The Protean Chartacter of Mathematics (contd.)

0:00 Among those permutations, there are the so-called even ones, the 60 of them, and that set of even permutations is a simple group, and that's fundamentally the reason, because you can't break down that 60-element group into a smaller group. That's the reason, it's called A5, that the Galois theory cannot be solved. When the Galois theory was first discovered by Galois, he died before anybody understood it. He died in a booboo. Nobody understood it in France even for many years until the wonderful book of... Oh, Chevrolet came much later. Jordan wrote a famous book on Traité des Substitutions, which as far as I've understood the history is the first one that really understood it. Still, the textbooks on it were mysterious, and when I was young and studying American textbooks, it was all mixed up with complicated formulas and the like. Later on, the development of abstract algebra in Germany. Under the leadership of Emmy Nurgut, who was an assistant to Hilbert and an associate professor of mathematics at Gehring, led to a real understanding of the Galois theory. You look not just at the roots, but you look at everything that is built up from the roots by addition and multiplication of the field. You look at the onomorphisms of that field. That means the mappings of the field in itself. They form a group, and the typical view of the Galois theory is that I have a field, and the rational numbers inside it, and it all sets in between the group, and all sets in between the subgroup. Then, given any subset of the field, I can look at all thoseomorphisms G, which leave every element of S fixed, that goes from subfield to subgroup, or given a group. A subgroup, a subset of the group. I can look at all those elements in the field which are left fixed by every element of the group.

2:30 So there is a back and forth between these two. The back and forth isn't exact. You don't go over and come back to the same thing. But if you look at those things that do come over and come back, you get exactly the subgroup. The subfields, those things that are closed under additive and multiplication, and the subgroups, those things that are closed under composition, so that there is an extraction from this junction, there is a bijection, one-to-one horizontal, and that is something that was later called a Galois connection and has later been developed in the program. Understood further by Dedekind, and later there have been many successive stages in the understanding of it, and for the fun of it, I wrote down Lagrange, Galois, who saw the notion of groups, who saw how the symmetries formed in groups. Remarkable that notion didn't come up early. The Traité des Substitutions of Jordan. Dedekind, about 1900, had conceptual views of it. A German mathematician named Steinitz showed how it worked not just for number fields, but for material fields. My teacher at Yale, James Pierpont, gave a colloquium on it about 1900. Emmy Erder, that wonderful mathematician, used to lecture with the words, Alles steht schon bei mir. Everything is already presently there. Out of her ideas and out of the lectures of Amy Lartine, von der Werden wrote that splendid book called Modern Algebra, which presented gamma theories in its conceptual form. Later, Amy Lartine thought a better way of doing it used the notion of vector space, connecting different ideas in the fashion that I emphasize is what goes on all the time. Hawking gave beautiful lectures on this. I listened to his lectures and Birkhoff and I copped them and put them in a book. Then Grit of Columbia, there's Birkhoff in the same book, saw that the same ideas applied to differential equations, not polynomial equations.

5:00 Jacobson saw that they extended to those fields called inseparable, Cauchon did more differential equations. The redoubtable Alexander Brodenty saw that the notion of a covering space was really much the same idea as a calendar, and wrote a general theory. Kenison at Clark University had a theory of action. Recently, in 1981, Joyal and Tierney had a general relation of Brodenty theory. A young mathematician in Tbilisi in Georgia put it in fine categorical form and just recently I finally got around to understanding something that I think Lavera understood long since, how in that original picture when you go back and forth you don't come back to the same thing but you can select those parts on each side which do come back to the same thing and in that fashion this so-called adjuncture comes out to be a bijection. So, there, straight out before you, is the story of this one idea that understanding the notion of Galois theory depended on the interaction of many different things, in this case, interaction between equations and covering spaces and other examples, so that the protean character of mathematics, the fact that it fits together, Mathematics is a protean subject in the sense that the same mathematical form comes up in many different places, unexpected connections in physics, in integrals, in the pentagon and the hexagon that I mentioned. The consequences of this are that mathematics has to be formal, because it isn't about this substance or that substance, but about the forms and common forms. But knowing about it tells you which ends in that things are dead ends, and that the understanding and the application interact with each other to make this flow itself.

7:30 Thank you. Thank you very much Professor MacLean for giving us the rare experience with some philosophy of mathematics. Before the riots start, we shall have a common sheet from actually more than four years ago. Confronted with such a wealth of lively examples, the job of the philosopher of mathematics, the most attractive one, was that he had to boil down this wealth of examples to some perhaps alien. We have philosophical concepts, quantum mechanics, and this is my duty and I start to do it. I'd like to compile my remarks from Professor McLean's lecture on the following two topics, the protein character and the network structure of mathematics. Forming in both mathematics form and function, I refer to it as an abstract function of it. The framework of my comments is given by the following general thesis. A philosophical theory of mathematical knowledge should be developed in a manner parallel to philosophical theories of scientific knowledge and quantum science knowledge. Because there are basic connections and similarities between the domains of empirical science, of common knowledge and of mathematical knowledge. All this looks parallelly to me. It is based upon an epistemological monism. Mathematical knowledge is part of the overall context of human knowledge and is not totally separated from other areas of knowledge.

10:00 The philosophy of mathematics should look for connections and contexts with the philosophy of empirical science and cognitive science. So, the philosophical theory of mathematics on information should be considered as an integral part of the general theory of cognition. More precisely, I'd like to argue for the following two claims. Abstract functionalism supports the parallelity thesis and second, the parallelity thesis can be used to explicate this negative form the parallelity thesis claims in mathematical cognition and human cognition, maintaining, for example, that in contrast to any other kind of knowledge, mathematical knowledge should be considered with some suspicion. The philosophy of empirical science are not supported and can be considered destructive if it does not become a protean character in the network structure and vice versa. I'd like to use the protean character of mathematics to elucidate how mathematics has a protein character.

12:30 Philosophical point of view. Would we find if we had more precise information about what those parts of mathematics, which are proteins, are? As far as I can see, there are two possible answers. One, the framework of the traditional philosophy of science. The second, more aligned with recent developments in cognitive science. According to a cognitive science-oriented approach, The first slide may sound like a rather trivial answer, but this is not the case. It is one of the central conceptual problems of cognitive science to provide an adequate explanation of what a concept is. A concept is not simply a subjective idea a mathematician had in mind. The basic question, what parts of mathematics have protein character, is the following. Mathematical theories are the parts of mathematics which have protein character. Again, don't say that this is a trivial under, because everybody in the field of mathematics knows what is the structure of a mathematical theory. This is not the case. Look at the corresponding problem in the philosophy of empirical science. There, a tangible conceptual problem is to provide an adequate account of what is an empirical field. The world, against logical empiricism, can be formulated as a criticism of the inadequate.

15:00 If we take into consideration the protean character of the linear account of topology, A philosophical theory of mathematics should be able to find something more, that is, a more precise explanation of what is meant by the intuitive claim, typical mathematical theory, several models, applications, interpretations. Once again, it is useful to invoke the parallelities and to look at the corresponding problems for empirical theory.

17:30 Common accounts of philosophy of empirical science have given up the idea An empirical theory has just one single universal application or model for interpretation, namely the world as a whole. Rather, as has been pointed out by Professor Smith, that usually an empirical theory has numerous and varied applications, models, or interpretations. An empirical theory has to be considered as consisting of at least two parts. First, we have the domain of its intended applications. And second, we have the theoretical or conceptual corner K. The theory makes the claim that the intended applications take the framework of logical aphorisms

20:00 The network structure of empirical science had been conceived in a rather deductive way. A classical example is provided in Ernest Nagel's book The Structure of Science. According to him, the various parts of empirical science are tied together in the network-deductive-reduction relation. Eventually, all of empirical science should be derivable from a single, comprehensive, basic theory. The case of mathematics, the classically deductive network of empirical science, corresponds to something like the Wabaki hierarchy of mathematical theories. According to this account, all the structures mathematical theories deal with are built up from the so-called mother structures, groups, order and topology. In this manner we could conceive the theory of Lie groups and the special case or sub-theory of the theory of different table manifolds and the theory of groups. More generally, according to this hierarchical network concept, all mathematical theories can be considered as special cases or sub-theories of said theory. It goes without saying that this approach does not fit well in MacLean's quotient consensus of mathematics. Present-day philosophy of empirical science has given up the far too simple picture of intertheoretical relations and deductive relations, as it has been propagated by logical empirics. Hence, the relation of, say, statistical mechanics to thermodynamics

22:30 No longer is considered to be a simple deduction in the sense of elementary logic. Rather, relations between real life empirical theories involve aspects of approximation, idealization and theoretization not being present in elementary logic. It has been a focus of intensive research in philosophy of science, that is to say philosophy of empirical science, To find out the structure of the inter-theoretical relations which exist between empirical theories, it can be considered a formal result of this debate that the usual logical relations of elementary logic both do. Subscribing to the protein character of mathematical theories, I get that the whole corresponding result was a network character of mathematics too. That is to say, we might conclude that the above-mentioned probability explication of the natural character of mathematics is nothing but the obsolete counterpart of the terribly whistled code that has been criticized rather successfully in the case of empirical science. A better explication of the natural character of mathematics seems to be offered by category. We could consider mathematical knowledge as a net of functorially related categories, but this probably is only a first approximation. There are far more subtle relationships between, say, the theory of numbers and the theory of complex functions than those which can be expressed by category-theoretical terms alone. But the intertheoretic relations between mathematical theories are far more complicated and of a quite different kind than the relations between empirical theories. Perhaps this could be used to distinguish between mathematical and empirical knowledge. But all these are, so to speak, empirical questions for an empirical philosophy of mathematics.

25:00 A branch of the philosophy of science is still in its infancy. I think McLane's personalism would help to bring to maturity such a car, at least in some steps. Before proceeding to a general discussion, Professor McVey will respond quickly. I'd just like to thank Dr. Moorman for his very insightful comments and his notions of things that still need to be done to flesh out my notion of the anthropobioteric. I have not explored sufficiently its connection with the cognitive science. And in particular, the connection with the work of Snead and others in the theories of physical sciences. Much more remains to be done. So, thank you very much for your time. Whether you wish to speak, would you please remember, this is quite a large room, so speak whatever language you speak in. Do not speak too quickly, and please give your name at the beginning, because I only know a few of you. What was my last question? Michael Radnick, question to Professor Plank. I was thinking about the distinction between mathematics and the empirical sciences, and I was thinking, as you were giving the initial part of your talk, I was thinking, well, lots of times in the... The development of physics, the idea of motion, could get more and more phenomena and something in motion in the analogies between the equations in various branches of physics. And then later on in the talk, you put up the network. You had a bunch of branches of physics in the network itself, so I'm wondering, do you think that there is a distinction between mathematics and physics in terms of approaching the parents of mathematics?

27:30 I guess I was claiming there was a distinction, and you'll point out difficulties in my opinion. And I would have to think about that some more, but you're certainly right. There are cases where, after Newtonian mechanics was developed, then the same idea applies to many other things. But then, I guess I would then have to claim that that was indeed a feature of mathematics. So I'm an imperialist for mathematics. I want to say more. I sometimes think that... Much of what is called physics or economics or biology, you would call mathematical physics economics or biology, and by that I mean you take a concept from biology or physics or economics and you play with it as a mathematician, but you don't give a damn about the empirical consequences. I think something like that was in your mind when you put in quantum mechanics and stuff in the network. My name is William from the University of Berlin. Probably the notion on the floor plays a very central role in your conception of mathematics. If your problem has to be developed in the systematic logic of mathematics, shouldn't it be the case that this problem should be somehow systematized as well, that one should build out a sort of theory of thought, a general theory of thought? And if this is the case, Is there no answer that you somehow fall back to the result of foundationalism where everything is a form of something that never was a form of anything? So you see my suspicion is that perhaps it's one thing. And your idea, to the very end, one might come to a sort of universal theory of mathematics, which would be just another alternative to the kind of universal theory we already had like that theory of mathematics.

30:00 No, I guess I haven't come to that point yet. And when it comes to Bourbati, to be sure, Bourbati had his mother structures. But any omitted sub-character. And for example, what about the omitted categories? I don't... there will be pieces that fit together, but I think that the nature of mathematics is open to admission. There is no permanent form. But thank you for your time. One minute, Tom. I think the topic of this lecture is appropriate. You should remember that we have copied, in the same way that we take all kinds of forms, which are one form and another form. And in your example, in fact, the form of the mathematical algorithm always remains the same, whereas the substrate, which is the tabla, can take the most advanced form. So this is, I would say, purely linguistic objectivity. Concerning the elaborating of quantum data of progress in mathematics, I think you stressed the study of historical architecture and I feel better understanding of some of these fields. I think you should have added the opening of new fields. Yes, I should have. I failed to do that. And the word propion, as you say, it doesn't fit quite exactly. I needed a good word as a title, and it brought out knowledge. I will ask you two questions. First one is, I know that mathematics is not about standards, but I think there is something new with the use of computers to prove some of the results. I am not thinking about cohomology. I am thinking about the new results of the research.

32:30 What they mean are services, the new competitively vetted minimum services of genus 1, which came two years ago, making experiments using computer graphics. The computer is an idea of outcomes of the problem. And then, of course, when we were able to give an analytic tool of the problem, the first idea of the tool... So, in a sense, it was a sort of experiment about projectors and how to plot them. The second question is about, in the summary of your paper, you said that Platonism is false but is perhaps convenient for mathematicians in the whole pursuit of research. And a chapter is dedicated to the executive of Platonism, and he is convinced that the most important result of mathematics is not invented by mathematicians, but just discovered because they are so already there. The same idea is in the book of Alain Comte, of Alain Comte, last year, Mathéa passé. I think the problem is not if this is true or false. Why do you think that this mathematician has this sort of necessity to justify, and they quote exactly the same part of the dialogue, I don't know, to justify their research? There are two very prominent mathematicians, like George Penrose and Alan Cohn, escorted, the same year, to two different groups, the same very famous part of the lecture, to say that they don't invent new theories in mathematics, but just discover something that is already there. The most interesting example in mathematics is not discovered and not invented by mathematicians. I don't think that the problem is that this is true or false, but I think what is interesting in your opinion is why mathematicians stand for mathematicians.

35:00 As to the question whether mathematics are invented or discovered, it's a well-worn old question. I don't think it casts much light on this. So I guess my answer to that is I don't care. So the notion of a group, whether invented or discovered, discovered too late or invented too late, nevertheless has its many applications. Groups crop up in the gamma theory, in the symmetry, in the beautiful symmetry here in the Arabic decorations in Spain. Whatever was invented and discovered, I don't care. It's there. No, I should have been more precise. I meant to reject only the stronger sense, in the sense that there is a single formulation, be it set theory or be it Heidegger, Ressler, Kempke, or mathematics, but that there should be some agreed-on formulation, yes. There's a legal validation I would make. Mathematicians need agreement as to what the correct proof is.

37:30 The difference between mathematical and physical theories is that physical theories are mathematical theories, but something else. So mathematical theories can be the pure study of some sort of form within the tendance of that form of realising something with the parcel of reality, whereas mathematical theories are, first of all, to begin with, mathematical theories that can be supplemented with some type of empirical evidence of those forms even realising certain parts of empirical reality. But one question could be whether you would agree with something like that. The second question is, I think this point here, that mathematics has that, for example, a character which has been spoken about, and that is very important. But at the very least, it is useful to differentiate between mathematical theories, like a group theory or a colloquial theory, I guess my intention was to reject that distinction, in that I think that the mathematical notion of real number pops up for measurement of one thing, for measurement of many different things, because there are many different empirical realizations of the one mathematical form of real number. But as to the connection with physical theory, I didn't try to get into that. Something has to be done. I have a very simple and very general question and I have a case for motivating it.

40:00 You started with a remark or a way of describing such people and now I miss some of the consequences from your point of view. Do you think... We need more philosophy of mathematics or less philosophy of mathematics as a form of your deities, of your protein character. And now my example, when in the 20th of the century, Hermann Weyer, who in 1918 wrote the book on intuitionism, and when he found that his way to the point of Karl Hilbert and his and Hilbert's formalistic approach, Exactly, I think, the argument is right, the one you have to write here. It is the only estimation for the formal objects that they have functions and applications. And so you said the main concept is that highly theoretical concepts are embedded in the process of... I think, yes, we need much more philosophy on mathematics. And many of the questions which have come up so far have been pointing out things that I didn't believe, many new investigations. As to Hermann Weyl, yes, I had the great good fortune of studying some with Hermann Weyl when I was a junior, and perhaps my dismissal of set theory comes from Hermann, because Weyl always said, set theory is not good, there's too much sand in it. And as to his continuum and the intuitionistic thing, that's another example of the protean character of mathematics, because the intuitionism which seems to be part of you has reappeared in geometry and the theory of an elementary totals, so that there are many connections. But certainly we need more of that, and I encourage more of you to do it.

42:30 Does your approach show the similarity between mathematics and fine arts? The second question is that do you think that the set theory is also the dead end in mathematics? Is that the best one? Do you think that the set theory is also the dead end in mathematics? No, I did not mean to say that. I think the set theory is not the foundation of that. But there are many useful parts of combinatorial set theory and the like there. As to mathematics and art, I have once given a lecture in which I claimed that the growth of modern art, Picasso and the like, was parallel to and influenced by the growth of abstract mathematics. I'm not sure I can defend that, but I claim it. Thank you. I'm Leo Korg from Tel Aviv University. You talked about our change in the understanding of a mathematical fact. First of all, you said that it was an improvement, but I would like to ask if you think that in all cases we improved our understanding of mathematical facts or maybe we are changing for better or for worse. Second, all the motivations for the calculus for this change in understanding that you mentioned were from inside mathematics. Doesn't it happen also that we have reasons for changing our understanding based on, let's say, the application of mathematics in natural science or elsewhere, or on the other side, philosophical conceptions, or any other kind of factors Do not count both inside mathematics. Corrie is correct. I did not mention sufficiently that many of the mathematical forms of which I spoke were developed because of their connections. He raises the question whether all progress is good or whether some of it is bad, whether things are better or worse. Yes, I think some things are worse. If I try to name them, I get in trouble.

45:00 You mentioned Ramon Biles' comment. It seems that there is a little misunderstanding I detected in it. Maybe if we reject that theory as a foundation for mathematics, it's because we think it's not a good enough foundation. It wasn't sufficiently thought out. It's not because we reject the idea of universal theory. I think, for myself, I think we must continue to strive for more and more universals. Which requires deeper study of more parts of that network, requires more thinking out of the interconnections, but it does not mean that we should give up the idea of unified vision. That's a correct formulation of your view, and you're a wonderful view in my mind. I don't altogether agree with that.