphism, Covey, and Ethereum. The choice of coordinates is the status of the fundamental field. It only makes sense if you have the idea that the choice of which of these is a matter of choosing something to do with what is observed, or to do with some true, correct representation of the spin-off field as being this way, let's say, rather than that way, unobservably. Because the observable only corresponds to the orientation of the spin-off fields to things like hands, particulars. And so now the issue is, well, what about the spin-off field with the orientation field? But the point is, you can use either of these two mathematics. it. Each of them describes both. That's the point. The choice of mathematics left, abstract, mathematical left, as opposed to abstract, mathematical left. And yet, it has nothing to do with whether the spinner fields are left-hand or right-handed. The only question to decide is whether the spinner fields are congruent or incongruent to some other hand and structure. And you can do that with either choice of coordinates. So I think it's actually innocuous. And the upshot of the talk is no violation of principle sufficientism. Everything has been successfully resolved. The only choice that remains is for those mathematicians who think that you really can choose abstract mathematical left as opposed to abstract mathematical left. you think you actually can, so I think even that choice is bonus. You believe that you can, right? And that is like a choice of two coordinate systems, not two coordinate systems. The only coordinate system does to describe all of the reality. Look, there may be some things that are not so clear, and, I mean, one of them is, how have I conjured up mirroring as a symmetry where there was none? And I do think we have a good precursor.

57:30 Think what happens with translational symmetry. Now, of course, we have it in Galilean spacetime, Newtonian spacetime, McCoskey spacetime. We don't have it in an arbitrary general relativistic spacetime. In an arbitrary general relativistic spacetime, you've got a bump that respects the internal geometry, because we have some structure here. This is not a homogeneous model. Okay, but of course, and in that respect, translations of some particle distribution relative to the bump is not a symmetry. And of course, we look at the manifold, the metric, and the matter fields. If we translate both the metric structure, then of course we retain how a symmetry can be lost at one level, restored at another. How we account for this is the translation is not an isometry. So t drag along of g is not equal to g. This expresses the fact that the space time is not homogeneous. So, translations are not isometrics. But, of course, translations in this sense, as a diffeomorphism applied to both electric and the dragolimetric and the dragolimetric is asymmetry. And I'm saying the same is happening with hands, with mirroring. Now, what we've got in M is an atlas of charts. It is a condition on the axis of charts that where two charts overlap, in this region partial derivatives of one with respect to the other exist, and we say that the manifold is orientable if you can choose these charts so that the determinant of the matrix of the Jacobian is always positive. I'm imagining now a manifold, let's make it an orientable manifold, and you choose an charts which has this property, fine, and build up a theory on this basis. Now you could have built up the entire description on the basis of, and now let me put a little symbol R for reflection, using the oppositely handed charts throughout. And you could have constructed a metrical structure which is oppositely handed.

1:00:00 I mean, this is most obvious if you imagine the thing embedded in a higher dimension. I'm not sure if it's true. Is it the case that only a four-dimensional general altruistic space pan would be played from five dimensions or higher? I don't want to make sense of mirroring as, you know, as a higher dimension. The idea is, I think this must be true, one can construct an identical, mathematically speaking, isomorphic representation using the opposite behind the charts throughout. And then the statement that this is a theory where marrying is violated is the same, that marrying is not an isometry, meaning that this is a representation of one, and this This is an isomorphic representation, but this is not the solution to the equation. One is and one is not is what tells you that mirroring is not an isometry. This thing and this thing represent the same reality is the restored a priori mirror symmetry. Well, suppose there were just as many people with their hearts on both sides. and relax to wear it on one side rather than the other. So if something was to be said, maybe that there is an asymmetry that you can use to identify one as it goes to the other. Yes, they need to be handed objects. No, no, no, not just handed objects, because there are handed objects even in the universe where everything you see in the mirror image is still handed objects. Yes, but you only have to pick on one handed object to dignify it. Yes, but how do you figure it out? Oh, well, you would have to have a definite description, I'm saying, that was right. Um, well, yes, I would, yes, let's have this, absolutely. Yes, yes, yes. Okay, so in other words, something more has to be said. Yes, but it's non-geometric, that's the point. It could be any old non-geometric fact about an object. Yeah. Or contact, is that what it is in the particular,

1:02:30 I agree with you on this reading of the directions in space, and I think you may have something there. But it's not the case of the inaugural dissertation, nor of the subsequent uses of it. In the critique? In the critique, it doesn't exist, in the protagomino, in the protagomino. It's very clear in the inaugural dissertation. I think there's no change. I actually, I'm sort of giving you maybe more than I really want to. I agree you have a point about directions in space, that really what Pound is getting at is how do we locate ourselves on a map, and I know that's your favourite way of putting it. And I think that it drops out of the equation as to whether or not being left-handed is a geometric thing. In some of the comments in that paper about the hands in space, there's the left hand and the right hand, and God will have to make a choice. Being left or being right-handed is a geometrical got to you. You have to choose one rather than the other, and how on earth is God's going to do that in violation? On the account that I'm given, there is no one. The left hand is not. The hand in space, when you otherwise empty space, is not met. There has no one. Well, the real choice is with the second hand of you, because it's a corner of your answer. And it's the world. You can make reference to any particular rule. Are you thinking that somehow it's been a rule out of reference to particulars? If you can access complete symmetry, then for every definite description, sorry. Let's describe this work a little bit more. A world in which there's no, there's norms to respect monosymmetry, would be a world in which for every left-handed organic molecule, you'll find a right-handed organic molecule. For every left-handed process, dynamic process, you'll find a right-handed dynamic process. in which, for every object, the mirror image object exists somewhere or other. But it doesn't mean that you can't pick out a particular. No, it might not have problems. A world in which you can't even pick out a particular would be a bizarre world, it would be two blacks, two spheres, or, you know, one of those. Well, how about two hands?

1:05:00 Yeah, so nothing else. Yeah, so there's no right hand, no left hand. Indeed not, there's just two oppositely handed objects, you can tell me, but again, no problem, nothing to explain, nothing further time lags. If that's it, these are just two weakly discernible hands, you can't make an individual reference to either. Well, okay, but the point is that when you said there was nothing more to be said, that's not quite right. It is. When you talk about something you called right-handed, there is something more to be said. No, no, no, no. Wait a minute. In a world where there's just two hands and nothing else at all, in the entire universe, There's no speech. There's no uses of speech. There's nothing more to be said. We're not thinking about it either. There's no way to call one thing right and no right. Absolutely. So there wasn't more to be said. No, but there's nothing then to be said. I mean, this is just two discernible objects. You know, this is like two fermions in a single state of speech. So there's nothing more... No, wait a moment. You're not getting my point. No, I'm not. So take a multi-various, take a variegated world. No, no, no, let's not do that. Well, I'm still happy to have mirroring the symmetry of the physical role. I don't think I'm talking with you about that. It's just that when it comes to something being called right-handed, it can't be done in the little world we're just talking about, and it can't be done in this world. So there wasn't anymore to be said, only about the difference between those kinds of worlds. Yes, but the crucial point is that Why you call something left rather than right is not geometrical. That's the crucial principle. I mean, it has some action. Well, there's a convention about words. You might have the view that there really is intrinsically distinct and geometrical. And then there's a convention about that word. But our position is a legitimate argument. To refer to any particular year in the world, which is not so symmetric, that you refer to a particular sort, then it doesn't matter what particular you refer to as a hand particular, you can call it anything you like, class everything respectively. I don't think Kant's point was this. I don't agree here. For Kant, what was happening in the recognition of an object as being left isn't that we just arbitrarily picked on some object and wanted a name.

1:07:30 To observe, when it's given an intuition, then one grasps the concept left rather than right. No, you just feel it first, the more that it is in congruence. Actually, what you can say more about this world is to have, it's orientable, and that's something important. I mean, you have a world where you have two things which are locally indistinguishable, these distances, but then not congruent. And that very important property of that couple of things, right, taken together, then you have just one hand, and you don't know really, you don't have this property of right hand. No, it's not an intrinsic property, but it's not an intrinsic property that is available to the understanding. No, but it's not available to the understanding. But it is an intuition. Not for intuition. It is not an intuitive property of the right hand to be the right hand. What can I say is that if you look at the two hands, you see that they are incongruent, but this incongruence is not conceptual. It is impossible to find a conceptual property, but it's the point. So, the intuition don't say that this is the right hand. The intuition says that these two hands are incongruent. They are not the same. Yes, but now I'm worried. They are not identical. No, but now I'm very worried, because... But this identity is not conceptual. This non-identity, this different, is not conceptual. Yes, but that's... And it is against language. No, no, but this is... But it's just false. Because the two hands do differ conceptually. Of course they do. As I said, I can get you to draw either that or that, and I can get you, once you've drawn, whichever of the two you've drawn, I can get you to draw an incongruent one. So I can get you to draw this right next to it, I can get you to draw that right next to it. And I can explain how these two differ in their relations to one another, discursively, conceptually.

1:10:00 I don't need any intuition for that. Yes, it was the point of past. You don't have definite description. If the point is this, can I conceptually understand this as opposed to this, the answer is of course I can. Of course I can. So the only issue is, can I conceptually understand this, as opposed to this? Yes, go ahead and do it. No, I can't. And indeed, I'm saying one cannot do that. It can't disagree. He's saying that it is individually that the difference is not conceptual in the individual. But of course the relation, you can describe it with concept, because of symmetry of congruence, of course. But it is individuals that are not conceptually different. So what Kahn is saying is there is no concept which will pick out this rather than this. But he is saying intuition will pick out this rather than this. Contuition will pick out the difference between the two. But you don't need intuition for the difference. You don't need intuition for the difference. Discursive concepts. Discursive, discursive concepts will be different. Ganz kling is that it is impossible to make the difference conceptually. Yes, conceptually, but it is given an intuition. Yes, but the difference is an intuition. The intuition, you bring an intuition to explain how you can refer to one rather than the other. Okay, sure. This is to say, what is it about this that makes it that, rather than writers given an intermission of the kind. It's not the difference as the difference between a cube and a sphere. I have a cube, I have a sphere, and conceptually I can say that they are not identical, because they don't have the same property. The sphere has no angles to say the right thing. But for the two hands, you don't have conceptual differences, but the intuitions, you see that they are different, because they are incongruent. In three dimensions, it is impossible to find a continuous path exchanging the two. For the artisan... You know, look, okay, and I repeat, and I repeat, but if I have two-handed objects... It is an homological problem.

1:12:30 And I repeat, if I have two objects, two-handed objects, whether they are congruent or incongruent is conceptually given. If I have two-handed objects, whether they are congruent or incongruent is conceptually given, stateable and discursive concepts, no need for intuition. So, what further thing cannot be grasped in concepts, discursively, answer whether it is a left or a right? But what is there that I've said is false. When intuition has said that this and this are different, that it is not a concept for difference, but when you have the difference, of course you can say that this is not this. Because here you have an A1, A1, A2, A2, A2. The pair, A1 and A2, and the pair, A2 and A2, are not the same conceptually. It's not helping me to put labels like that, though. When you know that this is not this, then this pair, of course, is different from that pair conceptually. But you need to know that this and this are not the same. And this point is that it is impossible conceptually. I don't need to know. I can try to make the difference between this and this concept. I have a vertical bar. That's what it is to single out one of them rather than the other. We are trying. I have a vertical bar. And I put on the top a horizontal segment, to the right, or to the left. So you need the difference between right and left, which is not conceptual. This is to pick out one of them rather than the other. And I agree that can't be done in concepts. And I further say Kant believed it was done by intuition. Yes. But that's what I've been saying. It is the alternative to go in this direction or in this direction.

1:15:00 Which is to say, to pick out the left hand in itself, as opposed to the right hand in itself. This is what cannot be done by concepts, and that is what is done by intuition, for Kant. But this is what I've been saying throughout. You seem to be disagreeing with me before. Now you are agreeing. This is what I've been saying throughout. It is by intuition that we see that a hand is a left hand rather than a right hand. That the left is not the right, but you can of course exchange the name left and the name hand. We're not talking about names. You can call left, right and hand. No, no, no, forget about names. Names as it shows. This trick is here only for showing that they are not the same. Oh, it's mostly frustrating. Just to see that two things are congruent, you also need the intuition or something, because how can you say two F are congruent without intuition? Easily, easily, easily. No, no, I can explain this guys exactly in what respect these guys are anti-congruent. I can get you to draw two anticongruent hands very easily, all two congruent hands very easily. Geometric construction. It's straightforward, that is not the problem. The problem is I can't get you to draw. Yeah, but your description will be the description of a geometrical act. That's right. I go in this direction or in this direction. No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, I won't get you, I won't get you. It's not the property to go to the left of the . Or going to the left. Or you would say, go to the opposite side, the opposite side of what? Or space. And here again, . Well, what are you suggesting I cannot do exactly? When you have, for instance, A2, and you are trying to . Yes, but that's not problematic. I mean, it's not problematic for me to get you to draw What I can't do, I can get you to draw this or I can get you to draw this. I can get you to draw either that or this. So what I'm saying is, to get you to draw something handed, to get you to draw something also handed, same handed, I can do it.

1:17:30 Oppositely handed, I can do it. All those things I can do. So I don't understand this. you're going to disagree. So the problem is how to draw that rather than that. And that's what I can't do. Because you need to know that you're agreeing I can't get you to draw this rather than this. You agree I cannot get you to draw this rather than this. Do you Do you agree with that? Yes, I don't understand. I can't say what it is about this. This is where the concepts get out. I can't draw this rather than this. And I can't say what it is about this. Here's where you don't agree with me. I say count. discursive concepts are not available, but intuitions are. And this is you denied. Well, you see, I think that you then think that means that there's an intuitive property or something of that sort. But you should, because, you know, you're trying to put concepts on the side of intuitions after all. And you shouldn't do that. You shouldn't say that concepts are by intuition that you can grasp left-handedness. Intuitively. Yeah, you cannot do that. thing as a concept of left-handedness to be grasped. But you have to ask, under what conditions is it possible? Yeah, sure, but I think concrete. Under what conditions is it possible for someone to refer to one rather than to the other? I do find this such a strange way to think. I just find it contrary to... It is very strange to say that we do not have the ability to recognize a hand as being a right hand as opposed to a left hand. I thought Kant was saying, given the failure of discursive concepts to do this work, something else has to be kicking in, because we clearly do distinguish right hands from left hand. There's a difference by referring them to ourselves, by relating to ourselves. So then how do I know which hand is my left hand and my right hand? That's not the point. I think the right point is that it involves something more than, say, local properties of this F or whatever, right?

1:20:00 There are some kind of, I don't know, global space or something we are embedding our thing in, right? He calls this intuition, but probably we could call it differently. And that's really important because it involves just more than those objects. And that he points out, that property or rentability is something which is not in that object but in this space. Well that's the direction in space. That's what he dropped. This business is the relation to space and stuff like that. Yeah, he called it the intuition. And instead it became that there are things given to us intuitively that we cannot understand through discursive concepts. I think he took it as obvious that we can tell a left hand from a right. He took it as obvious that if a hand were created in space, in isolation from anything else, it would have to be either a left hand or a right hand. I think he took it as obvious. No discursive concepts. Intuition kicks in. This is my reading of Kant. I don't deny it. But it's an obvious reading, it's a straightforward reading. You have a subtle reading here, you know, oh, actually, he denied it. that the hand in space was where the left hand would go right up here. But he kept on holding that up. He held that up as an example to reductio ad absurdum. He made a reductio ad absurdum ad absurdum. It's true that he gave up the point of view of the normal representation, for sure. He never gave up on the normal station. Really? I mean, who do you think, if not too critical? Every argument, except for in common counterparts, in inaugural dissertation is repeated in the critique, in the aesthetic, almost word for word. Yes, but for the space and intuition, not for real physical space. Not for, in the physical writings you were talking about in the space. In directions in space he still was. Yeah, but also in the inaugural dissertation I think there was a sort of halfway house, but he wasn't yet to the point where the transcendental aesthetic is in place. I think all right, he didn't call that space the form of outer sense, indeed, but he didn't call it the space of Newtonian mechanics either. His point in the orbit of the station as well is the sensitive origin of geometrical concepts. Geometrical concepts derived from intuition, not from the understanding.

1:22:30 And that, in that respect, stayed the same for our people. We have the relativity principle. It is impossible to give a physical proof of an inertial movement, physically the arctic. It's quite the same thing for the symmetries. There are properties of space, the symmetry of space, which cannot be described, because It is something of the same type as relativity in physics. It is physically impossible to prove an inertial movement, and Kant says it is impossible to to prove concepts, some properties of space, essentially a symmetry in the field. I find that difficult to have another access than a conceptual access to that. That's a point I agree. I think the parallel with inertia, No, I'm not saying that it is the same. No, it's not the same. But there is an analogy between the two. It is a problem of relativity, you have. The analogy is not able to grasp symmetries. Well, okay, but I would deny that. I mean, so, you're putting this into Kant's now. For every description of every single space, you will have the mirror situation with exactly the same description. But that's not correct. Maybe it's in Kant, but it's not correct, right? If you describe your space as oriented or something... But look, I mean, I really think Kant is a bit... Kant, it was a 3D, Euclidean space. I have another question. I have another question. Concerning the symmetry breaking. The problem

1:25:00 when you look at an explicit process, take the magnetism for it, when you have a magnetization. So you have an energy, like that, which becomes like that, this point becomes unstable, and the system has to go there. But the problem is that this process of symmetry breaking, the fact that the system loses, a symmetry, is described formally by a structure which is symmetric. So you have a symmetry of symmetry breaking, of the process of symmetry breaking. And for that, as you know, the response is fluctuations. You have somewhere an infinitesimal fluctuation, and so the system goes here, and phew! And this is a difficult point. Yes, it is difficult, I agree. And the breaking of parity, of course the problem of the breaking of parity is an example. Ah, well, I don't know that that's right. Well, you have an exactly symmetrical world where you can have another choice. So if in our world we have this choice, there is something to explain, to be explained. My point is that the fact that the process of symmetric weighting is self-described by a symmetric structure is a violation of Langley's Principle. You may be right about that, but it's not what I think is going on.

1:27:30 So when you say that God must choose, no. No, but this isn't the same, this isn't the same. This sort of symmetry breaking is where you've got a symmetric round state, which is unstable. Which is the issue here, when you've got a point here. Yeah. And this is not what's going on with parity boundaries. Now, of course, it may be in some grand unified models it is, well, maybe, but then I would like to see this. But this is a different sort of question. In these unification models, you have all the symmetries, and after when you decrease energy, you have a spontaneous symmetries. I grant that this is an important question, but it's not what I've been discussing. This is an important question, it's not the one I've been interested in. My point about the parity violation is the alternative situation, so here you have, does it go this way or does it go this way? And the point about parity violation is, and the way I've been putting it, is it congruent? Is parity decay processes, are they congruent to some other process or not? And that is not an example of this. Absolutely not. But, I mean, this is important, and it's something I've often thought about, and I don't know Of course, it's worse, because these different ground states are unitarily inequivalent. So there's some genuine mathematical sense in which they differ, and that I would like to understand. How do they mathematically differ such that they are... People don't seem to have settled on the sign in the Schrodinger equation where the plane waves are to be resurrected, plus i or minus i in the exponent. And then at some point, which I ignore, I ignore Wayne, but after Solveig in 1927, the

1:30:00 standard way gets entrenched, and I mean, what are people doing until then, really? So is it a convention plus I minus I, or is it something else, and do people then recognize that if it is that convention, that it is purely convention, and then just choose one? Have you got a view? No, I don't have a view. No, I would like to know. I thought it was Stonesfield, but I don't think that's right. So which sign do you take Stonesfield? Well. I think there's natural asymmetry, whereas positive and negative is not. I have to say that part of the point about, and this is how in the philosophy of mathematics there's a plus sign next to the line. You haven't thereby chosen one by doing the plus in front of the i. Now, what happens then is people say, well, look at the representation in terms of pairs of real numbers. And then you say the i is the pair, one is distinct from minus one. But then the point is, this representation is not the complex numbers. and a certain mapping from R2 into the complex numbers, and that mapping is what is under to find. So it's a very interesting question. I think that Kant's argument is essentially the same as the difference between I and minus I. Well, I think the difference is that talking about what is presented in experience, and with imaginary numbers, so it is a problem. So, I complain, it's a ghost plane, it's very intuitive. That's the representation between the space and the mapping.

1:32:30 Improperties, relations and so forth, and to find invariant quantities are real, and what do you mean by real? In between pure mathematics. We're talking about properties. No, no, my claim is invariants, not covariants. We're not talking about things now, we're talking about properties, right? And so they're saying length is not real, but space-time interval is real. Relative distances. I mean, these things are all theory-dependent. So, the symmetries of the theory. Yeah, sure. No, I understand how you're picking it out. I know what you mean by invariant. But, you know, in the context of relativity theory, you can still talk about length. Relative distances are invariant. Yeah, but why isn't length? Well, I think length is a relative distance. Yeah, but why isn't it? What do you mean by real? I mean, why do you think some problems are real, and others are not? Well, I was going to fill out a picture of reality in terms of objects with properties and relations. Who are the objects? What are the properties? What are the relations? And my answer is those defined in terms of invariant quantities. Where, take your physical theory, those quantities invariant under the symmetries of that physical theory. But you don't mean that there are no non-invariant properties? Yes, I do. You do mean that, there aren't any. So the analysis property is length. I can say length as relative distance is invariant. No, no, no, but length. As relative distance. Or as you give me a notion of length where I can understand the length as relative distance. What is the notion of length where it is not a little? Okay, no, I'm trying to talk about length in a very ordinary way. So suppose that someone is moving in a spaceship, right? and he measures the length of that on the side of that computer, and I measure the length, we don't get the same thing. Yes, we do. It depends on which it's actually moving, right? No. I agree this is a significant question, but my answer is no. Interval between events. And if you pick the same events, the event interview is the same. Yes, yes, but the length of the rod is not, right? Because we're picking different events.

1:35:00 In fact, time dilation affects the sun, but because we're between different events is the appropriate world. It's plain. The appropriate world is plain. But are you saying the length of the world is not true? The length of the world plays the interval between events? Yeah, yeah, yeah, yeah. That's okay. No, no, no. But now, why, I mean, I understand why you have to say that one property and not to the other, but why do you say they're not real? Well, I, for one, it's all cases. So then I looked at the problematic cases, things like length contraction and time deviation. And exactly by looking at those problematic cases, I became convinced that the principle does work. Only in their enforcement is real. So that's just personal biography of why to arrive at this. I mean, how to, you know, I think the issue is, I suppose, a fundamental reason for it that characterizes the physical reality, then it better not characterize the physical reality. But why covariance is not possible? Covariance is the covariance, you know, mere covariance wants to exactly change on the symmetry. What's covariance? Yes, they change covariance. But what's important about covariance is that it has an intrinsic meaning. Covariance is an intrinsic meaning. And you can see that when something has an intrinsic meaning that you will. Whether a quantity is covariate or not, perhaps, but it's not. And I'm taking a very simple slogan, if you like, and I take the challenge to show that this is adequate. And I show it is adequate in a lot of cases. You may have a story about what you mean by work and what you don't understand, how you classify what you know, how healthy you know. Yeah. Yeah, I mean, it's important to me to have that back up. I mean, I'm not sure what argument I would use. I've just given one, which I think could be quite a good one, actually. But what makes it really convincing is to show that structure, mathematical structure,

1:37:30 of the trends of that mathematical structure and invariant of the isomotor is what characterizes the physically real. But also something I believe in, that what makes me believe in, is the fact that I see it working physically or physically. That's what they're dealing with, right? But... I think I could have said... I mean, what a realist means by far. I mean, I'm a bit of a realist, too, but, you know. It's, for me, what is the furniture of the world and how is the world arranged and so forth. Now, if I can specify that in terms of objects, relations, properties, properties in relation to the circle that I will use to fill out this world of mine will be the invariant ones. And the resulting descriptions you may or may not find adequate, let's say, observable. I would essentially say that all observable quantities are invariant, but not all invariant quantities are observable, not directly observable. But I could fill out a story of the world in terms of the observables, which would be invariants, So a limited class of invariants. And if I then say this is adequate, I think you know what I mean by that. So now I'm not a constructive person. I'm saying there are also unobservable quantities, properties, relations, things. And now in saying what they are, I will use only invariant predicates, properties, relations. And the surprise is why can't we mathematically? Well, some find this surprising. Mathematically, do we not only deal with the real quantities? Why do we mathematically have to talk about covariance? There are different, this is relative, I mean, certain properties could be invariant with respect with one class of transformation, not the other. So, and then if you interpret as real, you mean you get kind of more real. Yes, that's exactly what I was thinking. So, real is a theory-related world. Real according to this theory. This is what this theory, this is how this theory... This is the world of the theory.

1:40:00 Ah, well, so it's not real, in your sense. It's not something that exists independently of the theory. I suppose theory is virtually accurate, so it's a rather crude theory any time you say. I mean, now I mean yes, that's right. If you think about invariance, like a group, then you can also imagine, I don't know, semi-group, or, you know, why the class of . I've only got as far as . Thank you very much. Gosh, it's still running.