Categorification & formal axiomatic method / Fred A Muller: The Suppesian revolution

0:00 Thank you. Welcome to the afternoon session. We have two speakers in this room. The first will be from Paris. Thank you. Categorification and Formal Exramatic Methods. Categorification in the sense of category theory, two words later, and humanistic model of science. Okay. The official definition, the first principle of classical model of science reads as follows. All propositions and all concepts or terms of S, meaning a theory, concern a specific set of objects or are about a certain domain of beings. And in my talk, I'll try to show that that principle, which really seems to be fundamental, was much troubled and much discussed in the beginning of the 20th century, so it's really hard to say if really science today sticks any longer. Just about some side example, which I'm not going to explore in my talk, but think about quantum mechanics, for example, and the fact that it requires interpretations, which troubles many physicists who think that really good theory shouldn't have interpretations, or at least different possible interpretations.

2:30 And even if we think about formal sematic methods in Hilbert Tarsky's sense, just the fact that we might have a notion of theory which is differently interpreted also seems to be probably incompatible with that very principle. So we'll see how that topic is much discussed and developed in discussion between Frege, who really sticks to Aristophanes, the classical model, his discussion of Frege with Hilbert and some followers of Hilbert, and that's the first historical part of my talk, and the second part, I'll try to make an alternative proposal on how to push further, if you want, this non-classical feature of science, to enlarge Hilbert's project with the help of what we have today as categories here yet. Okay, I decided to start just with a little quote from Aristotle at the very beginning of Stereoanalytics. I'm not going to probably analyze this passage because otherwise it's just taking the whole time. I read this Ross paraphrase which I think is much more helpful than the first phrase is not translated, it says that all knowledge, science, everything can be learned, comes from some earlier knowledge, so we need something like parentheses. And then he basically says, with regard to something we must know, before come where they are, say, OTST, and in regard to others, what they are. So that fundamental opposition which plays really a very important role in the whole posterior analytical, OTST, TST, which makes a lot of headache for the reader because it's not very, very easy to don't that but i think that uh how say uh we could interpret it as more or less than that way that this what is is something like assertion it's it's sort something about a subject matter meaning it gives us a subject matter so example that the

5:00 Aristotle gives you the tertium non datum, excluded middle, so it's not actually a natural statement of existence such as an object exists. But the second example he gives of triangles is quite clear that he means that we give something like definition, so we here oppose things like action, postulate hypothesis to definition so something which is asserted from proposition to something which is merely stipulated. So I just I'm not absolutely agree with the comment which Professor Barnes gave long ago in the 70s that he tried to represent the distinction more as like between propositions and terms. I think it's much more between something like meaning you know something stipulated and something And indeed, just with the principle of classical model of science, it's rather easy to imagine that we have a subject matter somehow given by, say, some very incomplete description, something like brain or whatever, and we still don't have a good theory about that, right? That's more or less the usual situation which we use writing research proposals and stuff with such methods there is no good theory so we should make a theory but the reverse situation which actually already somehow in aristotle also imaginable right that we have something like a theory but we don't know what is about and if we again think say about quantum mechanics is not absolutely unrealistic the situation yes there is certainly good theory but it's very very questionable and not even in philosophical talk but even in physical talk say what is the theory But I analyzed a different example, which will be the example of Lobachevsky geometry, that was exactly the case and very explicitly where theory was created, it was not absolutely unclear what it was about, and then there were some proposals made by Beltrany what it was indeed about. Yeah, so here, for one of my talk, I'll talk about that historical example. And then I'll touch upon this discussion with Freddie, Hilbert and Corbett, who was Hilbert's followers in this respect.

7:30 And then I made an alternative proposal. I tried to show that Hilbert's solution was only very partial. And so with category theory, we can treat what I think is the same problem in a much more general way. Of course, I will not do any technical work now, I just give general idea. Okay, now I come to this historical example about Lobachevsky geometry. I just try to mention, I don't stop at this quotation from Whale, because after this Hilbert view, it's quite common in books to think that non-Ecclesian geometry obtained by kind of playing with axioms, right, we just think, we change the axiom parallel to its negation and we get something else, and that's true that Lobachevsky worked more or less that way, but the thing is that there is another lag, say, of the history which probably is more important, because that first approach, axiomatic, playing with axioms of Euclidean geometry started actually in antiquity as an attempt to prove the fifth postulate, of course, the final progress, for example. So in this sense, Lobachevsky is very, very traditional, just that they change their mind at certain points, and instead of thinking that they are going to find a contradiction, they just say, OK, we are no longer expect a contradiction with just exploring new territory but in a sense it's just the same world but the other way uh the story is of course who i also i have no time to explain exactly but it was today's uh commonly referred as a curved space somehow he he really first and actually for very practical work in your day he he got this ideal uh curved space he wrote this Soffices and Riemann in 54 formulated that as a general concept for, you know, and dimensions. And Beltrani actually in the beginning of his career, he just followed directly Gauss's work, not through humans. He worked in and proved some important theory. And at the same time he read Lobachevsky,

10:00 his colleague, Nicole Normale, who somehow moderated all his work, translated the time is working French and published in this issues published in the corner my soul that's how no and and that first paper of 68 he just discovered that if you take a particular surface which he called pseudo sphere which is a surface of constant negative curvature. And we interpret, saying anachronistically, of course, Lobachevsky's straight lines as geodesic of that surface. So we get Lobachevsky's geometry, and his first reaction to the discovery was, okay, now there was a good theory, but the best thing about that theory was that it left subject matter, so there was a theory without subject matter. But I found subject matter. It's actually a geometry of a pseudo-sphere, it's not something else. But actually this answer was unsatisfactory even for himself. The principle reason was not because he didn't allow to generalize to three-dimensional case. While Lavachevsky explicitly developed also three-dimensional geometry, not only Plankiev, right? He couldn't do anything similar in three-dimensional. Even in two-dimensional cases, in fact, it didn't work as well as Bertrande first believed, because actually you cannot obtain really a model that way. You obtain, anachronically speaking, a partial model. You can represent only final segments of Lobachevsky's line as final segments of geodesic, not the whole thing. Then he thought that he could sort out that problem, but Gilbert was late in 1993, if I'm not mistaken, proved that it was not possible. It's not possible. There is no surface in Evklitian space such that every geodesic, You know, the infinite one would represent the largest percent. But that very year, already, just a question of, I don't know, weeks.

12:30 He read, in French translation, Riemann, Habilitation for Trump. And he found a better answer. He found a better answer, and the better answer was that, in fact, it was just a remaining manifold of constant negative courage. And at that point he apparently just could absolutely give up this talk about interpretation, so he found the subject matter. And indeed this answer, which at least even today for people not particularly interested in the foundation and stuff, I think for physicists and mathematical physicists, it's still quite a good answer. But, of course, given that we know, well, what is Riemann in manifold, and that's, again, a question of definition, because Riemann, in his practice, he gives, you know, kind of genius idea, but nothing like real mathematical definition of his thing. Anyway, that was, in my view, the most spectacular, but actually not the only example. Mathematically, one theory can interpret differently. Another example, it was very much related to the story, I just had no time to talk about that, was about projective geometry, of course. Where you have this duality between lines and points, you can just replace in any proposition word line, straight line by word point, and you just get another proposition which is true, which gives somehow idea that in fact what we call lines are not really, it's not important. We have algebraic structures, kind of variable, right, which we can just replace in some way where it works. Bertram himself, he had in mind when he talked about this interpretation of Lobachevsky geometry, he thought about representation of complex numbers on Euclidean planes as a point of a plane. So all that examples are how still made rather unternable this standard view which we discussed today in the morning session, that actually any proposition, mathematical exception, but quite on the contrary, quite a basic example, that there should be something like how to use a certain domain of beings, like points, numbers, something very different.

15:00 It didn't work any longer in mathematics like that. We had a lot of examples of that, when we had to switch the subject method but somehow keep the theory. And Hildur Grundlagen, of course, is a very important reaction of that situation and a particular solution. solution to that question, how we can formulate a theory, say, up to interpretation, how we can formulate a theory, what a notion of a theory we might have, leaving interpretation free, say, allowing for different interpretation, okay, and that is rather often quoted passage from Hilbert's response to Frege, I'll come just up to Frege's objection, that you probably never understood it properly. So he said, you say that my concept, for example, points between are not univocally fixed, and for a few it's absolute sin, so you cannot do anything without that. But surely it's self-evident that everything is merely a framework or schema of concepts to give their natural relations to one another, and that basic elements can be construed as one piece. If I think of my point as some system of other things, of law, law, actually, and cheating sweeps, it's just one system. It's a little mistake in English, in German. And then conceive of all my actions as relations between these things. Then my theory, for example, if I were in one, will hold all these things as well. One should, has to apply a new vocal and reversible one-to-one transformation. I just want to note that it is reversible because that would be my major point of critique of Hilter's view. Okay, and now just very briefly to Frege's objection. Here is his letter to Hilbert where he criticized Hilbert's approach after reading Grunewagen. Sorry, it's written before. Your axioms are kind of something that is a function of definition. I do not approve of using the word action in a facilitating sense, something like definition.

17:30 In definition, nothing is assertive, rather than something he stipulates. So we see here, and that, of course, is a very important feature, a very important novelty of Hilder, of course, right, that he stipulates his axioms. Sometimes he stipulates axioms. Axioms are no longer absolutely true, right? And for Frege, for example, the whole business of proving that actions are consistent, the system of actions is consistent, has absolutely no sense, because for him actions are true, right? And he has consistent. And otherwise we have this confusion which Frege never accepts. The other proposition action principle theory must contain no word science whose sense and reference is not already completely fixed. So we've seen already how Hilbert replies to that. Then actually Frege writes a very long reply to that, but Hilbert just stops the discussion. It's very interesting, for example, he continues, and so the discussion continues anyway. So that's already, I'm afraid, to reply to qualsals arguments, which mostly elaborates on Hilbert's argument. Let's continue our investigation of qualsals pronouns. They'll read it, of course. In this way, one second of formal inferences can sometimes be interpreted in different ways. What can be interpreted is that it has a sign of a group of signs, also the university of science, which we must obtain at all costs, exclude different interpretations. But an inference doesn't consist of signs, there is no room for different interpretations. And here is important because somehow he needs an account somehow for what's going on, right? And so he says, interpretation, in Hildur Karlsson says, is nothing but an inference from general to particle. But in fact, you must quite rewrite the whole thing, the whole Gudenlaggen to make it work. But it's rather afraid he wants to say what it really should be, right? Just a little detail, interpretation appears in English translation. They talk about Deutemann. So it's just a little bit more like a question of meaning, but in a way it's good for my arguments. Probably that we have the interpretation here. Okay, so it's more at the end of the historical part.

20:00 And now I just, I shall assume that Hilbert was right or frankly wrong, just as a matter, probably historical matter. So there are softness on the very, it's very valuable. Critics, I don't mean at all that. It should be disregarded. But I just now shall argue that Hilbert's response was only particular. We can go much further and probably should in, say, the same direction, which is not quite the same as we should say. Okay, so, according to Hilder, to build a theory up to interpretation means to build it up to isomorphism. We've seen in the death letter, he's absolutely explicit about that. By isomorphism, I mean now, a reversible transformation, okay? So, it's also rather noticeable here that he speaks about transformation, not just about whatever, different, say, structures. He thinks of transformation and he requires that they are reversible. And just a little remark, in a sense, that gives a room to open the door for restoring, in a way, Aristotelian view. Just remember how Grundlagen starts, right? Let point be one kind of things and line another kind of things, etc. that the fact that there are different kinds there is important, right, but I shall not touch on that point here. But this allows, finally, to just say, okay, mathematics is no longer about lines or points, it's about things, right? And an appropriate notion of things, that would be the notion of abstract sets. So finally, again, there are plenty of ways to say it's much more technically, right? a good semantic for logic, etc., but I think this opens the possibility to that development that we had in the 20th century, the sets every mathematical, to saying that actually ultimately the whole mathematics is only about sets. But now, just to remark that, yeah, first one problem, there is a problem of categoricity,

22:30 which Hildur just didn't see when he wrote Hildur Gangladen, but he somehow discovered, I still don't know historical concepts, he didn't manage to discover them, but somehow between the first publication of Hildur Gangladen in 1899 and the second in 1903, he discovered, he saw the problem, And that's why he introduced his first technique kind of axiom, right, which kind of second order axiom which postulates, which makes his theory categorical. but this notion of categoricity was introduced by Bledler and as a matter of fact okay the most useful theories like Cermelo Franklin etc they are not categorical and in a sense that so of course makes a problem and even today People, I don't know, have different attitudes to that. Many people just ask why we should require this categoristic after all, right? Why should this foundational mathematics block, there are many suggestions that probably, why we should rule out these non-standard models It's a little bit like in 19th century, French Academy ruled out non-Euclidean geometry. So what's wrong about that? But I think a very simple answer is that if we just give up this categoristic requirement, categoristic requirement meaning that all models of formal theories are either more, we just lose the whole framework, we apparently lose the notion of formal theory, because what does it mean in this case formal so my attitude is indeed that that problem is in a sense artificial that we really should give up this categoricity but in my view we should give up with the whole formal method in that specific Hildosense and that's why because this is principle now to build a history up to interpretation is to build it up to appropriate morphism

25:00 and not necessarily isomorphism. If we go back to Bell Climbing now and try to combine these two papers of Bell Climbing, right, one which says that Lobachevsky geometry is really about pseudosphere and the other is saying that Lobachevsky geometry is about ruminium manifolds. What he showed? He showed that one manifold, which is Lovacevsky plane, is embeddable into another ruminium manifold, which is in space. And this is not at all isomorphism right it is monomorphism not really monomorphism but it's kind of morphism which is not about an isomorphism okay if you now come to hilbert's own examples uh from brumladen i mean arithmetical models of geometry of course they are not isomorphism indeed you know because and that's something hilbert would certainly accept when he says when he makes this Does that mean that he makes a formal theory, which in a sense makes arithmetic and geometry, I don't know, isomorph, identical in a sense? Of course not, right? So we have some, we can't construe geometry or arithmetic, but certainly not the other way around. So if we just a little bit change point of view, it is not at all an isomorphism, right? and what kind of notion of morphism we can take and natural suggestion which comes today of course we could just think about category theory i mean it's it's very very weak axioms on morphism i have no really a lot of time explained like like now but what we require is only that our morphisms are composable, right, that every, that there are objects, something like, think about Riemann in manifolds, there are something like projections, translations, morphisms in general sense between them, right, and every object is kind of identity morphism, and there are very single, like, associativity of composition, and basically that's it.

27:30 So, coming back to a geometrical example, we would say, to define the notion of manifold up to morphism, yeah, probably not a good way to put it, I do not insist on the same thing, I use it as a translation, right, to construe notion of manifold as up to morphism, morphism of manifold, which is a well-defined thing, right, which is a differential transformation, would be to cost you a category of remaining manifolds and all that other thing. But here, of course, it's a little bit tricky, because how can we talk about, say, category of remaining space? One thing, the simplest way to do that is construing independently remaining manifolds set theoretically, whatever, then define this transformation between them and say, put the two things together, saying this is a category. But that's not interesting, it's not a problem here. And here the problem is rather to start with a general notion of category, right, and specify, specifying such that, that it could be identified for that particular category and not the other. And there are a lot of, there is axiomatization of this kind achieved rather, how to say, easily, that's a category of sets. It's Bill O'Reilly made it in his thesis in 63. We can really characterize categorical sets with many faults that doesn't work very smoothly, So it gives actually good reason to a little bit change the concept to make it a better work categorically. But at least it's, in my view, what I just now explained is a good reason to do that. Yeah, just one question which I can also, very, very briefly, what about, what about logic then? Right, because still, okay, I just got this, like, better with, like, hand-waving, but how can we do logic? And, okay, that's a little bit open question, because there are a lot of work today going on in category logic. But one thing I just want to touch upon is called functorial semantics, or equally proposed by LVIA, and which probably we should do this otherwise, because it's still too much resembling classical axiomatic method, but still it performs very new features.

30:00 What we do here? We use a category as a syntax first, right? So we write a theory more or less as a category, right? Then we take other base categories, which is more like a category of sets, so it's more like task and semantic. And that, as models, we look as frontons from this transformation from here to here. And then we very naturally define a category of these frontons. And so the first thing, already in that setting, which is in a sense where it's still very traditional, but we absolutely get rid of this categoricity problem. It's absolutely unreasonable. It is quite soon to require that we just up to isomorphism to have just one model, one category. feasible and absolutely not needed and what we really need is just having that that category of model of models having some reasonable properties to make it manageable and in some rather weak conditions we might have again this theory just embeddable in that category just jona delambo so we we can regard our theory as a particular model which is could be called as michael lavise does that we have one theory, one structure, say, one construction, we generate the rest. And that, in effect, very much resembles what we have in Euclid, say, when, say, circle and line just generate a vast number of figures and problems. Alright. So, just to make my point a little bit clearly, I distinguish between what I call form concepts and category concepts, right? And from concepts, concepts of that kind that all items falling under concepts are isomorphs, right? Sokos, for example, I like that, right? Or, say, singleton, etc. But if you think about something like more general, like not a singleton, but just a set, right? Or not a particular group, like symmetric group of elements or whatever, but just a concept of group, right? Those concepts are all category concepts, which means that there

32:30 They are not isomorphs, they are different sets, right, not isomorph sets, but there is still a very natural notion of transformation of mapping of functions between sets, which makes it into a category, and rather remarkably, the concept of category of set itself is a category concept, because we might easily have category of categories, and on the other hand, something like form concept, the concept of form is not a form concept, So there is good reason to suppose that form concept is also a category concept, but that of course depends how we define form, or something like figures. Right. And just concluding remark that that approach, which of course needs much elaboration from the logical part, but in a sense it also allows us to integrate history. Rather, remarkably, more or less the same period that I just touched upon, Beltran, etc., this idea of humaneutic as some kind of universal science, but in humanities, as opposed to, you know, whatever, street sciences were much put forth by Dilté and other people. And I think Dill-Tay just made absolutely wrong assumption that it's something which has nothing to do with science. So he reserved for what he called guidelines and stuff. But historically now we see he just didn't understand mathematics in his time. Because it's that important, or the importance of notion and interpretation exactly, more or less, that time happened in mathematics. But still, I think that that kind of approach advocating, among other things, would allow, say, to integrate history, in a sense, of mathematics into mathematics itself, which seems to me a good idea. We would just take the example of Pythagorean theorem, right? And here is how it stands in Euclidean, here is a kind of Burbanki-style textbook where, okay, it just has nothing to do with this, right? So I think it's rather hopeless to try to figure out a kind of formal theory over and both of these two, right, which would have these two things as, say, different models or something. On the other hand, it's quite reasonable to translate, we can still translate one formulation

35:00 to another. There is a good notion of translation between them, and hence the idea to somehow make this of translation of martin as basic and mathematical rather than say it is that's usually just about very quickly on that last bit. And I wanted to hear more about the critique of Diltai. What do you understand the fact that he is? No, I mean, he just assumed, he tried to create methodology for, yeah, for guys in business shopping. And he used the notion of human ethic on interpretation as something very specific for guys in business shopping. And I just tried to argue that as a matter of fact, it might be as much important in science. Do you have a question? Well, I just had a question about your interpretation of Hubert. What do you think about these now quite common ways of reading Hubert as analyzing phenomena with axiomatics, for example, by Leo Corey? Because it seemed to be quite different from what you were saying about it. I'm not sure about that. Actually, group lighting was a very specific word because, in a sense, it didn't capture at all what happened in geometry, say, at the time. you know, did that all. Of course, he didn't, he would need to go, say, Klein, Felix Klein, gave me a very, how to say, embracing synthesis of what was geometry at the time, and Hildred was somehow a philosopher, but I don't know how important, because of its methods, not because of its contents, right? And, of course, the question which remains open is how well these methods actually apply. And of course Dwarfs of Weblen and all these people finally with set theory somehow improved, made sense. But in my view, it's quite reasonable to come back to that point and to do it otherwise, as I just suggested, this category theory.

37:30 It seems that when Hilbert talks about interpretation, you don't interpret a theory with another theory, because there are there are different things, unless, right, you have something like, let me see, um, Veltero's translation of Veltero's in geometry in terms of geometries and a big thing, but it's not in so morphism, so Helot is fine, reducing one theory to another, but this is not, this is not two interpretations of the things that divide. Yeah, yeah, I agree, but it's something that we have, say, two theories, right? A and B, say, non-formal theories, and then we figure out another theory, F, which somehow generalizes upon, which works as a, with respect to which A and B are both interpretations, and that we require that A and B are either more. Ideally it doesn't work, but we would like to have it. That would mean that F is categorical. So that's his project more or less. Which he himself realizes only partly because he doesn't have, say, he has geometry, Euclidean geometry, but here not really arithmetic. Geometry is not either more arithmetic, but he figures out kind of specific arithmetical construction here here, where isomorphism works. And my suggestion is just to think differently. Here we have say geometry, say gradient geometry here as mathematics, and what Hilbert says is that actually he interprets one in the other, but we cannot do this normally in the other round. Or even if we can, it doesn't mean that these two flashes cancel each other. So I have a completely different reading. If you take your diagram on top, A and B are not interpretations of F. They are just theories that are inducible to F or embedded in F.

40:00 But F, its interpretation, it gets it from its own domain of application, its own ontology. No, absolutely not. Because exactly Huygens points in his discussion to Freud, that there is no ontology, that his notion of formal theory is a formal theory which is promiscuous. And he says there's even something evident for him. And in mathematics, of course, there are plenty of examples. And it basically comes to something simple, that we can say, we can represent a circle by many different images, and that would be every time the same circle. I think he thinks about it more or less in that way. So we have notion of theory, which can be differently interpreted, but all this interpretation, somehow it's, I don't know, personal matter or whatever. And, yeah, and that's what constitutes the idea of formal theory. And my first point, which doesn't work as a norm, exactly, because reasonable theories are not categorical, yeah? Because if there is no isomorphism here, the whole scheme somehow blows up. Yeah, probably, I don't know, that exists on that point. And second point, because his own examples are not real examples of isomorphism, so if we just change notion, which seems to be very innocent, right, we just, instead of isomorphism we take whatever morphism, but in my view we just change dramatically the whole setting. I'm pretty interested in this point. Thank you very much. Thank you. going to make a sustained effort to exhaust you. If you have read the abstract of my talk, you will know that my talk is not going to be an historical talk. This is a talk about philosophy of science. I assume that you have, well, you have knowledge of the logical positivistic view of scientific theories, which I call Ancien Régime, the formal linguistic conception

42:30 of science. I've only one transparency about that, so just to call it to your mind again. And then I'm going to talk a little bit about the Sephysian Revolution, or the informal structural conception of scientific theories, and then with an allusion to a certain political philosopher, the third part will be the main part of my talk, Reflections on a Revolution in Stanford. And that's because Patrick said he says, well, Stanford. Well this is my only transparency on the ancient regime. The formal linguistic conception of scientific theories developed by the logical positivists is roughly we have axioms of a scientific theory in a first order formal language. In the vocabulary there is a distinction between observational and theoretical vocabulary and from the axioms of a theory we make deductions and at some point we reach the observation sentences expressed in the universal language of science and the truth of those we can immediately ascertain by an appeal to our sense impressions. So here's the soil of sense experience, and by means of semantic rules, they give meaning to these observation sentences, and that's roughly it. Now there are more finer points to tell about this, which I'm not going to tell, because as I announced, I assume that you are familiar with them. Okay. Now, and then came Patrick Supes in a sequence of papers in the 1950s with J.C.C. McKinsey. He axiomatized Newtonian particle mechanics as a set of structures in set theory. Now, that goes roughly as follows. He started with, say, a Newtonian particle mechanical structure consisting of particles, space, time, mass, and force. Serpice himself was

45:00 surprised that when consulting textbooks on classical mechanics, he could not find a decent discussion of the language in which these theories were posed. So he was looking, well, what are the fundamental of primitive concepts? What are the defined concepts? But the scientific textbooks don't work like that. And he thought this was a much more appropriate way of dealing with these textbooks. Well, there are a number of requirements, of course. The number of particles is finite, space is represented by R3, time is represented by the real numbers, mass is represented by a positive real number, force is represented by a real 3 vector, and the particles obey Newton's laws. Now, his idea was now to formulate this rigorously, yet informally, in the first-order language of set theory. Hence, Suppy's slogan to be In chapter 22 of his Introduction to Logic, to exhumatize the theories to define the set theoretical predicate. Not the most appropriate place for a philosophical manifesto, right? The final chapters of a logic textbook. The papers on this, they appeared in the Journal for Rational Mechanics and Analysis or something. That's also not an appropriate place to do philosophy or science, but we can find another medium for that. Now, if we translate this a little bit, this is not yet the language of set theory, but it is sort of a halfway house, right? So we now could say that, look, that chi, or whatever, is a Newtonian particle mechanical structure, if and only if there is blah, blah, blah, blah, such that chi is this polytuple, such that the cardinal number of b is finite, there is an a and b such that b is a closed interval, mass is a function from the particles to the positive real numbers, So that's a position, is a function from the particles and the times 2, R3, which represents a space, etc.

47:30 So this is just writing things a little more systematically down, what you can find in textbooks on classical mechanics. And here we see Newton's laws, which is actions minus reaction, those are the internal forces, G. Here is Newton's law of motion, with the sum over all the internal forces on J and all the external forces, and that should equal the change in momentum. Now, this of course can all be spelled out in the language of set theory, yes, but there is no point in doing that. In particular, rigor doesn't demand that. We can convince ourselves that it can be done in just the same fashion as we can convince ourselves that it is possible to walk from Copenhagen to Cape Town without actually doing it, right? And so, well, what I think of as the important point of suffice is that if you are formal, you are rigorous, but you need not to be formal in order to be rigorous. But this is a rigorous characterization of, well, part of classical mechanics, Newtonian particle mechanics. It is completely rigorous, and yet we don't have to formalize. I mean, this is really more than enough. If this is enough for mathematics, then it is enough for philosophy or science. Now, from this characterization emerges a more sort of general picture of signs or scientific activity, you know. Well, here is the observer, yes, he reads from a measurement apparatus as certain numbers. Savvy treats reading numbers from such measurement apparatuses as collecting data for a data structure, right? Well, those are always rational numbers, and what you

50:00 usually measure is some relation on some n-fold Cartesian product set of the rational numbers. So, for instance, here is time and here is a position. If you think of Newtonian particle mechanics, and this is simply a finite number of space-time points. Well, Suvi's idea to connect what we see to what we construct theoretically by means of an embeddability relation. So the data structure should be embedded in some structure in the set of structures that is supposed to be the extension of the predicate I just showed you. There is a minor complication that this extension does not exist in set theory because it is equinomerous to the universe. see this from here see all those this quantifier here so if this is only required to be a non-empty set then the extension of this doesn't exist so then you can actually prove that the theory doesn't exist because if the theory is supposed to be a set of structures and this set does not exist then the theory does not exist So you find all these formulas like this in Sneed's book, The Logical Structure of Mathematical Physics, the famous book of Sneed, that should have been called The Set Theoretical Structure of Classical Mechanics, because that's what it is about. And it has all these sort of definitions, and all these extensions that don't exist. So the book is full of non-existent theories. That's a bit of a pity, but it is easily repaired. The only thing you have to do is you have to write here this. So you're simply going to pick your entities only from a very tiny bit of the set here at a good universe. We depict it like this. And so this VW plus W, well, it sits here.

52:30 So you're only in this point here, so this little bit. And that's more than science will ever need, by the way. You don't need more. All this is redoneable, at least for the purposes of science. So, and then they exist because by means of the Auschundrons axiom or thermolode, you get a subset from this set. Well, that's it. Now, so this is once more, this is sort of a picture that is now emerging. so I've drawn it a little bit different now, so we have an object in the world, we perform measurements, we do experiments with this object, we get a data structure, we embed it in some structure of the theory, and for this structure in which we're going to embed this, we have to search in this theory which is the set of structures so search and embed and measure that is sort of the of science, according to this view. Now, what I have here, the brown line, everything that is in the brown line here, that is all expressible in the language of set theory, see? The only anything not expressible in the world. Now, what we now can say is that this structure that we find, in which we embed the data structures and these data we obtain from the particular object in the world. That structure is a mathematical model of that object. Saffis himself has asserted on many occasions that he was inspired by Tarski's conception of a model. But that is still a little bit different, see? So let us recall Tarski's conception of a model. If I teach it to students, I always write it down like this. It consists of a triple, right? Here is some structure in the meta-theory, so that's usually another set theory.

55:00 So that's a structure, so it's a domain with a number of relations or sequences. There is a designator, so that assigns terms of the language to items in that structure, and there is the satisfaction relation, that makes sentences true or false. But this thing is a model of the theory. But here the structure is a model of the object in the world. world. So although Sibis was inspired by Tashkid, it can't really be the same thing. And as you can see in the way I have presented this to you, it isn't the same. Now, although it was conceived of in the 50s, it took quite a while before it latched on to philosophers of science. And Van Frassen is an example of a philosopher of science who advertised it and developed his own sort of view on the way of construing theories in this fashion. He followed more or less David Pett. but nowadays we can see that among philosophers of science this is almost now the received view so once the other view was the received view but if you now read received view it refers to this well this view of characterizing scientific theories has a number of advantages it is claimed and I'll list them for you we all suddenly do this, rather than what earlier philosophers of science did? Well, I think the first and most important reason is that actual scientific theories can be and have been treated successfully. There is a long list of success. Sneed's book is a treatise on classical mechanics in all its varieties, the Hamiltonian mechanics in the Lagrangian mechanics, etc., etc., quantum mechanics has been done, various geometrical theories, relativity theory, Subbis did himself, learning theory, and he did relativity theory,

57:30 theories of probability, exchange economics, chemical theories, theories of parliamentary democracy, et cetera, et cetera. It is an astounding list of successes. As the previous speaker said, he quoted Stegmüller, who indeed said that if you want to have any success in the the previous view, the formal linguistic view, you have to be a super Montague to achieve that. Now, actually, Montague himself, he was his own super Montague, so to speak, because he succeeded in axiomatizing in the, according to the previous view, a part of classical mechanics, actually. But that is, I think that is the only known example. And so I think this is the reason you can characterize scientific theories rigorously, and you can actually do it, right, without having to form lots. I think that that is the main reason for its success. And that's, of course, an advantage. You want to achieve something, so if you buy it into this way of doing things, you get them success. Another advantage is that the whole of all scientific theories and all experimental outcomes are available in a single domain of rigorous discourse, and that's the universe of set theory. Three, all mathematics ever needed by science is available there. I mean, if you want to axiomatize quantum mechanics, say, by means of a formal language, you also have to axiomatize all the mathematics that as needed. So you have to axiomatize the real numbers, the complex numbers. You want to have Hilbert spaces, operators, etc. But soon you would rather have the few axioms of set theory because then you can make everything, right? So this is extremely more simple to do. So now once you adopt set theory and you can make all the mathematics for all the theories

1:00:00 you're going to axiomatize. So, that is undoubtedly an enormous advantage. More advantages. The formulation of the theory is irrelevant. The class of models is directly defined. I'll come to that. There is no need for second-order logic. It was only a few years ago that a paper and analysis appeared, where someone tried to argue that second-order logic is really needed for science, and absolutely no reference to Serpice or anyone. It's not needed. There is no need for a distinction between theoretical and observational vocabulary, a distinction that is notoriously difficult to draw. It facilitates inquiry into inter-theoretical relations very easy in a single domain of discourse, again. It proceeds wholly in set of scene. It is rigorous and informal, and it facilitates representation theorems and the identification of invariances. And that last claim is established here in Seppi's his own book, Representation and Embarrassing in Scientific Structures, from 2002, and he got the Lakatos Prize for that two years later. Some of the things I've said, and I will say, are to be found in this review of the book. Okay. Now I'm going to talk about the five points of my summary in which I'm going to doubt some of these claimed advantages. Okay, so here's the first one. The liberation from a particular formulation of the theory. Well, here is one philosopher of science who claimed the advantage, this particular advantage. from Frasen in Laws and Symmetry. In any tragedy, we suspect some crucial mistake was made at the very beginning. The mistake, I think, was to confuse a theory with the formulation of a theory in a particular language.

1:02:30 The first to turn the tide was Patrick Zuppies with his well-known slogan, The correct tool for philosophy of science is mathematics, not metamathematics. Now, but, but, my criticism of this is, although it's difficult for me to make sense of what he's saying here, but my criticism is this, it is that the signature of a structure, right, or model, fixes that of the formal language in which theories can be formulated, made true by the structures. So it is difficult to see what sort of freedom in the language language that you have. All these languages are more or less, in a definite sense, isomorphic. So let me explain this a little bit more in detail. Now recall Tusky's conception of a model, and Suthys was inspired by that. So if we are going to take this seriously, And suppose the structure here has two different domains, and it has one subset of either of these domains. It has one binary relation, another binary relation, and one ternary relation. Well, and this simply fixes the language of the theory, because it must have two sorts of variables. one monadic predicate, two dyadic predicates, and one triadic predicate. And that will be the vocabulary. And conversely, if this is your first order formal language, then the models will have this form. So what freedom do you have to choose your language? Apart from the fact that everything proceeds in the language I've said to you. So, what might be claimed, and what is an intelligible claim, is there is an independence of axiomatization, right? Because no matter how you axiomatize a theory, if the theory remains the same, then they have the same class of models.

1:05:00 But you have exactly the same formal linguistic view, right? It doesn't matter how you exumatize it. The sets of sentences you can prove remain identical. You have the same class of models. So that's sort of a dubious advantage. How adequate is the informal linguistic view? So suppose we were to characterize a part of quantum mechanics. So for the sake of it, quantum mechanics consists of structures of the following type. We have a Hilbert space, we have a state operator, a time dependent, a Hamiltonian, and a probability distribution for energy values. Now it is a corollary of Gleason's theorem that quantum mechanics, characterized in this fashion, is irrefutable, because every set of relative frequencies of this form, so here of energy value, and here it's relative frequency, can be embedded in some model of quantum mechanics. In another paper of myself, in which I proved this. But if quantum mechanics is irrefutable, then it is not genuinely confirmable either. So the conclusion is that quantum mechanics has no empirical content. Quantum mechanics is not a scientific theory. And that seems an absurd conclusion. So something has gone wrong. But this proof is correct, you see. It is a single corollary from Gleason's theorem. So here something goes wrong. But this is a characterization perfectly in agreement with the Serpecian way of doing things. So the correct conclusion is, this characterization of quantum mechanics is not adequate.

1:07:30 The set of structures must be restricted much more. But how? This is more complicated. Now the relation between the structure and object, between the structure and the world, I mean, all we can say is that such and such data structure can be embedded in this and that structure. But the aim of science is not winning this embedding game. I give you a data structure. Can you embed it in your structure? Oh, I'm going to look. Yes, here I have one. I can embed it. Fine, success. But the aim of science, I mean, is that winning the embedding game, or is it to understand the world, to produce knowledge of the world, to find out what the world is like? Well, unless one is an empiricist, then one can be happy with this as the aim of science. right? But if you want to find out something between this object and this here, then the structural view doesn't say anything about this. But then the... So if this only makes sense, this aim of science, for an empiricist, then the informal structural view is not neutral with respect to the realism debate. And that was always one of its contentions, of course, that the enterprise would characterize what a scientific theory is that should be independent from the realism issue. So perhaps it is, perhaps it isn't. The fourth question is, how can characterizing scientific theories as sets of set theoretical structures throw any light on the meaning of scientific concepts? Is it one of the aims of philosophers of science, let me put it this way, is it one of the aims of philosophy is to get clearer, and consequently one of the aims of philosophy of science is to get clearer about particular troublesome scientific concepts, say the concept of a

1:10:00 quantum mechanical state, then it doesn't seem that fixing on the mathematics is any help. So that the mathematics of quantum mechanics, that seems very well understood. Big texts on mathematical physics about that. The interpretation issues rage on. It seems only concerned with how scientific concepts are represented mathematically. So it is not very helpful. And finally, it cannot possibly give any complete picture of what science is even if we set aside my previous complaints, for the following reason, you see. And this I call a sea of stories. When presented with a data structure, for which set of structures is it relevant? So let me illustrate this. this is a set d, n, k and it consists of what is it the posituples here so this is a natural number and here are n rational numbers so here are four examples of that in which n is 2 so I have ordered ordered pairs and I have only 10 of them Measure the length of an image of a stick and its distance to a length 10 times. So you have this distance as Q1, and you have this length as Q2. And you do it 10 times. Measure the current and the voltage over a resistor. So you can say, now the first is a voltage and the second one is a current. the travel distance of a free-falling object for 10 points in time. And finally, measure

1:12:30 the height of a mercury column at 10 moments in time during the day. So these are all sort the same data structures. But if I only give you this set of ordered numbers, you simply don't have a clue what it means by only giving these numbers. What does it mean? A story has to be told how I have obtained them in order for you to know to which theory you should go, for which it is relevant. So this is relevant for theories about electric circuitry. This is relevant for, well, the part of classical mechanics for free falling objects, you know, So, all data structures relevant for different theories, which cannot be discovered by staring at these structures. So, everything, all these data structures, as well as these theoretical structures, as far as I see, they all float in a sea of stories. And that whole sea of stories is of course, I would submit, cannot be captured in the language of set theory. In any case, if you want to do that, you probably have to be a super, super von Neumann. Well, that's it. That's it. That's it. I like very much this content of your talk, it should not come to any surprise to all the long time friends of that surprises, however I have major qualifications as to how these And maybe this is a, maybe I would almost urge, I would urge this to, on path when I see him to rethink the formulation in that theory, the reliance on set theory, axiomatic set theory, you mentioned Jeremy Wolfrenkel theory in particular.

1:15:00 I think that should be absolutely avoided. That leads to a major problem. Perhaps I can, instead of trying to go into the details, I'll give a kind of historical reason why these ideas were formed in a way they did. I think they patrialized very early that the logical positivist idea of philosophy of science as the logical syntax of the language of science is how it was restricted, and began to do something better. However, what he really wanted is a language of science which can operate as its own meta-language, and he wanted a model theory, and at that time, model theory didn't exist, and the best approximation to that, best replacement, was set theory. I sometimes call it the Californians, the Poeman's model theory. And that's how the formulation originally happened in the, by reference to cell theory and not some other, some other way. And so I think this explains the, historically, the use of cell theory. I'd be happy to give you more details, but I think, if I may put it in a form of a part, I mean, axiom and et cetera is Frankenstein's monster. It's taken over what is no business of it. But anyway, let me just state this and try to expand it. One for a small comment. So the book for which he was awarded, the Lakatos Prize, which is a few years old, you still find the slogan that is always associated with it. To axiomatize the theory is to define a set theoretical predicate. I mean, if we would go to some other set theory,

1:17:30 I think that all these objections I get will stand. No set theory altogether. but then what's left of this view I can experiment it's not a comment at least in the latest of his books there are references to Burbaki project and the whole thing looks like the attempt to make the same thing but not in mathematics and to empirical sciences and actually I have a question and another question Professor Hintik already clarified what's going on with Cermelo-Frankel, etc. Of course, you can't deduce something like topological theorem from Cermelo-Frankel. You can take it as a basis, then add some new axioms, which could well be second order, by the way, to make something like topology, and then hope to deduce something, but not deduced from the axioms of Cermelo-Franco or, you know, the rest of mathematics. You would correct me. I think you would explain that too. But what you can have in set theory, that is all the mathematics used in science. So that's sufficient for this enterprise. Of course not, if you use anything like functions, you know, Topolo-Hilbert space, you wouldn't deduce, it's absolutely crazy to think spaces from Sermelo Franco, you can hope to take Sermelo Franco as your basic set theory and then, construe Hilbert space as a structural set, by Bulbaki-Stat. But that would mean that you add axioms, of course. No, that's what Bulbaki are doing anyway. I can test that. I can give a set theoretical predicate for H is a Hilbert space, if and only if, and I give you a list of, so a conjunction of all sentences in a language of set theory, and I can simply prove on the basis of the axioms the existence of Hilbert's basis in the domain of discourse of set theory. I mean, that is really an elementary exercise. I refer to von Neumann's book of 1932, where this essentially is proved. You don't have to add axioms to the axioms of You don't even need the axiom of replacement, you see, because replacement is needed to go beyond that.

1:20:00 But what about axioms of this theory of Hilbert's Paces or something, so they're all redundant and they all follow from axioms of set? Yes. But okay, then another remark in time is that still, of course, this Burbanki approach is contestable and today is probably no longer in fashion, but at least it's true that a lot of mathematics in the second part of the 20th century was made in that framework. And on the other hand, that kind of translation of physical theories like quantum mechanics would never, as far as I know, taken seriously by working physicists. So that's all exercised less that kind of philosophical view, translation. And when you say that characteristic was not applied... What the positivists did, you know, so these sort of rigorous formalizations that was even taken less seriously. Yeah, but, although it was somehow rooted. The philosophers of science need not be taken seriously by scientists. No, I think they should. They have their own agenda. They reconstruct scientific theories. I'm not agreeing with that. So you think they agree with you? That everything should be done in category theory? You think that's what physicists would agree upon? I think this is a crazy idea. No, but I think they are working today. They are motivated. They are trying to apply and say categories in physics is something that inspires, you know, physics. Probably it wouldn't be... Yes, of course, yes. But if you are doing a kind of inter-size, it's absolutely... I know that, I know that. My claim would be that any category, any category theoretical concept that has an application in science, say physics, because I know these applications, be translated in set theory. Of course not. How about the Arrows-only theorems? Could you do the Arrows-only theorems in set theoretic form? Yeah, I think so, yes. Quite an advanced category of theory now is totally applied in physics, and of course no way you can do it in set theory. Otherwise you can make

1:22:30 kind of, how to say, this philosophical translation, but it wouldn't ever be anything workable I don't think he could do aerosol. But then still, but suppose we change sort of the formal framework. So these objections that I level, well I think they are more or less philosophical objections against doing it in this fashion, so the question is whether such a move would help. Do you then get more clarity about scientific concepts, etc. You then not need a sea of stories. I think you will still need it. So for the most of the objections, this will not help. I will try to reinforce it. Oh, you reinforce it. Yeah, yeah. Okay, I think we should move here. Thank you again. Thank you.