Sketches; A philosophical introduction to categorical logic

0:00 It is not necessary that. So of course, to prove something necessary, denying necessity, well then, well, you're in there with a chance. The ticket to answer or the ticket to leave? The ticket to leave. I find that many people are going to him now. At least since 10 years. Especially from the philosophical side, and he's not analytical in style, but he's analytical in spirit. Yes and no, in the sense that, of course, someone who spoke about philosophy as a river of time would like that philosophy to become a river of time. We have these ideas that I need to present to you. Who is that? From IT. Yeah, yeah. Something like that. Yeah. But, you know, later we developed these ideas in terms of, for example, starting from, not pre-pre-starting, but enticing the inner shift. But not for the reason that we want mathematics to be... Depends on the fact that human mind is created in an unconstructed way, this kind of machine.

2:30 But, you know, it's related to the basic structure of the universe. Because you create a kind of chronological game of dialectical characters. That's my, that's my point. Only on this point you can argue and you won't. Trying to find what works for you. What could be the best theory about this framework? You can find the right answers, but the right answers cannot be found if you separate the structure of math from mathematics. And you cannot find if you try to reduce mathematics. I like to think of the mind, the mind planet, like a tool, like a hammer, or like a light bulb. I mean, there was nothing in the world like lightbulbs or hammers before human beings came along, and yet, the hammer is nothing but a reflection of the laws of physics and the means of humanity which is part of nature itself, same way with the lightbulb. So why can't we just accept that, of course, the number nine is not a human. It serves us as a tool in the world because there is a nineness to what we know about the time. Well, which was there before anybody looked at it? Which was there before anybody looked at it? You know, the 99-1, 99-something. Hmm. Which is why I wanted to ask you about the smooth numbers, the so-called smooth numbers in SDG.

5:00 And do you think of those as getting a little closer to the 9-ness or whatever of whatever is there before? Before being is reflected in thinking, um, number, counting numbers, or, it's a further tool, but in some ways it's a superior tool for some, in some respects, or, yeah, that's, that's, yeah, that's exactly my, my, my question. No, no, no, that would be. That would be objective, no? No, but can you explain to me why you think superior to the concept of natural numbers as captured by the Peano axioms or the other concepts of structural progression that we've had for the last century or... I think it's very interesting because it... You know, it helps me to understand what it is about the topos of smooth spaces that is particularly important from the point of view of your philosophical position. I think that people like Colin have a very sort of philosophically naive conception of how the foundations of mathematics are. ...to be conceived. He seems to think that once you've said that concepts of mathematics are protean and the boundaries of its subject matters are permeable, these deep and beautiful interconnections which mathematicians are able to see, but you can have different shifts of perspective, but nothing under... That is just something, that's the show, that's a human creation, but you can't go below the level. In other words, I don't think he thinks in terms of all this as the reflection in thinking of being.

7:30 It's just something sui generis. It's just a voice in the conversation of mankind, and ultimately, I'm horribly afraid the position he takes is going to be driven in the end to say it's just sociology. That's a necessary condition if you're going to talk about mathematics, because the YouTube is to see what mathematicians actually do. What he's criticizing is spinning theories out of thin air. I think that's an absolutely important thing to say, and I think for the audience that he's addressing, there is that. It's, you know, it's especially important. He used to say, well, having said that, you, it's okay, on an Aristotle point, but you have to go against the experts, the first thing you do. And it's a sort of common sense point, isn't it? The first thing you do is to see what the actual, you know, what the consensus human view on these things are. In that case, I may have done Colin an injustice, and I apologize for doing that. But that doesn't mean that what the experts say is ipso facto what the case is. No, no. I mean, he sounded to me like he was flirting with that as a kind of paradox. Well, yes, and also when we were arguing in Cleveland, he was saying that he couldn't see that there could ever be a foundation, or that any point of departure in mathematics is better than any other. But that was really a side issue. I want to ask Bill about why he thought that the smooth numbers represented a better, or in certain aspects, better for grasping certain aspects.

10:00 There's a different kind of conception of structural progression of features of the use of numbers than the normal. What is that? What is that term? Leave the nuts. It's just a fragment. We can put something in contrast with the Brouwer and completely free speculation. We can go to the next number and make it just a perfectly, absolutely free thing of pure thought, not denying that there's any constraint that the real world puts. The fact that we can, in fact. Yeah, right, yeah, that's right. Somehow, there are always going to be some physical limitations, so if we want to have an accurate idealization, the idealization has to permit taking that somehow to some extent into account. If it's completely, well ultimately it's a poor idealization. You say, well, I can get natural numbers by forming stroke units, but, of course, you can always imagine yourself standing at the blackboard with a long, but if you look at the problem globally, which is, after all, the problem, I mean, the problem of natural numbers is that in the history of about 17, 2009, 1943, it's a problem about the tail end of your natural numbers. Then, you know, the stroke calculus business is completely unphysical.

12:30 Because we don't know what, we haven't, I mean, we don't know what limitations they're actually on, on physical operations. The point is, the point is, you see, that these guys say, well, never mind, it's just that if you got the idea that if you wanted to, you could buy those, not that, but the trouble is, the problem in the natural number sequence is with the infinite tail. So no matter how far out I've gotten the natural number sequence, the mystery remains, as it were, in the future. Okay, is this the kind... I mean, you're saying that alternative versions of natural numbers will maybe give you a handle on this problem. Is that the sort of thing you're saying? Yes, yes. Even if there are alternatives, there should be more and more accurate ones. Less idealized. In the book I came up with this, there are literally only ten of them. ...idealization of what's going on there, entire pure gravity, even approximately correct, provided it really is the main thing, all the while realizing that all of a sudden other situations don't even end anymore. In a somewhat similar way, there is this process of counting, the urge to add one more. We're talking about the motion of thought, actually, instead of the motion of...

15:00 That's about the motion of thought. This is what we want to describe as logic after all. The laws of the motion of thought. So there's one fourth. Yes, you can have a theory where that's the only fourth, but it seems to be, you know, really quite an extravagant theory. What about being able to make them more and more accurate ones? They're taking you into account. Such as? Such as the lack of money to buy more chalk, did you see? Yeah, okay. I mean, of course. Of course, people say, well, that's crude, you see. It is all crude, but I mean, there should be a range of possible theories between that utter crudity, but real, problem and the... It is a crude, but it is a practical problem. Yeah, and the completely idealistic, there is nothing but the universe to go one step further. That's all there is to the motion of thought. Always remembering that the thing is to actually account for the motion of thought. I mean, obviously, you're getting outside, in any of these, you're getting outside looking at this thing as it were from the outside to try and determine what the consequences are. I mean, I've got that impression before that people are, I mean, there's a, the way mathematics, God, you know, I haven't got anything to say. I've been trying to get a couple of colleagues to join publications here on stuff working out on utility and aerosols and numbers.

17:30 You know, every time I make a suggestion, it's taken over. These are colleagues in the philosophy department or in the math department? Yes, that's exactly what I would have guessed. Okay, you know, I mean, you know, I found some passages, there's a, there's a, in Eustace's definition of Eustace, there's an odd phrase that it seems to me is always trampling. I think you're, literally it means each of the things, each of the things that is puzzling in the context of Eustace. That's how some Eustaceans, the scholars of mathematics, refer to a domain of discussion, a domain of discourse. So I wanted to say, let's translate it that way, because then we can make sense of these definitions. Oh, well, you know, we can't be sure. I mean, after all, you know, and I said, I said, well, look, suppose, suppose somebody in 5,000 years time comes back and reads a modern logician and sees the term universe, and sees the term universe, right? There's going to be a scholastic argument over whether they meant the entire cosmos. Well, in fact, you're using it as a technical term to describe some limited domain, but no, no, you know, you've got to be absolutely certain about it. Well, that seems to me a recipe for not ever saying anything interesting. But I guess bottom line is, I don't care, I mean, bottom line, I don't think maybe we can know exactly what they are. And, you know, my reading of it makes them say something sensible. The other reading of it makes them say something unintelligible. Maybe they were saying something unintelligible. But, you know, I mean, why restrict yourself?

20:00 You know, why hold that open as a possibility? Why not, you know, come down on one side and say something else? Well, I feel a certain dissonance about publishing under my name a careful analysis of passages in Greek, when, you know, my Greek is very, very rudimentary, mostly, you know, and I've had to have you guys help me. But, I mean, I think that's the corruption of the institution. ...that makes it impossible to enter into a master of those. I mean, I think, you know, there are a few people in the industry, and I'm with you on that. Well, why can't there be several styles of translation? I mean, the style that you have, you see. As long as it's fairly marked as such, I don't even understand. Before, we were saying, well, this is the attitude I take toward translation. That's my attitude, you know. When I've ever actually published a translation process, when I read Drachmann, for example... In other words, you know basically where these people are headed, but anything Goebbels said, even if it sounds innocent, it's obviously got some legal intent, right? They need you to do the same thing as Rastner. He says something that other people, like you say, you know, what's that, right? That's what other people do. But I see, ah, it really needs to talk about a natural transformation on the identity end of a function and so forth. If I can find that interpretation, that guides me to reading everything else he says in such a way that I see it opening up, you know, incredible richness of content.

22:30 Yeah, I think that's a reasonable way to... Well, and you have to admit that you may be wrong, but so what? So what? At least it's interesting. I think it's true, it's correct. In other words, modern language is a more concise way of expressing the profound insights that a great man has. And if you use it, that will help you to understand his insights. Better than anybody else ever understood them. Why not, right? Yeah. Absolutely, absolutely, they're wonderful. No, but the kind, yes, sure. But there's a place for strong, there's a place for powerful speculation, particularly as Bill says, when it leads you to see the tremendous wealth of interconnection and depth of Grassman's ideas, which is pretty. And you interact with them. Yes, just as with Euclid and Aristotle. Well, it's a fine line to walk because you've got, if you want to, if you want to shake up your own ideas, you can't go into a, on a great author of the past knowing in advance what he says. That's right. But then, granted all that, granted all that, you've got to somehow, I mean, you know, you want to try to get into his way of looking at it in order to get out of your own. Yes, absolutely. I was just thinking earlier this evening in the restaurant when Bill was speaking about our understanding of is, how that way of thinking of set abstraction in the context of discrete vibration helps us to understand what is a in natural language.

25:00 It means immediately that it came to mind what you were saying about Aristotle the other day, about how Aristotle grappled with the problem of kinds and is a. And what he was trying to express by saying that being is not a genus, that being cannot be spoken as a genus, but only when he says about the unit. That looks like a throwaway line. It sounds like a mundane one. He would have seen something that he missed. For Prager, there's only one way. That's what I was struggling to express the other day when I said where did this where did this false generality in the notion of variable, that kind of false generality in the notion of object that is built into secular ethic Platonism come from. It came from a distortion. It gives philosophy, this field of philosophy, the most important role that the education in doing philosophy has been so strong during the last century, long century, that this action of the changing and the violence has been something lost. You said that we have a philosophical culture which is essentially historical, but this history has become such that it has been cut out of the present reality, whereas I think that in your case you put the accent on the need of coming back to... This is the historical origin of philosophical coding, since in general, this is my internal view of things, philosophy is here presented in a simpler way, with a list of subjects, topics, to be confronted in a simpler way.

27:30 Whereas, if you put the accent on coming back to this law, let's give a sense what, if you don't give this sense, it's just something trivial or something that nobody would ever notice. So, the needs of different cultures, of different traditions. At the same time, it could be just opposite one to the other one, but it depends on the genre you wish you mean. Well, I mean, I think you've got it. I think the whole... That's what we used to do. Thanks very much.