Discussions, incl. FW Lawvere (contd.)

0:00 So far, it's an extremely good way of introducing the crystal fly at all. I don't think there's a false note in it. Seriously, no, I think it's exactly the way to put it across. I think it's pedagogically brilliant. And from that analogy, you can at least give an orientation, kind of guiding ideas for what's going on when you're discussing things under deed of quantity and space. I'd say it's a wonderful idea. When did that come to you, to use that of naturality? I thought of it when I heard about category theory, in a rough way. Well, the fact that it's a commutative square, and the fact that it's really naturality, because there are these two different interpretations of the same abstract idea, and so forth. Once I get started, you know, the essence of it, you see, is if there's not just one performer, continuing with the musical example, if there's an orchestra, you have one of these little... Diagrams that I considered in my qualitative distinctions paper, a very special kind of topological space, where now time and sound take on, they're actually two quite different kinds of objects in the category in which that little diagram acts. Sorry, the little diagram is now, I multiply the arrow, which was the Cartesian product of that, with one of these dots unrelated to each other, but all connected uniquely to the...

2:30 You take the Cartesian products, you have a little sort of wedged cylinder which is the actual category which the naturality takes place, but leaving aside for a moment the previously discussed example, just looking at this versus individual thing, the time, and in this case the sound, or more generally the states or whatever, the states of sound, well the states of the system or whatever. There are actually quite different objects in the category. Already, just looking at pre-sheaves on this time, a little diagram, large topos, but within that there are two almost directly kinds of objects, and the time and the sound, or in other examples, the time and the space, or even the state of collective thinking, whatever, they're quite different in the sense that the... You see that the time of the conductor, it's fantastic, is actually synchronized with the time. We can imagine that each player has his own time, but it must be synchronized. So that means that the action of those arrows on the time is each one an isomorphism. Yes, and of course in the real world example, when I was at Goodhills, they're building the conditions which are allowing for that synchronization, like the time delay in the acoustics. Whether the concert hall they're playing in has got perfect acoustics or whether there's a problem that they have to, whether the orchestra has a leader in addition to a conductor, which would give one a kind of, no, I think it's absolutely fascinating. This can be, depending on the nature of the category, if we, if we not just abstract SES as a background, but some are more cohesive. Well, that's exactly what I had in mind, an all-restricted one could, one could introduce those examples. Even if we say that they're isomorphisms, those isomorphisms could in fact be time delays. Yeah, yeah, yeah, exactly. So here we receive the news, you know, a microsecond later than...

5:00 So actually the invertibility is not necessarily a total identity, that may involve more or less in a way of time delays and the like. So that's the nature of time as a diagram of that sort, rather degenerate in a way, because if each other is isomorphic, then one is reduced to the degrees of freedom, for example. What I always like to point out that sociology, psychology, and so forth completely leaves out of the count is that the state of the system, thought of as the major node there, restricts to individual states of each part. So the sound that the clarinetist is producing, the sound that the violinist is producing, these are all different. The state of the whole is essentially the ensemble. The state of the collective is essentially the ensemble of the state of the individual. I'm taking also mental states as another example. But just because as an object of the ensemble of individuals, the law of motion is not. If there is a law of motion moving it ahead, then the change in the collective state depends on the collective state. It's not that the individual state depends only on the... In other words, these little arrows basically as projections of a big product. So that's what I mean by a condition that is almost opposite to one in the other case as they all become isomorphisms. Well, can I illustrate also the particularity of the case of abstract sets here by pointing out the very restrictive conditions on the product in that case, again illustrating it from the example of the musical performance by the orchestra, that it would be a very, you know, you can even, yes, the case where it's, anyway, a third page to the top of splits as a full product would be a very unnatural condition in the case of an orchestral.

7:30 No, this is wonderful. Third page on this collective individual contradiction. It's a special case with my paper on qualitative distinctions. Well, that's a tremendous application. I think a natural, because you naturally, you even already said the performers. Yeah. So you naturally think of, well, a collective performer in music. Not necessarily a single one. Sure. So it's a natural expansion on page two. To say, well, actually, the performers may be collectors, but also there's a unity of distinct things, and so we have not just a discrete set, but this kind of a diagram, a discrete set, to which one point is that, with arrows pointing to it, and all these things. All this, you see, all this in principle is something that's understandable by anybody. Yes, yes, absolutely. It doesn't require... Even I can understand it. It doesn't require prior knowledge of categories and modules. No, they will grasp it. Just how fundamental... No, grasping the general idea would cause anyone, as a serious person, to want to go and learn. But it's going beyond grasping the general idea in a vague way, because we're writing down precise equations of what natural reality actually means, and... yes. Yes, yes, but I mean, so once you've given the idea, okay, that was badly expressed, but once you've given the decisive... I would agree with your use of the term general idea, just to see that the general usage has become a degenerate form, where general idea means automatically a vague and useless idea. But here, this is an idea which precisely would lead people to want to go and understand, in its scientific particular, the tools that we have actually now. I just need to run up and get my wallet. Yeah, it's OK. It's all right. Let me take a second.