EPSA Opening / Transcendental Arguments Brouwer Bar Theorem / Mathematical Knowledge

0:00 Thank you. Well, good afternoon everyone. It's a pleasure to have you all here. We thank you for your presence. It will take place the opening session of this conference fundational of the European Asociation of Ciencias and Ciencias. For this, he has the floor to the President of the Filosofía. In the name of the Mr. Decano of the Faculty of Filosofia, the professor Juan Manuel Navarro, and in the name of the President, the Faculty of the Faculty of Filosofia of the Faculty of the Faculty of the Faculty of Filosofia in Madrid. I hope that you can meet with your respective centers of work, or, as we say in Spanish, I hope you can meet with them in your house. of the Asociación Mundial de Filosofía de la Ciencia has been a great satisfaction for all the professors of this house. And lastly, I hope you have a great success in the work that you are going to do during these days of Congress. And that is it? Aprovecheme, no obstante mucho trabajo que seguro van a tener, para conocer un poco de Madrid, que seguro que merece a la gente. Acabo reiterándoles la bienvenida y deseándoles una feliz estancia entre nosotros. Muchas gracias. Thank you very much, Vice-Rector Carmen Acebal and Vice-Dean Antonio Benitez, who is here

2:30 representing the Dean of the Faculty of Philosophy, and first of all I want to thank you both here at the opening conference of the European Philosophy of Science Association and joining with your words of welcome to all the delegates to this conference. We welcome you to the Faculty of Philosophy and Universidad Complutense. We hope that you will have a very productive next few days. But as Antonio Benitez reminded us, we also hope you will have time to enjoy the city of Madrid, because it's definitely worth it. I would like to say a few words about the history of Complutense very briefly, as a very brief introduction to all of you, to the university. This is a very old university, one of the oldest universities in the world. in 1293. Its first site was the city of Alcalá de Henares, just about 50 miles outside Madrid, and that is the origin of the name of the university. Confrutense is the city of Alcalá de Henares. The university moved to its present campus in the 1920s under the name of Universidad Central. And as you know, the EPSA conferences, and Mati Sintonen will be saying more about this in a minute, are inspired partly by its North American CASSEN, the Philosophy of Science Association. and it is a happy coincidence that the Complutense campus was also in turn inspired by North American campuses and particularly was modeled after the University of Chicago. The building itself is one of the oldest, the building that we are in, the Public Health Philosophy building, is one of the oldest in the campus as well. It's over 75 years old now, and it has had a very important and distinguished history in both intellectually and socially, in the whole context of the social history of Spain.

5:00 Many important and distinguished philosophers were trained and taught here, maybe most prominently, Jose Ortega y Gasset. For many years, the Faculty of Philosophy at Complutense was the only center that awarded a degree in philosophy in the whole country. And during the Civil War, the building itself was the site of some very hard fought battles between the international brigades that entrenched here forces just across the river Mancenares in a siege that lasted nearly three years and that has left marks on many of the university campus buildings, including our own. Now, I'd like to say a few things about the organization of the conference. Perhaps the most important thing is that you should take a look at the badges, the accreditation that you carry with you. There are batches of two colors. The blue ones are particularly good, because these are the food batches. These are the ones that allow you entrance into the restaurant downstairs for lunch, where lunch will be served between 1.15 and 3.00 every day. The black badges aren't as good, but they're still good enough. They allow you entry into all the sessions of the conference, of course, and they also allow you to enjoy the coffee breaks, which will be served sometimes in the foyer, just outside, next to the registration desk. But on occasions, the coffee will also be served downstairs in the restaurant, outside in the terrace. We've been lucky to be enjoying very characteristic Madrid autumn weather, it's sunny and crisp, and we thought you might all enjoy being outdoors for those coffee breaks. So the morning coffee breaks in particular will take place downstairs. So you will have to show the blue badge when you go into the restaurant, and you won't have to show your badge virtually for any other purpose during the conference. There is another purpose for which you would need your badge, and that's if you want to use the computer room downstairs in the basement. It's in the basement along all the way to the right wing of the building. And in order to gain entry into that room from 1 to 8 o'clock in the evening, you will also have to show your badge. It doesn't matter whether you have a black or a blue one for that.

7:30 Okay, let me see. Besides the computer room, we also have managed to get wireless access throughout the building. The signal is particularly intense in the main entrance hall, so if you have your laptop and you want to connect up to internet, it's probably best if you go to the entrance hall and sit in one of the benches just there. There are instructions on how to connect in your folder, and also in the notice board, which is just next to the registration desk, will be putting notices throughout the conference. Check the notice board just outside this room for an info when you come in. There will be steward dresses at the registration desk throughout the conference. So for any questions that you may have, or any needs that you may have, just feel free to go and ask them. If it's something they cannot sort out, then they will be asking us to do so. Generally speaking, I just want you to feel free to ask for any help to any of the Complutense members of the staff, registration desk when I really need anything. And I guess I need to close with a number of important words of thanks. First of all, I'd like to thank our funding bodies who have generously provided funds for what is really a new and untried venture and initiative. and funding bodies are not always keen to fund new and untried ventures and initiatives, but in this case we've been backing to a very generous backing for which I am very grateful. In particular, the Spanish Ministry of Education and Science provided a very generous grant. I'm very grateful to that. The Vice-Rectoret of Research provided as well generously a large grant. And the Faculty of Philosophy itself, headed by its dean, Emmanuel Navarro Cordón, provided the use of the main rooms of the building for free,

10:00 which allowed us to bring the registration fees considerably down. I'm quite proud that registration fees for EPSA 07, the first conference of EPSA, are below the standard PSA registration fees. I also want to thank the Department of Logic and Philosophy of Science who provided logistic support. And then I want to thank all the members of the Steering Committee of the European Philosophy of Science Association, these three, four of them are here, for their support to me personally over this very long and complicated year, and for putting up with all kinds of requests, often rather imperative on my part, and sticking to all kinds of difficult deadlines. I think the steering committee of EPSA must think of me as a rather posse and dictatorial monster. So I look forward to being able to reveal to them my true nature in the future. Finally, I want to thank all the invisible people that make a conference possible, and there's always a lot of people that work very hard to make a conference like this possible, and rarely make it to the big posters and the publicity. Starting with the porters and the cleaning staff in the building, who have been most dedicated, evening we were all busy cleaning the holes and the blackboards and whiteboards and I'm sure they will be doing their best in the rest of the conference as well to keep the building running. The staff at the restaurant will be handling catering and the brilliantly efficient Patricia Gonzalez from El Corte Inglés who is not here but must be outside in the registration desk, taking care of registrations, who has been wonderfully helpful. And then most importantly, to thank my small group of collaborators and PhD students who have been absolutely brilliant, helping out with all kinds of important tasks, and particularly during the last two months, such as the design and maintenance of the conference webpage, the mining of the conference email account, the preparation of documents for the conference

12:30 very important things like that. Pedro Sánchez, Giñaki San Pedro, Isabel Guerra, and Albert Solé. Thank you very much for your help. Finally, I want to thank you all, or maybe I should say thank us all, for being here. The enthusiastic response to the Kofi papers really astonished us. It's wonderful to know that this association is born with such a strong and enthusiastic backing. I am sure that this is an association that leads to future and many more conferences like this one. Thank you. Thank you, Vice-Rector Asipal, dear representative of the faculty, dear colleagues and friends of the Philosopher's Science, it is my pleasure and privilege to welcome you all as President or so far President of the European Philosopher's Science Association and on behalf of the theory committee, or provisional theory committee as we like to call it, to this conference of European philosophy of science. It is a historical importance, I think, because now we do have an organization which gathers together all philosophy of science in Europe. In a sense, I can say that this is coming to the roots. So we have a few intensive days ahead, So I shall be brief. I shall say a few words about the glorious past of philosophy and science, and try to give an argument as to why we lead EPSA, or European Philosophy of Science Association. Well, you are actually the believers, so no argument is needed, but here it is in a very rough form. Very briefly, philosophy is a very European thing, or in Europe, that's where philosophy in the form that we know of it came into existence, that is systematic, critical thinking about the nature, scope, and sources of knowledge, as about the problem of justification, as

15:00 As well as, of course, the other side that is about moral issues and the way that the society should be wrong. Of course, that is not what philosophers of science are professionally in the business are doing. But certainly, we can say that in the form that we know philosophy have, it is a very European thing. As to science, we could say pretty much the same thing. to say that it all started in Europe, because, of course, there was critical thinking, there was the mass in the data, there was scientific reasoning outside Europe, in other cultures. Yet, in the form that we now know, whatever philosophy of science, the roots are, of course, here in Europe. So we had the classical Greek way of doing mathematics and their views of empirical knowledge. we have a Jacopo Zabarela's views about how the disciplines should be organized and how to be refined, there is the theme and evolution of science or the scientific enterprise, and how it eventually developed into the disciplines. And of course we have something we could call the scientific revolution, and it of course started in Europe. Now, it is arguable that that there was no such thing as the scientific revolution. That is, there was no such coherent, so to speak, parallel events that could be called the scientific revolution. We know that this argument has been put forward. But certainly something did happen in the 17th century and a bit earlier that eventually resulted in the new way of seeing natural philosophy and separation of science from philosophy. The way that the scientists emerged from natural philosophy was of course a messy thing, and we cannot say that it started at any particular moment. Nevertheless, the rise of science was so important that a great historian of science, Bukhari, Dr. Jeremy Bukhari, Herbert, said that the scientific revolution was so important that it was about to show that many of the other events that occurred in Europe around that same time and earlier were a minor series of incidents.

17:30 and the reason must have been that science certainly and especially one of the technology became an intellectual source for the advancement of the culture much more important according to Butterfield than just about anything else and I think that the sociologists argued that the science in the form that we know it based on critical thinking and based on open discourse has been an extremely influential force behind the social and economic success of our civilisation, that is, European civilisation. Of course, now science is all around us, is permeating all around us, and of course the main reason why we should be interested in science is that one cannot understand the world in our days without understanding science and without being able to have a first time knowledge of how it operates and without being able to see what this place in this society is. Now, I said that there's no way of saying exactly when science began. Now, the argument that I want you to say is this. We have philosophy, which is pretty much a European theme, We have science, which is pretty much a European thing, so philosophy and science should be a European thing. But then it occurred to me that it's not a very good argument, and actually it's not an argument at all. But nevertheless, it so happens that, of course, the birthplace of philosophy and science was also in Europe. One of the landmarks, I think, at least according to Nicolas Jardin, was the debate about, well, let's say, as we now put it, about scientific realism and the way that we should represent planetary astronomy. The incident was, the incident where Johannes Kepler seeming innocent and influential defense

20:00 of realist methodology in astronomy came around. Tycho or Tycho Brahe had proposed a new world system. However, in Tycho's opinion, he had been plagiarized by a certain chap by the name of Nicholas Reineberg there, also called Ursus. Now Ursus was not a nobody but an influential man, the mathematician of the Holy Roman Empire, and Ursus would not have a swell of such occupations. He said Kepler was right off the mark. The problem that Ursus was big wringing back was that Kepler was not saying anything new. All of these things had been said before. Furthermore, he said, there are umpty possible ways so to speak, saving the phenomena and giving an account of the movements without actually holding on to this or that particular theory. Now, if Jardin is to believe, here is one of the beginnings of philosophy of science, which is consciously made about epistemology and the logic of the arguments, and what what's going on, but something that holds these views at a distance and looks at the logic underlying these arguments. So this was, according to one of you, one of the places where philosophy of science began. So this was the idea of having just mathematical astronomy and convenience as the case, the possibility of saying something that was physically legitimate and real. Now, I won't bother you with all of these, as all these are of course extremely controversial things. But if we move on a little bit and along, we can realize that of course the development of philosophy of science and the emergence of this ISP is certainly a European thing up until now or up until the early 20th century. And I'm of various scientific movements, I'm skipping a lot of history, I'm referring to the scientific philosophers that came all around Europe, especially the Vienna Circle, but also in many other places. It is not a complete coincidence that EPSA, or the European Philosophy Science Association is actually founded in the Vienna Circle home base. So we have a

22:30 Vienna Institute which is actually our address and home base, thanks to Friedrich Stader who took the initiative and had this organization accepted. So we are actually working under of the laws of the Austrian not-for-profit organizations. Now, why EPSA? I said the philosophy of science was born in Europe, it's not just the science and philosophy. But of course, now that we've been spreading around, there certainly was a need to get the act together. So a number of people gathered about a couple of years ago in an informal reading in Lisbon and they decided to have more informal settings later on in London. And this happened on September 27th in that London School of Economics. And I think maybe four or five of these family fathers are here now the rest will be coming later on. So we established the provisional theory committee, as I said, the idea of a provisional being, we are provisional. We are inviting all of philosophers of science, not just in Europe, but all around, to join in in the enterprise of advancing philosophy. So the idea we had was that we would be having a greater visibility in philosophy of science in Europe. So that was certainly one idea. We do have a lot of organizations and we do have the sister organization PSA in the USA, but we thought that it would be grand to have an organization here and continue in Europe every other year, as the idea was. And, of course, what we wanted to know, wanted to do, of course, is that we wanted to have visibility towards the European Union and the funding source is here. Money is important, of course. We want to advance philosophy and science by being able to raise money and being able to have better and better conferences. Now, from the start, everything looks extremely good.

25:00 Now that Mauricio is setting the standard, I hope we are able to live up to the expectations in the future. But of course, another idea was to increase transaction and information flow between philosophers of science and Europe. So this, we thought, would be an important way of advancing philosophy of science. Of course, we would not want to exclude anyone, so everyone can join in who feels the idea. I'll simply read, and then I'll wrap up this thing, the general overall aim, which we had in the announcement, which was, I think, wrapped in Vestatis Psyllos. The general early name of the EPSA is to promote and advance philosophy of science in Europe. We do this by furthering their contacts among philosophers of science in Europe, by ensuring that information related to philosophy of science in Europe is regularly circulated, by supporting, on the international level, progress in philosophical studies of science, and by promoting the public understanding of science within the learning and educating the public. Now we are moving towards the first conference. I just want to thank, first of all, of course, the Law to Law Organization Committee, without which nothing of this would have happened. I want to thank Mauro Lovato and Miklos Lenny for chairing the program committee, and of course I want to thank all the colleagues in the provisional steering committee for providing the service. Of course, now that we are here at the university, I would like to extend our gratitude to the University, not just to the University, but also to the Faculty of Philosophy for providing us these wonderful surroundings and for these premises. So, we have a few intensive days ahead, so let me still thank you all on behalf of the provisional steering committee the invited speakers especially the keynote speakers

27:30 the people who serve in the program committee which means more or less the who is who in the european philosophy science it is a great pleasure to welcome you all i hope you will have Thank you very much, Mr. Sanchez. Thank you very much. And for the foreign speakers and assistants, thank you very much for coming. It is a pleasure to meet you in our university. And finally, I wish you enjoyed your stay in Madrid. Good luck. Thank you very much. Coffee is already waiting outside. Each show is allocated 30 minutes for presentation, including discussion. So you kindly ask to finish the talk after 25 minutes. If you really want to have some time for discussion, then please do leave room for that. If you are over 20 minutes, I'm going to raise my hand, indicating that you are now biting into the discussion time.

30:00 Okay? So please, don't be surprised. The first speaker today is Mark von Atten, from Paris. The title is Phenomenology and Function Depth of Argument in Mathematics. Please. I'd like to thank the organizers for giving me this opportunity to speak to you today. so this section of the meeting is actually I saw on formal methods and what I will speak about is rather an informal method but I know that it will all the same in this talk I should like to take a look at one aspect of Brouwer's proof of this so-called bar theorem of philosophical interest because it is based on a phenomenological consideration of proofs as mental objects. Brouwer uses these considerations in an argument that I claim is best considered a transcendental argument. First, a work on the terms of Bartheorem and transcendental argument is in order. Now, the actual interest of this talk will not be the content of the the bar theorem, but rather the way Brouwer proves it is proof strategy, but I will say something about the content of the theorem all the same. The bar theorem is a theorem in intuitionistic mathematics about trees. A bar is a set of nodes in a tree such that every infinite path through the tree intersects in. So you have a tree, you have nodes in the tree and so on. Then a bar is a subset in the tree such that every infinite path intersects, has a node in common with the bar. An ensignate bar, right? The bar of bars. Nothing is nothing than the bar of bars. The question arises whether a bar admits of a well-ordered construction. For the development of intuitionistic analysis, it turns out to be crucial that such a well-ordered construction be possible. The content of the bar theorem is that this is indeed the case. It states then it contains a bar that admits of a well-ordered construction. Classically, that's completely trivial,

32:30 but constructively it's not trivial at all. The theorem was proved by Brouwer in 1927, and this proof is the only constructive one found so far. However, its validity has been questioned, for example, by Gödel, and we will see the reasons why. Now, a transcendental argument for the purposes of this talk is an argument of the following form. So there are two premises. First premise, I have mental experience E. Second premise, it's a necessary condition of the possibility of having experience E, that C, conclusion, therefore C, holds, right, some condition C, where the necessary condition in the second premise is not, or should not be, merely a matter of logic or the meaning of the terms, but rather a claim to what Kant called synthetic a priori, knowledge. Now, the structure of the remainder of this talk is as follows. First, the role of the so-called proof interpretation of intuitionistic logic in the Bartheorem will be explained. This is followed by a presentation of objections of Gödel's to the proof interpretation that would also invalidate Broward's proof of the Bartheorem. And finally, I argue that Broward's argument for the Bartheorem is a transcendental argument in this sense, and therefore an argument that delivers what the proof interpretation requires while circumventing Gödel's objections. The bar theorem is a universal proposition about trees and sets of nodes in trees. For any tree and any set of nodes in it, if the set of nodes forms a bar, then the tree contains a bar that admits of a well-guarded construction. So at the most general level, the constructive way to demonstrate it is prescribed by the so-called proof interpretation, made explicit by Kolmogorov and Heidning, but already used by Broward. That is, one has to indicate the construction method that from any given tree and any given set of nodes in it will produce a proof of the following proposition. If this set of nodes is a bar for this tree, then this tree contains a bar that admits of a well-ordered construction, right? So you can instantiate the universal quantifier. And this latter proposition has the form of a simple implication, if P, then Q. All right, so according to the proof interpretation, means I have a construction that transforms any proof of P into a proof of Q. So again, according to the proof interpretation, the way to demonstrate an implication is to exhibit

35:00 a construction method that will transform any proof of the antecedent P into a proof of the consequent Q. And indeed, the central part of Brouwer's proof, the Bartheorem of 1927, consists in exhibiting a method that will transform any proof that the tree is barred, into a proof that the tree contains a well-ordered bar. But the conviction that the treatment of implication in the proof interpretation is genuinely constructive, conviction held and still held by all intuitionists, was soon challenged by Gödel in a lecture from 1933. Now Gödel did not speak of the bar theorem in particular, but attacked proof interpretation in general. I do not know yet whether Gödel at the time knew Brouwer's proof, which had been published But in any case, if Gödel's general criticism is correct, then there must also be something wrong with Brouwer's proof. Indeed, in 1972, some 45 years after Brouwer's proof was published, Gödel still wrote that unfortunately no satisfactory constructivistic proof is known for bar induction. This is in his Dialectica paper, second version. Gödel probably was introduced to the proof interpretation through Heidens' talk at the Königsberg Conference in 1930. Gödel's qualm with it, as he voices it in 1933, is that the proof interpretation does not satisfy a condition that at the time he considered necessary for constructivity. So here's a quote from Gödel, 1933. He says, Heidens' axioms violate the principle that the word any, as you see it here, a construction that transforms any proof of defeat, Heidens' axioms violate the principle that the word any can be applied only to those fatalities for generating all their elements. The totality of all possible proof certainly does not possess this character. And nevertheless, the word any is applied to this totality in Haydn's actions. Totalities whose elements cannot be generated by a well-defined procedure are in some sense vague and indefinite as to their borders. And this objection applies particularly to the totality of the intuitionistic proof because of the vagueness of the notion of constructivity. And shortly afterward, Gödel would add another objection, saying that this clause, the claustrophication and the proof interpretation, is actually impredicative. Because you're referring to all possible intuitionistic proofs through the use of the word any there.

37:30 And Goethe, in effect, is saying that the general notion of intuitionistic proof would only be constructively acceptable if it forms a totality that can, so to speak, be generated from below in an inductive definition. Now, an intuitionist will readily admit that there is no definite construction method for the totality of intuitionistic proofs. That is because we can never exclude that further introspection will lead us to new mathematical objects and new principles of proof. Indeed, the very methods of bar induction is an illustration of this possibility. Brouwer discovered the method only after two decades of experience with intuitionistic mathematics. So the intuitionist will not deny that the notion of construction method is open-ended. In fact, it's the very first thing that Haydn says when he goes on to give his interpretation of intuitionistic logic. Rather, what matters to the intuitionist is that we recognize a proof when we see one, as Kreisel once put it. For if we do not, then it is for that very reason not an intuitionistic proof. So this is the classical, no, classical is not the canonical understanding of the word proof, or the original meaning of the word proof, that it's a sequence of stats that convince you of the truth of the conclusion. So the cause for implication is not to be understood as quantifying over a somehow given totality of proofs. It's not to be thought of as referring to a domain of proof that somehow exists. Rather, it expresses that one has a construction method that whenever proof of the antecedent can be used to transform that into a proof of the consequent. Of course, the question then remains how, in the absence of the complete characterization of all intuitionistic proofs, one might convince oneself that a certain construction indeed will work whenever one has a proof of the antecedent. There are two strategies that come readily to mind. The first strategy consists in using only the minimal information that any correct proof of the antecedent, indeed is a proof of the antecedent. So, that is, you use only its conclusion. The construction method asks for an proof interpretation, in general, operates on proofs as a whole. But in a given case, it may be that to transform a proof of the antecedent into one of the consequent, no information is needed about the inner makeup of a proof of the antecedent at all. Such a transformation is independent of the details of a proof of the antecedent,

40:00 be the guarantee that the construction method will work whenever we have approved the antecedent. Where this first strategy can be applied, no epistemological problems arise. And Goethe's principle criticism of the proof interpretation, of course, does not really target such cases. This is of great practical importance because nearly all actual applications of proof interpretation successfully use this first strategy. And to reflect this fact, Ben Rijtenberg has proposed to modify the proof interpretation as follows. In his version, a proof of an implication consists in a proof of the consequent from the assumption of the antecedent, where an assumption is treated as a black box of which we have no further knowledge, and in particular no reference is made to the proofs of the antecedent. The Barthes theorem, however, has not yet yielded to this strategy. The second strategy is to try and find necessary conditions on proofs of the antecedent that, even if they will not be strong enough fully to determine of the antecedent because that is impossible are nonetheless sufficiently informative to construct a proof of the consequence from. And of course the first strategy, which I just discussed, is merely a special take of the second strategy. The necessary condition appealed to in the first strategy is that the conclusion of a proof of the antecedent should be that antecedent itself. That's a trivial but nonetheless necessary condition on proofs of the antecedent. But, as said, there may be cases where this seems too little to derive the consequent from. In such cases, one may try to find non-trivial, necessary conditions on proofs of the antecedent. And one already sees room for philosophical disagreement here, namely, disagreement over what kinds of necessity can legitimately be appealed to in establishing the foundations of constructive mathematics. Broward, in his proof of the Bartheon, appeals to necessary properties of the mind. Now, Brouwer's proof of the bar theorem is, as far as I know, the only example of an intuitionistic proof that employs the second strategy. Brouwer found a way to extract additional information from the assumption that we have a proof of the antecedent of the bar theorem. In particular, Brouwer claims that any proof that a tree is barred, when analyzed into sufficient detail, can be decomposed into elementary steps that come in only a kinds. This analyzed proof Brouwer calls the canonical form of a proof that the tree is

42:30 barred. Through the device of canonization, the potentially great variety in proofs of the antecedent can be reduced to the one specific structure of canonical proofs, and this specificity of the structure enables Brouwer, in the second part of the proof of the barred theorem, to devise a construction method that transforms a canonical proof into a well-ordered construction of a bar, as was required. Now Brouwer does not explicitly say in his paper that his strategy is that of a transcendental argument, but consider how Brouwer presents his claim that to every proof that the tree is barred corresponds a canonical proof. He does not present it as an empirical claim, for then there is no guarantee that it also holds for proofs that we may nor does he present it as a claim that is based only on the meanings of the terms involved for the claim that any proof that there is a bar in a tree admits of a canonical form is certainly not seen to be true just on the basis of the meanings of the terms proof, canonical, tree and bar in fact one is reminded here of Kant's example that knowing the meanings of the terms 7, 5, 12 and plus is all by itself not enough to see that 7 plus 5 is 12 Instead of analyzing the meaning of the term bar, in his paper, Brouwer considers how bars are given to us in intuition, that is to say, how we construct bars. To this he adds another introspective observation as to what types of mental acts are available to us in determining the properties of bars. So Brouwer is shifting attention from the object, the mathematical object itself, to the acts in which we construct and contemplate and analyze that mathematical object. And in that sense, Broward's proof depends on a phenomenological analysis. Broward's reflection on the mental leads to the formulation of a necessary condition, namely, nothing can be a proof of a bar unless analyzable into a mental proof of the specified canonical form. And it is this plane that makes Broward's argument the transcendental argument in the I put up on the slide. As such, it applies to any possible proof of the antecedent, even though we cannot have a construction method for all its possible proofs. The remaining

45:00 more straightforward part of Brouwer's proof then consists in showing how to transform the just-obtain canonical proof into a well-ordered bar. So note that Brouwer's proof starts from a mathematical premise, the three-inch bar, then makes an explicit passage through the realm of properties of the mind, and by referring to the mental acts that are available to us in determining the properties of bars, and then arrives again at the mathematical conclusion. For Brouwer, this passage through the mind does not, as such, make the conclusion un-mathematical, because for him all mathematics is mental to begin with. And he was very well aware of this aspect of his proof when he said when he made the point in a more general way in the Cambridge lectures after the Second World War there's a quote theorems holding it intuitionistic but not in classical mathematics often originate from the circumstance that for mathematical entities belonging to a certain species the inculcation of a certain property imposes a special character on their way of development from the basic intuition right so there you see the reference to particular mental acts that we have at our disposal and then from this compulsory character properties ensue which for classical mathematics are false and then he continues striking examples are the modern intuitionistic theorems that the continuum does not split, I know the full function of the unit continuum is necessarily continuous. And for the Leibniz theorem you need the Barth theorem, so there's an implicit reference to the Barth theorem here. See, I'm almost done. All together 11 minutes. Alright. for contrast, one may think of Leibniz. Leibniz the objects of geometry are objects in God's mind. But Leibniz also said that this ontological fact does not make it impossible for atheists to do geometry. So what Leibniz is in fact saying is that in doing geometry, specifically ontological properties of these objects are of no interest. In particular, Leibniz nowhere proposes to exploit

47:30 specifically ontological properties of mathematical objects to demonstrate mathematical truth. but that's of course exactly what Brouwer is suggesting here he makes a specific appeal to the fact that mathematical objects and also mathematical proofs are first of all mathematical, mental objects and then he says well that means that if you have independent information about these mental acts, qua mental acts that you can use this to perhaps arrive at And in a longer version of the paper, I hear go into reasons why Gödel would not have accepted the bar theorem, even though Gödel himself, in the course of his life, came to widen his notion of constructivity. In particular, what is striking in Gödel's case is that whereas in the early 30s he objects to the proof interpretation, he's saying he's categorically denying that it is a constructive interpretation of logic because it's impredicative, whereas later on in the Dialectica paper from 58 and onwards he refers to the very same impredicativity of the proof of interpretation to defend that his own notion of computable function or finite height is constructive so what he first uses to beat the intuitionist with he later uses to defend his own position in the didactica paper he's discussing his notion of and then he says well he says it's really constructive and it's impredicative but that doesn't keep it from being constructive because look at the proof interpretation that the intuitionists use that's also impredicative but there he clearly accepts it as constructive The only thing that he does claim there, of course, is that he will say that the notion of computable functional is more evident, and in that sense, constructive to a higher degree than the proof interpretation. So that's a difference in degree and no longer in kind. Also, later in life, Gendel made some explicit remarks about how he saw intuitionism as a kind of a priori psychology. and there need be nothing wrong with APR psychology, it's just that from Goh's point of view, it's no longer pure mathematics because besides

50:00 the objects of mathematics, you also talk about the mind of the mathematician and the mathematician's ego but of course for Brouwer, that's exactly what mathematics is about, it is about the mental activity of doing mathematics contest that the crucial claim that Brouwer arrives at through reflection on the mantle is an evident one, or at least a correct one. But I do not propose to discuss the question of correctness of Brouwer's truth of the Barthesville right here and now. I only wanted to make a point about the structure of the argument. Thank you. But I'll end it for discussion. Questions in arms. Well, one small question. The transcendental argument, you called it a transcendental argument just only because it has this form and resembles Kant's arguments, or is there a little bit more to it? there are various types of arguments that have been called transcendental arguments but this is certainly one of them which you find in Kant but also in Descartes for example this Coquito argument is also one of that form so I should say to the extent that traditionally that kind of argument has been called transcendental argument. Brouwer's proof of the Bar Theorem is also a transcendental argument. Yeah, but then the word transcendental is of course used because there is something that is somehow transcendent, eh? Our experience or we're talking about the conditions that are considered in the knowledge. Yeah, it's an argument that refers to... What is then the similar thing here? But a Broward's argument for the Bartheorem is an instantiation of the argument's schema where you refer to conditions of possibility for some experience, for some mental experience. Okay, yes, thank you.

52:30 In fact, there has, of course, an analytic philosophy around Strauss's book, for example. there has been discussion of transcendental arguments in particular about knowledge of the outside world the physical reality there are certain reasons why within intuitionism where you're speaking of mathematics but then also mathematics considered as a mental construction the validity of transcendental argumentation much more or at least reasonable to assume, than it would be in the case of arguments about physical reality. I didn't go into those things. May I ask one? What makes the non-constructed rule of the theorem non-constructed? What specific feature is objectionable for a conclusion? Well, the easiest way to prove it, so what the Bartheorem says is that if you have a certain set of notes in a tree then you can have a well order of that set. Now, classically, you could just say, well, of course, by axiomal choice. So that's the axiomal choice, which is objectionable. That's the axiomal choice. Yeah. I see. Yeah, the bar theorem is certainly one of those cases where you have a mathematical theorem that classically is completely trivial. It's a one-line proof, basically. And constructively, either you have to do a lot of highly non-trivial work to do it, which is what Broward did, or you fear that it cannot be done as constructed as other different factors say. Now, these trees are countable, are they? the nodes are countable. Well, yeah, well, so the bars are also countable, right? Yeah, so you see the axiom of choice here for a quite small cent? Well, yes, from the classical point of view, that would be the thing to say, but intrinsically, it's not, to say that something is countable is making a stronger tangent when you do this class. Right, and for the theorem, all that you need actually is, there is a technical condition on those bars that I didn't go into for technical reasons.

55:00 A technical condition is that the bars should be decidable, otherwise you can devise counter-cancers to the bar theory. What you really depend on is the fact that proofs as mental structures are well-founded structures. The well-ordering, and hence the well-foundedness, of course, but the well-ordering, obtaining the Bartheism, where that well-ordering comes from, you copy that, so to speak, from the well-ordered structure of mental groups. My question, I was just wondering why an intuitionist would be bothered by the classical proof of that theory? Yeah, but it's because the classical proved that there's not an Indian order. Yeah, I'd like to see an axiomist choice. Sure, but still it's a very restrictive application of the axiomist choice. Still intuition is simply unacceptable. Yeah, yeah, yeah. It's probably better now. Yeah, so intuitionists are very, very reasonable, like the same, so... Well, they consider themselves for the dogs. Good. Any other question or remark? Just a quick one about your love, about Leibniz. This perhaps takes us rather away from Brauer and from the subject of this talk, but I'm very struck by Leibniz's refusal of, you know, to insist on that the ontological status of mathematical objects should play any role in his methodology of proofs. It does seem to me that in that respect he is taking aim if they can't, who clearly does regard the ontological status of mathematical objects as absolutely central to the method. Yeah, the reason why Leibniz insists from the ontological point of view that mathematical objects are ideas in God's mind is basically that in mathematics you don't talk about actual objects but about possible ones, But for Leibniz, existence has priority over possibility. So a possibility should also, in some sense, exist. So he says, well, they do exist, but as ideas in God's mind.

57:30 So it's to guarantee that these possibilities exist. And then with Descartes, of course, he had at least this conceptual discussion over whether the truth of mathematics are subject to God's will, yes or no. Whereas for Leibniz, they are not. For Descartes, they are. Thank you. Thank you. Thank you. The next speaker is ,, University of Sevilla, the pilot of Mathematical Knowledge and Interplay of Practices. Okay, thank you very much. I guess I'll be dealing with an old-fashioned thought. Sorry, I don't need to say anything. I think we don't have to work. Okay, the talk is going to be much more general, and to some extent I decided my aim was going I want to be simply to give you some idea of the approach to an analysis of mathematical knowledge that I am working on and developing for a book. And the title of the provisional title of the book is actually the book, The Talk, Mathematical Knowledge and the Interplay of Practices. To begin with, let me say some things about the notion of mathematical practice. It's, of course, an important notion for many philosophers nowadays. More generally, scientific practice is regarded as important in the case of mathematics. We could point to many examples of this, starting, for instance, with Kitscher's well-known book, The Nature of Mathematical Knowledge, 1984, where the idea of a practice plays a central role in his reconstruction of the development of mathematical knowledge. Or, to mention things that are coming soon, Paolo Gocosu is in a book that will learn

1:00:00 the title The Philosophy of Mathematical Practice, and will be published in a few months. Here for my presentation, in order to make things short, it may be a good idea to contrast my viewpoint with the one of the kitchen. And I will, as soon as we have a lot of what the teachers are proposing, just to remind you, a practice is defined as a group of elements of the ingredients that were a language, problems, some problems of results, some methods, and some metamathematical views. And the general style of this kind of construction is very much similar.