Conversations after FW Lawvere lecture / discussions at JL Bell's house (contd.)

0:00 Specific examples of the smallest levels, but it does show, it does show that in general, these are the ones that are at the highest level. You're talking about that there is a first order formal theory. I'm giving you a formal result here. Right. The formal result is that we can take a definition which has as a fragment the elements of theory, and which includes all the standard theorems. So now, if D.F. makes sense, I claim if theory makes sense, it makes sense. Now, you're going to deny that. But I claim, yes, if you know theory, you're going to make sense.

2:30 They don't say that the foundation for mathematics is this formal theory, which it is not. No, I want to, if I'm going to give a formal statement of a result, I'm going to have to give a formal answer for it, right? But if it's a real result that applies in this case, you ought to be able to paraphrase it in real language that non-logicians would understand. Except that I just said it a minute ago. I said you believe all the claims I make, and you believe that they apply the theorems that I assume. No, what I would be disposed to utter the same sounds you do. But in doing so, I would not be making the same claims you make, because otherwise, you wouldn't be arguing. Otherwise, you'd be saying, okay, so that's the element of your category assessment. Do you know that what he's saying is meaningful? No, I'm just... You can't tell, because you don't know whether he means it the way you would or not. You can't tell him what he writes, whether he means it the way you would or not. You do have to agree that the things he said would be true if he meant them the way you do, and you agree that they imply zeroes. You can't tell whether he means them the way you would or not. You'd have to go talk to him then and ask him. Well, that's a... I mean... There's something wrong with the claim that's unintelligible. You know he's a member of the Baha'i Faith. Yeah. So he believes some one of those strange things. No, because there's more to these things than just uttering certain propositions. You're going to embed it in a story about what's going on. You know, you're saying that all I have to be able to do is say the same things you do, and we agree. Well, I just don't believe that. I can embed it into what Russell believes, and you better agree. And all three of you will do accept the embedded versions of what you believe.

5:00 I agree to a paraphrase. Yeah, I guess I have to be careful about Russell. But certainly, yeah, you, Axel, Betelgeuse and Heise, well, not Betelgeuse and Heise, but von Neumann's functional set theory. Not quite new foundations either. Well, that all just goes to show if we would all agree, then we would all be understanding you in different ways. Separate consciousnesses. I mean, we would be understanding you in different propositions. Yeah. But I mean, I actually don't even believe that. I don't think it has anything to do with formal systems. Precisely what I suspect you're thinking, it doesn't. When I say there's a one element set, I'm not asserting backwards E, X. I'm saying there's a set that's got one element. What is that element? It's got one element. I didn't say it. Oh, well, I mean, if you don't want to say it, that's okay. But if you think that it doesn't have any properties, it's something like a point, only it doesn't even exist anywhere. If you think it's a sort of bare one, I mean, I just think that's stupid. I think it's the identity function on the set, which in fact I haven't said it, but I do think that. If one member is the identifier, well, then you've clearly got a different number of sets of names. Yeah, but the claims that I've made, you agree with. Well, I mean, that kind of situation can come up in a lot of cases. All that follows from that is that I can't believe that your theory is inconsistent. I never said it was. But I just, what I do say is that as a beginning point, we're thinking, as it were, ab initio, about mathematics as an intelligible. I can understand it as an axiomatic theory. I can't understand it as something that were given directly to the mind. Yeah, the only way we disagree there is you want a beginning point to be something that I teach and at the end of it was the bottom of the whole product. I want the beginning point to be something that gave me some ideas and gave me some ideas and gave me some ideas and they got into the point. I don't care that the beginning points remain in abstract mathematics as long as... But I want some kind of story about what you're doing when you do this. And how you justify it. What your definitions mean. What your first principles are.

7:30 Well, we agree that there are one element sets. In fact, there are one element concrete sets that we said is only a moment. We agree the concrete sets have products. Thank you for your attention. This is what the ordinary people do. They know the order of parrots. The order of parrots is like the order of parrots with their names. That's what an order of parrots is. We know that. But, as long as we stay with concrete tests, we're going to have trouble believing in an infinite number. I don't really think there's an infinite number of tests. But I also know that to know the type of size of a particle, it's necessary. Well, it doesn't matter if you play the horn in my concert, that doesn't mean I can get somebody to speak these features. It doesn't have any properties at all. No, and it certainly doesn't follow any set. But this leads me to a new idea, the idea of an abstract set. An abstract set. This is creation by re-description. Well, by abstract. I mean, you can, you can taunt at an encounter, but I don't want to be in that conversation. Well, what do you do? You take a set of chairs and then you somehow bleed away all the problems in the chair. No, no, I don't take the set of chairs and bleed away anything. I say, I'm not going to talk about these sets of chairs now. I'm going to talk about abstract stuff. What is that? Well, I can explain it. They have a definite, an abstract set has a certain number of elements. But what are? Well, they're just elements.

10:00 Okay, and now... Are you getting the elements from the one element? Yeah. Okay. Don't say anything. Well, I mean, I could, yeah, yeah. Yeah. Yeah, I mean, really what I'm going to say is that sets, they have functions between them. They're just these sets. They have functions between them. Function now no longer means what it did for concrete sets. Function is now a primitive term. What does it mean to be a primitive term? That means I've just got to swallow it and I don't have to swallow it again? No. And then we're going to go on from there. Well, I can tell a story about it using axes. But I have to invent that story. There's something in between. Using it completely undefined and unintelligible and having it primitive. What does it mean to say these things? No, you can explain these things with analogies and big talk and stuff like that. And this is supposed to be at the bottom of... and I'm supposed to prove things about them? Yeah, no, no, you use the axiom, you use the axiom. You'll be finding cool things in my finds, but you didn't say what my finds are. That's right, that's right, and you can draw little dots on a paper and you can begin to build up a concept of dots. And it's not just an empty word, there's nothing between an empty word and having it defined. Well, you... Frege, Frege used to talk about the difference between defining and explicating. Please. I'm keen on having primitive terms that they can be extricated. People outside the system can chit-chat about it. I can make a certain amount of sense about the tradition of geometrical string law. You know that the size of the disjoint union of concrete steps is dependent on what their elements are. So you know something about already concrete steps without worrying what their elements are. So now let's just talk about abstract steps. We can only explicate it. I know what lions and tigers and bears are. Now let's talk about blizzards. What are blizzards? Well, can you tell me something about them? No, they're just, they're rather like lions and tigers and bears, only they don't have stripes or teeth, or claws, but they're animals. I can tell you how abstract sets are like concrete sets. They have functions between them. The functions compose and include identity functions. But how can they function between things that don't have elements? This makes sense. Well, of course, I mean, okay, I mean, now you've got me. In the standard way. I can always ask how. You had kids, right? You can always ask how. How can you ask me how? The point is, how can you just...

12:30 And now this gives you a type of structure that will interpret them. And now what I'm saying is, it doesn't matter what structure you take to interpret these things. So the question about what species they are. How do you interpret them? I've got the axiom. I assign certain relations to them. Here's a binary relation. Here's a binary relation to the structure that needs to be interpreted. There actually is a lot of informal text here that gets taken into account. But here I say, look, there are functions between saps, between exact saps. They compose compositions associated with them, like my daddy. I can tell you what this means in terms of how to work with it. I can give you the motivation in terms of concrete saps. You know what functions between concrete saps were. And we notice that they did contribute to the composition of other saps here. Okay, we also knew that for lots of purposes it didn't matter what their elements were, so now we're going to say the rest of the story. Well, I suggest that you write these actions down the way, or I could produce a model and concrete that. Except that one of the actions is going to imply that one of these has more than one element. Well, then I won't do it. I can make sense of what it is and how that would have to be ordered. Read them on the surface so they don't have them. And then notice there are infinitely many different interpretations. I haven't got one interpretation. I'm not talking about any particular object, but I'm talking about one. Yeah, yeah.

15:00 But now when you say there are infinitely many interpretations, now I really am at a loss to know what an interpretation is since what we're talking about now is my actual understanding of set theory. It's not a form of theory. It's precisely because you haven't. It's precisely because you haven't got the clear notion of sex in this, that the idea of sex is not clear, that it can't be. Well, no, no, it's clear enough. I can interpret any first-order theory in this. Any consistent first-order theory has models in it. I can prove it in fitness. What I don't have is another universe of my real sex somewhere that I could interpret this one in. This is not a formal first-order theorem, right? It's the concept. It's just the concept that you hear, that they're sets of groups. You can talk about groups, and you don't care what the elements of the groups are. But that doesn't mean that there's a group with no... These elements have no individual peculiarities any more than to say that the personalities of the girls in the chorus don't matter, not to say that they don't have personalities. So I misspoke myself a little bit. I don't know that the elements of these steps have no individuating properties. I don't know any individuating properties. I'm not going to posit that nobody's ever delivered any of them. I don't know that they don't have them, though. It may be, somebody could discover, that one of the two element sets here is radioactive. I have no idea what that would mean. You're talking about some unspecified model of your actions rather than thinking of your actions as describing reality directly. There's nobody getting up to it by saying, well, they might have properties, but I'm just not sure what they are, because then I agree with it. I mean, then I agree that it's a perfectly intelligible notion. It's an intelligible notion now without talking about those, because I don't have to talk about any of the great properties at all. I don't have to even talk about them.

17:30 Now, I haven't really used names, age, anything. I can tell a perfectly intelligible story that would apply to lots of parts of the situation, but I can't add to the story. Oh, by the way, this guy didn't have a name, he didn't have a face, he had no individuality, personality, so he's no longer a man. And the story is no longer... And that was a big difference between then and after that. We know there are other differences between then and after that. You should be perfectly happy with the following. Yes, each of them has three different types of particles. That doesn't matter. So, I mean, they do have individual particles. Yeah, but not individually. Well, that's the other thing, yeah. But there aren't any two of your sets which have exactly the same numbers. In this set theory, no two sets have a common number at all. And the numbers have no features, except for set A and B. So a three element set, those three elements have no features except that they all belong to this. So why torment yourself to imagine such a thing? Because you can make a perfectly straightforward account of it by using conventional ideas. Well, yeah, because... Just like Hilbert says, why torment yourself to wonder whether a point is a tiny dot? Because I can give an asthmatic therapy. And there are two answers. I don't torment myself doing this. I quite enjoy doing this. I torment myself arguing with people who don't do it yet. And the second part of the answer is, sure, I could tell a story that would satisfy you. In fact, I could tell any of several stories that would satisfy you, and the choice between them would make absolute zero difference in their relation to my story or to working mathematics. So why are you going to tell me this deterministic story about a cumulative hierarchy of sets instead of an anti-well-founded hierarchy of sets? I don't need to worry whether it's cumulative and well-founded or anti-well-founded. It doesn't matter. It doesn't come into mathematics and I don't need it in my story. So I just leave it out.

20:00 All you need by the isomorphism is that every set of axles is the same size as a well-founded set of axles. And they say a few times, for example, we're going to represent propositions like this. And it doesn't really matter how set it is, but we will assume that in any set here I've represented propositions. The chance of exposure includes a set of theoretical representatives. No, no, I think that's true. That's really perfectly ad hoc. But once you've made that assumption, if you want a set of propositions to refer to itself, it can't be represented by a well-timed set. The only module of that assumption that's pulled out has been there, really, to get this role. I'm going to use this because I'm authentic. Meaning in what sense? There's no other motive in the wire for using non-wire in that sense. What about just the self-referential proposition? That's what I mean, but it's only once you've decided that the representative of the proposition should include representatives of the things it refers to that self-reference... But it's only when you decide that indexing should be some kind of iterative membership that self-indexing becomes a problem in CF. Nobody really thinks indexing should be a problem in CF. There are plenty of non-well-founded relations. Everybody knows this. It's only when you anticipate a model by some iterative membership that you can't ever miss it. Pauline, do you know how to get back to all of them? No, I don't actually. Because John and Matthias have left. Do you know them? I don't know. I'm interested to get that Robert knows all of them. And you guys can come back when you want to, but I mean I don't want to push you, I mean if you have to stay, but if you want to push on.

22:30 Do you want to do this? No, I think it's probably best if I come back with you. I think things are winding down. You know that curve he wrote down there? Well, I think it's a good conversation. All right. Curve. That was the worst thing ever. Thank you for your attention. Do you have a good back button? No, I don't. I'm on the back body. Oh, well. I was on the back body earlier. I know. I know. I know. I know. I know. I know. I know. I know. I know. I know. I know. I know. I know. I know. I know. I know. I know. I know. I know. I know. I know. Thank you for watching. You have to come back. I think we'll do it. Assuming everybody gets off tomorrow, which I think they will, then you come back here. Stay the night downstairs, the three of you. That would be great. Thanks very much indeed. Just a bit of music in the beds, that's all. Yep. And then Bill wants us to come down. Thank you for your attention.

25:00 Thank you for your attention. No, it is a job, I think it's not. I left the box somewhere, but I want to just go on with it. Why don't you just grab a pencil? Yeah, I will. I mean, where I'm sleeping, I would give you one of my handkerchiefs. No, that's okay, I've got plenty of them. Well, we pretty well adopted that, don't we? We did indeed. The cover is from a three-piece regular paper. It's quite nice. Which book? Testimonial of what? Of conceptual mathematics? Oh, yes. Certainly. No, it's a marvellous book for learning. Yes, it's a very, very good book. Well, it does popularise, but it also does lead in... No, even in that role, we were talking about a good way and a bad way. Oh, well, this is a very good way. Yes, of course. Don't talk too hard, or you'll never look back. Oh, I'd better. You'll see tomorrow. Thanks for a very great day. Very, very good. I know I've bettered. I'm sure I've bettered. I'd probably better go home. I must go and find my case, though. Well, you're going as well. Sorry, Bill. Thank you for your attention.

27:30 Right, Jesus, and my attache case, wait a minute, who did, well no, it's got some stuff in it, it's got my like change of underwear and stuff, I really do need to, I had it in the car, no here it is, it's alright, it's right here. I'm very sympathetic with those who like to keep math close to the throat. There was something about the form of the piece by what you could develop. Well, you know, but I guess, you know, Bill would say, you know, he disagreed, that in fact we get the really deep ideas are the ones which... Do you have an ultimate basis although obviously ideas about various physical structures. I mean, I certainly don't think he thinks of the elementary theory of category sets, the categorical set theory, which himself, of course, has developed in the way that Colin does, in a strongly postulational way. No, I don't think he would buy that at all. He'd say this comes out of a constant set... When he says that they're the limiting case of what he calls cohesive active effects, he really does think of them as coming out of geometry, and out of the geometry of the real world, in a way which turns out to lead to, I mean, even if that turns out to be batty, it has led to something of mathematics.

30:00 What joke was that? You had about 4D. Oh, no. I had time to climatize. No, I'm just suffering from the... With the bronchitis. No, bronchitis and the antibiotics. Now, they are very fast. I think they do knock you. They do keep you feeling very... Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Well, I still claimed the booze when we got to John's. I did have a beer in the ground pad, but I haven't had anything apart from that. Did you like anything? Like an Aspen? No, I think I'll be OK. It might knock my temperature down a bit, but I did have a couple of Aspen at home. It didn't do any harm. Oh, this is your backyard.