Unification & explanation, algebraic geometry / discussion (contd.)

0:00 I pointed out with those cases, like the previous case, for three years, different argument happens for different classes of five, and classes of problems will require different argument happens. In short, whereas with respect to the specific problem in this capital maximum of polynomials, all three proof strategies can rise to a certain uniformity in a treatment of statements falling under that class of problems, only the strategy that relies on the past pre-statement of the decision procedure can be extended to all of K, in such a way that the single argument and actions apply to the regeneration of all scales. Thus teachers' model of explanation needs to have inclusion, and thus causating after that the systematization of the explanatory one, one which in practice does not enjoy the appropriateness of explanatory that teachers' models would seem to distort upon it. The conflict with mathematical practice emerges because Branty has different intuitions, and favored as explanatory increase those of type 3 and rejects as non-explanatory those of types 1 and 2. And in particular, to surpass one, I doubt it would be anyone around that would claim that they have any claim to explanatoriness because what kind of insight can you get from just finding the algorithm to get the one at the end and say, okay, now it's 2. And so I think it's a serious problem, we think it's a very serious concept of teachers' model to end up claiming that exactly those concepts will turn out to be the best an accusation, i.e., the explanatory score that we should refer on explanatory grammar. I think that's just not going to work. In the rest of the paper, which are not longer capable, we also try to see whether we can to do some partial work, for instance, making a primary distinction between groups of type 2 and 3. What does it project concerning 2 and 3? Would be one better than the other? Of course, it turns out that it refers, right, that if the model works for 2 and 3 and projects that the groups using the transfer principle are more explanatory than the ones that are uniform the algebra, well, that's yet another level of conflict with branches and so on.

2:30 But we're still working on the details of what we can do today. Thank you. I'm wondering, I'm not quite sure what it is you're saying, but it just might be, it might be, but you think an algorithm, and what you want to know is how something being an instance of an algorithm that it compares to its having a certain pattern implemented, that's what we call it. It doesn't seem to be particularly polite that that would be a good satisfaction. There might be a lot of interesting stuff going on inside this algorithm. do you want to bring out in terms of that? Is that essentially what you're saying? Well, one thing we're saying is certainly the way the knowledge has a sense. It doesn't do a bad job at determining what might be allowable if argument happens. And nothing in what teacher has been presented. In fact, what he presents suggests that our proposal of of seeing that very simple argument happen, as we leave an argument, I think, is perfectly within what people are, in fact, thinking about, except that, of course, that wouldn't be focusing on this particular example, because I think it would have seemed that that would have trivialized something like this here. So, in that sense, that's a good proposal that I'm making, in the sense that if you want to rescue teachers more, that might be go to say, OK, well, let's just see if we can get around your counter-examples by moving on appropriately the notion of art in the past. You know, I think you might end up having a lot of words today, because I think most of these facts will look at all, and I think there might be other examples of that, but he is very aware, I mean the teacher is very

5:00 aware that these arguments happen, that you have a lot of flexibility in how you choose an argument. When I said, you know, from Socrates in Greek, you can, you know, take off Greek, you can take off Socrates, you can get two variables, you can take off, you know, leave And imagine this in the schema, for instance, that, you know, it loves to put down this argument happens and you make an argument in which it will determine the notions above. That is the main paradigm of what the argument happens to you and how it works. So you have a lot of flexibility that there is nothing in itself that you think anything to assume that this is perfectly appropriate part of the time. So I'm concerned, so let me try to represent the picture, and I'm concerned that you played a very little bit here. That's logician's word. No, the Grumfield statement, the statement after, is a statement that every semi-algebraic function on a semi-algebraic that achieves the maximum. So that's something that can't even be expressed in the right way. That's fine. And then there's... Now remember, we're systematizing K. Well, but that's what I'm saying. We're not systematizing K. So Grumfield, the issue is that statement. So it seems what, I mean, Grumfield has to say, no, no, what's the best group of that mysterious statement? You deny it. Yeah, but, um, so we have to distinguish two things. Of course, people in mathematical practice are not worrying about what the teacher is hoping to do, right? They, they have the theory. Right. But the point is that these theorems, from the point of view of the teacher, represent systematization of specific individual knowledge. So, for instance, you can look at the situation now as, you know, suppose you have grown about, I guess one way to do it, first discovering that, you know, x cubed has a maximum of 0,1, then you move back to the situation, you see, well, that's a general theorem, and you prove a general result. Now, why are the theorems by Brownfield systematization, in the sense of teacher, because now from the general theorem

7:30 by Brownfield, by infant creation, you get all the tools in K, i.e. the real close field theory, that also the algorithm gives you. In that sense, the systematized, remember that in the right notion of systematization, you can have premises in K star, which you cannot express in the bold K you are trying to systematize. And so here there is a very simple argument pattern, for instance, from Branson's third group. You just take the theorem and then you give an extra rule of instruction by saying instantiate on specific functions and intervals. That's a systematization of that particular problem. But is it fair to characterize both fields? So when you compare evaluations of exponential power, and is it fair to say both fields? Well, well, no, wait. Certainly, we want to say k is the curve of both fields. I imagine that people working on real algebraic geometry or semi-algebraic geometry That's why people work on this. They try to... That's just reading, because of our knowledge of K, there are general statements about models of K, and those statements are not... Yeah. Okay, but you see, you're getting stuck here on a distinction that I perfectly well appreciated, which is the theory of real cause field is expressed in the language of personal It's not, right? Like the theory of groups, like the theory of groups is expressed in the very basic theorems of group theory between groups cannot be even expressed. But that's not the point. The point is this, that no matter what Brownfield is going, and it's certainly going after much more than simply the elementary theory of both theory, the outcome of this work on the whole general thing will also be a systematization of the elementary problem. So we say in order to make this model of Kinshaw, which is extremely bad, I mean that's part of the difficulty of doing this stuff, trying to find an example that is approximate in intention what you might be thinking about, we say, okay, specify now as a body that you want to systematize. And that's what we take to be simply the set

10:00 of elements of the theory of real force. We don't need the intention of Bramfield. In other words, we don't need that, you know, that Bramfield's explicit intention was to systematize that part of the body. All we need is the fact that an effect of the systematization of all of the algebraic semi-algebraic theology functions also, according to teachers' categories, as a systematization. So remember that the notion of systematization is a very trivial one. It's just a psych of argument that starts from elements in K or K star at this point, down to K. So any work you do on any psych of statements, if you have proofs around, which is true thing, that's always systematization. And that's why, for our grantee's work, count as a systematization, and then we can ask the questions we can. Yeah, this is fascinating stuff, I'm not sure I understand it, but I'm looking at this sort of a much more whole story on the block. So the first thing that strikes me is this undefended claim of uniqueness. Right? Why not a problem with that? Another thing that strikes me is it's going to be very hard sometimes to count the number of productive patterns. Think of a baby example. I mean, you could wipe with some problem in the internet, right? So there's another pattern that deals with an implicit function with that respect to why. But you would never teach that at the beginning. And you can pass through, you know, quite a long amount of universities, a problem that I can agree with, and never discover how you differentiate, you could count that as one pattern or two, and different audiences, different things. I imagine that sometimes time is a more serious point, that if some deductive, some chiturization, can I call it that? Yeah, I think it's a good term. would count seven, and then some comes along tremendous theorem, and that number rack was count four, okay, at least count six. So, that said, I mean, you have a nice indication of how Kitcher's theory fails. It's still the case that Kitcher's theory is precise enough to be worth 14, which is already a merit.

12:30 So I'm a little bit inclined to want to keep it alive, and I guess in which I write your paper, you want to keep it alive. We want to give it the best shot. We're very generous, though. Yeah, yeah, yeah. So what I'm saying is, why not think of this as maybe a partial order on different theories? It does a good job here at explaining why this theory is better than that one. But maybe not such a good job over there. And we get a set of overlapping criteria for good job, bad job. In other words, you're suggesting that this unification criteria might work in some cases but not in all. Is that the second suggestion? That's what you say, what I'm going to jump here. There's some position in the capitalist world that I would personally find helpful. But I then want to topple it to the non-positions that I subject historically. Right. Right? And they simply disagree. I mean, you've done a good job on showing that the one theory that teachers would prefer is one moment of perfection out there in the real world . Right. But it might still be the case that you, if you allow for these various impositions in that we can get a kind of tree, I think, that we could locate our, we the story, we can locate our mathematicians on. I see. Well, but then the cost of that, the fact that a novel becomes very annoying to you, because you can then retailer it every time, depending on what things you want it to, you know, come out. Same. I think I want to come back to your first one, which I think is a very serious one, the problem about counting and counting. If you think about the two criteria I gave you, both of them are numerically very hard to use. The set of consequences of a normal body that you want to systematize is usually intended. Already that is a problem. how do you distinguish two systematizations that, in fact, derive infinitely many sentences

15:00 in both times. The second argument pattern, it was implicit in what we said. We make it explicit now. The only reason why we were able to pull off this work, and this perhaps the treaty we're talking about, is that we had a situation where we could count. We had one argument pattern, and we could make a very broad argument that said every must be less than or equal to, in terms of quality, to this one, because it's got to have either five too many patterns or infinitely many. We guess that in the standard situations it's always infinitely many. When we said, how would the pattern of polynome for polynome was applied to Brauer's fixed polynome? Obviously, you need a different pattern there. You will need a different one on another class of problems. Usually, you end up with infinitely many patterns. Now, the risk is probably great also, that depending on how you play with the details of how you determine an argument happens, you'll have different factors, you'll start adding different numbers showing up. In the most extreme case, where you don't replace anything, right, you might have certain results, or then you can have the other extreme case where you just look at the logical framework all the non-logical content out and then focus on that. So, these are all problems for them all, not for us. In other words, when I say we're generous, I think what we try to do is to give it its best consideration. You know, we think this other example shows one very important thing, that counting patterns doesn't seem to be really valid when it comes to accountations. It's not on a key case like this. Now, could it be the case that there are perhaps situations where in fact the mere counting of patterns might be not to account for this one? We do not assume that. That's why we say there might be a lot of situations in which then all it could still be salvaged by that disease. All right. Do you have a very short one? I'm not sure how to sort out an intuition theory. Suppose that we had a decision in a given area,

17:30 but it was nice and clean and short. Many of my thinks is the number one one that's really the right one. Yes, that could be a possibility. The problem here is that the decision procedure is infeasible, and we know it works, and it is infeasible, but because it's just so complicated, you can get it. I don't think feasibility is what's going to do it. It's also unfeasible to determine for an arbitrary file whether it holds or not, or not, and the strategy works in the same way. Yeah, of course, yeah, you couldn't turn that one into a number of prints, but something like B-mobility, though, in this respect to what we talked about, to what extent does explanation of how in the time of B we are, if we can grasp the algorithm in some sense of practice, then we can not be able to grasp the algorithm. Right, but then it's not the number of parameters that ought to be, right, yeah, yeah, in fact, no, there could be very well a case of explanation where, in fact, we start from the algorithm So we understand this algorithm so well that we, you know, that's the explanation for us. That's fine. That's fine. The algorithm is well understood. Very well understood. Well, except that what you don't understand is what's happening to the particular statement. A particular run. In a particular run, they say that you can... Exactly. That's what you don't understand. Okay. Thank you. Thank you. Thank you. Thank you.