Discussions, incl. M Wright & A Peruzzi on Lawvere's paper

12:30 I define the reference, in the case where it is always possible about the nature of the, which I do not believe is valid, it seems that some kinds of natural systems, the model of which any other model is a quotient, has many of the properties of inanimate and animate. This is not the place that the interested reader may consult those of 1985b for a fuller discussion, to have another chance to consult.

15:00 He also remarks, and this is the interesting part, where he makes this connection to category theory. I don't know how much category, no idea how to do mathematics on, because we put the philosophers like this. But he seems to me to be very good. He says, indeed, the modeling relation, it's... He then goes on to say, what are the ramifications for understanding of the modeling relation within mathematics itself? Sort of the modeling of the relationship between... And he then makes very introductory remarks about modeling in mathematics. Since 1945, an entirely new branch of mathematics has been developed, initially to formalize the developments in methodology of topology initiated by Poincare, C. Eilenberg and MacLean, 1945, and in fact is to be regarded as a general theory of the modeling relation within mathematics. This concludes my category theory in depth here. This is a six-page article. Spaced with this, by going into category theory in depth here, I must refer the reader to one or other of the excellent texts which now exist. Suffice it to say that in category theory, the active agent of comparison between different kinds of formal or influential structures are functions.

17:30 Now this is interesting. The formal counterpart of natural law in this purely abstract setting is the existence of non-trivial functions. And then he goes on just to make a few introductory remarks about natural transformations and so on. Now, this is interesting. Let us, for example, look briefly at the original motivation for the development of categories. And that, in fact, that's just that one side remark. They don't know that. One of the most basic problems of algebraic topology is the classification problem to be able to tell that two given topologies are homomorphic or not. The basic question of algebraic topology is whether there are enough models to discriminate any two topological spaces in some kind of effective way. Fine, okay, which we also know is directly related to the accident choice. It appears that sometimes there are enough such models and sometimes there are not. And when there are, we can think of, we can build a largest algebraic model of each of the spaces involved and settle the classification question by comparing these largest models, which of them is a quotient or some model of the other. But when not, there is in effect no second invariance, his values settle the question. It will be seen that this is an abstract reflection of the question that I have raised in the preceding section of it. It will be seen that this is an abstract reflection of the question that I have raised in the preceding section. And one whose answers should have a most profound effect on our understanding of the relation of biology to physics.

20:00 You do not have, in the case of biological systems, this condition, which he also remarks elsewhere, is that you have a separation of the notion of spatial transformation, and that in biology you do not have a separation of language, theory, and ideas. I don't, but I also see how one could perhaps make it more precise. It's just that it was in connection with this remark of Bill's about the significance that he has. Level zero, discrete, non-discrete, discrete, chaotic, in the case where one has induced by the dynamical action, and it does seem to me that there's a notion of cohesiveness induced by dynamical action, particularly in the case where you have what he terms a mixing up of the structure of the domain to be viewed as internally varying within itself in a way which is not captured even in the case of what I understand is the point he made in these comments. Whether this would have any connection with the And finally, the definition of state with our ability to treat the system as a state with a dynamic from which one thinks of as actually our ability to separate out dynamic dynamics and initial conditions. I think it is a profoundly interesting problem.

22:30 Essentially, yes. Yes, but precisely his point, the determinism, in fact, one of the points that Rosen makes as well, is precisely the way that determinism, the structural classical dynamics, actually hangs together with what he suggests, there is a direct connection with the action of choice. I agree completely with this kind of, with the relevance. My way of thinking through the problem was different. Do you want to use that?

25:00 Oh, sorry, we're getting all mixed up. I'm having a question about the copy of these. Oh, is this one? Yeah, thanks, I was thinking I needed to... I mean, sorry, I promise I'll start. The key notion is not simply that the functions are the same in the end.

27:30 Take for example the of course it is easy to say how we pass from the forgetful. Once we realize that this construction is just an example of the general construction from the forgetful function. Here a way of generating what we have indicates where this is the category of space.

30:00 Yes, I remember.

32:30 Which of course brings us back to what we were saying yesterday, and I'll interrupt it and go on, but I'm obviously closely connected with those remarks about the transposition or possibility of arriving at kinds of abstract, particular mathematical objects, which effectively ignore, which effectively really look only at one, at one direction. Before we go any further, can I just, I'm sorry, very, very quickly, I'll get that out of the way so I can concentrate properly.