Conjecture & natural properties (contd.)

0:00 This example, just as a first observation, and then I'll move on, this at least illustrates how inextricably these judgments as to the naturalness of proper definition of basic concepts can turn out to be embedded in mathematical method. so let's say we're then asking just internally to mathematics right now leaving aside metaphysics just saying if we're now sitting within mathematics and asking according to its standards, is the logeomer symbol artificial or natural and how can we tell the answer is one, yes it's natural and two, that's a mathematical question to which we can give a substantial answer sort of work out generalized reciprocity and say yeah why the genre symbol is natural, it's an instance of the art symbol. But why should we take such internal classification seriously? Okay, I should say, when did I start? I'm wondering how much time I had. You started 55 minutes ago. Okay. Then you have to attend the clock there. oh I see that's when it opened up and you began at 18 minutes and 5 5 past 18 oh ok well then I will skip over real notes or sorry prime notes Sorry? Sorry? Was it going to be only a generalization or was it going to be about generalizations? No, just about prime numbers. Well, sort of, implicitly about generalizations. no, the mere observation was going to be there's a simple definition that we learn in school of prime number and a definition that when we're doing Algebraic number theory, we learn to be better, and it comes to be accepted as the definition of prime numbers,

2:30 which is different from the definition you learn at school. Very simple. I can talk about that in a question period. Also, I've got a paper. This discussion is part of the paper that's posted on my website. You can get to it from the Michigan Department. Check that out at your leisure. there. Okay. Now I want to say, why should we take this sort of thing seriously? I want to say, not only is this sort of, you know, you look and you go out and you ask a mathematician, you ask a mathematician, why is the general symbol natural? And they answer, well, geez, look at, it's an instance of the Arten symbol, and you know, the Arten symbol is natural. You say, well, what? So natural, well, the Arten symbol, well, we'll reduce some facts about the land ones program. But then you say, well, okay, that internal to any practice, you're going to have these judgments of what seems natural and what's not, right? People who read a lot of Harry Potter books will have firm opinions about what Potter would do under certain counterfactual circumstances and they could even defend this with reasons, right? Why should we take it that the mathematical case is any different? And now, I'm not about to propose any general theory or a criterion for the objectivity of a judgment. Instead, I'm going to borrow a practical strategy for mathematics. If you can't solve the general problem, in this case the problem of objectivity, find a tractable special case and solve that. The foothold here is going to be the interaction between finding the proper definitions of concepts and the practice of successfully verifying conjectures. Now, the nice thing here is where a different kind of argument can come in for the naturalness of the Le Genre symbol. I think that this is, you know, I'll explain it generally. What's interesting about the Le Genre symbol is that it was proved, or sort of say, it came to be known by quasi-empirical means. Well before it was proven. So Euler kind of did Euler analyzed a bunch of numbers and made a conjecture and he was wrong.

5:00 And so he reanalyzed the numbers with different plastications and conjectured again and he was wrong. I'm not sure exactly how many times he conjectured but at any rate finally he hit upon what was essentially the Lajara symbol analyzed the numbers he was interested in that way made the conjecture it was correct, which he could prove by verifying cases and then verifying further cases there's this idea that somebody might have, that there's a saying in the case of a greening group there's this idea that being able to successfully predict interacts with the property being natural, at least in these very simple inductive cases, by saying, you know, emerald, green, green, green, green, I predict the next one is going to be green, is a good prediction because green is probably being proper, correctly chosen, or as you may say it's a natural property, that the next is a bad prediction. If that's the picture, then given that the fact that Euler was talking about something necessary and people talking about emeralds and making predictions about their greenness are talking about something contingent, doesn't affect the reasoning at all. The induction as a pattern of reasoning doesn't depend for its correctness on physical causation. The property is supporting Euler's claim. a correct inductive reasoning of the same claim to naturalness deriving from projectability that Green has. This is consistent with the observation that mathematical properties don't participate in causation as, say, a sitting shoemaker in the standstill. I'm referring back to something. I skipped over it. Okay, now there's much more about inductive reasoning in mathematics that we need to better understand. But the one thing we want to and we want there are certainly differences on in empirical reasoning and mathematical reasoning. But we don't want to underestimate the affinities, at least in terms of practice and conjecture of verification. Now just one more observation and then I'll end. three minutes at a time.

7:30 One of the reasons that Goodman's new riddle is such a compelling philosophical set piece is that it links in a simple and clear way The question of whether or not a category is homogeneous with a judgment whose rightness or wrongness is outside the control of the judger, right? No amount of reclassifying or pragmatic revision of conceptual schemes will make the next emerald examined be what's called in the current vocabulary blue, right? The color of an emerald is a contingent empirical fact, but the reasoning doesn't turn on a contingency. Since Euler's records of the computations that prompted his conjecture display the same pattern. Examined cases, made conjectures, tested them against an expanding series of examples, where in the case of computing arithmetic inequalities, the examples were so to speak hard, unyielding. what I sort of want to suggest is that this be alert to the fact that this kind of induction, this kind of conjecture, it is pretty rare, right? It's very rare that what you do is you examine cases in mathematics. You examine specific cases, you conjecture a general result, and then you have, like, actual numbers to compute to validate the cases, right? Typically, the conjecture is more subtle. of success are more subtle. The question of whether or not an acceptable proof will be found using this. So, you know, someone might conjecture when they see the concept of scheme that this would admit of support a proof of the Bay conjecture. That's, you know, that's a more social observation. The question of whether a person will find the proof, whether it will be written down or accepted by people. There's a lot more sociology involved in that. But what I want to emphasize is that there's also a core of hard facts, so to speak. When you're even making questions about things like whether or not a particular concept will be fruitful, whether or not a particular pattern of argument will, a style of argument will

10:00 eventually lead me to an acceptable proof. These are conjectures that can be right or wrong. And though to some extent they depend upon social interaction. They're not completely within our own world. And so to that extent, they have that foothold on objectivity that Euler's case did with a nice hard computing number. That's what I have to say. Did you say at the beginning that this position was, I thought you were saying that I was, that this was quite an unusual position to hold, and that you were putting it into line with one end of the extreme that I presented things being, on the one hand completely natural solutions being transparent and obvious to them Oh, I certainly didn't if that's what you meant if you meant that solutions had to be transparent and obvious what I was saying, what I am saying and I took this to be one of the things we're counting as crazy is that when you say this is a that the Lajonro symbol is a natural mathematical function. You're saying something objectively correct and not merely sort of correct. I wasn't at all of it because, well, you know, in my work, I mean, I precisely see this, the middle line of this approach I kind of adopted that's actually allowing you to get it. Natural entities. You may be wrong about it, but in the context of historical... Then I take it back, then I take it back. Then I take it back. I should be known. Never argue with somebody trained in psychoanalysis over an assessment when there's something that's very zero. Sorry, sorry, okay. Perhaps I was merely sort of in a juvenile way trying to make myself seem more radical. And in fact, many years ago showed that you need to be trained in the practice to appreciate what's natural. The notions of... Well, sort of, yes, and sort of, no. I mean, I could, I could, to appreciate the full story, but, you know, I could go to my mother and sort of say, you know, with this, find it for her and say, you know what, this is completely natural, because when you, you know, using it, you can make all sorts of crazy predictions and they turn out to be correct.

12:30 And she might? Pardon me? Well, and she might say, oh yes, you're right. Yeah, yeah, sure, yeah, that's nice. She probably. I think it's an example of problematic nationals. Okay, good, good, good, absolutely. Actually, it's about natural problems, right? Okay. One reason to find people that are probably in some ways, it makes sense of all the nature, but probably another reason, I don't know who they are talking about, is that actually they are involved in, say, well, I'm a man, and I hope I'm a man, but, so that, there is a sense in the future, they come out of a lot of life, right? Yeah, yeah. But, say, if you have a great, not even too late, let's say, countries that go underground, so like that, right? They can put down negative, and actually once it goes by people, I think it's one of the people that they call Greek, you know, like, Greek without language, sorry. So it's algebraically a very bizarre thing, and that's, let's say, Oh, yeah, yeah, no, no, I'm okay with that. Well, I mean, just like the elliptic curve turns out to have only a kind of indirect connection to ellipses, it turns out. Same way it could turn out that the natural numbers are quite unnatural. You know, just as you know, I guess many of the people go into the Ecole Normale Superior, or you know, not normal. Maybe a little injurious. Oh, really? Oh, okay. Well, in that case, they've got to be... All right, take it back. The Vatican speaks like a catheter on these matters. But no, that's a lovely observation. It is very interesting that the natural moments turn out to be unnatural.

15:00 Maybe the framework dependency of... Sorry. ...the framework dependency of... ...of... ...of... ...of naturalism. It would be a way of... ...like... ...another... ...small... Okay. ...but it's... ...so very... ...and it may be useful for me... ...when these people... ...have a business called... ...co-safe... ...co-safe... ...co-safe... That's... When you say this, do you mean people like Sagan Schumacher? People in the United States. Yeah. And philosophers in the United States. People so are asking. Is something that should include, there was a time or a night of time when many philosophers were trying to get rid of them to work for that. And actually, if you avoid calling it a concession, we call it the way of possibility. You call it a Marko process. And you still associate with the same meaning that people need to associate with. You find that in mathematics. Well, I'll maybe expand on this a little bit, because I've got to say that by temperament metaphysics, I find baffling. I find it baffling when you do it. I find it baffling when the arguments people give in it, and I find the fashion choices of people who do it baffling, it's just baffling. And so I find you know, it's for me learning a natural language late in life. I'm doing my best. There is one pattern that seems to me to be kind of promising, in that, yes, there are a couple of papers by Jaguar and Kim, where he tries to go back to an Aristotelian idea of causation, where he wants to understand causation as just one of many general kinds of dependence. And he uses the fact that there are possibilities of mathematical explanation as anchoring this idea that it could be that just as you have conceptual dependencies of certain types that indicate that certain principles are in some way or other prior to others, so too you have causal relations and they all should be seen as instances of

17:30 a more general scheme. And then if that could be worked out, then that could be extremely useful from what I'm saying, given what I'm trying to say. But on the other hand, if causation is seen to be essentially between contingent events, then it's going to be a stumbling block. I realize that doesn't sort of really, that's an expansion on what you were saying. Yeah, I absolutely love it at all, but I just want to say, for a pleasure, because it seems to me that it's being able to do a great deal of work and that some of the things that it's being able to do are perhaps something different. Take the example you gave of scheme, and clearly it does indeed, which is part of the central test of Jerry Bird's purpose It does turn out to be a key part of a conceptual machine, which led to the proof of everything in the form of a matter of their own, and the results of what was seen, which had been recognised to be outstanding, but it also had another consequence. It turned out that which was the first advocate to be discovered, which did not possess a so-called theoretical country of a captive set. So the scheme could have another consequence, but I don't know a practical way, and it supplied a reason to question the adequacy to our understanding of the able captive integration of that kind of knowledge, that it should rest ultimately the interpretation of the figure and kill, et cetera. That seems hard to be, certainly not a reason to think of it as being very formal. It doesn't even connect with this aspect of projectability in quite the same way. Well, I'm not. So maybe one needs to have something. Well, I'm certainly not. I mean, one of the reasons why I described this as a football is I'm not going to suggest that this will explain everything, right? Only that it gives you one way of saying in a, sort of, say, skeptic

20:00 skeptic-proof way you know, these judgments that seem that clearly are judgments internal to the practice well, maybe I'll step back and sort of set up the specific work I think it's doing you have this intricate mathematical practice in which people make judgments of naturalness and then you ask them, well, why do you think this one is natural? And they can tell you, like, why? You know, my colleague Brian Conrad in the Math Department of Michigan, if I hadn't gotten on a plane to fly here, he'd still be telling me why the Lejeanre symbol is natural. And so, you know, but then somebody's going to say, okay, okay, okay, look, you know that, sure, you have this elaborate practice, and people give reasons, and people recognize those reasons, and people argue about it, and it's all very systematic, and so forth, but there are lots of practices like that, like, look at, you know, imagine you have got these people who do these Harry Potter fan fiction things, and they, you know, read all the Harry Potter books, and they dress up like Harry Potter on Halloween, and they go to all the special parties with Harry Potter people, and they write all the stories. And then somebody will say, I think that if Harry Potter had gone into such and such a room in such and such a book, he would have done such and such a thing. And then people say, no, he wouldn't, because Snape did this. And then you have these elaborate reasons, right? This is a huge battery of considerations that people are arguing, and it's all very systematic. So somebody comes back and says, look, why aren't mathematicians doing anything more objective than these Harry Potter guys, right? to is the fact that there's this extensive systematic practice which involves giving and asking for reasons and the other one. And I would say, look, well, here's one thing I can point to point. There's more, but, like I say, as I said, I'm identifying a special case and solving that because I can't solve it generally. Here is something. We make predictions, and they can be right or wrong. And in some cases, at least, like Euler's predictions, those are exceptionally hard and unyielding. In other They're a little bit softer, but nonetheless, that is one thing that distinguishes the mathematical practice, the judgments in terms of that practice, from the very popular. So that's all, that was the sole job of the method.

22:30 I absolutely agree, and the only thing I want to add is that the examples you focus on, and clearly this is a key aspect of the term, particularly, I just want to add that there is also a case, an example of a team, perhaps one that I also Okay. I will say one thing, and this is just a sociological observation. I may be wrong on the facts, but it's just an empirical observation about the mathematicians I hang out with and the ones I read. but it seems to me that there's, you know, mathematicians are kind of, they have a respect for the bottom line, right? It's going to prove things, right? And so any amount, and so, but that sort of, and what I want to emphasize is that emphasis on, you know, I use it as my word to conjure with fruitfulness, the emphasis on fruitfulness does, I'm trying to emphasize, does involve an element of prediction, right? But also it means, say somebody comes along and says, here's this great way of setting things up. Right? Okay, okay, okay, you know, a little thousand-bar of gluten. If you think it's good, I can even see some advantages, but in the end, it's got to deliver the goods. In the end, it's got to result proven. If down the road, it doesn't do that, then it's just going to be kind of a side show. And so that's sort of the one thing I would point to about any kind of improved gluten understanding, sort of the mathematics sort of angling in the bottom line. I understand things in a few ways. You have different things of the truth definition, and you wonder if they have certain specialties. These things are appropriate with respect to certain, let's say, structural conditions.

25:00 For example, a definition depends on the right object. The proof is approved, there is no gain, the proof is regular and that. And you say, okay, they are structurally appropriate, but some one of them are natural, some other ones are not natural. They are a property of appropriateness, but some of them have special virtues that they call naturality, and some of them are not special virtues. and you say the fact that they have or they have not a special virtue, it's an object in fact, and we have to look for it using different methods, among which conjecture, mathematical methods. Yeah, I mean that's one of its methods. Okay, so, now, it seems to be very clear. But, you began with definitions. And then, you take as an example, a legenda symbol. Now, it seems to me that the question of a legenda symbol is not a personal definition, because there is not another way to define a legenda symbol. I said at the very beginning, there are two different phenomena that one might call stipulative definition of a new symbol or a new object, new function, and redefining something which is already defined. I was going to use prime numbers, the example of the second, but because I just prepared too darn much, I jumped over that one. Two completely different things. So in the case of a legend of symbols, the question you can call definition if you want, but if you call a new definition, it means that the question is, is the natural object Yeah, it's a natural logic. And in the case of right now, it's the natural definition of... Yeah, I mean, what I might also say is that you could say, you could give a definition of the Le Genre symbol using the Arctic symbol, and just say Le Genre symbol is a certain special case of the Arctic symbol, and then say that's a natural definition, whereas the original...

27:30 Okay, so I imagine that in the case of Le Genre function... Yeah, Le Genre function. You can give two different definitions... Yeah, but I also do want to say... A priori, the problem of knowing that a function or an object is natural or not, is not the same question, but asking if the definition is natural or not. Okay, okay, that's true. They are related, but they are not identical. So, then you can argue that, in fact, there is something similar, something similar to two problems, and you can use the same methodology in order to solve the problem or not. So, I think that's why. Now, if you put that, now, the question in this context, it seems to me, the following words, that you cannot, it seems to me, perhaps I am wrong, but it seems to me that you cannot limit yourself to use natural, as in order of correct, or, you say, it's not correct. Because in this situation, what is relevant, it was a subtle logic, and I agree with you, a subtle logic, is not natural. It is natural with respect to certain angles. We are going to say, if you want to say that the fact that there is a certain definition, a certain function, a certain rule, is a certain view. Okay, this field is natural, it is objective that it has or it does not. But to call it naturalized in general. So if you want to be natural means to have a certain field, you have to explain to me which field. And correctness is not a good field, because all of that is correct, otherwise it would not be structurally appropriate. The truth that is not correct is simply not the truth. So, you are open to the context where all is out of it, all is correct, and you pick up a certain problem, a certain virtue. So I think that you should say which virtue is in any case in question and then ask for the fact that it has or has not spoken. And so, the only thing that I wanted to say is, these things, I put it this way, I think that your status is very, very important, because I think that in a sense, in a sense, this is the very essence of what the philosophers do, look at them and wonder, not if this is better, if this is worse, or not, say, okay, this is Virgis, and it's our role to do that.

30:00 But we have to be, I think, more clear of it. Yeah, okay, I will sharpen this, because it seems to me, again, this will depend a little bit on what the background metaphysical story is. If you want, like, one view would be, right, there's a claim, it's not a claim that I'm in much sympathy with, but it's a claim that, you know, it's not crazy. Somebody might say, look, what counts is, the whole idea of naturalness in general is one that turns out to be sensitive to things, right? That if you're talking about chemistry, then crystalline structure is natural, and if you're talking about physics, then mass, you know, and position of momentum are natural, and, you know, if you're talking, and so that being natural is sort of relative to a general set of objectives of aimly, right? And in that case, you know, then I'd say, and then it turns out that that's true in mathematics too, the, you know, being prime in the sense of if A divides B, C then A divides B, C is natural in algebraic number theory, and it's not, and the other one is natural in, say, elementary number theory, you know. But then you're, but that's a view that you're implicitly rejecting. So you're saying, no, no, no, it's a hard fact about what's natural in the other world. It may be, it may turn out that what I have to end up doing is kind of just retreating on the terminology of it, because I will have learned that there is an important difference between natural, in nature, and natural. And I'll use another word like proper, or, you know, correct is tricky because of correct reasons, but, you know, proper, or, you know, I don't know, good, you know, unfortunately there's no sort of proper is about the closest I can come to a word that's used in practice, this a lot, and would sort of have the same resonance. But yeah, I mean, that's a hostage to fortunes. It depends on the gentleman of physics, it depends on it. But if, but what I'm sort of, I'm kind of

32:30 retreating here to the how should I put it, the epistemological side, saying, well, here are sort of questions about how we gain knowledge about these things. And if it turns out that our gaining knowledge of those things. You know, about gaining knowledge of the property of being proper and the property of being, in the case of numbers and the properties of the genre symbol and being natural in the case of the physical world, you know, turn out to be that we have generally the same patterns of reasoning involved in deciding whether something is natural or proper. Then I'll say, okay, I can live with the fact that the underlying phenomena are different in a certain way. You know, if it turns out the patterns of reasoning are different, too, well, then they'll be disappointed, but there's actually one thing I would mention, though, that is a complication, and it just requires me to think more and more work. I mean, one thing that does happen is that you'll have theories of mathematical physics in which deciding what the natural properties are in the mathematical formulation will translate into deciding what the natural properties are in the world. And so that might serve as a bridge that I can exploit I'd need better examples, and I'd need to know more. But anyway, I want to emphasize, when you say that what I'm saying is extremely important, I could not agree with it at all. My thought is that if you take the naturality as an absolute poverty, then there is the danger that this sort of search, that it seems to me to be the essential search of the philosophy of mathematics, is converted into a sort of religion, as my friend would say. Because you say, okay, it's big, dear, to the day one. No, it's not our work to say, what's a good, dear, to the day one? Where would mankind be without faith? Where would mankind be without faith? But, I don't know, probably here. Anyway, yeah, I mean, there are a lot of things I'm not going to be able to figure out You know, I just, I have to know how this interacts with physics, and it's just, it's part of it, this is one reason why there are so many people working on questions like this, and we're all talking to each other, because there's a lot to learn about it.

35:00 Just one question. Oh, yes. All of this story about the mass and government and this process is quite clear and extremely convincing. I would like to say just one point. When you think of the complexity of economics, there are cases, when you have several speeches, multiple objectives or different diverging projections. Let's say, for example, well, the way to generalize Koomer theory, Koomer psychology theory, along a lot of different lines. And here you could have also a way to tell you some analogous finding out and building the limitations along the product lines. Yes, yes. Devising this system of modules, so to speak, and the mainstream along the data in lines, building this concept, using a sector that comes in building this notion. So, in a way, and caricature, from the point of view of each side, The other one could look as gerrymandering, in a way. So, if you go along these lines, perhaps, just a question about the notion of non-divocity, or non-divocity of objectivity. If you go this way, even in the deep meaning of what you said, is there so much difference with concepts that David spoke about in the first talk about this? You could have different ways of generalization, different ways of finding naturalness. I certainly think that but what I want to suggest is I mean I certainly think that there are going to be different ways of generalizing that will allow you to that will be fruitful to use the magic word and keep things back

37:30 and that's sort of one way in which you know even before accelerated the process I've realized that had to, you know, appreciate the Feierstrauss' virtues because it just, you know, turned out that even though people were, you know, you know, people were sort of like fighting like cats tied in a sack over the way to do complex analysis in the 19th century, it turned out actually, once you generalized, right, well, in some ways, as a vile pointed out, they were kind of doing the same thing, and also they were doing, you know, they had different virtues going to be lots of ways of doing it well and it may be that that will indicate points where people will disagree about naturalness it may then be that there's a point in the future where people will understand the things even better and say well either they were both right or they were actually doing the same kind of thing but but but what but on the other hand they're also going to be good ways of doing it badly but the point is that even if there are lots of ways of doing it well. The point is it need not be there's a unique way of doing it well in order to make contrasts between ways of doing well and ways of doing it well. So you might say the trouble is that history only remembers the victors, right? So it's hard to come up. So one example I sometimes point to is this, what was attempted as a unified account of Weierstrauss and agreement that was put forward by Prinsheim in the early 19th century, or late early 20th century, and he described it as explaining things, and he attributed these virtues to it, and I think the judgment of history is that he was just wrong, he was just mistaken, right? And the reason we feel constant in saying that is that it didn't go anywhere, it didn't lead anywhere. And so we would say, well, it turns out, yeah, there's lots good ideas that kind of latched on to aspects of, you know, mathematical reality, to speak provocatively. And the fact that there wasn't a unique way doesn't, doesn't undercut what I'm saying, because I can still make distinctions better and worse. You see? Yes, but I'm not sure I understand what you're saying now, because, okay, you're not just

40:00 a good thing in that thing, did you say? Well, yeah, okay. The good thing, and you say that. The fact that there could be different ways, different ways of generalizing, different ways of finding naturalness within the good ones is irrelevant. Is it so irrelevant? I mean... Well, I sure hope it's irrelevant. and we've got some talking about it. Does it lead us to a commercial activity? Well, let me just look at it. It could still be the case that, let's say, for example, I don't know, by generalizing the Wehrmstrass, by following things out of Wehrmstrass' way, you can predict a whole bunch of surprising facts about it. Look to curves, and then generalizing Riemann's way, you can predict some surprising things about counting intersections you know, different what occurs in general, and, you know, at this, at least as far as this talk is concerned, you know, obviously, you know, just the general kind of imperialism of ideas that I'm going to try to explain absolutely every time, but right now, I just want to focus on the fact that if it's, if it's really good, it will, if it's good in the right way, it will interact with a successful prediction. I do have to borrow some ideas from David because obviously everything is going to allow you to do some prediction it's a successful prediction of interesting important things things that are worth knowing so knowing 2 plus 3 equals 4 allows you to predict 2 plus 4 equals 2 plus 3 equals 5 knowing 2 plus 3 equals 4 really allows you to make some surprising prediction but this is definitely an issue I have to be more careful doing both of those things be careful of that thank you

42:30 So, I'll just leave the thing on there. I'll just leave the thing on there. Oh, also, can you email me the computer that I've got here in a way I can't use the computer? Yeah. Because they're just typos. I mean, there's nothing nicer, but, you know, they're little things. Small. Oh, thank you. Very bad. Thank you. Very bad. I don't know. I don't know. I don't know. But we're going to have to go with the rest of the world. But each time you have to put a table, what's going on? You put another table? It's very good. We can change our artistic culture. No, it's for that. You changed my... No, it's just for the function. What's the function? What's the problem? Because it was the same character? Like if it was another screen? Just because this is the function of two screens, and this is the function of an screen. If you want to put it there, you have to do the function of two screens. It changes, it changes. Okay. Okay, so here... I think I... Oh, I think I... Okay, okay, you will move there. Okay, okay. You have to ranger? You have to ranger? Yeah, I'm going to... The trick is not... You have to ranger that. Thank you very much. Thank you.

45:00 This is the same thing. The thing is important. It's really important. What does it mean? What does it mean? What does it mean? What does it mean? What does it mean? What does it mean? I mean, I'm afraid. Look at the... And this is... And so I joined this one and here... And this he was... All in the room is there. All in... Thank you.