Lecture (contd.)

0:00 So it is a comparison between Hilbert program and logical empiricism. The scope of Hilbert program is mathematics only. The scope of logical empiricism is the whole of knowledge. So therefore I divided it into mathematics and science. Hilbert program is simply a research program. It's fair, but it's not important at this moment. Logica empiris is a doctrine, as far as mathematics is concerned, and research program as far as science. At least precisely define research program maybe. What does doctrine mean as opposed to research program? Research? Maybe that. Doctrine is something which you accept and you believe that it must be true. A matter of faith. Ideology you can say, yes. But program, it is something which may be connected with some amount of faith. You know that you try to achieve some aim and you know that there is always a risk. Maybe the aim is not achievable.

2:30 In this area, but not in philosophy or mathematics. It's not my territory, but I have the impression that logicism definitely failed and they needed it if they were to classify mathematics, pure mathematics, as analytical. So, yes, in the 30s, Karna, for instance, he was a kind of logist. What do you say? Logistic, yeah? Kind of logistic. But it was not very pure logicism. Like, let's say, Frege. But it's a very complicated question. When I say that it is doctrinal in mathematics, I mean Carnap's writings, let's say, in time just before or after the Second World War. And I mainly have in mind his paper about theoretical objects. So he simply says that the problem of universals is me. It is their solution of the problem of universals. Do you agree? A patron, a patron. Yes. So it was a kind of research program, wasn't it? Yes, it was a research program, which failed.

5:00 All right, but Hilbert failed as well? Hilbert's program also failed, yes. So maybe part of it, but part of the research you did right? Don't you think so? I have an impression that Carnap first was a kind of logicist and next he abandoned logicist because there were many difficulties within this doctrine and he shifted into a kind of radical formalist. So his latest patron we could say was Hilbert but Hilbert ill understood. Because Hilbert never attempted to say that mathematics is a formal science, formal knowledge, according to him, for Hilbert, formalization of mathematics was, according to radical formalists, there is not mathematics beyond formal mathematics. Non-formal mathematics is, at best, a pre-scientific study of mathematical theory. So the final proper form of mathematical theory is formal theory. That view has nothing to do with it. I change the picture because now I have this last transparency.

7:30 Detail. Heritage of radical formalism in contemporary philosophy. Identification. Precise equals formal. Seems to me the most radical formalist in the world. With format, definition with clarification. Clarification of a notion is a process. One can define a notion if it is clarified. One cannot create clarity by means of definition. Definition is a means of expressing this clarity achieved on other means. Metaphysical concepts may be not all, but contemporary philosophers use the term understanding. In science, for instance, there is a thesis that science consists of formalism plus metaphysical interpretation. All those discussions concerning the interpretation of quantum mechanics are exactly in this field. And the last one, there are some, is the notion of limitative theorems.

10:00 This is Gödel or Scholem-Levenheim. Theorems are called limitative. In this sense that those theorems determine a limit of human cognition. They determine a limit of expressing our knowledge informally. I had no time to elaborate those points maybe during discussion. How about, is it your first conspiracy, your thesis, and now we see how... That's right. What I promised and what I did. I'm not quite sure. For me the words, the problem of universals, I think is their redness and about red things and... Not only is this problem of universals in physical world, but there is a problem of universals in connection with empirical knowledge, but there is also a problem of universals in connection with mathematical knowledge. Would you like to just spell out what you mean here about, in this context, the problem of universals? Are they numbers, natural numbers? Are they real numbers? Are they, let's say, Hilbert spacepoints? So, could have been, I mean, could be the problem of the ontology of mathematics?

12:30 The existence of certain ideologies? No, for sure, those problems are not solved, but nevertheless it is a problem. And there are two solutions, two possible solutions to this problem. Platonist solution and constructivist solution. There are two solutions, possible solutions of this problem. Nobody presented a proof that one solution is okay, another not. Okay, you characterized the problem, okay. So, in every science, and in mathematics in particular, you have some abstract terms, and you can ask, what are the references of those terms? Have they references at all? And if they have, what kind of references? So, both Platonism and Constructivism claim that the terms of mathematics refer to objects. But there is some divergence between them. What kind of objects they are. That's the problem with universals. The problem with universals are existing independently of human beings. Yes, but we teach in the... If you construct it with interpretation, they are constructed by human mind. Thank you for your attention.

15:00 What? Yes, I don't think it is... Platonistic position. No, no. By the opposite, no. I said that now, in the Platonist approach, the objects are existing, and you are discovering them, like mountains, and for the constructivist, you have to invent them, construct them, and they exist in this sense, but only if they are not contracted. Just to go back to your view, you said Hilbert thought that Platonism was okay, and you promised us that in fact it would be much more for Platonists than it appears at first sight. He was not a radical populist, I agree with you, but in what sense did he think that Platonism was okay? Do you think that the aim of the Hilbert program is simply to show that? You said you were going to show that according to the Hilbert position, Platonism is okay. He was convinced. It's a kind of reconstruction. I was not talking with Hilbert. It's a kind of reconstruction. So what I say is simply this reconstruction seems to be...

17:30 It seems to be plausible, because Hilbert's program is an attempt to show that all those infinitistic methods of proof are okay, and they make sense only how you can know that infinitistic methods of proof are okay. You can know it if you know. If there are all those infinite objects, if there exists actual infinity... No, sorry, I think you're mistaken. I mean, certainly he wasn't a radical formalist. For one thing, he believed in the existence of numbers, but the way he talked about them, I searched on a piece of paper. Well, I mean, he didn't say anything more on them. He didn't say the existence of the cosmic level. So he's not a radical formalist, I agree. He didn't accept... The infinitistic method means of giving us information or enabling us to deduce certain theorems about number, but this does not make him into any kind, because these are instrumental, he had an instrumentalist attitude, an attitude to idealism, to ideal ethics, you know, and to pretty instrumentalism, and that is the, that is actually the originality of the position. Right, but it was, I could say, a temporary instrumentalism. I could say he was a kind of Platonist who used such phenomenological brackets. He was able to remove them after the proof of his program, after the proof of conservation principles. It was everything contemporary. There is some evidence. You can look for historical evidence and you can look into the program better. Let's first recollect the famous Hilbert saying concerning the Cantorian heaven.

20:00 Could you interpret it otherwise than Platonist? Heaven is the profusion of methods at the disposal of the classical mathematician, which he did not want classical mathematics to forego. It doesn't mean the heaven exists if it consists of a platonic heaven corresponding to certain linguistic statements in the proof. So you interpret it only as it concerns methods of proof? He said it. Oh, not only, you know, it's not only conservative. You know, he said, if I remember correctly, sometimes as I read him, he said that in fact there are certain theories of number theory which you could not prove without using the infinitistic methods. But the actual content of these lower level statements, which are not real, they aren't real, but they are above and above. No, no, no, no, no, only in the real case. It's decidable. There are no levels, but you see what I mean. They are about whole numbers. They could not be proved without the use of ideal entities. We introduce ideal entities only in order to deduce this result should be otherwise uncluable. But nowhere does he say that the reason for introducing the infinitesimistic method is because it goes for something somehow. He actually denies, specifically denies that. He is not, of course, a radical formalist, but he is certainly not a totalist. If you are going to say that Hilbert was non-consistent in his annotations... So maybe he was just in a position? Yes. He's useful. I understand it. I understand it, but the problem is, was that his conviction forever or temporarily,

22:30 till the moment when the proof of conservation principle could be found? That's the problem. Unfortunately, the proof was never found and we know that it is non-existent, so therefore the discussion is a bit difficult, but without hope that there are platonic objects, I think that it is... My reading of Hilbert is not that he introduced ideal elements only in order to simplify proofs which could otherwise be constructed. It is to increase the amount of things you could prove at another level. I don't think he was banking on a general conservation theorem. I don't think so. I can agree that this is, for instance, von Neumann's position, but not Hilbert's. He said it. I mean, he said it in... He said many things, but those things are inconsistent with the program itself. So, maybe I could be that radical, formal, inconsistent with Hilbert's program. Inconsistent with Hilbert's views. Hilbert's program, I don't know, Hilbert's views and... What opinion? I don't know exactly what is Hilbert's philosophy. I know better what is Hilbert's program. And knowing what is Hilbert's program, I want to proceed to Hilbert's philosophy. That's my strategy.

25:00 Simply, if you accept radical formalism, you cannot formulate Hilbert space. But what is radical formalism? Could you define radical formalism? It makes no sense! Why not? If you have crisp position... Well, you wouldn't, presumably, why could you? You wouldn't need to cast away all the mathematics, no? You cast away all the system and then you use far-ratistic method to prove that it's consistent. And it is complete. And why not? Excuse me, if you think that mathematics, if you think that mathematics, the whole of mathematics is a pure formal science, what sense makes distinction between things? What is the sense of this distinction? No, no, I'm sorry, there are two levels here which are, again, which are confusing. The actual mathematical theorem, the mathematical theory, is a pure formal system, which has a certain... According to radical format, or according to you? I'm talking about the value of formalism as reconciled with Hilbert's program. Mathematics consists of... I think I'm saying something which is too very low. It's a formal system in which there is a sign which we usually interpret as not. It's only a hook. Now, you can then... your meta-proofs are not, of course, formal. They are about the sequence of signs. And you can actually set up as the following aim. There are so many of these purely formal systems that you cannot derive within them P and not P at the same time, and that of P and not P you derive only one. I mean, why not? What is wrong with that? I tell you, if you have a formal theory, if you prove that it is consistent, so you cannot find formula A that is a theorem and non-A is a theorem. In order to make proof of mathematics and such...

27:30 Call it meta-mathematics. Doesn't matter. But it is nevertheless mathematics. And you must know that it is not producing false results. So I can take any crazy theory on the meta-level in order to make this proof. In some cases you shall believe me and in some cases you shall not. It depends what I take as a... I understand. Your idea about radical formalism is not that somewhere there is an interpreted math. You see, a radical formalist is somebody who denies to formal systems any reference, to systems which are given as formal languages any reference. All of these elements, occurring in a form of theory, are irrefutable or abstract. Now that is impossible. Now, Hilbert was not even that. I agree with you. Hilbert was not even that. Sure. But this does not mean that his program, as you put it forward, is inconsistent with this radical formula. We will view that it is a schizophrenic attitude, saying that on the level of science there is quite other philosophy, on the level of meta-science quite different. Which the inconsistency seems to be. There's rules of chess and there is chess, I mean, I don't know. I mean, you know, I mean, and you can argue about the rules of chess without attaching any... No, no, no, I mean, I'm emphasizing one morality on days of mid-week and another in Sunday. Very consistent, actually, you know, and practiced by everybody too. So, I mean... So, maybe, are you convinced that radical formal...

30:00 I was prepared to convince you, of course, if that is accepted, that Hilbert's program is simply involved under the condition that this is accepted. Repeat once more what I understand, what radical formalism is. It simply says that the problem of universals concerning mathematical theory is universal. Next one, let's say... No. So you don't know that you have two terms in each term? One. One. Moderates? Moderates. Moderates. Moderates, formalists, it is such a point of view according to which there exist constructive objects. So all those objects are non-existent. So mathematics is formal as far as it refers to ideal engineering. That's moderate format. Name that methods of...

32:30 You know, they don't produce them in the realm of construction, but some of these proofs are not reducible. It's something to consider in extension. Some of these proofs exist only by means of ideal entities. And in fact, Hilbert, if I remember, comments on this, what is the value of such proofs? Because you talked about melody. What is the value of such proofs? I think they show that the proof of the contrary is very, is very improbable. I think he says somewhere in the Bon Blanc de l'Amérique sometimes that if you prove by non-constructive means some film at the lower level about real entities, then you render improbable the refutation. That's what he said. I mean, I'm not saying, which is very, very dubious. Agreed. But at least it shows that he was, he did not say that the reason why you actually obtain a so-called correct result, he didn't even call it correct, is because your non-constructive methods correspond to platonic entities. No, where do I see this? I agree with you. He read it, he wrote it. In fact, he pondered about how is it that he has any confidence in a non-formal result which is obtained by formal means. I mean, I don't know the exact passage. If you give me time, I'll find it for you. All right, yes, it's in Hilbert, I'm sure it'll be in the introduction. Or else in Hilbert, I don't know. You see, he did write about this.

35:00 A, of course, in the case of a conservative extension, there's no problem. There's no problem at all because it's only a short term. But in the case of a non-conservative extension, then of course, the problem you pose is exactly what all asks oneself immediately. What confidence? He himself does not do this explicitly. It is not because, you may say, the National or the English, of Hilbert's program does not go through without criticism. That you may be able to say. It's certainly not Hilbert. He was speaking on philosophical approaches and therefore he was straining down on philosophical methods. So what I'm talking about is, he cannot be in complete agreement with all his negotiations. Accept. We're looking for an entity because I've already found out there are many atoms in the world, isn't that so? Some people believe in being a piece of the universe but aren't so focused. It's very hard for somebody who's not statistically defined. Alright, I guess that is an argument. Well, I'd have some sympathy with what Mischel was saying here, but what he actually said in philosophical discourse may not have actually fallen out as true, but there's evidence why, you know, what he said in philosophical discourse may not have actually fallen out as true, but there's evidence why, you know, what he said in philosophical discourse may not have actually fallen out as true, but there's evidence why, you know, what he said in philosophical discourse may not have actually fallen out as true, but there's evidence why, you know, what he said in philosophical discourse may not have actually fallen out as true, but there's evidence why, you know, what he said in philosophical discourse may not have actually fallen out as true, but there's evidence why, you know, what he said in philosophical discourse may not have actually fallen out as true, but there's evidence why, you know, what he said in philosophical discourse may not have actually fallen out as true, but there's evidence why, you know, what he said in philosophical discourse may not have actually fallen out as Omega and two to the omega. That's all there is. I mean, if you can prove it. If you assume the opposite, you can deduce it's not a matrix.

37:30 You see, that you can do. There are purely formulas that does not commit you to any statements. You may say you had something else in the back of his mind, maybe some kind of a model. But you can prove that is possible without any commitment to it. But what about a matrix? Why can't you identify it? It's unsubstantial. Yes, yes. You can tell me, why did Poitier bother to do classical analysis when he was an intuitionist? No, he was ready to show some tension between this sort of hurtful idea and the other part of the issue. But exactly that kind of problem he actually commented on. This is why I found it so. Because it actually obviously occurred to me. Of course, you use ideal entities, you use something about low down the scale, what promises do you have? If you didn't have these ideal entities, you couldn't get them. But he didn't answer, a completely anti-Keyboardist answer. Maybe I say one thing, that it seems to be quite evident that Hilbert programs an attempt to beat constructivists by his own weapon. And if that is right for him, he is not alone, but to take for granted the assumptions. It means that for him constructivism is not enough. It might be also a problem. It's almost a present idea what you are trying to present here. It's just my impression that what Hilbert tried to do, he tried to establish the practice of the working mathematicians, and you can say that the majority of the working mathematicians even today are mathematicians themselves, but if you take it seriously, it doesn't mean that Hilbert is a scientist.

40:00 Mathematical philosophy or physics called patience. So that's a very different thing. He definitely had a different approach. He tried to institute those methods. We can establish statements about mathematics. And that was quite all right in your presentation, as it seems to be mine. But to conclude that Hildegard, he was a patient. It's just too strong, because it was a genuine approach to the whole problem, the soul of the problem. Platonism, I don't say that dogmatic Platonism, such kind who was able to appreciate criticism, constructivism, to see that at least prima facie it is strong, it must be answered. So I see the Hilbert program from this point of view. You can conclude either to Platonism or to Radisson. Can I try a question here, if it will be helpful? For the sake of argument only, just let's go along for the moment with the idea that the object level, mathematics, its streams, uninterpreted, the whole of mathematics, the object level, I mean the actual world, The program is somehow to prove you had a T. The T then, I don't know about the whole of mathematics, but this T here is to be understood in that way.

42:30 And then there is to a meta-level. The problem is to of A and not A out of T. I simply claim that there is no difference between that you cannot adopt such a schizophrenic policy that something is quite different on the one level and quite different on the meta level. So if you accept on the meta level that Your mathematical reasoning is not formal, that it has some reference. The same must be, must be honest. No, I don't understand that. I mean, in Tarski very definitely things do occur, or may, are permissible at the math level, which was not permissible. You can't say his stuff here, but he must have it down there as well, because... So I don't see why, correspondingly, a difference does give, I mean, this sort of uniformity piece. All levels are the same, but then... I don't see the rationale for that. Maybe in one level was... and the second level, God level. There could be a reason of... but if this... on both levels, people are... You're using the same word. I mean, I told you, in chess, you're not there by... you're not there by committed to interpreting the same word.

45:00 I'm giving you a counterexample. You tell me, ah, yeah, but this is mathematics. We call mathematics and the meta-level and this and the meta-level. Therefore, they must be continuity. We could use different words. Then what problems, I mean... I agree with you that you can abstract with situations from reference, yeah? You can disregard reference. You can. But it does not mean that you can do it all the time. For some particular you can do it. I think he was not schizophrenic, he was not Platonist, this is the other thing, he said that there is a sort of reason that we are underweight in the real part, so he is somehow realist about sub-parts, overtly non-partilist about sub-parts. Or the rest, like the ideal statements. I don't see what, but there's no schizophrenia in that mathematics only becomes a formal down frame when you bring these ideal statements in. There is a part of mathematics that isn't extremely explicit, of course, in that it's not simply formal, in an elementary number theorem. And the methods that you use in metamathematics are to respect the methods that are available in elementary number theorems. And he spoke explicitly, sort of concrete, non-formal, about those of mathematics and this small, very small, tiny kernel of ordinary mathematics, but then explicitly non-flatting after this development of mathematics. Moreover, at the Meta-Level, he would not accept any of the results.