Varieties of mathematical explanation (contd.)

0:00 I mentioned that case of Cranston's, now all the foundational concepts in the world. That was an attempt to reconfigure the whole of Cranston's. But here is where I see the tension between the local and the global, and this has been addressed in the literature. Suppose you start from a global reorganization. You find what you think are the central concepts that are going to give... You start writing down your proof. Well, the type of proof you were calling for was basically the definition that you're making. But now the question emerges. This reconfiguration. You might give a proof, and then you might give an alternative proof of the same result, in which those conditions that you mentioned are all satisfied. But, I'm going to make so many detours. I'm going to end up with a proof of 2,000 numbers, which I'm just doing it for the sake of the argument. But just taking a global view, it's not going to... But that seems to me a serious theological problem. In other words, in my view of things... See, the way you were expressing things, it was basically, look, if we get a global organization right, basically the local will fall. But if that's the idea, what I'm saying here is that you will not be able to get the criteria for distinguishing explanatory versus non-explanatory just with a global criteria.

2:30 Now, there are two ways to go about that. Still hope to get some criteria. Or, what I actually do, clear-cut criteria for really the cases of global reorganization. Those, to me, are the very best. I tend to look at these cases here as two... It's almost difficult to analyze and to really get some sort of criteria. Of course, the example still works for what you want to do, right? We're using it as a counterexample to a theory that was proposed. But when it comes to providing a positive theory of explanation, my hint is that one should go for the global view. You see, your idea, as you were explaining it, reminds me very much of certain models that were given, starting with Aristotle and then with... The idea was this. Look, there is a realm of truth. Some are the grounds and others are the concepts. And this is an ontological order. Logically speaking, we might actually find proofs that go both ways. You might appeal to something lower down on the scale, use it as an axiom, and get something that should be higher. In other words, there is logical reversibility sometimes. Assumptions and conclusions. Bolzano would say that it's not right. There is just one order in nature, and if you get that right... Again, I say that this model fails exactly because in the end we are unable to say why certain theorems act as their grounds and always have their consequences. In other words, we lack any intuition whatsoever that will allow us to make that conclusion. All of this just to say, though, that this particular model's improved structure has been looked at and it still doesn't quite solve the problem.

5:00 Basically doesn't solve it because we have no criteria whatsoever that can be operational for deciding what is it. So this was a long answer for saying that you have touched on a cluster of topics which is absolutely central. Just as a point, but coming back to your examples, it's exactly this problem about lack of this globality, what probably Bernhard mentioned, because I agree that there are a lot of senses of what this explanation, there is no one that's novel. No sense to look for a single one but still I would say it's sort of sort of illusionary explanation in both cases probably right because in your first proof it refers to the fact which is basically less obvious much less trivial I mean uniqueness of representation and product of primes so this second proof depends on the fact which much... Which is not clear at all. And similar thinking with the second example, right? You're seeing things epistemologically. In other words, the prime factorization theorem is something we established pretty much down the line. But this kind of intuition, you see, they're ontological. They say that certain properties are ontologically quite central, and they are the characterizing property. Epistemologically, you might arrive at them very late. Very late. I'm not saying it's the right thing. So that's not an objection. The fact that the primary limitation theorem is a, you know, it was an explanatory proof, epistemologically speaking, because it comes on the fewer axioms, you know, that you get, not according to a Platonist view of syntax. See, you have to distinguish here. There are two ways of looking at explanation. There is an epistemic way and there are epistemic theories of explanation. And there are ontological theories of explanation.

7:30 The epistemic theories say explanation is deeply tied to your understanding. The ontological theories say understanding has nothing to do with explanation. Explanation is an ontological move. Aristotle, Bozzano, Frege, in certain, Bouligan. There are very deep problems with that position. It's wanting to attack it for the right reasons, but the ones you are mentioning are not the right ones. All the elementaries, in general, belong to the same heterogeneous. Organize the proofs, therefore, even if far from each other. You have the step. This is the case, for instance, in analytic numbers here. The analytic continuation of the z-atoms gives you a lot of information on the distribution of primes.

10:00 But you have to code an analytic proof. And then you use imaginatively the zeros. And so, Cartesian, right? The idea that's simply concatenated logically in different areas. Quite every interest is necessarily global. The great, great scientists that were not manned, and for me, the power, the fundamental qualities of Giordano, an extraordinarily important amount of detail. So you would still see those proofs as the primary bearer of, as opposed to it, because I heard Pierre Cartier saying that it's basically the reorganization of the whole body. Steps, architectural or archaeological. And I think that the fundamental error of the logic is that the idea is concatenated. The fundamental hermeneutic dimension is that you interpret this theory in another theory. Well, as a genius, the white school, the essential of the...

12:30 If I could, though, one point only, I don't care very much to define the logic, but I think that hermeneutical dimension I'm talking about existed in that project. After all, it's not like you find the natural numbers already in logic. All those structures have to be translated into logic. And you better have an hermeneutical criterion... Logically, it's a cut elimination. In the logicist program, you have to define what the natural numbers, the real numbers are in terms of logic, and that requires an interpretation of theories inside the logic theory. So that aspect is there, but I don't particularly, I'm not defending here the logicist approach. I actually find that a problem like this one, it's exactly interesting, completely beyond what the logicist approach is. There is a very deep and in exactly the same sense you cannot reuse mathematics. But why you cannot reuse? Essentially because you have a semantic level, semantic association with many different semantics, many analogies. But I think it's exactly the same problem here. So the explanation cannot be reduced to mere logic. Good. That's what I meant. Just to add something more, I entirely agree with Jean-Pierre Cartier's approach.

15:00 And I think that the key point where we are radically departing from Descartes is the fact that for him the reasoning was a chain, first point. Those elementary components had to be very simple in order to be intelligent. That's where I really built the parts in these two points. First of all, the elementary components may be very complex in various ways. Jean mentioned, because you may refer to another theory, a very complex one, which is brought synthetically by translation to speed over time, which remains to be elementary. That's where we have microphysics that is very complex. You have to face that. And that's also true. It's natural. That's where it's natural. A complete formalist failure of giving an account of induction. I mean, mathematicians understand perfectly well a very elementary step which says, look, I have a set of numbers which is not empty. Then you have for a century, in order to avoid synthesis of a variety of practices, it is an invariant built in.

17:30 So the reference is not only to other theories. Well, at least one thing I might probably would like to answer that by saying that the kind of problems you're trying to solve by doing a logical analysis are not the same that you can address by doing the kind of cognitive work you're coding for. But I think we're all involved on one issue. Logicists or logical reduction processes are not the ones that we can hope to, or at least, there are a lot of problems, I think, to do with our understanding of mathematics and our doing mathematics that simply cannot be accounted for by means of logical analysis. That seems to me, and that's what I've been arguing for here. Now, where we differ might be simply on how much does the cognitive component... In other words, I can very well see that somebody might say to me, look, explanation is an explanation, you better, I have no objections to it.

20:00 But, since explanation is not such an instrument, that gives us a more stable view. Which one is right, I don't know. At least we have basically, we have the picture of what can be done and what we should go looking for. It is somehow, or explanation, true that many of the most univariately cultural, or clear-boundary children in this country... We'll take just two minutes, and we'll go back to the example, because there I think there is a kind of objective, frankly. If one looks at the proof by contradiction, it doesn't teach you anything about other numbers. It doesn't even tell you that other numbers, like the root of B, or this root, or that, could be irrational. And then you have this fact that if you know that a number is a square, it gives you an algebraic information for the commutative fractional expansion, which implies that it has to be given. Take a cube root. It's still an open problem to know what it says in the continuum of fractional expansion. It's a problem that has been open for 2,000, 5,000 years, and today we don't know anything anymore than we did. But nevertheless, if you look, there must be so many rational numbers, all of which have an infinite. And then I look at this very tiny class.

22:30 Very simple algebraic rules, it has been enormously enlarged by the fact that I can see how many irrational numbers are and that there must be some algebraic rules which imply that it's infinite. So I think this is, to me at least, according to my culture, it's much more explanatory because it produces a world in which this phenomena inside the very... A great part of it is devoted to programs in the symptoms of reconstruction that completely avoid proof cycle, exactly on this sort of ground that they basically gave certainty, but they provided no evidence whatsoever for why the result. You're carrying in a long tradition of... Okay, thank you.