Morning talk / Part 1

0:00 You don't start from where you've got to. How do you do that? This is a place, of course, where typically philosophers of maths do want to start, but we couldn't start from this space. Well, that's a different point from what Bill's making. It's making it less simple, I think. But your question was where do we start from? Well, first of all, the domain comes in categories, and it's not one huge world with all the main basic characters in it. They come in categories, and some of these categories are space-like, and some are algebra-like, and there's a lot of them, too. And this is all... How do you make sense of the old... I claim that in the fundamental operation of mathematics, there's a universal algebra, because that's the subject that seems like one of the circuits in the attack method. I did that. We have discovered algebraic structure. If you make categories, composes, whatever, structured in a way that you don't know what you're investigating, if you make this operation in the past as totally abstract sets, if you manage to extract that, well, like any country, it has an algebraic structure. The natural, the natural part, the natural and the more vivid part of this. So you get a category of mathematical stuff that's built over here, which is sets of the power of abstracts of all of this stuff. I mean, it's the most profound, the most profound theory of any other type. My virtue, in fact, that you extracted these abstract sets, not just one of them, but from a certain category, could be a fairly small category, but nonetheless, it's a category, i.e., it's not math.

2:30 And so it makes sense to talk about natural transformation. And natural transformation is not just an underlying setting, it's there, it's there, it's there, it's there, it's there, it's there, it's there, it's there, it's there, it's there, it's there, it's there, it's there, it's there, it's there, it's there, So really, these abstract steps are the underlying steps, not just of real objects, but also of abstract algebraic objects. And with those, there is an algebraic theory, linear operations, archives, all. So this is, because you plant A, you work this. So you've automatically got, because of where it came from, open to naturality. One in that group is saying that you wouldn't have a particular analogy with this, space x and operating x, and the natural transformation as a component, and that naturality says that we have a map of x and y, of x and y, just the same with naturality.

5:00 This is just spelling out, but you know, reminding us what it means. It's a very strong condition, so what one finds typically is that if you actually started with... A mathematical category. Even though those are very huge categories, there's only this whole number of naturalities and strings. It has a tendency to strain out the conceptual definition, but it has a strong tendency to be viewed as a form. A simplest example I like to consider is the category of sex. The sign is the square of the word sex. And that's what are the natural transformations of that group of facts. It must be the, it must be the, one or the other of the projections, that's all there are. Likewise, back to the diagonal, so like, although the set is big, you're asking for naturality overall, almost a very small number, they are just a definable thing. You know nothing about the product, except the defensive direction, so naturally. Well, you know the product is the thing that's the product, but you know nothing about x. So generally, you see, this same square, we've got the other way, it's the definition of homomorphism. We were given, structured by the alphas, we need a definition of homomorphism in between them. So we're just trying to turn that square around, turn it more or less the same thing. Well this is the reality-based structure of any particular process, distracting.

7:30 That's the theory that we need to treat those actual objects. In other words, it's not just hand-waving, it's a mathematical operation. And there's not, if this is algebraic structure, there's also other kinds of things. It's the same idea. The doctrine of algebraic theory is general algebra. I'm still puzzled about where one starts. I can start with E, armed with a machine. Oh, mathematics. How do you penetrate that? Well, what would they see? They're supposed to imagine that E's coming from reality. Describing alchemy may be a little difficult. The whole question of the history of mathematicians is a subject. You conceptualize it as a... But how do you start? Engels says you start with being, that it's there, and then you go in from below and understand it. But you can't start with... I don't see how geology might be included. What I can't see, I mean, in order to make sense of that diagram, you have to have a category. What I can't see is how you, I mean, you're being vague about what that category is. Yes, that's a problem. How to, how to, how to, the vague idea is that we know, we have an idea of what it means to compare, let's say, mathematicians.

10:00 That's what the work has been to do. If we understood what a math is, stipulated what a math is, and then there will automatically be Aristotle's qualms. It sounds like something about yesterday. Aristotle has this idea that what you do except for Aristotle, you just, but then you conceptualize, you have to consider yourself that. Under a particular... Man qualms. Man qualms. Man qualms. So then... You have some, a man for a man is not the same as a man or even, it doesn't even capture what we naively think about like the same man that's standing before us today and lives in Buffalo, New York and has lived in Italy and so on and he's got such and such friends and all these things but man for a man is, we think of you as a man. But we think away all these connections. And those connections are what are key. Those connections are in your mind.

12:30 And their relationship, their serious relationship. Because every particular is enmeshed in the whole of something. We have a particular, we have a limited, you know, and with the problematic aspect of turning the concept to morphisms. If we knew that, then we could apply a general doctrine, a big particular doctrine, and get an abstract general. In a very general way, the relation between abstract and this kind of a general concept, a general concept has only these two aspects, like the fiber category, the fiber itself, the system itself, the concept is never just the class of the concretes, it's always just the abstract. That's all very fine, I can make a lot of sense out of that mathematically, but the interesting note, because this is a double dualization, the original particulars of the inclusion we have, that's literally a double dualization. It sounds to me like it's related to this question you were talking about just a minute ago about free and bound variables, free variables, what the status of free variables is. If I say that, he'd be a group. The logic of left, the logic of sister point, say, left?

15:00 Left is a little bit like the point. No, you can do that in first order logic, which is... Well, it consists of putting the variable in the hypothesis. Like you say, you take us through a hypothesis. Right. This is a group. It's a pre-variable hypothesis. Thank you for your attention. But that gets back to this abusive notation where you use the same letter to denote the underlined set. No, no, it's not about that. It's just a case. What I'm saying, what it illustrates is that left is some kind of logical operator that has not been formalized as far as I know. Well, you could even say let this be something that turns out as logical. But you know there's nothing. But in the end, because you get a contradiction, you know there is, let's say, a puzzle about it. So you're talking about what can be proved. It's not a matter of a statement to prove. By introducing this left thing, you're entering a richer structure than you had before. It's a matter of a richer concept. And another, that's why Wolfgang Pitta, in fact, he could have added that the signature of any particular discussion is richer than the signature of any discussion. The concept of bogey somehow is the only place I know where this is going to be tested. One concept implies another.

17:30 Well, I mean, the simplest example is with any number of ways. Neighborhood systems, bogey. I agree with the neighborhood systems. So in either case, I'm producing concepts. So you're doing essentially the same concepts. I mean, there are statements which are auxiliary to them. The main thing is the concept of bogey or the concept of bogey. It is a puzzle. They're not just properties. They're not just properties of the set and the world. Oh, it's a property of a topological space, which is a... I mean, however you define topological space. But if you take it as a topological space by definition, then, of course, you can't just say the set is what it is. You're like, the set can't be the topological space. It has to be the set together with this... I prefer to call it morphology. But morphology, you're imposing a form, morphing. So the ones you impose automatically pop up, including... What seems to be the case in the case of, say, the alternative definitions of a topological space is that they all inhabit the same world of concept. You can define closure via the definition of topological space using an open set and then you can then you can define the closure operator and you can recapture the notion of open set that way, but then neither does mathematical practice.

20:00 Mathematical practice involves typically both those aspects. Constructing more involved concepts. Statements of logic, what I call logic in the narrow sense, is focused entirely on the latter and not considered formalized in the first part of this series, except for Bourbaki and Farley. I don't like Bourbaki's formalization, but he's the only one I know. Well, which formalization are you talking about? The attempt to... Well, the structure. The attempt to... He talks about constructing one kind of structure from another kind of structure. And there are usually statements that go along with it. If topology is closed under intersection, you'll compare it to quantum math. I'm just trying to say this business about this. All you have to do is give me a category, and I can do this. This is formally what I call the big categories. Oversets, opposite, and semantics. Categories, they're natural transformations. These are totally arbitrary categories, but in theory, natural math can do that.

22:30 Well, these are actually adjoining species, so in other words, given that there's a map into the structure of the matrix, meaning that the big algebraic category, the natural number on one side is the exact number, but on the other side, the category is determined by the cohomology theory. And the cohomology group is on its basis, and it turns out that the cohomology group... All these natural transformations, some of which are rather complicated, but they all, they're all interesting in variance, the non-isomorphic, the two-spaces, the two-spaces, the isomorphic, the rings, the not-as-groups, but this is a much richer structure than the ring. It's kind of the ultimate amount of structure, distinguished spaces. And there you see there's this slightly surprising fact that even the large categories of the small numbers have the power to measure up the doctrine of finite products, which is the mode of power, which is the capacity of gravity.

25:00 There are very several others, not real homework, and it does express this idea that the abstract has got more structure than nothing, yet less than class, precisely because of the... Naturality and all the relations between the mathematicians are incompatible. This idea of the relations between the actual mathematicians and the natural ones are really the ones that come about because of the fact that we view them as natural. It does seem to relate to Richard's point about the quay, the quon, and the Aristotelian relation of the quon. Well, Aristotle's understanding of what all of science is, that they're all studies of the same subject, what distinguishes natural science from mathematics, that's our, we ask that the natural sciences study science of what that is. Well, physics, according to your studies, is being, quad, moving. Moving, moving. Yeah, and arithmetic is being, is... Geometry is being, quad, moving. Okay, whatever. Right. So I can say you, I can continue, which means I don't think... The abstract general is a structure, what I'm calling a structure, a natural structure.

27:30 The semantics then is a concrete general, a response to them, but the ethos are much more general. We only had identity going down that way. The E's are much more general in the categories that are precisely captured in biology. We get the economical inclusion of the particulars. I think you've got a more general notion of structure than Klobocki, but if you took Klobocki's notion of structure, his notion of structure is what's sometimes called a signature. His is far more general in terms of the kind of structure. I'm just talking about the bridge structure. I say that there are many doctrines. The doctrine of finite products is about a general algebra, which is not really that general. It's about finite products, nothing more or less, over and over again. Key is attempting to deal not only with the first book of the library, but even the higher order, in the sense of the richness of the kind of stuff. But the curious thing is, he seems to be... Much more precise than his... In the very limited context given rise to the abstract category are included in that abstract. Well, I say included. Of course, this arrow may not be the worst. It's certainly not, well, typically non-bonding.

30:00 The thing is, if you start off with an algebraic category, then of course you can get back to the same thing. It's a claw, of course. When I say being, I mean being clawed. So, people claw, I mean, mags and organic law people, people claw writers and books. I guess there's a subtle distinction here. Without, before we have a formal theory, we know that certain things are related. Because of their, the claw such and such implies that there's a preferred kind of comparison between instances. So it seems to me that you're either here in this general dialogue. It's not so much you're referring to some sort of existing book, but you're saying it's a kind of challenge. If you're going to handle this stuff... So that you can set the thing up. It's not as if these things are just lying about waiting to be picked up. So it's kind of like a challenge if you want to be able to recognize the possibility of doing it. That itself is a result of past science.

32:30 There's still, in a sense, I may be misunderstanding you because I'm, but there's a sense in, but there's, it's naive. If you're teaching, say, first-year students, you're teaching about relations. It's not as if you, I mean, it's a, it would be an odd thing to say that you're going to, that teaching on set theory, you're going to teach everything in, everything in mathematics. I'm making a serious point here. When you, if you look at categories, you see things that you're quite familiar with. I mean, for example, you've got, you've got maps, and you've got, so it looks like... There's a sense in which you're just doing the same kind of thing you do when you're playing one group so far, or a topological space. You're doing the same thing. Well, I mean, you're doing the same sort of thing. You're saying, look at certain things and certain operations. And the operations are definite. You can tell when two operations are like two compositions, or a composition and a third operation. So in some general sense, it's already, you're still doing it, except that now you've watched it yet. Insofar as the theory of the 20th century, they found ways to define homomorphism.

35:00 No, I'm just thinking about the notion of category itself. Oh, as an algebraic structure. Yeah. The fact that it is a super-preferred algebraic structure, that takes a lot of structure. Not quite in this sense, but you see, as I said, there's another doctrine that's slightly different, where instead of finite products, you have finite limits, equalize them, and then it turns out that the abstract general, any small category, finite product, or the corresponding concrete general, is all video, finite limits, and very concrete sets. So that includes the category of categories. If you're teaching group, use this abstract general. Basically, it's just an underlying domain of the final operation. You're hanging in the air and you say, for example, or take a set of letters, and this is what you mean by, this is what you mean by the identity. You've got these sort of simpler examples. Actually, for a student to be able to do those exercises, he has to perform the operations. And notice the structure of it. And then...