Discussions 5th Tape, incl. FW Lawvere /

0:00 It's all right with you. Oh, I was trying to go by train. I prefer to go everywhere by train. By car or by bus. Yeah, I... Exactly the same thing. I see. At the same time as you. Yes, that's exactly what I did, too. I was just a terrific man. I'd say, too,

2:30 one of the many, many things I learned yesterday is to retain them. I was thinking a little about what you said about the five structures down here, about the five structures down here. You know, I've heard something about the Lamb-Rutherford shift, you know, the early kind of creators of the Grand Old Man, from these lines.

5:00 The quantum mechanics is, if I'm not wrong, is it the lamb shift that is the number that's supposed to determine the propaganda, the QED, right?

7:30 Yeah, it's the lamb relative shift. Actually, it's now overtaken by something, some number that they actually determine in general activity, which they now have. How many places the lamb relative shift is determined is an impressive figure. Trace of correctness doesn't prove anything about the conceptual foundation of the theory. No, I don't know enough about how impressive a claim that I've not achieved is. What would you say about the conceptual foundation of the theory has been vindicated? You have to have one number. It may have been difficult to obtain this number in the sense that I know these asymptotic procedures in themselves are very laborious. I think you're right in itself and I don't know how many of the, I think that the people sincerely, I don't know, rest. They just were, they were just resting there quite loudly the day, as it is with your theory, which they could have said before.

10:00 The analysis is essentially that the government has a bogeyman, which means some arbitrary thing, but it's been a number of cycles. I'm honest.

12:30 Yes, I see. They give you the ideological issue in one direction about the older one, and the other direction about the other one, a wonderful one. I hope it could be expanded in science and co-science. I never heard of that. It's an inference problem. I see it as something so exciting. Yes, I noticed that Heisenberg was brought into, Connes talked quite a lot about matrices, the history of physics, and then, I think, perhaps brought in, seeing how, you know, confidence of Heisenberg was, but then, he came along and lulled the physicists back to sleep so they could actually think about them.

15:00 No, no, no, that's amazing. Do you want another coffee? Yes, please. I could be wrong about it. He was almost exactly, no, he was before. He's right, yeah, he's that. I think he was in my, I think that the static wave of fate was 1936. I wouldn't claim any, you know, expertise in that. But I think at that stage we were both working prisoners. At least you made it sound like, you made it sound like his method of deriving it. Yes, yes, you are.

17:30 By thinking upon the subject constantly. The Chrysler story by...

42:30 I was going to say, this all seems to be just gossip, really. Yeah, personal gossip. I've read several of the articles. So, Riemann Dyson was asked to respond to this. No, I didn't know about any of that. But I'm sorry to learn that Dyson has become a... Well, actually, Miles Tierney, I remember, Miles Tierney was part, this was several years ago, was part of an organization in New York which... They attracted people by claiming it was going to be a scientific culture, but the leading figures were Dyson and the other very prolific science fiction writers, Isaac Asimov, Freeman Dyson, and one or two other people of similar fame, but then Miles Tierney was for a time one of the foot soldiers in this organization.

45:00 Until Dyson started openly, this was before he actually published, but in the meetings of this organization, he started, you know, talking about his Buddhist experiences and how that was far more important than any physical nonsense. So Miles, of course, quickly got out of it at that point. But was somebody, Miles, did any of his abilities as a mathematician do any of that company? Well, as I said, no, he was misled. I mean, he thought, oh, Diamond Dyson is a physicist, and, you know, one or two other, and, you know, I mean, his... In other words, they thought it was something like the Viseu Foundation. Yeah, yeah, that's right. So it was spreading... That's right. ...a clearer understanding of science and theory. He did go selling newspapers one time with me as a Chairman Mao's May 20th statement to the people of the world, their rise and defeat, the U.S. aggressors, and all their running dogs. So that particular newspaper where they reported on that, and you saw him in the streets, because, sort of, you know, as a liberal radical protest, because people were arrested for the crime of selling newspapers, he, you know, ceremoniously addressed you. So much for the First Amendment. Yeah, that's right. So he knew about, you know, he knew about the discussion of the philosophy of science. Anyway, he was, you know, he told me later about how they had these leading speakers in the time of the 19th century. I've just been reading. I don't know if this is what I wanted to ask you about. A bit like Alberta yesterday, hopefully. But yes, of course, I'm rereading this. It must have been very exciting, that period of creation.

47:30 The guys were the first to build it. I was there seeing an incredible richness of these ideas. It must have been one of the most exciting times in any period in any branch of science in the last century. Believe me, until when I was about, before I actually started studying books like this, I believed that too, because that's what I've been told. And that the great payoff of those It takes your choice whether you take the subjective idea as we have the objective idea based on either of them. The hybrid showed that the whole project of trying to unify mathematics and the unified understanding of mathematics was misconceived and dismissed to yield to the ineffability of mathematical concepts that would be able to touch all the symptoms and just try to be just user-resistant. It could be completed, but only if you accept that it is complete in the mind of God, the world, and the plane, because that's like Earth, which became a plane, and all the structure of the physical world was just somehow, you know, in place. It was a tiny little world held together by the structure of the ordinary, the cohomology, the structure of the new, and the real. No, I'm afraid I was in such confusion before I discovered just how... How deep and important the work in the geometers, of the real geometers like yourself and other people who have worked in the development of cascades has been and rescued me from this stuff. But that is a lovely paper. I wanted to ask you a couple of things, if it's not. I want to understand more deeply how this big category of space is there.

50:00 Universal covering spaces illuminates the position of the QD opposites, the metallic ones, within the ring of all spaces. Now, very interesting observation you make right at the end. About the idea that every type of quantity being a set could be maintained, but that you see that the case that domains of variation, which are isomorphic to a part of a type of quantity, which has been retained just as the definition of a particular special case, quantitative, as a special case of qualitative domains, now the qualitative domain, the domain switches, as it were, have this purely qualitative character, the... The non-tribulatandu would be an illustration of this, would it? Because of its, the case where because of the action of groups in the points of the space you can have a structure which is locally a space but then it's not, when you go to the coverings we're not globally a space, it's not globally just a space. Would that be an illustration of what you had in mind by a qualitative domain or have I misunderstood the notion of qualitative? I've obviously been to an earlier stage of the program, of course, so I realize things have moved on a great deal. Well, not as much as you would hope. Are the qualitative domains simply the ones in which the, you know, the two co-equalizers are not preserved, in which the action is, in a sense, it makes the case that the airfonder is not for space because you don't have preservation of the two co-equalizers in the topological category? Or is one...

52:30 And again, what is the relationship? It's not being quantitative, is it? In terms of, in what sense is it? This conjecture was explicitly refuted by Johnstone as an example, a very simple example. I mean, the category of sets each equipped with a single idempotent end of them is not a neat time to do that. Because, as I said, you see, really the broadest possible theory that I know about, in general, is based on... There's a site which has no idempotents. This would include the QD and would include the locality and hence the groups and the classical spaces. But just having a single idempotent operator always has to use some terms. So his claim is that this, you see here what I'm trying to do here is the idea of a small category of points with quantities. This may be an annual identification, at least something worth investigating. But he says, Johnstone's idea, I don't know if he ever really proved that y sub x exists, but anyway there's a small number of points, set value points. This is a much more, nobody's ever investigated this. Well that's what I was wondering, had anybody investigated this? You can have, you see if you have a general topos of domain, but you look at geometric morphisms, they realign. The sheaves on any topological space, well, there's an internalization of that. There's an object and you're given to those points whose sections are the equivalent to the continuous maps into the other space. Do you spell that out at the beginning? Yeah, it is. So the geometric morphisms from x slash u into y, there's a special kind of y.

55:00 Which is something which is normally a large category, but for these special y's it should be actually a set, and indeed the map enacts itself from u to a certain object, which is sort of a sheet of germ. Right, and the speculation was that the only ones which have a sheet of germ in that form are the entanglement worlds. You're saying that that's been refuted by Johnston's discovery. You know, there's only two or three points. So certainly in the category of sets, there is, if you take a three-point set, then you can represent where X is the category of sets. Now, whether he actually showed, come to think of it, that it would work for all X, probably would be all right there, too, if you could internalize the notion of a flat action of a... So investigating which topos as Y have that property is still an open question, if it isn't exactly the agent. It goes both ways, in other words, if you have, well here's a problem, if you consider a topos whose site has no item, perhaps a QD topo, is it the case that over any other topos, any other topos, X has only a set of augmented morphisms, parametrized points, whatever you want to call it? I don't know. What is the class for which this exists? It's worth, you see, since these people have claimed that all people deserve, well, yes, they've completely missed out on this question, whether, yes, I mean, if there were, you know, if, if there were, you know, qualitative, purely qualitative domains, which in a sense is too topological character for that example. Any classifying topos of an algebraic theory, in other words, if you take pre-sheaves on the left exact category, they're all pre-sheaves, they don't go to the topology.

57:30 So that's the classifying topos for structures of kind C. You sort of think of C as a theory, but the left theory seems to have got real humor and so on. Well, that's always about the class. That's always too big to miss. ... cohesion and variation, trying to explain everything in terms of variation and generalize it beyond all limit and in the space of rings, the space of rings, well it's too big to be a classifying topo, the points are all possible rings, the maps from any topos to it are the rings in that topo, this is essentially always a proper class, so this is qualitative, in other words you're parameterizing all the models of a theory. That's what I mean here by qualitative. And firstly, the quantity is a very special case of quality in that sense. We also have classifying topos, seeds of germs, but those are qualitatively smaller. That was the idea that quantity is a special case, and that among all these interesting classifying topos... They're the ones which always have shoes of jams, and those are the ones with internal shoes of jams than any other topos. In any other topos, and that would give you the quantities for the case of... Because, you see, you can think of inside x, the maps from u to y sub x are the variable quantities varying over u of type y sub x for any object. Now this y sub x comes about from another topo that is external and internal. You have what these variable quantities of that type are over you. Well, it did seem absolutely fascinating. It seems a very, very topical investigation. Because we're seeing how you put together the internal and the external view. It's very interesting. And, well, what you have been talking about is the distinction between quantitative and qualitative demands, with quantitative in the sense that there's restriction because of this special condition, the sheets of jazz.

1:00:00 These, of course, are all intensive ones. Right. So what I'm saying here in the last line is that, well, this is an example that's sort of well-known, so that's why I mention it to the people. In other words, the cohomology class of coefficients, this is called H1 of X with a coefficient of G, is represented by arbitrary geometric morphisms here, and yet it is a small thing, which we can internalize into X in that way, too. So it's a very, it's an intensive quantity, but it vanishes at every point, so just to point out that really these quantities are more general than what you might be used to, because if you are used to them in another context, cohomology classes really are intensive quantities, but again, using the word, qualitative characteristics, that's using the word in a different way. But anyway, yeah, that's a definite intuitive issue. Again, not really... And this whole trade-off between quantity and quality in other words is taking place there too because on the other hand on the one hand this thing is small enough to be it should really be a kind of intensive quantity and yet it's completely qualitative because it's merely cohomological it vanishes at every point. The dialectic between quantity and quality is even deeper than this suggests in here as well it's sort of implicit. Yes, begging, but it's, well, that's fair enough. Yeah, well, I'll get something. He won't probably come back for a minute or two. I do continue with one thing. I mean, I'm particularly interested in understanding, Greta, the case, the specific case of the domains with subsidence as a form of two topological characters.

1:02:30 Oh, yeah. The ones that, which, whether or not you relate, I'll do. There's not a case, because they're clearly, well, it's that they're varying internally because of the action, the way that it affects the quotient, the construction. That's what prevents them from being set-sized, in the same way. Sort of in addition, if they take the case of the classifying topos for rings, and this is always appreciated, you know, to talk about the class, the model there is talking about the class of the rings. This is totally absurd. It's not a class. It has to be a category. That's the very idea. Since it's a concrete general, based on an abstract general, the notion of morphism between specific instances is totally tied up with the idea. So just to consider it as a class is completely correct. But when I say two-topological character, I'm really saying that it's a category, really, instead of a class. But then, top is a two-category. Yeah. But, of course, there's a corner of it where you can sort of ignore that, and the sheaves on a house door in a topological space lose all the... Yes, and you can think of, yeah, when you think of the sheaves on the one-point space, as it were, as spitting into a corner of this, obviously, it's the extreme case of... The sheaves on a one-point, yeah, right. The sheaves on any house door space, the non-house door space, you have at least the ordering of the points by... One point is less than another. All the open sets that contain one contain the other. This is about reordering the actual points of space.

1:05:00 That's sort of a trace of the two-dimensional structure which provides even into classical topology. Yes, even when you're thinking of things in terms of, you know, defined by those set structures, rather than looking at them from a deeper level. In general, the geomorphisms from one turbulence to another form a real, real-life category with more than one mass between two objects. So when I say two topological characters, I mean the fact that the top is a two-category, perhaps between geometric forces, and this is a dynamic. And it's hard to understand the points of one of these so-called faces of classifying proposes. So a lot of this is correct, actually. It's not been developed. Oh yeah, so the idea that every domain is variation because it's not going to be every one, it's only a certain one. Domains of variation which are isomorphic to a chart of a type of quantity retained as the definition of a particular kind, quantitated as it does the same as the quantitative domain. But such would seem to reduce to the fact that Y sub X exists. Every type of quantity is a set, or put in more qualitative terms, that Y sub X exists is all X. And of course, Johnston thought he'd established that through establishing his eyes, but no, we'll talk about the locality anyway.

1:07:30 I'd almost look at the Q-detox as this exists. I don't know. I'm speaking for the greatest experts in the world, but I know the claims made. What's an example of a complete non-physicist? What's an example of a non-physicist? Anything with a psychedelic purpose. Anything with a psychedelic purpose. So if you take classifying topos for any non-physicist, I assume you have, look, if the site has strategic products. Yeah. Then you have graphs and maps. Graphs and maps are hidden codes. Right. So, basically, in the QDE topos, no site can have products. It won't have cartoons within the site itself. Right. It won't have the categories of math. In the topos, where the site is embedded into the topos and have products there, you think the product is too representable, but it won't be in the site anymore. It would give you some idempotence, but you would use the, when I say it, I keep taking it up, but that's a sort of very coarse byproduct of, you just can't have idempotence. The more precise thing that he found there was if you have a site wherever you map it. A site that thinks every map in it is epic, they're not going to be epic if they're co-constructed within the site itself. You have simply the cancellation law, that if any map followed by two maps makes the two maps become equal, then they're not cancellations. But as a trivial case, that excludes input. You could have x and x squared, or say the identity of x. The following acts are equal, but no, they don't get 1 equals x, so let's just write that down. If the 1 equals x, so which of course just gives you the QD case in the end.

1:10:00 For epic, that implies if x squared equals x, it implies x equals 1, so you don't have any input. So you don't have any input. Again, using the logical negation in the sense that ordinary people use it. It implies, you know, the property implies the trivial case, not that it implies falsity, because it's worth one of the idempotents, but that's how people use it. They say there are no idempotents. Everybody knows what they mean, they mean this. So it's not that the property of being an idempotent implies absolute falsity, it implies a trivial case. Yeah, that's very interesting. I didn't get to see many slides, so if you could go on and on. The proper kind of logic really applies to everyday language. See, this is another thing. The claim is that logic has nothing to do with ordinary language and has nothing to do with logic. People, the great unwashed masses, couldn't possibly be thinking rationally because languages are so completely inconsistent. Yes. Well, that was Frank's great line, wasn't it? It was, probably. Oh, very much so, yes. Anyway, that's just one small adjustment to formal logic which shows how I have no brother, a man could say, even though the definition of brother is another man, a man, sorry, with the same father and the same mother, so he himself is his own brother under that definition, but that's always understood. If x is my brother and y is x equals me, when you say I have no brother, you just have to... That illustrates beautifully the point about having a brother. It does, as it were, the difference in that case. Yeah, that's right. It's the trivial case. In a sense, you do have the even brothers, just that the even brothers are trivial. Any time that the category has products, if you have any math at all, it has a graph of h and h cross b.

1:12:30 The sub-object of a cross product in the graph. What is the graph? It's really a map from A into the product. Whose projection back onto A is the identity? That's what I'm just trying to explain. The graph is not only a sub-object of the product, it's actually a retract by means of the projection. That's the definition. This is one common, one sub-A, and that's the graph of that. The property that the projection back onto A is the identity of A. You find a point just above. Of course, the other projection you give back F. That's the virtue of the projection over here. You get the value of F. But notice that this being an idempotent, the mona, which has a retract by sub A, whenever you have that recomposed in the other direction, you have an idempotent. E sub F, if you like. Which assigns to any point in the rectangle, the point on the graph of that which is just on the same vertical line, once you're on that graph, you stay there, that's an idempotent, that's a non-trivial idempotent, that comes out of having products, except in the case of a poset, a poset could have products in the intersection. But then all of them have to be the identities of the concept, the sort of category with products that's not a process, basically, because of this graph construction, there are lots of idempotence. It can't have the calculation property, it cannot be a TV. It's got to be the site for a TV. Oh, it's over a TV, Thomas. Yeah. Oh, that's tremendously different. Really, that's cleared a lot. Of course, the thing to do would be to do some of these exercises yourself, and then have it much more internalized.

1:15:00 That's what I plan to do. I really do plan to spend some time now, this winter, doing some of that. So there are loads of examples of the topos of reflection graphs that are treated in our book. Besides, there's this monoid with three elements, too, which are idempotent. In contrast with the irreflexive graph, which is locally a space, it's an etendue, etendue means that the site is locally mono, yes locally, but of course not locally, whereas QD means the site is epi, everything is epi, but the common generalization is everything is locally epi. And then, of course, within that, you've got the more illustrative case where you've got epi-mono-factorization, which is, well, no, no, everything is epi-mono-factorization of horizon products as well. Anyways, most opuses are not locale, obviously, like the reflection graphs, obviously. But you got the argument about why the irreflexive graphs are, I think, trying to do a little bit of space. A little bit of space. There's three-point space, a positive, zero, and negative. There's three points. The belief on that is the same as the set of global sections, a set of sections over the positive number, and a set of sections over the negative number, and the two restriction maps, which are arbitrary, but then, if you... If you collapse it in a world of no-poses by identifying positives and negatives in the right way, then you still have two restriction maps, even though there's only one set of them.

1:17:30 So if you have two parallel restriction maps, that's the same thing as a not-necessarily-reflexive graph, which is a picture of that set of elements. And so on and so forth. Basically, any site has interposed. No interposed, anyway. So whether these still have only a set of points or whether they have a sheet of terms in any of their topos, because I think it's kind of these simple, basic things that I always wish those smart people like Julia and Bernd Eich would investigate these problems. They could probably resolve them very quickly. There's somehow this ideological spin now that everything is up to epsilon, metallic, and all that stuff, which allows them to put their heads up their asses. Yeah, yeah. She's fixing it, just saying. But then they sort of keep, they sort out variations so completely that they're not, yeah. The ideological crippling of the pre-investigation of mathematical assessment by the ideological phenomenon. I'm sure they can answer these things.

1:20:00 It would be very nice if it would all come down to idempotence, you know, if you have to put the quantifier, but every site, or there exists a site, which has no idempotence at all, or which always has idempotence, whether this could be the basic classes, which the positive properties of the growth of those idempotence would be the pi-zero preserved products and the omega is connected. So what if one could correlate that, which it is quite closely correlated, with the existence of plenty of idempotence in any site, and on the other hand, if the lack of idempotence could be correlated with having always a heap of germs, then this would really give a much more definite answer to my various speculations. And an even clearer deepening of the dialectic of quality and quantity. Anyway, that clarifies for me much deeply how this should-havely, non-trivial potency, in my case, connects up with this guiding idea of thinking of structure in terms of the relationship between the spaces and the universe, and how the space is a number of groups. The basic point is that there should be a relatively small subcategory of topology which would include all that. There remains a sense, which every toko says, it is a classifying toko, so in that sense it's a space of rings or a space of...

1:22:30 There's a sense to this, in a more general way, but I think one has to... It's sort of a purely conceptual space. As opposed to anything that could really be thought of... They're satisfied. Somehow if they can find a counter-example to something that Bill LeVere said, then they stop with that. They don't say, oh, well, what he really should have said is this, and go and prove that. That's what a real mathematician should do. But they're content with having sessions over there, unless you address them more in depth, so they need to pursue the matter further. It's happened more than once. Yes. Well, if you have the view of topos, there is just being a piece of machinery, but there's always a problem. Already work has problems. All of these have a profound conceptual basis and analysis of quantum variation, but I suppose you'll be tempted down that road and you'll think that Drosten tried to sort of clear up the problem there, but then again, I mean, you must surely put this point about the topos in the reflexive graphs. In 1986, in Cambridge, you distributed only a two or three-page paper for a distributed paper. I can't believe it. It was published in a sort of obscure place, namely, Bogota. Ah, yes, this is your qualitative distinction between some topics and some graphs. No, no, no, that's the one that happened in the U.S., America. We amassed computer science and...

1:25:00 What's the title of the book at the time? Topos may not be generalized spaces, so it's sort of a general behavior of John Stone, as exemplified by directed graphs. Opposed actions could load topos and prove that the localic topos were not satisfied. And I give the two examples that reflect an irrespective graph. Such simple examples, but showing that there is a bifurcation, it's not all simple because they're the same. They're all counted as black or something like that. Getting me all worked up here now, isn't it? Oh, I'm sorry. I suppose, I mean, where do you see the ideology coming from here? Is the addiction to the Q-D case because it makes things sufficiently set-like? Well, the belief in logic is very honest. Poor Murdoch and this incredible misguided... So there is this very famous Swiss geometer that De Sivre often refers to also, about geometric measures theory and so on, called Hethlinger. Hethlinger has used topological groupoids quite a bit in his investigation of manifolds. It should be accepted as a circle of mathematicians, establishment mathematicians, as a possible means to impress Hefler with the necessity of using the category theory that he has devised. So the pretext is that, in fact, a more precise statement than every trochosis locale is that every trochosis is covered by a locale.

1:27:30 This can in turn be encoded as a Dupoy, the locale in Dupoy. Dupoy is sort of a generalized equation. To give a Dupoy in the category of locales, this will give rise to a burden in topos by taking the sheaves on the locale. The equivalence relation comes from the arrows in the Dupoy, the locale of arrows in the Dupoy. There is such an error and two things should be identified. So the same procedure whereby I get the irreflexive graphs out of the three-point space. There's a three-point space, but then you could think of it as being the nodes of a groupoid, but it's just slightly larger to tell you what things to identify. In general, a locality groupoid gives rise to an exact diagram of two locales and then another topos and none. And so the theorem of Choyal and Tierney can be refined to say that every group of topos arises in that way. Of course, just to identify topos... Topological group law is absolutely insane because this would be like identifying a group with a group presentation. There are loads of such topological group laws. You give the same group the topo. In order to push this insane idea that topos are the same thing as topological group laws, he had one paper where he points out how to calculate geometric morphisms between the topos in terms of topological group law. The natural maps are present at the homomorphisms of locale groupoids. They're not enough to imagine one presentation to be expanded out, and the other presentation to be expanded out until they become the same, and then by that, they're not the same, but I mean, they can be matched together, and from that, you can object to it.

1:30:00 We go through the whole technology of these spans, the two-dimensional spans of locale groupoids. As though this machinery, you know, was the same thing as topos were. But now, you see, so now he starts making this big pitch to Heflinger. You like locality groupoids. Actually, topos are nothing but locality groupoids, and so you should pay attention. Now, I've heard through other sources that Heflinger's guys are getting sick of all of that because he sees through it. But Murdike, at least until he persisted in pursuing this, took such incredible lengths, incredible lengths, that he has a paper about homology of locality groupoids. Homology. This long paper never mentions the word topos. It starts with the locality group, I guess I'm incredibly, horrendously complicated, and then goes to great lengths to show that if you have one of these spans with the locality group, it will induce an isomorphism. So in other words, to make a long story short, the homology of the locality group, I guess, is an invariant of the topos. But instead of doing what an honest mathematician in any other field would have done... And say, oh, so in terms of topos, I can define this homology, in particular, if I care about it. No, he never does that. That figure needs a homology theory, of course, without a viewpoint, so he's going to provide that, you see, but without ever... So clearly the mathematical content that he knows about, but he's not telling, but it goes through topos. Instead you have to read through dozens of pages which prove that, so without ever being told what is so extreme. And anyway... The locale at Blue Forge has come up in presenting a couple of these insane zero-dimensional monstrosity spaces that logicians like. None of all like the locale at Blue Forge has those C-infinity manifolds that Ettinger has in mind. So even to identify the two words is formally correct, but you know, two extremes. But it's actually showing that because of projects like that, that these excellent mathematicians like Berthoud and Kerr diverted from it.

1:32:30 They do seem to be absolutely fundamental. There's so many fundamental conceptual understandings in terms of the priority of ingredients and definition of concepts. Now, Gavin Ray does on his website... The proposers are no longer needed. It's now been discovered that they are at least a little bit... Because they're just O'Callagrew parts. You mentioned this statement yesterday during the course of the conversation with Alberto, but I haven't realized how it's connected up with this. It's just basic. To an extreme extent, your ideology is blinding you. And it's also going with this line that, you know... Homology and, you know, pheromology are everything. Yes, absolutely. It's clear that there are extremely important aspects of structure where deep identity was provided. Yes. And in fact, this is, again, ideological propaganda is made for this. I mean, I have seen reference in print, too. The program will be an opportunity for the re-description of all the structure of mathematics in terms of homology and cohomology, which of course is a serious distortion of them.