Discussions / FW Lawvere presentation & discussion, incl. J Mayberry, R Pettigrew, D Bernardini, M Menni, M Wright (contd.)

0:00 ...dialectical rules that we make all the time, and there were one group made of this, etc., etc. So, and the same way with analyzing the, you know, I saw the actual racetrack, they claimed an Achilles on the tour. This is probably a curious attraction. But anyway, the point is that, you know, there you have all sorts of, several dialectical elements, but mainly the sort of objective. So you have the objective and subjective contrast in analyzing this notion. So the thing is that dialectical ingredients like that, it's extremely useful to realize explicitly that these are parts of the way that thinking develops. On the other hand, it's always a possibility that reactionary exploiting classes can use these things to create mysteries. I don't know whether they did that. The history seems to say that already in those times they were making these things into mysteries, as impenetrable. How could one possibly ever be many? How could many possibly be one? This is something inscrutable. We cannot possibly grasp this. You could make that of it, but you don't have to. It's the same way with motion. So again, this whole bit of the race and so forth, it has to do with, it goes back to Heraclitus, you see. You can't really step in the same room and say, oh, that's terrible, we're forbidden to even move, right? But you can make it into a mystery. On the other hand, you can recognize that these are crucial dialectical ingredients in the construction of concepts about matter and motion. So, I sort of extracted this summing up out of this experience, even though it's certainly not from Barnes' lectures. Well, Barnes should have known. Aristotle did kind of verbal. You can't ever say that, his sentence, you can't say something is moving. You can say something has moved. Okay. So you can only, as it were, describe things after the fact. Yeah. The tortoise is actually on the racetrack. What do you say? He's moving at a distance. But you can't, as it were, in the middle of it say, he's moving. Well, you'd have to cast the idea of his movement in order to get mathematical physics.

2:30 In my own writing, I use this moving, this I-N-G, because it's an awful lot. And finally it keeps objecting, you know. From the point of view of German or Italian, which are two languages that we normally have, sometimes you can't even directly translate it, and other times you have to make quite a certain locution. Whereas to me, it seems very natural. Yeah, that is odd that we use present participles much more in English. I mean, the French have to say, je suis en train de, instead of yes. Yes, yes. And maybe that makes it more clear what is actually, because, yeah. I am in motion. There's something that just escapes us as being odd, you know, when we use these present bar symbols. That is very interesting, that there's always a kind of... A preposition in French, you know, so you say, I am in motion, I am in a state of motion. And there's less likelihood of the distinction between I am... It wouldn't say I am translated in English, present participle. Like, in French, it's just... You're starting to change. You're starting to change. You're starting to change. You're starting to change. It's not really a parsnip. It's gerundus, yeah. I am walking, it's exactly in French, it's a train from there. Yeah, okay, okay. In French it's more... More elaborate. Is it the same in Spanish? Pretty much. I am walking. Okay. So, so, yeah, you're right. This is a state of becoming. Yes, it's a state of becoming, exactly. You can say it in that way in English. I mean, I am in the course of, but it's not such a... It's easier to say, I'm running. It's easier to say, I'm running, yes, exactly. Which tends to obscure the distinction between being and becoming. I agree, I agree, it's interesting.

5:00 So my crudely oversimplified history of Greek philosophy is that these paradoxes really... And then this was the basis of Plato's idealism. The conclusion from the fact that you can't do whatever is that you can't know anything about the real world. The only thing you can know, but since we do know things, therefore what we know is the forms and ideas. So somehow this became one of the main supports for this theory of Plato. But it was really, it isn't clear that Heraclitus wanted to make a mystery of it. I don't know, maybe he did. He basically, anyway, he discovered this fact about this, again, about basically about the states of being. So my crude idea is that this, the D, see this, is what... You've got a line on Parmenides, what Parmenides says, that there's only one thing. To say that something, whatever it is, can't be, can't not be. So you can't have something that's here today and gone tomorrow. So they hold it. But anyway, somehow this Heraclitian idea becomes implicitly, if we model it now, but it was implicit in Galileo and Newton, this idea that, well, you can step once, but you must do a little more at the same time. So in other words, it's like, as I was saying, it may be a Leibnizian monad. There's only one point, and yet the thing is itself more than one point. A single component can have a single point. And this is somehow the basic model of the states of becoming. So in other words, you are here, but at the same time, you're becoming. But the bit more doesn't consist of points in the same sense. I mean, that's the trace of motion, but it doesn't involve there being more than the one point in the component, yeah. That's been a puzzle to me, thinking about synthetic infinitesimal analysis, that the real numbers, in some sense, correspond to points.

7:30 I mean, admittedly, you can't always distinguish them. I mean, that's crucially different from the classical physics. But then the infinitesimals are not identified with, you know, the line is kind of composed, or curves are composed of infinitesimals, yet they've got points, they're sort of planar points on them, so the switch back and forth between these two pictures of the line width, which is... So you're saying that the linelet is somehow the carrier of the becoming, but then what is the point of where you are? Okay, and so because the linelet is in motion, as it were, or in statue nascendi... It's all an idealization, but this is the best we have. Okay, so it's at a point, but being at a point in this context... It's not quite the same as being at a point in a classical context, but it's exact and immutable. In other words, the typical line of these topos has these infinitesimal elements, but it's matched to the Dedekind reals. Essentially, by killing you, you get a quotient of the Dedekind reals. Okay, so you agree? People try to say that there's some extension of the real. It's an extension in that sense, not in the sense of inclusion, but in the sense of backing up along a surject. For some reason, the word extension in homological algebra is like that. It's not an inclusion, it's backing up along a surjection. Yeah, but I think that's certainly the case if you take any of these actual, like, topos or something like that. So the Dubuque-Thomas built on, I mean, what is the map actually? This is assumed, which we don't always assume, complex geometry, you wouldn't assume it anyway, but basically if the line has got a partial order in it, and by the way, it's not strict. Well, in fact, you could look at it that way. The strictification of it is synthetic and real.

10:00 We identify x and y, if x is less than or equal to y and y is less than or equal to x, then we get... Yeah, so there's a kind of, there's even, no, I mean, I say even, the secret of it is that there's a, you're saying the fundamental thing is not there's a confusion of identity, but there's a confusion of order. Yeah, yeah, that's right. This is the fact that the strict, all the models of ordering, which, you know, Koch and Kuhn and Reyes and these people have been able to come up with, they're never strict. The orderings are never strict on the line. And the so-called closed intervals are always infinitesimally open. There's a set of all things that are equal to A has some stuff that spills over above A as well. When you say that this isn't, these aren't additions, what, I mean, the way you've explained it is, it's coming from, it's backing up the... Well, if you have to think of it as... But I mean, actually, it's throwing away things, not adding. Yeah, exactly. No, no, that's right. This is just sort of reinterpreting it in classical terms. It's not an enlargement. There's no section. I saw a paper by John Bell, who was trying to explain the map from Euler and the reels. And he claims that also in the dedications it remains the invertible infinitesimal. He says that to go from Euler to Dedekin, of course, there are no longer nipotens. But in Dedekin, there is the possibility to have such a... But that's what happens in non-centered analysis. You get inverted quintetismals. But you have to include infinite... I'm very much... I feel very negative about that whole idea. This was a point, a selling point of...

12:30 Merdike and Reyes spoke that they managed to get these. But you see, why did they want them? They wanted, you know, in order to give the wrong interpretation of Dirac delta, Dirac delta is, should be thought as an extensive quantity. And it's this whole fiction, starting, of course, with Schwarz, that there's some kind of generalized intensive quantities, which there are not in many several different ways. There may be invertebrates, for example, with germs and so forth, but I don't think the Dedekin reals are ever going to have them. What is an infinitesimal inside the Dedekin real? Well, there are none. There are none. At least I understand it. The goal of invertible infinitesimals is really in order to pursue the goal of giving a justification for the wrong idea that distributions are generalized functions, and it doesn't serve much purpose other than that. Well, I mean, part of the aim of nonstandard analysis is precisely to make the... Infinitesimal is merely an ideal object which allows you to reason, to derive the classical Vedic-Indian consequences from these infinitesimal arguments. So there, the inversion is really because you're inside the larger dedication, you're inside the enlargement, and it's supposed to be exactly like that. So there's something wrong about that because they're being in that context. This was actually, again, there was some kind of fuzzy justification that, well, this is sort of like complex numbers. Yeah, or like points at infinity in projective geometry. Complex numbers precisely are not elementarily equivalent.

15:00 They're both rings, they're both fields, and so they have some properties in common, but they're not. I mean, it's just, in other words, it's taking a subjective route, saying the important thing is to have some. So you would think, for example, that the complex poles off the real line that are buggering up the convergence properties of analytic functions on the reals are not, in some sense, you're not really dealing in the, you're not really inside the complex plane. What you're doing is just allowing yourself to speak as if... That would be the non-standard interpretation of complex analysis in a way, yes. That's right. That's right. That's right. That's certainly not the way complex analysts think of it, but I guess they were keen to get this conservativity resolved because they wanted to make sure that everybody would agree with their theorems, if not with their truths. Yeah, yeah. But it's interesting, though, that there is, again, an objectively justified partial result of the Robinson type. Robinson simply asserted it's an elementary extension. This is basically the imposition of that idea. But this is actually a theorem if you consider only positive languages. Can you possibly use another marker, Bill? It's just a little bit too faint. No, the black would probably be okay. Well, I mean, isn't there something ephemaneous about not using chalk, by God? That's much better. That's better, but there's something, you know, having to buzz around with these damn markers. I mean, you know. Why not just use chalk? Then there's no problem about it.

17:30 It has a bit of a problem with a whiteboard, though. Yeah, I know, I'm teasing. So there's an essential geometric endomorphism of the topos given by... Well, the inverse image is simply exponentiation. And, of course, it has a left endomorphism, an essential morphism. Why do they use the inverse image? Let's say we're assuming these are based on some other type of a test. Do you consider any theory in positive language? So there's no negation? Well, basically, no. Positive language, okay. The languages, the theories that are preserved by arbitrary inverse images are those that, first of all, don't just have a set of theorems. They have deducibility as a binary relation of various types. Not just one universe, but other universes. And you're allowed to have existential quantification, misjunction, conjunction, and also true and false, but not generally universal in application. So, the turnstile is taking the place of replication and then, you know... Well, it's entailment. Yeah, yeah, yeah. The idea is it signifies actual inclusion of sub-objects. Yes, okay. It signifies inclusion of sub-objects. But then this is always relative to a particular, you know, if you consider a certain universe R, then maybe R to the N, you know, reach N, you have these sub-objects. And this signifies actual inclusion, whereas implication denotes a binary operation on suboptics. It gives you a third suboptic. And so there's a certain contradiction that people pointed out. These operations are meaningful when you give them to a place, but they're not preserved by arbitrary inverse images, only positive. But part of, I hope, Bolivia is going to give us next week. There's a lot more details about this, but any theory in this language, and that structure has a classifying topos, so if T is a classifying topos for such a theory, then a model, a morphism, a model of T, then E, has a classifying morphism, meaning that its inverse image is the actual model, and the inverse image...

20:00 ...to take the various types, for example, if we had one called R there, take it into an object of E, which was a model of T. That's the way they classify it. So you see from the way I've drawn the diagram that for every possible T, every model of T, if I take M to the power D, that is this, the actual object to the power D, it's again a model of the same theory. So exponentiation by D gives... And the extension, which is, quote, elementary with respect to positive. And, you see, this, and so far as classical theories are concerned, this is just as expressive, right? Because if you, you can, if you have a negation, a classical negation in the axiom or theorem, you can, you can just join that as a new primitive, and together with the, you know, axioms, So whenever something occurs negated, you are joining the negation and the new primitive and then join these two axes, so you have the same classical content. Every classical theory can be presented as a positive theory. Although of course it's important that not all theories are classical. But in particular I would apply the mark here that if I actually have, then that will also be preserved. So somehow the idea that the M to the D model of T, it's a matter of composing arrows, like everyone says.

22:30 It's kind of a version of Walsh's theorem, isn't it? I mean, M to the D is kind of like that. Yeah, maybe, in some way. Actually, there's even, yeah, I don't know how to put it exactly, but yes. The intuitionistic logic sort of spreads the thing out, so you don't have to actually divide by that ultra-component. It's already a Boolean value model without doing that, yes. Right, yeah, you can look at the Walsh construction that way instead. That's right, you can look at the Walsh construction. So in other words, it is an objective content, but the imposition in particular, the idea that the... Well, it looks like it would be a special case in this act. Have you ever seen the introduction to Robinson's second edition has got a series of remarks by Goethe Which are puzzling in the extreme, but I mean what first of all he says non-centered analysis in some form is going to be the analysis and then he says in the future people will be amazed that even though Mathematicians extended the numbers so that you had rational and irrational numbers. The logical next step of adding infinitesimals took another 150 or 200 years. Then he says, I think this has something to do with the fact that very simple propositions,

25:00 He has simply stated propositions in elementary numbers. He mentions thermostat. Of course, that's been solved by Goldbach, Twin Prime, these things. So little progress has been made in these things is due to this oversight. I remember reading that 20 years ago and thinking, what the hell is the man talking about? And finally, infinitesimals, taking infinitesimals seriously in an analysis problem, which, in a way, Robinson doesn't really do, because he carefully arranges numbers so that if you're nervous about infinitesimals, you don't like infinitesimals, okay, I can prove that, you could have proved all this stuff without using it. But the interesting thing is, how do you really make use of it? That's what Goebbels seems to be saying. He thought it was his duty to pose as a philosopher. I'll tell you a funny story about that. I started studying logic under a guy called Bill Boone who worked on the word problem groups. And one of those few people that Goebbels sort of took, because he sort of had his favorites, you know, people that he kind of cultivated, he told Bill that he didn't see how anyone could possibly, unless he were grounded in the ancient language, Latin and Greek, in central Illinois, the opportunities for getting Latin and Greek. I mean, I did Latin. Greek was non-existent.

27:30 So Bill moved his kids to a Jesuit school as a result of his advice from a group because they taught Greek and Latin. I sent my younger brother to study philosophy and science because at a certain moment I thought that was the most important thing as a result of his undergraduate career. Your results were wasted. It's a similar thing. I didn't get the advice too seriously. Yeah, maybe he doesn't. Thank you very much for your time. I don't know, I conceived this as something rather important, so I was hoping that, as I said, that Atiyah would volunteer to give a three-hour talk this afternoon. I mean, in other words, it's like a Quaker meeting, right? The discoveries, we say, are understood by anybody, and what questions do you have, and how does one field relate to another? Reef interchanges are long presentations, but it will give us a presentation about everything he knows, across our bodies and so forth, but we can perhaps, after some informal discussion like this, we can, each of us can come up with a specific proposal for what he or she, unfortunately we don't have a she yet.

30:00 We will have it. We will have it. Matthew, so what was your question? I don't think it will take three hours, but it will take a few minutes to study it properly. Good. Good. Perhaps we can do it now. Yeah, yeah. But it needs the definition of Eulerian, sorry, Eulerian real numbers. Well, I'd like to see what you've got to say on that. But you can give a brief expose of that, surely, the definition, before you start? I could, but it'd probably be better if you did it. Well, let's see what you have to say. Yeah, that would be helpful. The question is rather concrete, so... The point about the Eulerian reels is that it relates to this point about the orderings never being strict on the line, except when you've got the strictification that you have in the Dedekind case. Am I right, or that this point about the, I always have an invertible element, sorry. I came up, sorry, maybe I should explain this to you, sorry for that. No, no, those are the ones that work now, Bill, the ones that he's just put on the table, use those ones. They're fresh now. No, it just needs the ink to start flowing.

32:30 If you invert one element at a time, you have to change, if you think of the things as functions, you have to change the domain of definition and pass to the part of the domain of definition where a given function doesn't vanish in some strong sense in order to be able to find there the inverse, so that to think of ratios simply as a bar b, where a, b are two given things, is very, is, I think, so independently. In my story, I think that if you have a rational theory of ratios, you have to admit that, well, to specify domains, you're going to have a certain domain of things and another domain of things, so that you can talk about the principle of one being multiple. But this is the sort of thing that ratios are, as you want to say. To say that B is a multiple of A means there exists a math, R times A is B. And the space of infinitesimals is another ingredient of some kind of cohesion that's a basis for dynamics that we have.

35:00 So it's combining these three ingredients that the idea of the fronted space, D to D, these are the ratios of infinitesimals. And then, well, then more exactly we realize that precisely because of variation... Although the points of this might be the right sort of thing, the space itself is a bit bigger, and so you cut it down precisely by evaluating at zero. Oh yeah, I had that based on the fact that Latin languages have two kinds of points anyway, masculine and feminine. Something unknown in English is a big discovery for me. So La Punta is a D. And il punto n is that point. Clear? Is that clear? That's this idea. Again, if you like, you can find it. Leibnizian monad, perhaps. It's expressed by... So anyway, there's this one point, zero. So we can evaluate that. And then we can take the inverse image of it and call that r. So this is the definition of whether it's real. It's this object r. It's a part of g to the t. Which evaluates from zero to eight to zero, rather than. So if you liked slopes, ratios are given away. And this is automatically a monoid, so that the multiplication that comes at you first intrinsically by the construction is a category of the object.

37:30 Sometimes that R also has addition and exponentiation. The homogenous map is automatically linear. Again, it's common. I think it's what Walter Knoll often says, that we have something homogeneous, which of course, by Euler again, means multiplication by lambda does something. But that actually applies as linear as well. So from the point of view of abstract, when you're out there, you say, oh, linear math has to do with addition and scalar multiplication, which are two things that are clearly required, but in a cohesive background, if this is taking place in a cohesive background, very often the addition is there, sort of as an afterthought, and it's automatically preserved by anything that preserves multiplication by lambda. And this restriction from d to the d to r, that's clear in the picture, right, because we want the thing to be centered right at zero, and that would be better if it's slightly off zero, in order to have a commutative moment. Now, a remark that I noticed about this is that an object of the form d to the d and a y to the y is never a commutative. It's always a moment, right? But it's never a commutative. Never, never, never. Except if y equals 1. Because in any case, it's a fact about lambda calculus if you insist on it. No object can be an application of never commuted. Basically because if you had two different points, there would be the constant maps, and to say that two constant maps commute is to say that they're equal. So all constants are equal and you can internalize that argument. So another very delicate point is, and this would be the point where Alan Cullen and company could rush in and try and spread the whole thing, right, that R should be commutative.

40:00 As far as we know, R is commutative. Multiplication is commutative. And yet, it's constructed out of this thing which could not possibly be commutative. A more advanced interpretation of Heraclitus will have that multiplication of reals itself is non-communicative, but certainly nothing, including the mathematics upon which modern quantum theory is based, all of that always assumes that reals are non-communicative. So this is part of what has to be stated and developed. John Bell's 60th birthday. Is he 70 yet, by the way? No, I'm not even 70. You're not, okay. But anyway, for his 60th birthday, there was supposed to be a book, and I guess it's still in the works. So I have something about this there, because he is so interested in different models of the continuum. This seems to me a very attractive one. I discussed there something about the addition, what is really involved with the fact that addition is just, it has to do with the real, I mean, we have the idea that the algebra of a monad is one of the richer things in itself, things like that, but I don't remember exactly. There was a version based on integral calculus and another one based on differential calculus to justify and explain. Thank you for your time, and I look forward to seeing you again next time.

42:30 Whereas the usual description in the existing books defines D as a part of the line. So the really, extremely synthetic idea is that the algebra flows out of the geometry. Part of the geometry is that we can form function spaces. Having done that, we'll automatically create algebraic structure, even if the spaces we started with have little or no algebra. Even in the extreme, you could even say that this point is a property of D, rather than the structure, because there is only one. By the way, there's a paper by Max Kelley and Steve Lack about the property-like structure, so this dialectic between structure and property, which I've been emphasizing. They actually make one very nice formal attempt to explain this. I think they get 12 cases, which is kind of funny, because also Kant had 12 cases. This is a two-categorical way of organizing this dialectical property, which I'm not using here, but you should try to understand their work better. Yeah, so I found a way to express this commutativity. Another important property is that there's an attraction that doesn't quite go through the origin.

45:00 It does go through the origin, except in slope, by the chain rule. In fact, this is such a trivial case that that disappears, so express that. Again, this is not an additional structure, it's a property, which forces any more likely to be commuted to universal algebra.