Discussions, incl. M Wright, FW Lawvere, A Peruzzi & J Mayberry (contd.)

52:30 Why I don't agree with it is that independently from any way the world is, since any way the world is, is once again trying to change it. I can't think about mathematics unless as a result of the development of something that I don't like. Well, you have many options. These are consistent in many contingency objectives that we like to characterize at this point.

55:00 And at this point we can hardly move along trying to find what could be the best field that could give us a framework. But in any case, we can find the right answers. But the right answer cannot be found if you separate the structure of the mind from the... and you cannot find if you separate the... if you try to reduce the vibration of the mind... The use of the term contingent is used in a very contingent way. It means all kinds of things, and it's put in opposition to necessity. I like to think of the nine planets here like a tool, like a hammer, or like a light bulb. There was nothing in the world like light bulbs or hammers before human beings came along. The hammer is nothing but a reflection of the laws of physics and the needs of humanity, which is part of nature, etc. So why can't we just accept that, of course, the number 9 is not a human, and that is because it serves us as a tool in the world that there is a 9-ness to what we know about, about the products of human beings, which were there before anybody looked at them, which were there before anybody looked at them, and the number 9 wasn't. Hmm. Which is why I wanted to ask you about the smooth numbers, the several smooth numbers in SVG, and do you think of those as getting a little closer to the nineness, or whatever, of whatever is there before being is reflected in thinking?

57:30 So that's another tool, right? It's a further tool, but in some ways it's a superior tool for some, in some respects, or... So it's superior to the concept of natural numbers that we've had for the last few years? Yeah, that's that. Yeah, that's the fact of the problem. My question is... No, no, no, that would be... That would be... You don't have to admit that, then you come back home and... Objectiveness. No, but... Can you explain to me why you think superior to the concept of natural numbers was captured by the Piana axioms or the other concepts of structural progression that we've had for the last century or so? It's very important because it helps me to understand what it is about the topos of Smith's Basics that is particularly important from the point of view of your philosophical position. Colleen have a very philosophically naive conception of how the foundations of mathematics are once you've said that the concepts of mathematics are protean and the boundaries of it are subject matter, permeable, deep and beautiful in the connections which mathematicians are able to see, but you could 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. It's just something sui generis. It's just a voice in the conversation of mankind. Ultimately, I'm horribly afraid, the position he takes, he's going to be driven in the end to say it's just sociology.

1:00:00 I don't think he actually... I don't think he actually... I mean, he's talking to an audience of philosophers, and he's making a point that, you know, that's a necessary condition. Well, I may have done him an injustice. I think that's an absolutely important thing to say. So the audience will be addressing the terms of that paper, especially when many of the pieces have philosophers. Having said that, it's okay, on an Aristotle's point of view, if you have to, to go against the experts in the first place. It's a sort of common sense point of view. The first thing you do is to see what the actual, you know, what... What the contents of humans do on these things, I'm inspired by some of the experts on this field. Well, exactly. No, in that case, I may have done Colin an injustice, and I apologize for doing that. But I mean, that is to say that what the experts say is, if so, a fact of what the case is. No, no. Thank you very much for your time. 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 and conception of structural progression features and the use of numbers than the normal.

1:02:30 But what is that? What would it entail to you? Even that's just a fragment, yes. The contrast is that we grow in a completely free speculation, that we can go to the next number and at least reciprocally, have purely truth and a pure thought, not denying that there's any constraint with the real world. And you're talking about things like 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 you want to have an accurate idealization, the idealization has to permit taking that somehow to some extent into account. It's completely why we're looking into poor idealization. You can always imagine yourself standing at the blackboard with the logs, but if you look at the problem globally, which is after all, the problem of the natural numbers is a history of about 17, or 2,943, it's a problem about the tail end of the natural numbers. Then, you know, the stroke calculus is completely unphysical, because we don't know what limitations there actually are on physical operations.

1:05:00 The point is, the point is, you see, that these guys say, well, never mind. It's just that if you've got the idea that if you wanted to, you could buy another piece. The problem in the natural number sequence is with the infinite tail. So no matter how far I've gotten in the natural number sequence, the mystery remains that it were in the future. Okay, that is it. Is this the kind of... 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 have? I mean, if they're alternative, there should be more and more accurate ones. Less idealized. In the book, I came up with a story about the Maranello and the idealization of what's going on there, the entire complexity of it. All the while, we realize that in some similar ways, there is this process of counting, where the force is the urge to add one more. We talk about the motion of thought, I think. There is one force, and yes you can have a theory where that's the only force, but it seems to be, you know, really quite an extravagant theory. What about being able to make a more accurate one to do the count?

1:07:30 Such as? Such as the lack of money to buy more trucks, did you see? Yeah, okay. I mean, of course, of course, people say, well, that's crude. It's crude, but I mean, there should be a range of possible theories between that utter unity, that real problem, and the... It is a crude, but it is a practical problem. Yeah, and it completely ignores the theory. There is nothing but theory. I mean obviously you're getting outside in any not any of these things you're getting outside looking at this thing the outside you're trying to determine what the consequences are I got the I got the impression I mean I've got that impression before really what you want to do is is to shake people out I mean there's a, the way mathematics kind of, as it were, is set to a complete order of field, all these things are sort of, maybe these aren't the ones we ought to be looking at. I mean, go back and- That's one of those things that you mentioned, I'm sure somebody- Yeah, but I mean, of course- But not an instructive thing. There is something better. That's a worse approximation. And if you stick to these things-

1:10:00 There are a number of possibilities, even in a sense unhistorical, because you forget we arrive at where we are by guys solving a series of local problems. So there's a series of local problems and all of a sudden you find yourself where you are. And I'm assuming, you know, it's the opposite of rational wisdom that's more relaxing. I mean, that's the attitude that's more maximizing the role. And fossilized thought. I mean, one way is to go back and see where things came from. And notice, to your surprise, that people actually were thinking about quite different. You know, that this whole thing started. You know, it's like people getting a long, involved conversation and then ending up in a sort of stalemate and wondering, well, what were we talking about? Where did all this begin? And, you know, you've sort of forgotten where the thread was. I think that we all have to be alone by that experience, you know. But I mean, like, if this is about real numbers, I mean, where does it come from? What were the people trying to do? What did they actually accomplish by making these? Our monumental figures are, you know, I mean, they're not important figures, right?

1:12:30 Of course, they're given by God in the same way that the way of peasants. And of course, this is a much more sophisticated, deeper, but I mean, if you don't go back, then you're just imprisoned in the present. Now, this is a point, actually, about the, which I think is valid, about the power of tradition as a grip on men's minds. In a society with powerful organizations for developing intellect, in those 95 years, certain traditions have been established, and now they're built into... What Heidegger didn't have, is that traditions are something that are not made out of salmon. These are things that have at the same time been studied and that can be modified, that human beings have a way of not being imprisoned in the particular tradition they are aware of. So, whereas in Heidegger you simply find this incredible heaviness on your hand, and you can't do anything unless you accept it. Well, please, I don't need to explain.

1:15:00 I would simply say that first history is not made just by one tradition and of a still substance. And second, that human beings are related, of course, to traditions, but first not to just one tradition. Perhaps we have a struggle of traditions within our minds, within our lives, but just this multiplicity and just this historical nature of traditions may... All of these make us in the position of comparing, of evaluating, of judging, of taking decisions, whereas if there were just one tradition which I cannot do, I would not do anything other than to accept it. We have just mentioned there would be any reason of speaking and trying to persuade others of the truth of our assertion. So why did Ider said that to me? Well, I mean, I think in scientific matters, there's a strong case to be made for conservatives to say, you know, to be an avant-garde. You know, it's all very well for a paper to say I'm going to throw away the whole of the Western tradition. I mean, maybe some of us might like the results that come out of this stuff. Yes, well, no, but the point is in some cases, you know, I mean, you can't do that. You have to embrace something else. In fact, it's embracing the dark ages. It's just in a different form. So-called avant-garde is just, I think, a smokescreen. Yes, we'll see that again and again in 20 centuries. The point is, there's a whole trick to intellectual life. It's to know what to retain.

1:17:30 You can't, you know, a man who says, you know, I'm going to start physics and you're going to forget about Newton and all that lot, the picture of Einstein is to say is a kind of avant-garde. Yeah, yeah, yeah. Okay, now Einstein is one of the guys who actually is. I mean, I think there's two, there's always two sides. And in a way, this is why it's a collective effort. And these, it's not just... It's not just contemporary mathematics. We're not just in a collective effort with our living contemporaries. We're in an effort with the guys that are already dead, and we've got to think that we've done it. But on the other hand, we've got to remember that these are human beings that have done all this. They're fallible. That seems to me really what's behind your complaint about the real numbers. I mean, that is a fossilized illusion. And, I mean, it's not that it isn't beautiful. You can't spend your life investigating. But if you try to remember why these things work or explore them, do you think that there may be many other things? You won't get that if you just take pride in your code of ethics, what grasp the tradition has on you. Tradition presents itself, ultimately, as a kind of God-given, a God-given that you have no choice of.

1:20:00 Unfortunately, that wasn't the kind of subject we were talking about. In the previous talk, we were talking about science. So, at this point, what I've ever said is not right. What type had those kinds at the beginning, and the difference in what issues that's the kind? Well, I mean, if you don't want to follow his example, you don't want to follow his example, then what do you have to do? You know what I'm saying? That's good advice, and I didn't suggest that. If there's any point to the philosophy of mathematics at all, there's got to be an attempt. I don't understand. I mean, because I mean, that, that way it blends into mathematics itself. But you can't reduce mathematical, mathematical and mathematical mathematics to history. No, no. No. But you only, on the other hand, you can't ignore the history of the other. The history of mathematics is an essential ingredient of life. In the sense of, you have to change the story. But again, in a sense... Not simply... An explanation is not simply a description of the steps in which the world is something alive and new. You say that you explain the fact that we have found the universal field simply by describing the ordinance of the things that we have.

1:22:30 From that point of view, it's a disaster even if you get it wrong. It kicks you in the ear. I mean, in the last month, if we don't really know, we hardly know what we think ourselves all the time. How can we know what they thought, really thought? What we can do is say it and see how it hits us. I think Anglo-Saxon is only used by continental European philosophers. I mean, maybe the Comprehensive Walks will use more of the ideas written in them, you know, not just sort of... Here is an example of cohomology, where you can turn the subject into a kind of scholasticism, where you, you know, you've got a continuum of arguments on some hoary philosophical problem. If you brought these out, and your components went out, they'd probably be on the other side. And it's... Well, it's perfectly misplaced about our encyclopedia, isn't it, Mr. Steele? Mr. Steele says that Argyll-Argetina, he says a lot of this is just designed in order to not disappoint the realists, you know,

1:25:00 because they come down to see their, in his last, you know, the old-style examinations of the doctor, he had a public disputation. And so what you had to do, the trick was, you had to map all these known arguments. You know, not to necessarily make any progress on understanding the continuum, which is what he's writing in that article, but you don't necessarily make any progress on it, because at least your relatives are satisfied that you haven't been argued with in the science. I mean, that's the ultimate humiliation, is to be standing up and have to say, well, my God, you know, I haven't got anything to say in the past, but you're right. And that's still true. One of the worst cases is what I've had is that I've been trying to get a couple of colleagues in my district to write publications, you know, on some stuff, you know, on Euclid and Aristotle. You know, every time I make a suggestion, it's taken as a basis for an argument. I'm not marshalling all the reasons why it couldn't be worse, but my feeling is, you know, maybe... These are coming from the philosophy department. Yes, that's exactly what I would have guessed. Okay, you know, I mean, you know, I found some passages in physics, in Newton's definition of units, there's an odd phrase, which is translated, I figured, literally it means each of the things that, each of the things that is puzzling in the context of his definition. And I found some uses of this in Aristotle's metaphysics, where it's clearly a discussion, a domain of discourse.

1:27:30 So I wanted to say, 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 a modern location sees the term universe and sees the term universe. 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. 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 think. And, you know, my reading of it makes them say something sensible. The other reading of it makes them say something unintelligible. Maybe if they were to say something unintelligible. But, why restrict yourself, you know, why hold that open to the possibility? Well, you know, come down and mess up the rest of the answer. But this is the, I mean, the danger to the public's head. I'm glad if I want to do that, but that's not it. I feel a certain difference about publishing under my name, a careful analysis of passages in Greek, when you know my Greek is very, very rudimentary, mostly, you know, I had to have these guys help me, but I mean, I think that's the corruption of the institution that makes it impossible, you know, just to enter into a natural sort of, I mean, I think, you know, there'll be people who'd be interested. Why can't there be several styles of translation in the style of your education, as long as it's clearly marked as such, as a foreword saying, this is the attitude I take to a translation.

1:30:00 That's my attitude. I'd rather have to publish a translation document than I read draft documents. In other words, you know the basis of the way that people get it. Anything verbal, even if it sounds innocent, it's obviously got some real intent, right? And what that says is, that leads you to, this is the conclusion itself, it leads you to the same thing with Fraschmann. He said something that other people, like you, say, you know, what's that? That's what other people... I see, ah, it really gives me quite a lot of natural transformation on the identity and so forth. If I make that interpretation, it nides me to reading everything else he says in such a way that I see it opening up, you know, incredible richness of content. So I think that's a reasonable way to... Well, you have to admit that you may be wrong, but so what? Sure. So what? At least it's interesting. I think it's true. In other words, modern language is a more concise way of expressing the profound insights of the great man had. And if you use it, it will help you to understand it better than anybody else ever understood them. Why not? No, but there's a place for powerful speculation, particularly when it leads you to see the tremendous wealth of interconnection and depth. There's no point in really interacting with it. You try to make it alive by putting your answers in it and putting it in the present.

1:32:30 Just as with Euclid and Aristotle. That's an interaction. 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 great office of math knowing in advance what you said. That's right. But then, granted all life, granted all life, you've got to somehow, you know, you want to try to get into his way of looking 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 turning up ears, about the way of thinking of abstraction in the context of discrete vibration helps us to understand what ears are.