Discussions (contd.)

0:00 The origin of the notion said isn't. But listen, we can make, say, our notation. People had the idea of set, but they called them numbers. They had that idea before anybody even started writing things down. Sure, sure. And they had that idea because, and probably even, you know, I was going to say the higher primates and possibly even much of the animal kingdom grasped that. We can't imagine any way of apprehending the world, or maybe if we were a sort of nebula of gas, you know, in the atmosphere of Alpha Centauri, it would be different. We really can't... we can't... We can't imagine how we could apprehend the world at all unless it came to us articulated in a way which already embodies, I would say, topological and geometrical structure. Unless there were already closed and bounded components of half-disconnected spaces, which is why I'm able to pick out something as the member of a set. To begin with, maybe that's all true, but it seems to be like a lot of philosophical speculation. I don't think it's philosophical speculation at all. Weighing down a really simple idea. But it's not a really simple idea to a... Of course it is. It's the number of people... Only to creatures that have already... Where all of that transposive spatial structuration has already been lifted up to the level of syntax and language. If we were all sea slugs, we wouldn't be sitting here arguing about it. No, but sea slugs can still distinguish between sectors of their environment. But even sea slugs, I put it to you, even amoeba, can distinguish between different sea slugs. Sea slugs. It's like a primitive marine creature. Primitive marine... creatures that don't have backbones. I forgot what the technical term in biology is, so... Invertebrates, I'm sorry. Even primitive invertebrates. I'm not making... it's not a highfalutin metaphysical point, it's an absolutely basic point about...

2:30 It's not. It's a thesis about... No, this is the point you always miss out. It's a thesis about the way that the epistemologically primitive ingredients of any concept, including our mathematical concepts, have to be reconciled with what are the semantically primitive ingredients, which is the thing which you always focus inclusively on, and what are the, I would say, you know, the ontologically primitive ingredients. I know this doesn't... I just can't... Get it across, but I fail every time. But I still, my point, the notion of being a member of a collection is not an absolutely primitive notion. It seems to me epistemologically. It's one which, I don't know, all I can say is it's clearer to me. Yeah, it's clearer to you because you're a rational. No, but it's clear to me that the idea of ordered pair comes from our conventions for writing things down. Yeah, whereas the notion of being a member of a collection extension doesn't. But then you can make a case that the way we are writing them down just reflects our kind of deepest mental constitution. It might, but it seems to me that once you say something as its origin in syntax, then you're sort of fasting down on it. May be, but I still say that you think that the notion of being an element of a collection and extension is something absolutely fundamental that you can't you know you can't dig any deeper than that okay that's fair enough and I say that you can but we wouldn't even have this notion we would be able to articulate this notion. Okay, I still make the point that there really has to be sufficient topological and geometrical structure in the world. For us to be able to apprehend any notion like that, if they weren't... Sure, that's true. I mean... So, therefore, in what sense are topological and geometrical concepts not more primitive than these logical concepts, than the concept of being an element of a collection and extension? As I say, I say that unless we could apprehend... Closed and bounded components of a half-disconnected space. We wouldn't have the notion of being an element of a collection and extension. Because I've kind of got this simple-minded idea that I'd like to be able to understand arithmetic without having to go through...

5:00 Through geometry and topology, I agree. It's a very, it's not, well, you keep, with great respect, cut the rhetoric about it is really complicated. But in this case, John, exactly, you can make a point that non-order pair like 1, 2 is more fundamental than non-order n, which is kind of more geometrical. Yeah, but even with two fingers, which, by the way, is a vulgar gesture in English. Yeah, provided you keep it that way around. You have to make sure you keep it... It's okay doing it that way. Yeah, but you are aware of that. It's very important, actually. Seriously, otherwise... No, no, that's very rude. That's extremely rude. That means fuck off. That's very rude, and don't do it, because if you do it in public, you might get somebody coming and punching you. Yeah, but I agree. If I look at those two fingers, I see the index finger first, or maybe the other one. And if I look at something, if I look at a group of people, I sort of see... I agree. I don't think it has anything to do with anything. It's just the way we happen to see things. Arithmetically you just have this ordinal number first, right, and geometrically, of course, you kind of symmetrize the situation as you can. Cardinality is more primitive, I would say, geometrically, it makes sense. Cardinality... I mean, just the idea of a bunch of things together. Yeah, but the very... I keep coming back to it, the very notion of a bunch of things already assumes a great deal of topological and geometrical... I think I think well I hold to the view that all all our concepts are ultimately the transpose of spatial structuration so I think that abstract notions like numbers in fact ultimately emerge from a bunch of topological geometry. You couldn't have the notion of being a novel by Tolstoy unless you... Sorry, what? Oh, okay. Yeah, so you're going to say... You can't confuse them. And if you confuse, you think kind of geometrically, which is not very relevant. No, I agree. The more you know about it, I mean, I don't even know... Only an educated, literate person can have... You tell me, you tell me that we mustn't make our primitive concepts dependent on what you've described as high-faluting notions. And then you say to me that a notion which can only be apprehended or understood by an educated, literate person, namely...

7:30 Being, you know, being a member of the set of novels written by Tolstoy is somehow either absolutely on a par with or even more primitive than the notion which I'm appealing to, which is just a notion which I claim even primitive, possibly even invertebrates have, which is just apprehending a this there, this there, that there. Why is identifying this, there, that, there doing topology off right now? Well, because I think that it does involve being able to apprehend the notion of being a closed and bounded... what we have to describe in mathematical terms as being a closed and bounded component of a far-disconnected space. At least that's the way that it works in this... sitting in this room and being able to say that there are... But me doing that does not mean that I... Nine people sitting in this room, right? Sorry. Are you okay, Anders? Andrews, who hasn't contributed for at least two minutes. I'm sorry, Andrews, I'll shut up. Tell us, point out something. How many menus, now concerning the set of toast, for example, how many few menus do I have here? Well, it depends on whether you count copies or... Oh, that's a cheap Phrygian point. How many menus does he have in the ordinary language use of menus? He's holding two. And the very notion of being a copy of a novel is something which can only be apprehended by, you know, a literate person. If you hold up, look, if you hold up two books, which happen to be two copies of the same edition of the same... The same text. To a small child and say, how many books am I holding here? Do you think that any small child is going to reply, one, they're both copies of Wind in the Willows, come on. That's really, and that's what's so silly about Breger's point. Otherwise it's too easy. When he held them up I thought it was a trick that they were identical copies. I thought you were trying to trick me. I don't know why. I thought you were trying to say, ah, you said two, but there's really only one, because they're exactly alike. And Anders would make my point. One from each? Yes, yes. The difference between... And not only that, but I claim that your dog or your cat also apprehends, obviously not that you're holding two copies of Menu, but that there are...

10:00 Although obviously they don't have, they're not able to express it in language, but there are two things which my master has told us. But they don't have to comprehend that they're closed bounded. No, no, that's what I kept saying to you. I actually said it four times. Of course they can't express it in that. I have to say, if I'm trying to say what topological concepts it is that I'm appealing to and saying that I believe these are... More fundamental, I would have to say ontologically more fundamental, that these are prior, as it were, in the order of constitution of the world, then those are the terms I have to use. Of course I'm not claiming that a non-language using animal or a small child understands the words that I've used, which is why I try to... To boil it down to the absolute crudest possible level by saying a this there a that there, which a small child would understand. But you're saying that being able to do this, this here, that there thing depends on beforehand being able to say, ah... No, no, no, not being able to say. No, it's not being able to say, sorry, being able to think. Of course not being able to say. Not even being able to, well, I have to use the term being able to apprehend because if you think the thought implies... I'm not going to assume that in the least, no, I'm not sure. Well, then in that case I do say that, yeah. I do say that I think it does depend on that. I think that all our concepts are, when one digs to it sufficiently, when one actually examines semiogenesis, the semiogenesis of our, both of our, the concepts, our everyday ...carving up, you know, the reference of our terms in the world and also the meaning of our much more sophisticated theoretical terms. I do actually believe that it all rests on, essentially, on the way that we are able to transpose what are effectively acts of spatial structuration of our environment. Okay, so this thought that something is closed and bounded comes before... It comes before saying, oh, so there are two things, right? I mean, you actually have this... I think it's as far as that, well, I'm not prepared, I'm not too happy to start talking about the order of experience because I don't know, but yeah, I think it is, I think it is, well, no, no, no, it's precisely not because this is prior to, this is about a level prior to, which is...

12:30 ...already has to be assumed before you can give any account of semiogenesis, of the origin of semantic content in the first place. But if you're going to say epistemologically more primitive, yes. Yes, because unless one could just distinguish the one thing from the other, you couldn't possibly have the concept of there being two of them. And because all our evidence is that... At any rate, you know, mammals, certainly antelopes, let's say, on the Serengeti who are trying to escape from lions, are certainly well able to distinguish and act in ways which are clearly crucial to their survival differently when they apprehend that there are two or three lions stalking them rather than one, but they're pretty lost when there's a pack of seven. And all the evidence is that probably anything about four or five. It's something, you know, cardinality is more than, it is alleged, I don't quite believe the story, that there are primitive tribes in New Guinea that can't count beyond three, but you know, I don't quite believe that, but one can certainly, the story is conceptually coherent, so I think for animals it probably is as if there are, you know, one, two, three, many, but that doesn't... But that of course doesn't all... which is, again, precisely my point, that the topological... And geometrical information is actually epistemologically more primitive than arithmetic, which is why I think mappings are more fundamental than the notion of being an element of a collection extension, which is all unique conceptually to get. The kind of set theory that he is saying is informally present and has to be present in order for us to have rigorous definitions of structural notions and to get any other mathematical concepts off the ground to begin with. You don't need more than that. But I say that even for that you actually already have to have implicitly all this topological and geometrical information. You have to be able to process this topological and geometrical information.

15:00 The only thing I question is that, are you not getting something that's kind of an infinite regress of conceptual, you need to kind of underpin, you need to underpin a rhythm? It is a regress, but I think the problem is it's not an infinite regress, because since we live in, we all live in the one world, you know, I think the regress closes on itself. I mean, what, you know, one man's infinite regress is another's, do I say, kind of, you know... Holistic version of a consistency condition. For me, if I look at two things, the conceptual level that you're talking about of identifying them as topologically distinct things, that doesn't happen to me conceptually. No, I agree. And I haven't claimed that that does happen. Well, when you say conceptually, you mean to say you are... When you say it doesn't happen to you conceptually, you mean you obviously do not. Any more than you solve differential equations when you're peddling a bicycle. Of course. And of course it would make absolutely, that's absolutely not what I'm claiming. Of course it does. Otherwise what I've just been said about antelopes on the Serengeti or sea slugs would make no sense at all. Small children would make no sense at all. But I do claim that small children apprehend that as too... Before they learn to speak. Oh, absolutely. Yeah, yeah, yeah. Well, in that case, that's all I'm claiming. Which we are, implicitly. Because I'm saying that I think the issue of... Because the whole point is that John's position, which is to put... I would say, I don't want to put words in his mouth, which is to put all the emphasis on the logico-semantic ingredient of what it is to be a foundation for one thing, for one concept to be primitive, to be absolutely unanalyzable in terms of further concepts, which is, and in judging that, I say he's putting all of the weight on the logico-semantic ingredient, don't I? That ignores the whole dimension of semiogenesis, of how it is that we came to be language-using creatures in the first place, and how semantic content itself is formed in our minds. After all, as far as we know, we're the only species in the universe which, one doesn't know, the clicks and whistles that the dolphins employ may well be so fantastically rich as to constitute, but setting aside the business about the dance of bees and all that stuff. But let's accept, for the sake of argument, that we're the only language-using creatures in the universe.

17:30 Well, unless we all believe in intelligence, unless we reject evolutionary biology and say we're not the product of... At some point, this capacity emerged. I can understand the observation that there are closed-bounded objects out there being a profound discovery. As far as a physicist is concerned, as an mathematician? Well, I don't think it's a discovery. As I say, I don't think it's a discovery of physicists. I mean, don't you think it's an absolutely evident fact about the world, or at least the world as we experience. It might be that the ultimate ontology, of course, is some kind of process ontology, allo, you know. Oh, dear old David Bohm's ideas about the hollow movement, and that even the idea of closed-bounded components is actually an illusion which emerges from some deeper or primeval level of pure process, but that's a separate issue. It's a physically bounded idea that you have going on here, and I don't see why that would be a candidate for a... This used to be... No, that's absolutely my project. Oh, absolutely. I should be completely open about that. Oh, yes, yes, yes, absolutely. No, I think that the first principles of physics and mathematics come together as a unified package. I'm absolutely a naturalist. I'm, you know, an unrepentant... But why would you want to cash out mathematics as it's practiced today in terms of the contingent, I would say, physical facts that first sparked people talking in this way about... This is a story which leads from the circumstances in which we came to be language using animals and concept forming animals in the first place, to the circumstances under which mathematicians of the present day, including geniuses like Rotendiek, or indeed Bill sitting over here, actually form their concepts. Otherwise, where do their concepts come from? Either they are apprehending a completely autonomous realm of ontologically independent intelligibilia, the pure platonic realm. I don't believe that.

20:00 You know, what's the alternative? One doesn't have a naturalistic account of the origin of mathematical concepts, even including... I mean, one has to tell an extremely complicated story, which we're not remotely even on the cusp of being able to tell in any detail. We can tell a rather good, very detailed story about the history of mathematics, as you were, in turn. If your project is to naturalize mathematics, then... Well, it's not my project. I just think it's a project which lies probably on the distant horizon because we don't begin to have a sufficiently deep or unified understanding of... What is the payoff? The payoff is to have a unified ontology, I suppose, in which mathematical concepts are the things we refer to when we're doing... But an ontology of this sort is exactly the one that physicists are searching for. Well... That's what they're looking for. They want to know what fundamentally is there. I don't understand why mathematicians write that in order to legitimize... I didn't say they needed it. I didn't say they... I'm speaking now as a philosopher of math, not as a mathematician. Mathematicians don't need it. They can get on perfectly well, as they are for the most part, although I think the deepest of them, you know, the most powerful of them, have usually been more interested in these questions than the average mathematician. I think the, but it's sure there have been many very great mathematicians who have never, the question of where do mathematical concepts come from, what, how do they fit into the, how do topological vector spaces fit into the first, has never crossed their mind, and they were regarded as a totally naff question. ...and say, oh, you know, shut up, you know, don't bother me, get off. That's a perfectly legitimate attitude, and for pragmatic reasons, it's probably the attitude that most mathematicians need to have. But even as a foundationally inclined philosopher of mathematics, why would you fail to be satisfied with simply the observation that there are things which are distinct, and we can talk about how they form sets and all sorts of interesting mathematical structures. What physicists might later tell you as we discover more and more about physical work as a mathematician?

22:30 Oh, because I think that the answers to the second question might impinge on the first. I mean, I think they might. They might. I don't say they would. I mean I don't know, obviously physics is a very very long way off and may never in fact, I mean maybe the pessimistic meta-induction is justified, maybe that's, maybe there's a methodologically inescapable reason for being structuralist, you know, for being structuralist about, not I would say ontic structuralist because I think the problem is there is no satisfactory general notion of structure which is what we've talked this afternoon was about. But methodologically, and certainly as far as epistemic structuralism is concerned, as an observation about the methodological situation in which physics is for the foreseeable future. I think it's almost a chasm. I think it's almost, I mean, it's almost circular. But that doesn't mean that although it may be postponed, you know, I think it certainly would be postponed long beyond my lifetime or possibly the lifetime of anybody alive, that one doesn't continue to feel the tug of the question, what if? What if, in fact, physics were to converge on an ultimate theory, and what if we were satisfied that we really had got a hold on the ultimate ontology? Would we not expect... I mean, in what sense could that... In what sense could that account be exhaustive, complete, integrated, be in any sense a satisfactory candidate for an ultimate ontology or for a conception of the world as a kind of metaphysical unity, unless it accounted for mathematics as well? I mean, the only way that mathematics could remain completely detached from that would be if the ultimate ontology itself was one in which the mathematical entities were themselves the ultimate reality. Which of course there have been plenty of people out there like the Tegmark and the panstructuralists who hold that view, the idea that all possible structures exist.

25:00 But that's the only way I think you can reconcile... I just find it very surprising though, the idea that continual discoveries in physics might hope to inform matters on the foundations of mathematics, because as far as I can see... The foundational concepts in mathematics seem to be things like identity, negation, plurality, and things like that. Plurality is a perfectly relevant example, because supposing that foundations, supposing, I'm just using this as an example, supposing that the direction that future physics took, supposing... ...that it was in the direction of, I mean, way away from anything like string theory or M-theory, and supposing that it was in the direction of some quite sort of radically novel account of space-time, which it does seem, obviously, our concepts of space-time are likely to undergo very dramatic... ...conceptual and structural revision. They are undergoing it, but supposing that what they converged on the end was not some form of, you know, something more mathematically sophisticated than we can currently envisage, or only the very brightest people working on categorical quantum gravity can envisage, and which one could only make sense of in terms of a kind of purely structuralist account of the ontology, but actually on something... Much more. ...kind of metaphysically led, and something like, you know, Bohm's ideas about some kind of ultimate underlying process. Supposing that was the way that you made sense of the kind of structures that we have to deal with in quantum field theory. I think there's a lot of very interesting ideas coming out of geometric quantization here with algebraic structures in place, Underlying, all-encompassing, this business of the implicate order, which I'm sure you've probably read about. But supposing something like Bohm's notion of the implicate order actually turned out to be the ontology that the physics converged on? Well, that obviously would have implications for our understanding of notions like polarity and identity.

27:30 So, I mean, that would certainly bring with it... I mean, how would you account for structural universals like number and set in a process ontology? I mean, you would have to. It would be very interesting to see how people set about doing it. I mean, some metaphysicians at least have spoken, so that's yours, I beg your pardon. I'm not saying that will happen, but you said to me in what way could issues in the foundations of physics possibly impinge on issues in the foundations of mathematics, and I'm saying I think that's the kind of answer that I would give, that I'd have in mind, yeah. Double check, you going okay for confidence? Yeah, I'm okay. Can I just have an ordinary white coffee? Thanks very much. You completely missed the coffee order. Ah, yeah, we were burbling away. I'm sorry, I was burbling away to you. But I do think that the, I mean, my perception is that as, you know, I think the greatest scientists of the 17th and 18th century thought, as Galileo and Newton and Euler all clearly thought. And Riemann thought, and I think Einstein, and I won't give a long list of names, thought, and this guy sitting over here thinks, that the first principles, the ultimate ingredients of definition of the concept of mathematics and physics actually will turn out to form a unified package. I mean how tightly meshed the unification will be and you know it may be a long way off but I believe in the unity I believe in the um the unity of natural philosophy I feel like I kind of believe in the unity of nature and and and and I am a naturalist so therefore I believe that the first principles of mathematics will be part of that. You know, it would seem grossly unusual if we ended up devising a system of mathematics which had made no contact at all with the first principles of what the universe was like. And even John over here, I can say, put words in his mouth, he's listening to me now. Even John, in his book, his excellent book, which I hope you have read, actually remarks in the little passage about the serpent in Cantor's Paradise, the possibility of the serpent in Cantor's Paradise, that one of the reasons for suspecting that there might be a serpent in Cantor's Paradise is precisely the fact that... No, the transfinite has actually never really shown itself to be relevant at all in the formulation of physics, to anything in science. Which would put it sanditiously, but simply, if there were transfinite numbers out there, why didn't God make use of them when he made the world?

30:00 Okay, that's a perfectly good metaphor for what I'm saying, except I prefer not to talk about God in presence of my atheist friends. After Kantor... Oh, Kantor thought he did. Kantor thought he had done it. He thought that he thought that he thought that he thought that he thought that God made space and made use of it in making the physical world. Yes, he did. No, but Cantor did think that. He thought precisely that the Alice would actually turn out to hold the explanation for what the hell was it called? There was a distinction between ponder and motive. Ponderable matter, and what was the other? Yeah, but the point is, he didn't make, even he didn't make use of all these transponderable materials. Cancer didn't. Well, no, of course he didn't. Well, that's what I'm saying. But he did claim... That's what I'm saying. He did acknowledge that one of his motives in developing Ethereum... Maybe that's why we're the lowest level of existence, because there's some kind of neoplatonist hierarchy, and we're the finite level, and then the next is common. Well, I can't tell you how very deeply I do not believe that. There's a paper published by Grattan Guinness, right, which was rejected from another, his paper, published actually about 15 years ago by Grattan Guinness. Oh, Cantor's paper. Oh, Cantor's paper. Is that the one about the ponderable matter? Yeah, that's the one. Yeah, somehow, if I remember correctly, he hoped to say explain the difference between living matter and... Well, that's curious. I didn't realize that he also thought it would have an application to understanding the nature of living matter, but he did say, he did say explicitly that it would, he thought it would have an application to understanding the difference between what he called ponderable matter and, you know, celestial matter, the stuff that the… No, I don't know. Can you hang on a second, I'll just wait a while, because I can't hear you. I can't hear you when he's talking across me, sorry. Right, right, right, right. Okay, right. A hundred years later. I must read that. Sorry, I'm just going to say it.

32:30 That's the ethereal matter. I'm sorry, I just couldn't think of the term. It's ethereal matter. He thought that the Alephs would be implicating an explanation of a theoretical matter. But what we know about physics, or think we know, my argument is much more convincing. If those things were there, if all those trans-finite cardinalities were there, how come God didn't use them when he made the world? Well, that's the same question, as far as I can see. It's just dressed up with a theological metaphor. And then somebody would come along and say he did. Well, exactly. No, but the point is, my point is that this question obviously pressed on Cantor's mind as well. I think the answer he gave to it was, well, the answer he gave to it was obviously a very tough one. No, I do think, I mean, I do think... A non-starter. We may be made less theological, but those cardinalities and possibilities, why come they're not realized in the world? Yes, well, okay, so which turns to the point, so I'm saying, so even old Mabry, as it were, sees where I'm coming from. Yeah, yeah, yeah, yeah, yeah. She wanted to explain why we come to a natural number, why are the natural numbers not realized in the world? No, he's agreeing with you. I agree they're not. I agree they're not. Well, certainly not as a completed discrete infinity, they're not. No, of course they're not, as you pointed out so brilliantly in that lovely little talk you gave in Cambridge 20 years ago, when people go on about fractal geometry and say, oh, well, the coastline of Scotland is fractal. Not when you stick your microscope down into the sand of the beach in Stornoway, it's not... You go there and try it, which was more or less what I had said in this lovely talk. When you get down to the level of molecules, you're talking about binaries. You don't even have to get down to molecules in that case, you just have to go down as far as grains of sand. That's what he was doing when he was killed. Exactly. Exactly! Of course! You could actually name the number of grains with the universe. Which obviously shows the enormous power of the Greek notion of Erythmos, which nobody disputes.

35:00 So powerful that people end up believing that purely arithmetic ontology will actually account for the whole of mathematics and the rest of reality. He said, you know how far you can travel with an ox in a day? On an ox cart in a day. He said, you could travel for 50 years in an ox cart, if you could go straight out towards the celestial sphere, and it would take you more than 50 years to get there, traveling steadily in this ox cart. Well, I mean, that's not too bad an estimate. It gives you the idea that it's something unimaginably large. He could have said, of course, 50 million lifetimes and still have... You mean his paper on how many grains... Yeah, it's the so-called sand record. Yeah, yeah. The sand record. And he figures out... But we calculated the number. Yes, that's exactly what we were just talking about. That's what we were just saying. But it's still... That's yours, isn't it? Sure. Thank you very much indeed. I will, actually I will have one bit of a run of show, thanks. Thanks a lot, cheers.