Morning Discussions, incl. FW Lawvere, Colin McLarty, Leo Corry, Angus MacIntyre, John L Bell (contd.)

0:00 This is incredibly corrupt, right? Incredibly corrupt. I'm sure he didn't, you know, only come to that instruction, obviously. But he was a very, very good guy. So I remember we were discussing on the phone, Howard and Washington, me and Buffalo, precisely this question of the classification, you know, so we had this report or application that we'd get with, or maybe we were even discussing. And finally, I summed it up by saying, Well, my purpose is actually to totally destroy your classification system. That's my purpose in life. So that all these things are really one thing and they all flow into each other and it's not going to live forever. So they work according to the AMS classification or they have their own? Yeah, they use that. Again, I don't know exactly what they do now. I'm pretty sure Daniel Scott knows this. Daniel should. You know, after, you know, they, the NSF made a decision not to support Keiserberg theory. This was even announced at Missouri by Bill Thurston in 93. Who was that? In 93. McLean and McKenzie were sort of united, and so we had a joint meeting of universal algebra and category theory and misery, under the course by the NSF. So here we're all assembled, the newest algebraists and categorists of the world, not everybody of course, but a large portion of us. And the director, he's an NSF official, he comes up and says, well, the NSF has decided they're not going to give you guys any money ever again. You know, words to that effect. But literally, words to that effect. And did they experiment? Did he explain why?

2:30 Well, I have to explain why. When was that? 1993. 1993? Yeah. Now, I knew that something like this was going on only five years earlier because they had resolved it. They stopped automatically. It wasn't just my, you know, version, who, Isabel had an English version, of course, you know, the initial one, I didn't mean to say it was just because it's in, but it was cut off, and the NSS instead started to do massive funding for attempts to destroy the school education. Even our book was a byproduct of that, not because I didn't have direct money for it, but a dean of ours wanted to cash in on this million dollars for performing education, so within that, he gave us a little bit of extra money with it. We hired an assistant. We were committed, you know, two professors teaching one course that does that main conceptual rule. But it's within that that we were able to produce the conceptual math. Because he asked us to produce something on discrete math. And we said, no, we don't want to do that. But maybe we could do a framework which includes continual and discrete at the same time. The starting point, there was a concept. And as you know, they've gone on to some incredible things in the field of education. I made one big application. I was advised, I think in fact by Al Saylor, to make an application in the field of computer science. So I made a big, very detailed, in all possible aspects of computer science, everything I had done and all this, very impressive application. I thought it was rejected out of hand, just because the method seems to be, you see, that the NSF has a stable of reviewers, you see.

5:00 And you get a number of different people reporting on your reputation. But if at least one is negative, then it's negative, obviously. This is the highest standard, honestly, in Britain, too. I mean, it's utterly infuriating because you get to see the reports, of course. I mean, typically people are so busy that not everyone who's asked sends in a report. But, I mean, you can get. And this is discouraging any new idea, essentially. You get two, your assistants get two excellence, and you get one person who might even say very good or something like that, and sometimes that can be very hard to kill the whole thing. I like what Mark and your application came up with, and one thing that strikes me about this, you see, is that, and this sounds like totally, you know, subjective, but I claim it's very objective, the good report. In the sense of being scientifically analyzing what the hell is going on, those are the ones that are positive, and the bad reports are the ones that are just negative, are very, very bad in the sense that they haven't even, those are the two that they say, they haven't got it carefully, but he doesn't like the field or the approach to begin with or something like that. Or he's told by the NSF to make, you know, I, I, I was turned down so many times in Canada on this game. Really? Although in Canada it's actually... I think this situation certainly was better for category theory, of course, in Montreal, although I don't think it's so good now from what I've heard. But you write a book fast in those forms. Even in my own university, I applied for a grant. I've been rancid in writing this book in the continuous. I was turned down. I think that, you know, from a historical point of view, these things are more influential than you may think at the beginning because... You know, if there is a good idea, it will find its way. We spoke about Kantor. And today, more than in the past, you know, you need more than a good idea. You need the support, you need the professorships at various universities and things like that. And you can make a very detailed analysis of how this good idea was not allowed to...

7:30 And then the matter of classification is related to this, because the other... This is also very prominent in the European, a lot of experience of Germany's thing. First of all, bureaucrats, these simple classifications, to send the applications to, maybe only to two reviewers, and it's then somehow stuck with them, because if you complain about it, then you're accused, and I've been several times accused, you know, of political incorrectness, because I deny... A person who was giving an opinion at the table at any, that means since they had never been anywhere near the area, but they use these classifications, send them to them, and frequently massive international clients, major figures of them, have gone down because somebody, let's say, a physicist from Spain, who will judge an algebraic geometry, or a Greek person, who's never taught anything, they need a moment, you know, this can be, can be, for as long as it's not a review, but go back to Breitling. Sometimes bring the forewords in to evaluate the activity over a few years or something like that. I gave an extreme example. And it was fascinating. I went there with McPherson and Tubbs, who had been funding some other group. So there was, you know, five people. We got a look into it. I got a look at a rejection of proposals around that, and the guy who rejected it. There were two, again, there were two excellent reviews on one. The guy who rejected it was the one who said, This kind of stuff can never be any more splendid, one of the most remarkable interactions between logic and the remote outside world of mathematics. Whatever I imagined, I really thought those people were schmicks. I asked them the day before, what's this idea? But this guy had killed one person's chances of working efficiently on the subject matter. Just an ignorant comment.

10:00 One ignorant comment. And the pretense of expertise, I remember one report, I don't think this was unarmed, but somebody else had a proposal which involved, among other things, working with certain categories designed to include things like the category of fields, air and wind. So, in particular, these categories didn't necessarily have the terminal object. There's this big expert on category theory somehow, who says, oh no, that's not the way you do it in category theory. In category theory, if you don't have a terminal object in your category, you first join a terminal object, and then you go on. This is just so crazy, with all the arrogance of, I don't know, it should be done. That's in category theory, from some smattering that the person has heard somewhere. It's the principle that in correct geometry, you first assume that all equations are true. Yeah, if there's a terminal field, then... Yeah, a field with one element. Yeah. It's interesting, it was supposed to be a field with one element. And the notion of scheme, we find over the 20th of May, hopefully getting better, but the only approach we have to it, except for some very far Japanese approaches, I still can't understand, the only approach is via a second group, the first one we can do, but the second one we can't do. Tinker enough and you get some definition, which then turns out to agree with the special cases that Tiggs had found about sort of numerology, where they're still sort of finite fields, and so it's quite a fascinating piece of... So Soule's work, Peter, just... ...seems to have had a detail through some of the very interesting and relevant sociology, and it's kind of touched on, and it's touched on, you know, over-minimality in terms of policy. And from what I was able to listen to downstairs, I think we've pretty well exhausted the philosophical exposition with which we all began about an hour and a half ago.

12:30 Would it be a good idea to use the remainder... I'm just going to go back to the water and set the volume up there, I believe. Yeah, well, okay. I was just thinking, you know, looking at the time, probably another hour before lunch. Would it not be a good idea to go on to... The connection between the O-minimality program and tamed topology and the overall, you know, the sort of the many problems, the wrong turnings in the 19th century as to the solution to the initial crisis of geometric intuition that codified in, sorry, in the dedicated piano construction. Short discussion about Shannon's conjecture. Oh, it's very good. With the documents. It is, it does look good. And with what has been saying in many ways along the way about. I mean, the rest of the things that Mr. Chani was saying was probably what he seems to have with his students for a new approach to the transcendence of mind and so on, but it may be just too fast, and I suspect that we will have a mutiny.

15:00 Now, I'm just waiting for him to finish sorting out his bloody hotel room and everything else. If he mutinies, leave it to the general staff to deal with, okay? I think we should have had these in our discussion. What is schedule-subjective? Schedule-subjective is... When I was in January 1960, when he was working with Senator Johnson, and I'll say it precisely in a second, but... It is a... It creates a kind of thing I now call a schedule machine. I mean, you... It's a principle which yields that the process is essentially... Everything we ever imagined can be true of the transcendence, of the sensitivity around e and pi and i and so on. That's the first wave of things around e and pi. And over the years, you know, one has never been able to, for example, And isn't this a sub-conjecture of the algebraic relation of the opposite part? Essentially. I mean, the statement is the following. The idea is, I mean, this of course is basically the exponential problem. The idea is that exponential functions are exponential. A functional equation gives us a means of converting linear relations into something which doesn't look just like numerology, but these numbers are linearly independent.

17:30 At the start, there are no linear relations, and take also the exponentials, the n-exponentials. When channels converge to a transcendence degree, the maximum number of independent things can be extracted from that sequence of two, at least n. I mean, it is a kind of thing to go with them. You never send it back. But basically you can get some things in one line. It presents an E in one line. Potentially you've got a pretend degree of these four things, so it's certainly the exponential. You can play games with the Euler relation, E to the pi equals minus one. So lambda one times two S are an E-algebraic number? No, complex numbers. Right, right, right. Totally arbitrary complex numbers.

20:00 Complex numbers, okay. Totally arbitrary complex numbers. And in the two-line genesis, you'll get the E to the E. This is the way, insofar as it involves values of the exponential, and of course you cannot use it. It yields, for example, Baker's Fields Medal winning results, not with the numerical ones, but it meets the basic results Baker obtained in most linear functions of it. It yields the most advanced results known, which are due to Nestorenko, which give things like e to the pi. This is again cranked up with a Shannon machine, in a sense it looks like numerology, but it's... So just so I can tell you about it, so the list 1 already shows that e is constant, and the list 1e will give us something we don't know yet. That already gives you something you don't know. It gives you that e and e to the e are algebraic and independent. You'll start with, since E is transcendental, you've now got to pair it with linear dimension, at least two, by taking, adding to the list E, and you already had E, you know, you get transintegrated these two, now what? Out of the three numbers, 1, E, E to E, 1, 7, you're not contributing. So, this is the typical kind of cut, you know, cut out one number in the middle. I am, so consider the following decision, you know, suppose I've got...

22:30 What I start with is the denotations of these terms, such names, you know, groups and extra, such that E transforms additionally. Equation. Equation, yeah. So we call these E-ring commutators. It's almost the same thing. It's the same thing. You see, in turn, what you can do is you can, you can even do this for functions, for terms of many variables. You sort of identify the three algebras by an independent construction. Right. And you can, if Shannon is true, and then it's... How does the degree of transcendence translate into an algorithm with certain properties? Well, it has to do with the kernel of this map, the exponential ring. So there's a certain ideal which is the kernel, so the transcendence measures how big that is.

25:00 There's a lot of things your calculation doesn't have to rule out. And then you've got the map into the reels that sends the one that could be. Is there a kernel? I mean, is there a kernel in this solution? If there is not a kernel, how do you do it? Well, you need some sort of independent look at the free, the free thing. Now, various people may have constructed the free thing. I don't know, but Jacobson once said something to me that hinted he might have thought about things like this. Anyway, finally, the reason I, a long time ago, read things, and I mean, it's utterly simple-minded. What do you do? You start with the polynomial ring in n variables, mainly e of 0 is 1. That's all you can process, which you'll make. The first time round you've got, say, the polynomial ring, or what you might just have said, you've got something free in the ring category, and you've got the exponential defined only on an additive summand. The thing is visible just naturally. So what do you do? You've got the thing defined on an additive summand, and you've got the other, you've got the complementary summand, but not yet succeeded in defining the exponential, but it's an additive group. Now, the exponential, when you do define that, will transform into a multiplicative group. You can either do it formally, you can do this nice thing, you sort of pretend that you take this additive group, call it gamma, and you form, like, t to the power of gamma. Yeah. It's like a group algorithm. It's technically very convenient to think of as... Well, literally, I mean, the polynomial of the word itself... That's right. ...is the monoid irregular. That's right. I mean, all of this can be done... The monomials, I've often thought that the word monoid probably was abstracted from the term monomial. ... because the monomials can be multiplied, you see. In fact, in that case, it's the free...

27:30 Yeah, so what you do to n, you see, you've got... In general, you can take any monoid and you can iterate... Yeah, exactly. This is utterly like... You've got this kind of a circle called gamma where you have not yet defined e of gamma is the most simple-minded thing you could ever do. It will at least give you the function equation. And now again, you've a visible... There's a sort of augmentation ideal of the rest. You do it again and again and again, and then you can readily prove that this thing is the free e ring on the set you chose to start. If you start with z, you would create, e would be t in this case. Now you can prove regularly that the chain rule is a differentiation of an exponential. And then you've got this kind of hierarchy of terms now, it's like generalized. And then you can readily prove the kernel in terms of an induction using shingles. Quite by accident, I have a copy of my paper on the upstairs. It stayed in my briefcase. It's really quite simple in a way. This is an interpretation of the numbers that we built, the sort of explicit numbers built up from the node.

30:00 There are a number of different ways by which a subject, by kinds of basic things, has succeeded in defining an exponential. I mean, it's not easy. And then this guy, Gonifer, it turns out that their exponentials are entirely equivalent, and his equivalent also. I had a student working on this stuff, and he was constantly having to work and wave the object over his eyes to get it to seem the same way he wanted it to seem. I was interested in that. Oh yes, in order to realize abstract quantities as a bird-side rig of something, you need a pretty tight category. Yeah, yeah, exactly. So let's ask Paul. Anyway, we have a channel, but it's turned out to be a remarkable, motivic version of the conjectural group. And this is in a quite detailed form, just a singular piece of cohomology between them, which is generally the field which is...

32:30 And this can be generalized to one mode, which are things about the level of elliptic curves. It turns out that there is a conjecture which is standing out and implies Shannon's conjecture and also implies analogous things for the periods of the Weierstrass elliptic function. Basically the only transcendence relations are the ones that are staring you in the face from a functional equation of an analytic function. So Shannon somehow, and the number, you get similar inequalities in many, many other situations now. Now I guess that the proofs of the Transcendence of Pi, which is somehow more conceptual but a little bit less computational than standard algebra that we ever know, is a standard in paper, which I certainly recognize, I imagine, they talk about as an important number, and then the next important class, you take a bunch of semi-algebraic, and you, the corresponding forms, you integrate them around semi-algebraic, over semi-algebraic.

35:00 Well, you see, the thing which is surprising is that I think it's completely unclear whether one of them, this came out the other day, he's using only form, he's hardly using, the way he's doing things, he's proceeding alteringly, he's hardly using any of what he's doing, except he's using integration by parts, but in the form. Consider this as an idea explicitly that any equality period can be obtained by... And they give one example, and they also show this thing that Cartier was referring to, he referred to the body curves. The first guy who managed to do an intelligible proof apparently had done on the irrationality of Zeta. So this again is at sort of the same level as Shannon. It's something to do with, in the case of these basic numbers that come up, the qualities between them are somehow... From the things we've already observed in the function equations, maybe integration by parts. Have these materials, like you, for example, appeared? I think it has appeared, but it's more readily available off the IGS website. And just look at the dedications and what they say about it.

37:30 But I think it did appear as something quite important. Maybe a fresh trip for somebody important. I don't think I'm not in with it, but it's readily accessible. It's like a completeness theorem of logic, in the sense that we have this concrete model of complex numbers. The language of addition, multiplication, and exponentiation can always be interpreted in there. And the obvious relations are associativity, the exponential law, and so on. So it's just a question, have we completely axiomatized this? In other words, have we added enough relation that the kernel will therefore collapse to nothing? So, this would be the kind of proofs that Cartier is looking for in computationally? Oh God, yes, yes, yes. Strong computations. Tremendous, meticulous, and they involve constructing large numbers of polynomials and auxiliary variables. And some of it is very tricky. We have to use sometimes very serious, complex analysis. These proofs are very difficult. Forms completely non-memorable for an outsider. Yeah. And the kind of conceptual things could come from schemes and... Yes, yes, yes, yes, yes, yes. I mean, Kosevich and Zagier relate all of this to motive equations. I mean, also related are a couple of things around Bershman and Dyer. Bershman, no. The Clay Institute, no. Big ideas and combinations of approach, like going very far. And then calculations. I mean, they're going further than the schemes in the sense that they're passing to the molders, which seem that it's not literally a cohomology theory.

40:00 I mean, you have correspondences, and then you basically forget as far as that's molders. Any time you have an idempotent map, you pretend it's a projection, a summand, and use it informally. You can say what it is, it's just all the nice properties, the categories. This is the intention. This is the intention. In other words, just like David Hummertorff in topology, David Hummertorff is known to play that. In fact, people who have played in that game, Peter May and so on, certainly consider they're good at the formulas and torts. It is somehow, of course, Wachowski decided to play in both games. There are hybrids to malfunctioning, where somebody uses this to organize the beginning of a proof, then calculates the way through. Well, I mean, I'm not entirely sure. Daniel Bertrand, who comes in twice to Cambridge, they have new proofs. I'm not absolutely sure, I haven't seen them, so let me know if you're familiar with them. Bees and hybrids. Oh, yes, absolutely. I understand that it has to do with the zeta function. Yes, yes, it's true, it's true, it's true. So, in other words, the zeta functions are, I mean, this is spelled out immediately, and it's also connected very much to polyorganism. So it's typically a case that you get, but you don't get it.

42:30 You don't get exponentials coming out of these periods. The exponential period thing, you would close the periods under, but it might integrate things involving the exponential function. That's much less interesting. Experience number one, in the sense that, sometimes the number five eliminations which work with the logarithm is primitive, but won't work with it. And it's the case also with the logarithm. Yes, that's the point. You see, we don't know whether some way in which you could go over something utterly weird might get E. Like, it's unlikely. You can, and there may be some things that look like, yeah, it's, but then it's over, it's a multidimensional thing, it can be, so, does it suggest that the inverse trig functions are more fundamental than the 5F or all the functions? Yeah, yeah, I'm not going to say that, I mean, but this is also the game, I mean, this is always, you make these big choices, this was the key step forward in the elliptic functions, I mean, at the very beginning, this is how they made the progress, I mean, they managed to do inverse functions, I mean, that's a, certainly my own experience in, We have theorems, definitely, but there's a quantified elimination where you find logarithms, they're simple, they're primitive, and you can insist that it's got a singularity in the origin, as it has, but a case in which we're simply adding that and being prepared to get the consequences of very partial functions gives you one, but there's definitely not one if you try to work with global. It's highly surprising. It's certainly brought up how it brings in many.

45:00 Could you say a little bit more about how it connects with the... First, the indication of existential quantifiers, so constructive, and then looking at it by essential use of Shannon's sense. And then subsequently, I have a theory that seems impossible

47:30 to get at. Anytime you have such a rigidity, then you're likely much integrated. I'm not sure if they've made any... I mean, it's early days, I think. ...algebraic, geometric things, considerations, and so on, and he managed by cunning methods to get a definition of integration into these things, which you can then specialize, I think, to explain certain units, I mean, they're very beautiful, but what about the rationality of certain p-adic pointers?

50:00 There, first of all, should be proven rational functions of p to the minus s, where p is the point of question, s is the complex. And then I, one of the next students, proved a weak uniformity in p that you always get the same shape of, if you're integrating the same shape of function into the same shape of domain, one can see that you always get the same shape of rational function of p to the minus x. I mean, it uses some curves and waves, but I mean, the uniformity is attained by kind of ad hoc, and it is much, much tighter than that. So this formal integral can be specialized into the uniformity of these things that we have detected as actually coming from one global formal integral. Issues of positivity and so on come in, it makes sense really. I mean that's going on now, we hear quite a lot about this at the New Pinsir, Trusovsky is trying to go even further, he's working with Kasdan, so Kasdan has come in from outside of the moral theory because this, this approach does promise to make rigorous or instantiate various intuitions people have had about some kind of global zebra function. You remind me of something in a way.

52:30 That's simple down to where it is. But the problem with Tarski's High School Algebra. Oh yes, yes, I wanted to mention that. Which Wilkie may have injured the attribution to work. Well, I mean, you know, maybe the idea of using an equational theory is not one of the best ways to get at this. But anyway, more or less an equational theory. What I call finite topos. Typical example being finite sets raised to a certain power, finite creases on a finite category. Very combinatorial theory. In most of our work on I've directed a number of these and we've always been just looking at the rig aspect, but actually all these examples have exponentiation as well, so we get exponential rates, obviously, but what isn't obvious, in non-trivial cases, the exponentiation is defined by naturality, so that it's not defined point-wise, so the question is... Does that make any difference to the equation? In other words, it's wide open. In other words, if you look at the algebraic theory of the exponential rate associated to one of these concrete categories, are these always the same, more or less, or can they be radically different in terms of how the exponential rate of multiplication is done, because it also sounds very much like it, particularly when you say this.

55:00 In other words, this is a huge supply of rather conveniently accessible models for the high school algebra. So we thought, we thought immediately we should try one of these models and maybe this will serve the same sort of purpose as Loki's, which is quite different. Loki's thing is not at all objective, in other words, you can't. I don't know if I can construe it as being an abstraction of something. But would this make any difference? It may be that all these toposes are enough like finite sets that really, you know, it's all like, it's all just like John's arithmetic, but on the other hand, from a geometrical point of view, they certainly are very different, and so... Various things, as you know, or perhaps others don't, I mean, Wilkie, Tarski had to deal with the functions on the integers, and then the binary thing, x to the power of y, and we know, we know from high school, some equational rules about these things, and Tarski wanted to know if the identities, so Wilkie gave examples of things which are true, but require cancellation to prove, basically, and so there's a lot of equations. A bit mysterious, and then I proved it much earlier, an embedding into a lyric structure. And then communication with some answers was connected. He constructed some finite models, like sort of ad hoc methods.

57:30 Now there's models where the original axioms would be true, but some of these rookie things would not be. I've never been able to understand these examples, they were just far too ad hoc, but maybe one can... I don't really have any idea how to begin to answer my question. We don't normally think about possible, you know, any rules other than the 10. And you know, the interesting thing is kind of when you take equalizers and pushouts, you see, it's not, you know, not capturable in any obvious way by mere addition and multiplication. That's the one thing that's most valuable theory. It's difficult to relate to the categories in this equation of theory how, because most of the way you understand the categories is through more operations than just the three. Imagine what Shanderwell did do one thing. Namely, he proved Shanderwell's conjecture. Yes, yes, yes. In a sense. He devised a precise analog of all the ingredients of Shanderwell's conjecture, which you could interpret in one of these finite topos. But just a minute, this was his starting point to try to solve this problem from a topos point of view. He started with... He started with both. He had his hand on those conjectures. Oh, the original conjectures. Oh, that's 1960. Without any connection at that point with topos theory. Oh, of course. It's clearly not. Purely number theory. Number theory. Any issues he observed? Even in one variable, and even if you're just thinking iterated exponentials in one variable, x to the x to the x squared plus one, blah, blah, blah, for these functions in some sense, there are no hidden, there can't be any hidden identities, and that's the sort of version of Shannon.

1:00:00 I mean, you can prove it by analytic means. But Schoenberg, who's configured with the variable which he had spotted, the function is in one variable. Consider the relations of one function over another at infinity. I mean, these functions are hardly implicit, but these functions are ultimately monotonic in there. So you have the notion of the plus-planet by way of growth at infinity. And in fact, so you have a linear order, and what's surprising is you move well over it. The Kruskal theorem thinks that Harvey eventually got into the argument of the plus about it, but Scolom conjectured, and it's never been refuted. But the ordinal of this thing is epsilon zero, and it contains a lot more than the norm, and it's very hard, I mean, people can check out, if you take a particular function and look at the things below it, if the function's given a sufficiently evident way, you can sort of figure out that the ordinal of things below it is more or less what it should be, and the game is approximately, not exactly, take your x and replace it by omega. And it's been released in an awful lot of things. Actually, that's very interesting because that ordering was DuBois-Raymond's theory. This is precisely his theory of infinitesimals and infinitesimals. I don't know whether he remarks anything about well-orderings there, but of course he's really interested in getting infinitesimals. But he does remark on the very strange and intricate order properties of this realm. That's right. Orders of infinity. That's also very interesting from the history of math, because he's trying to set up something in algebra, and even to integrate and differentiate, and he just can't quite get it to work.

1:02:30 It doesn't work because Hardy is not in possession of the motion of valuation, which certainly existed of course at that time, but not for too much before, but it certainly would not have penetrated into the English, I think. No, no, no. The Germans, strongly German. But it's also an important key, actually, to deal with this matter in the intersection of Hardy fields. Some were on the, exactly on the orders of religion and religion, because the crucible is to take functions of real people and have them close to the good branches. And really the abstract algebra, he says, was not worked out properly until the 80s, 1980s, by Rosenberg. He found the axioms eventually depend on something like L'Hopital's rule of infinity, and Hardy is working in a very unhelpful way. This is an interesting line of development. Oh yes. In the 1880s, he would have, because he had these arguments with Cantor. Cantor didn't like him, of course, to do a remorse theory. I'm not sure. I mean, often mathematicians... We're much more appreciative of what Kander had done than vice versa. There is one other connection here which is, I'm not sure if this is so. You can, suppose you take, I mean I can write down a term for a real exponential function. It can involve a great exponential effect. Say I take one and define it with a rational. So I start with, I always just use rational operations and the exponential, So this will give a function of infinity. So when I write down two terms, and I want to, and I know because of the idea that one of these functions dominates the other, how do I decide that?

1:05:00 If only neural methods use these channels, I mean this is rather surprising that something originating in transcendental number theory is needed to... The numbers of the complex numbers turn around into the real numbers. Yeah, well, I mean... So did we ask you something about if you use it and so on? Yeah, yeah, sure, sure, sure. In fact, we also... Is there an easy explanation of how the complex theory can affect the real theory? Yeah, well, it certainly affects it very much. And our decidability proof of the real exponential diffusion channel at some point, we have to get into the complexes and do some... Some kind of serious, whiny number calculations, and we don't know any other way to do it, and the corresponding thing happens in the proofs I do for the bias transformations. At some point you can write it. The other beautiful thing is the connection between E and pi, which are both real numbers, is mediated by the number squared when it's what. I got an Italian student recently to check, and it's not trending, she just finished it. The only relation between them that you need to get all the others the only relation between Are you using algebraic and exponential algebraic relations? All the exponential algebraic relations are between e and pi. Is e to the i pi minus 1? Yeah. Well, don't call a boon sham you'll exist. I think this is rather beautiful. I mean, I... You must be true, but I couldn't prove it properly. Like, the mother had to do it. She's done it. So, they tell me it's just a vina terzo. Isn't it still open? It's available, I think, now. She's coming back to see me in Kimmerich in about two weeks' time. I'll get her to send what she's got. Yeah, yeah, yeah. I've checked. It's very good. Yeah, I think it's beautiful. Sorry, isn't it still open? No, this is assuming, Sean. Ah, because it's a good fight. No, no, no. But you've no hope, otherwise. For all we know, you know. In fact, we don't know that your fight is not rational.

1:07:30 What? We don't know that e over pi is not rational. What about pi? Isn't the other one that says it's pi to the e? Is it pi to the e to the pi? e to the pi is transcendental, but nobody knows... Mr. Rankin proved it, I think, that pi and e to the pi are also really independent, was it? Probably not. But not pi to the e? But you can prove that pi to the e is transcendental using Shannon. It's not so easy. Because you end up doing things there like i to the power i and stuff. It's a bit tricky. I did that in my chanel machine too. That was the most difficult of them all to get. I want to ask you something about this. I once heard Barry Mazur speaking about the role of conjectures in modern mathematics. And he was speaking specifically about number theory. And he said the following, he said, it was before I don't know how much it really... He calls them architectural conjecture. He was thinking of, especially of the Taniyama-Shimura, but also of Baylinson conjecture. And he said, the Riemann conjecture. We take them with them. So, I mean, we know it has no... But we are willing to hear their saying about Shanoel in fact. Yes, of course. I mean, Shanoel saw that there is some very simple... I mean, it may turn out to be false, although I can only imagine how one would ever prove it. I mean, that's something we might discuss also. He just got hold of some fundamental architectural principle here. I mean, this one little inequality that explains everything that we could ever want to know about E and Y, for example. Whether it's true or not, it's a plastic thing, because it's not artificial. You know, it's not, it's not on any of them. In other words, there's an object, I mean, Maser is confusing the objective and the subject in there, right? I mean, there's always been a logic, the ejection theorem. To prove A implies B, you assume A.

1:10:00 Objectively one is proving that one kernel is contained in another. I put it at the lowest level. A person wants to build a mathematical career, this student. She is willing to put a lot of effort. ... into a proof of something that perhaps is wrong, I mean... but people will accept it as good mathematical work anyway. Anyway, so this is... I mean, for many reasons apart from that Nelson, the Euler thing is widely regarded as, you know, one of the most beautiful equations in mathematics. I mean, if this is the only thing that one needs to know about everything else but quantum mechanics, then one has certainly, I mean, one can deduce it. I keep asking myself, what could be the shape of a proof that I want to example? Well, but you see, it's not so easy. I mean, you have to find two different ways of representing these numbers. Is there any case known, that's really my question, is there any case known in mathematics where people somehow entitled to an extensive experience in the area had suggested that certain numbers were algebraically independent and it was good to be independent? I don't think there's no arena case. Why don't you ask the more general question, whether people are entitled to believe a strong, important conjecture that proved to be false? I mean, it's just that the case of this one seems to be a bit more, but it's constrained by, it's always been related to, I mean, that's about the only way we can ever imagine it. If these numbers are periods, for example, and that you, you know, you do, you find out that there is some hidden geometrical connection that will reveal that two verticals are dependent, I mean, one can vaguely imagine that happening, but I don't think it ever has happened.

1:12:30 I think that would be something very rare, but of course he didn't believe it if there were infinite of them, so of course it didn't change much because whatever is proved for regular primes remains so and people keep looking for non-regular primes, so this was an expert who didn't put it as a conjecture. To a certain extent, people took it as, yeah, well, proofs of Fermat's theory will follow this, because it will be very easy just to take out the non-regular case. So it turned out to be different. Yeah, no, I mean, it's tricky, but I mean, I did somewhat deliberately open users here. Right, I mean, Kummer at that time perhaps didn't really have a right to an opinion. He discovered things, but there was no, but better to say, he didn't offer it as an opinion. He offered it as some hope or something. I mean, for example, the case of this Hodge conjecture, I mean, the cohomology, the classes that are represented by Christ, I mean, the original formulations of this were incorrect. I mean, they had, they, and they were overturned by, I think, Hegel or something else. Grudnik refers to it somewhere in his work. The original suggestion was that these cohomology classes would be occupied by integer combinations of factors. In fact, it's known that that is not true. I mean, that you will need at least rationals at the beginning, maybe even Hodgson. Hodgson is a strange character. He was an academic lecturer as well. He worked closely with lectures. They both of them go. There's an awful lot of things wrong where they had insights far beyond what everybody else could see, and you didn't get it quite right, but it's held up in the form of the rational groups which are quite a lot of them, although I don't think too many people expect it. There's a feeling there that we just don't know enough examples for it. Is that the picture of that paper, the hard conjecture is false for trivial reasons? That's right, yes it is, that's right. Hodge was a native of Edinburgh, he was educated in Edinburgh, but then he was in Bristol for a while, and then he went to Princeton, where he teamed up with, and then he was eventually a professor, he was the head of the 1958 International Congress in Edinburgh, but he was a very unusual sort of character, very much like that, at least for the first guys to get any kind of cohomology.

1:15:00 And it's one of these uncanny situations. You've got these penetrating theorems, which are true, whose original proofs were just clouds. But it's uncanny that something that's not on its face, likely. Well, the people who discovered this had no utterable reason to believe it. I mean, there's a similar thing that you often see, for example, in Italian Algebraic Geometry, where as long as you're not dividing by zero, and commonly enough, there wasn't a zero in there, but this is something else. This is a proof. You can't add any conditions and make it a proof. You can't define the terms in it. It's just... No, but look, there's something simpler. Ruffini's proof of the impossibility of the 20th. I mean, it is totally wrong. And not only totally wrong, no one understood it. And do you think there were reasons for someone to believe that this would be the result? And nevertheless, he published it, or people knew it, and people started to accept the fact. You know, they said, well, I didn't really understand. Then came another proof by Abel, which also has its problems. And then comes the general proof by Galois. But insolubility of the Quintic, the Quintic hasn't been solved and there's only two possibilities, it is or it isn't. No, because the very idea of insolubility doesn't exist. I mean, now you look at...

1:17:30 They're Lambert, Lambert conjectures is kind of... That some equations might not be solvable in radicals. In general? Yeah. So now the question is, is it true for the Quintic? The Quintic is the one they haven't solved, the question exists. He says, ah, no, to that question. But Planck-Wray, coming up with Planck-Wray duality, the question didn't exist. There are lots of alternative answers it could have had. Yeah, okay, you cannot compare one to one, but I mean, I... The institutional setting is also interesting because, you know, I think of people... Definitely trying to make a career on these kind of things. What things you are going to undertake and what things you are going to undertake. Of course, this seems fascinating and anyone who is close to that will want to take this problem. But you have to be careful because you may go, you know, like no one is going to write a dissertation on the Riemann conjecture because you assume that you will not solve it unless you know. You try to look for things, okay, you are an established mathematician, you... You can take risks that a young mathematician cannot take and I think it plays a role in what kind of things people are willing to undertake and it seems that there is today more legitimation. To go into things that are unknown, into things that are based on conjectures and questions and so on. Yes, and he makes a saying, of course we've always had the deduction theorem, you could always prove if, but there's a bunch of this's that we're now willing to publish conditional proofs on that we weren't before. You know, we were talking about Bourbaki and I don't think that anyone would, not only a young person, I would say even an established mathematician would publish... A proof of if that thing, if that conjecture is correct. I mean, probably people did it, but not people that were under the influence of Bourbaki, because there everything has to be very clearly established, and perhaps, you know, in the office of some professor, they were telling to each other, I know that this is true, but from there to publishing it and becoming public domain and things like that. Trying to crystallize architectural principles for the organization of mathematics. The problem is that it has the opposite effect too, you see, because the stuff is based on conjectures. Then there arises a small community which believes that this is a fruitful way to move.

1:20:00 And so you have to work in that community. I mean, nobody who's close to that kind of professor is going to suddenly take up the, you know, the exponential rigs of the puncture categories. Even if this might lead to some important results, since it's solid mathematics instead of the shaky kind, you know, it won't fit into the community. So the community develops in some sense. Yes, of course, in one sense it's more adventurous, but in another sense it's less so. Continued hypothesis provides an example of where they were locked into that, and why even today they just simply don't get the kind of theoretic insight. And also, you know, inside the community, you also, you know, I think a person, what community is this? Okay, I am protected within this community going this, but is it in Cambridge, is it in Jerusalem, or in another place, you know? But sort of what you're saying is, if we get a conditional proof, if the Riemann Hypothesis is false, then this happens, later we discover the Riemann Hypothesis is true. This proof probably still has some content. Yeah, it says that one idea was contained in another. Yeah, it says something. Yeah, okay. If you are willing to look at it from the third point of view. Why not? But some people are like, ah, you're wasting your time. I mean, this might turn out to be a really important significant corollary. You might be able to extract what you knew that the premise was for, to extract something that you hadn't noticed before in the proof. It shows that this situation does imply this situation, even though the zeros of the zeta function never give rise to this situation. Nonetheless, this situation exists and it implies that one. It just doesn't bear on what you thought you were doing. And one way to find this is by abstracting, finding the most general statement of what you're doing. Well, the theme has been repeated over and over and across this discussion since the beginning, especially over this international program.

1:22:30 I think this will be an actual place to wrap up this morning, but before you all dash out, Mr. Chauviz, we promised him we'd go back and have dinner in his club, and from there we'll go on to the castle, so get your walking shoes on, and we're going to reconvene here at 5 o'clock sharp. Oh, okay, and then I want to... Hang on, hang on, just before you go, sorry, there, go ahead, yes. No, I need... What happens tomorrow when... I'm going to tell you all about that after lunch. Everything's arranged. What time now? Buses, everything. I'll tell you all about it. No, no, no. What time is it when we have lunch? Well, now, basically. Well, I've got to go pick Megan. I have a little problem, so I want to join you. Okay, we know where it is. Yes, at the cab, of course. But you know... Well, I'll leave the key in the door just for a minute, won't you? But you know exactly which one we're talking about. There's a bar on the corner. Grab her. Yeah, so it's up there. It's really cool. It's called Ron the One Hunter. Okay, Ron the One Hunter. It's true. It's the one right, you know, where you turned, it's where the traffic light is, where if you go up onto the square, and turn right, it's the bar right there on that corner where the traffic light is. Okay, so I get that, and I get that by going up there? You go up there, you do the, through the arch, you do the very tight 180 degree turn, then turn right, and up the corner where the traffic light is, where you turn right onto the main road, it's that little bar there. Okay. And there's parking space near there. We'll go up there in about ten minutes or so. Okay. That's good. Let's give ourselves 10 minutes just to draw that, okay?