Lecture

0:00 Universal, intellectuals, foundations of mathematics. And it's going to be the end of the chapter, as you can probably predict. Does it need any further adjustment? Maybe not. Thank you. I'm really pleased to have the opportunity to speak second time in LSE. Thank you very much, John, for this opportunity. I found it difficult to me to invent precise title of my lecture, so instead of the title, I present you topics which shall be covered, or maybe theses which I defend. Sorry, I'm disordered. I was working with this overhead projector and I had some difficulties. There are my fuses. I am fond of the Hilbert program. I think it is interesting, even now. I say that the Hilbert program is radically different from the so-called radical formalism. I will explain later what that means, radical formalism. Radical formalism is blatantly false. It is a kind of historical point that radical formalism is the main doctrine in logical empiricism. And the last thesis is that radical formalism managed to diffuse into contemporary philosophy in an unconscious way.

2:30 To present the Hilbert program, I must present the context of the program, mainly the discussion between Platonism and Constructivism, which are two points of view concerning the problem of universals and the foundations of mathematics. So in the beginning I show you a short comparison. In comparison of those two, two different solutions of the problem of universals. I put stress on similarities between those two opposed views. They both, they both that mathematical sentences or terms refer to objects. They are objects. The objects have following features. They are beyond space. According to Platonists, they are also eternal, beyond time. According to constructivists, they are becoming objects, becoming things. They are simply constructs of human mind, and when the construction is executed, then they become to existence. Of course, those objects are some connections with human mind. According to Platonists, it is only cognitive connection.

5:00 It is not able to change those objects according to the cognitive connection but also existence. Maybe most important from the mathematical point of view is the view on the nature of infinity. According to Platonists, both potential infinite objects and actual infinite objects exist according to constructivist potential infinite objects. Mathematical knowledge, starting from what is knowable, according to Platonists, both constructive and non-constructive objects are knowable, according to constructives, only constructive. Another similarity is that, according to both points of view, mathematical proof is a mental construction. There is no difference between them. There are two types of proofs within Platonist's point of view, namely phoenicistic proofs and infinitistic proofs, according to that point is not very precise because here is a difficulty, namely that there are many versions of constructivism and I was unable to present this point in a very precise way because the notion of

7:30 Constructivist proof changes, I could say, in every version of constructivist, it is a bit different. So actually, Finites proof is a kind of proof which is, I can say that Finites proofs are accepted among all parties, all versions, and also, of course, they are accepted. The sentence is true, it means that there are some objects and the sentence says that there is such and such. Excuse me, then you are not classifying intuitionism as a sort of constructivism? I try. Which point do you think is... Why do you say that intuition is not a constructivism? Because there is a difference between truths and intuitionism. Yes, that's what I say. There is a difference. But, what I say, that in mathematics you have constructive proofs and also such proofs which are non-constructive, yes? And in constructive mathematics you have only constructive proofs. Excuse me? Truth is a difference? Why? Maybe we can discuss it later, but I try to convince you that it is the same kind of thing. Just one last question, but let's leave it if you do. I have no point of the idea of an infinity for Platonism. This is something we do, mental constructions.

10:00 This terminology, Hilbert terminology, I use here Hilbert terminology. Even if it is not maybe the best one, but I'm going to present Hilbert program, therefore I use... What does it mean, infinitistic proof? It does not mean that the construction, the mental construction is infinite. No, no, no, it is not the case. It is always infinitistic. It is infinitistic in this sense that, let's say, the material of the proof is... Let's say infinite set. So we are doing some operations on each infinite set, infinite object, let's say. This is already taken care of by nature of infinity, potential and actual. Why do you repeat it at the level of truth? You see what I mean? It's the same nature of infinity. It's actual infinity which gives rise to the infinitistic aspect. Yes, that's right. I repeated this in order to show a connection between some presuppositions concerning objects and some consequences concerning proofs. Connected with some assertions concerning the role of mind, namely accept thesis that human mind possess some abilities, non-mechanical abilities to obtain knowledge concerning those objects, namely the ability to perform mental construction, constructed this assertion.

12:30 But in Platonism there is another concern, ability to have an insight into non-constructive realm of object. Yes, but the constructionist would have the intuition for example for an array. The intuition is as early as an insight into an array. Yes, but for sure it is not insight into non-constructive domain. Maybe there is much more to say on those points. I don't like to talk too much because I think there are a lot of known things. So my point simply is that the Hilbert problem was created in a situation when Those two points of view concerning mathematics were discussed. You could say there are philosophical theories concerning universals, but those theories have strong consequences concerning mathematics itself, concerning validity of some proofs and truth of some sentences of mathematics. So that is the context of the Hubert's problem.

15:00 Context of ideas, I don't care, so-called historical context, which seems to me unfortunate. It seems that Hilbert accepted following basic convictions. First, that the whole of classical mathematics deserves to be deindicated. Because classical mathematics was criticized severely by the constructivists. And what more, Platonism is okay. Second, Hilbert's conviction is that constructivist criticism of classical mathematics is convincing. And third point is that philosophical mathematical dispute must be decided by mathematical means. We can say that this last conviction reflects the fact that he was a kind of positivist. He was not very sure that such a philosophical problem as the problem of universals could be solved by philosophical discussion. So he preferred to find a solution of it by means of mathematical arguments. And that's the, let's say, beginning of the Hilbert problem. I mean that it is Hilbert's conviction. It is also my conviction, but do you have a proof that it is true? Do you think it could be decided by voting? Is that the truth of the Platonist? No, he's simply asking you what you meant by saying that Hilbert said Platonism was okay. Hilbert's famous for saying that the only thing that's real is the mathematics and that's the number. And even that is the funniest sort of saying. But everything else is an ideal element, which are purely fictional and not to be treated realistically.

17:30 You know why the Platonists treat it that way? No, I don't understand you quite well. What does it mean, okay, or you ask me? Well, there are two things. Is okay somehow sort of... I just wondered if you had some sort of special meaning for okay as opposed to... No, no. Okay is okay. It's correct. But how... It seems so wildly false that I don't... You must have something else in mind. It is wildly false as a historical reconstruction. Historical plan of our children. Yes. I agree with you in one point. Namely, it is... It is widespread belief that Hilbert was not Platonist. In this sense, I can agree with you, but later I present some arguments that he could not be formalist in this sense of radical formalism. Excuse me? He's not so sure. He's not so sure. Yes, after reading his paper on the infinite, you can have such impression. Yes, that's right, that's right. The problem with Hilbert is that he was working on philosophical-mathematical problems, but usually he played down those philosophical problems, so they must be extracted, and it was done mainly by Kreisel and Pravitz, so I'm, what I'm... Talking about here, it is not my original thesis. I repeat here thesis of Chrysler and Pratt at this point concerning basic communication of Hilbert. Controversial reinterpretation. Non-standard interpretation is more of a platonist than is generally... There may not be a bit more touch than neutral about this, but is it okay in the sense that

20:00 paganistic types of arguments, I mean infinitistic types of arguments, are okay within a proof which leads to the existence of ordinary and critical objects? He need not be committed, Hilbert did not He thought he could not be committed to deterministic ontology if you use infinitistic arguments in a tool. That is that everybody accepts this. Can't you, for the time being, the okay is to be understood, I suppose, in this non-committal way, I don't know. The best way to follow is accept it with a discreetness for a time and see... How it fits in the whole picture. That's it. That's it. Well, it pushes him into more playfulness in the order of itself. Okay. Starting to the Hilbert's program. The main idea is to use constructive methods of proof in order to convince constructives that non-constructive methods of proof are safe. That's the main idea. The Hilbert program is elaboration simply of this idea. There is a dispute between two parties, so constructivists say that infinitistic proofs are non-valid. So the idea is to use constructive methods of proof, which are accepted by constructivists. So use such methods, accepted by them, in order to convince them that also non-constructive methods of proof are valid. In a sense, which shall be explained in one way or another. Do you mean metapruthy? That's why you should put it. You don't want to do that. No, no, no. I prefer not to use the term meta.

22:30 Yeah. I mean, if you had put metapruth, to use constructive metamethics or methods of metapruth, I could, you know... No, no. I... This is the description of Hilbert's position. No, it is an attempt to broaden... To broaden certainty concerning the infinitistic proofs into the area of infinitistic proofs. That's the idea. The term meta was invented a bit later. It doesn't matter. I mean, it depends on what he intended by it. Didn't he invent the word mathematical mathematics, you mean? He didn't. But the system has so many meanings, so I am afraid to use them. Yes, then you beg the question as to how starting from constructive methods of truth you end up with non-constructive methods of truth. I mean it's a very difficult thing to solve. How could you break out of the circle? It's very difficult for me to imagine. Sorry, what is difficult? I mean... Ah, I see. I am sorry. Constructive methods of proof are considered to be sure. So, we can use them in order to prove... That non-constructed have also the same properties. It's quite easy. And now, there we are in voice. Sorry for the ignorance in this, but there seems to be two things. One is, the things which hitherto Platonists had proved by non-constructed methods could be now used to replace through the same result, but with a different constructed method. Thank you for your attention.

25:00 I don't think that the term meta explains anything in this context. I don't think. So, what is necessary in order to implement Hilbert's program? Necessary is demarcation between constructive, or so-called finitistic, non-constructive, or so-called infinitistic methods of proof. This distinction is used in classical mathematics. It is known to every mathematician, but it is not quite precise. So it must be made more clear. But more urgent is the second point, namely the demarcation between sentences which make sense for constructivists, so-called real sentences according to Hilbert's terminology, and on the other hand, such which make sense for Platonists only, so-called ideal sentences. Such sentences, which refer to infinite objects in actual sense, make no sense to constructivists, of course, so they must be called ideal terms. Roughly, it is quite clear what is the meaning of this demarcation, but it must be made also a bit more precise.

27:30 So, now I can say what is the objective of the program. Objective of the program is to prove by phoenicistic methods the following conservation principles, provisional formulation, for every real sentence a, if p is a proof of a, then a is true, is true by constructive standard, that means there is a constructive or a phoenicistic meaning of the term true in this context. In other words, this idea, which was on the beginning, would need more precise formulation. Here is the definition which makes very clear where is the line of division between real and ideal sentences. It has the following form. Universal quantification in the beginning. Those variables are natural number variables and this formula is decidable for any sequence of natural numbers k, 1, k. So we can say that the real sentence is almost decidable sentence. It's Hilbert. No, certainly not.

30:00 Yes, yeah. Certainly, he tried to interpret this in a sort of normal way. No, that's... Maybe I can, if we have enough time later, I can present arguments that it is okay, namely that the universal quantifier could be interpreted in constructive ways. I can present you arguments, but maybe not now, because I am afraid that I have not enough time. Making this distinction, the conservation principle could be formulated in a bit more precise way. If A is a real sentence, P is a proof in a real sentence in a given theory P, P is a proof in the same theory T, and P is a proof of the sentence A, then A is true. It is not a final formulation of the conservation principle, it shall remain the same. And by true you mean there is a valid proof of it? Can P, I mean you just said P is a proof? A proof, yes. But any? Any sort of proof. Not necessarily constructed? Not necessarily. So this sentence is non-trivial when it is non-constructed. Because when the proof is constructed, it is trivial proof. But what is the... maybe I show you once more.

32:30 Now we have the following problem. In order to prove conservation principle, we have to prove something about proofs, all possible proofs in a theory T. So it seems to be a difficulty because... It is not a mathematical object, so we cannot present truth concerning something which is not a mathematical object. Truth in itself is not a mathematical object. So, it seems to be a serious stumbling block, but fortunately it is not so serious, because it happens that theory Could be represented, theoretically, could be represented in such a way that, so TF, it is a formal theory, it is a mathematical object, which represents a real theory. I'm not going into details of that. It is not quite to formalize theory. Maybe it is axioms, let's say, pretty easy, but not so. And it is not mechanical. This notion of formalization involves a notion of translation of sentences of the theory into well-formed formulas of formal theory and representing proofs by some mathematical form.

35:00 It could be best if you could achieve such formal ticket that such equivalence holds, but usually it is not possible to have such equivalence. So such minimal adequacy principle is required, not this one. So the final formulation of the Hilbert program is following. The proof must be found of the following conservation principle. For every formal proof in formal theory PF and for every real well-formed formula in PF, if PF is a proof of AF, then A is true. Yes, this fact could be easily proved. Maybe that you cannot find in formal theory such a well-formed formula that this formula and its negation are both theoretical theories or non-contradictional.

37:30 Colin, what is the terminology in logic? Maybe it is now evident that the Hilbert program involves Platonism. Is that clear? Not yet? I'll tell you why. When you say true, you mean true under some interpretation. Through, it means through by constructive standards, finitistic standards. So, it means that there is a finitistic proof of A. And this is according to the consensus that we have. If, I don't know, it's news to me, I may have missed something, that if something, something is consistent, that the system is consistent, then there is a moment in which, is that what you are saying? I think we say that. If theory fulfills the conservation principle, it is consistent. And the other way around. The other way around. Where true means not simply that it has a model. I see. Then I don't understand. So simply the proof of conservation principle must be finitistic, not any proof. Any proof does not make sense, because before the program is accomplished, as long as it is not accomplished, you cannot be sure that infinitistic methods of proof are okay.

40:00 Somebody says that all that you can require is a statement that involves what Herbert talked about as ideal elements, that it not lead to any inconsistency. If the only thing you can acquire from the introduction statement to that idea of mathematics is that they preserve consistency, then you might want to express that as saying that that's all that truth in mathematics consists of, at any rate, for the whole of mathematics as a whole, not just the real part of it, the whole world. But that wouldn't mean that you're like recapturing Plato into the original study or redefining the truth, so that it means consistency. Could you repeat this question? If you redefine, if Hilbert's position, as I understand it, is that first of all you've got certain, you've got statements mainly about natural numbers, finite statements about natural numbers, things like three plus five equals eight minus a square root of three, that are absolutely true, guaranteed by the global intuition. However, if you stick with what can be guaranteed by the global intuition, then you only get... A very small box of mathematics, namely... Well, you are speaking about decidable sentences, yeah? Yeah, alright. 2 plus 3 equals 5. Decidable. Right. So now, in order to... and you get a... if you stick with that, you get only a very small box of mathematics... Yeah. ...you get a very weird logic. So in order to give yourself... Excuse me, what logic? A very strange logic. Strange logic, yes. As in the institutions. Yeah. In order to give yourself classical mathematics and classical logic, you extend... The only requirement on which is that you don't, by introducing the ideal element, introduce anything that's provably false, and that reduces the requirement of consistent. So then you could go on and say, which seems to be what you're doing here, to say that, look, that means that Hilbert's position is that mathematical truth reduces to consistency and therefore he's flattening it, because he believes in mathematical truth, but he's re-characterizing what mathematical truth is.

42:30 But it's not that you can get that out, is it? No, certainly not. It is not that you cannot prove. It is simply an attempt to show that infinitistic methods of proofs are correct, and such proofs could be done only by means of finitistic methods. Is that okay? I can change it. Is that okay now? I told you at the very beginning that true means true by constructive standards. How I understood character, you set out that A entails that A is not formed. Would you say that? I find it. Ah yes, the translation. What do you mean here? In the picture? Yes, it's that line there. This is translation, this is not implication. Procedure of translation of... A sentence in the way for formula. Well, they enact it, but it doesn't go to the point because then it can't be balanced. Because what? Because it doesn't give A sub A. It cannot give you the two. Well, what? If A is the right-hand side, A is the lower form of formula. I don't see how not you could get from there to... If it's going from the right-hand side... In the formula... This is not consistent or non-consistent. Because if you had A and the translation, then there was no problem about the translation.

45:00 You can certainly get A subscript F. If you've got A subscript F, the problem seems to be how do you get A? If you are able to translate it this way, you are able to go back. You can go back. Nothing is proof as yet. It's a question about your claim about the fact. I don't understand. Let me just ask the question. I say that a theory, a system, a formal system is consistent if and only if each of its elements, each of its theorems has a finalistic proof. Does it stand? That is not true. Because in conservation principles only real well-formed formulas are involved. What are the real ones? Real, they are such which are translations of real sentences of theory A. I come back. Thank you. Real means corresponding to something which is intuitively mathematical. No, there was no definition. There was definition. I show you. Sorry, do you suppose that the translation is one-to-one? From 1A to 1A? Translation? Oh yes. If there is given formal language, it is unique translation.