The Protean Chartacter of Mathematics

0:00 I'd like to introduce a fellow member of the mathematical community who needs an introduction. It's very common on these occasions that the chair will name you with one introduction, and that's somebody who doesn't need one. And I think these are usually too sensitive to paradoxes. And we deal with that. So, I shall complete my own introduction by proceeding directly to the title, which is the Procedural Character of Mathematics. It has been a pleasure. Thank you very much. It's a pleasure for me to be here in San Sebastian. And I'd like to begin by thanking the Department of Logic and Philosophy of Science. No, the general thesis of the conference is the sense in which mathematical structure is related to scientific theories, and in connection with that, I would like to advocate a view of the philosophy of mathematics different from the standard one. So partly I think that the old triumvirate of intuitionism, logicism, and formalism has long since gotten retired, and the old experts like Wittgenstein, Quine, and Putnam never knew enough mathematics to really talk about it. Well, I think we have to look at what mathematics is and how it fits, and so my thesis is that mathematics is protean. Meaning, it applies to many different things. So, for example, one and the same mathematical idea or mathematical structure appears in many different realizations, of course not just physical but other similar scientific realizations, or put differently, that mathematics describes forms which have more than one possible meaning.

2:30 And this thesis is based in part, if we focus well enough, in part on an examination of what actually goes on in mathematics. And I will begin with evidence of that sort. An example of particular mathematical notions shows many different interpretations. To some extent, some of these are taken from a book which I recently wrote called Mathematics, Form, and Function. But the thesis here is more explicit than it was there. So, first, then, of the various edicts. So I mention dynamite arithmetic as this protean character. A character that applies to many different sorts of things. That's a better topic than I can read from yet. So that when you look at numbers... The natural numbers are really not unique. They can be the ordinal numbers, first, second, third, or the cardinal numbers, one, two, and three, and there are two different definitions, but they're the same system of numbers. Multiplication can be different sorts of things. Namely, it has different interpretations, but I mention only a few of them here, that area or cost equals number of items times price or length. The laws of arithmetic or algebra have a common name, but they apply to different things, so the well-known associative law applies to addition, it applies to modification, it applies to other operations, such as the tensor of products, which I've written down here, all men are a product of you, so the same idea, the same mathematical structure of an associative operation. They are all cropped up in different forms. And this illustrates by thesis that mathematics has a protean character. Now I go on to other parts of mathematics.

5:00 There are different ways of defining what similar triangles are. There are different ways of picturing them. And later I have a whole bunch of triangles. The right that are all supposed to be similar, they are perhaps carefully drawn, you can imagine that they are, and the definition can be done by ratio of sides, or by angles, or by a more, by use of symmetry, by the operation which carries this triangle back into this triangle and around. Similarly, things like the most original, the first original theorem of mathematics, well known as the Wegerian theorem. It has an arithmetic form, the one that led the Paramounts last year, where you have all the squares, like 3 squared plus 4 squared equals 5 squared. Or it has a geometric form, like the famous thing from Gilbert and Sullivan. I'm teeming with a lot of news about the square on the hypotenuse, even some of the squares on the clouds. And just as an illustration, when you come to prove something, Like the Pythagorean theorem. There are different ways of proving it. There is no unique proof. I have sketched here two of the proofs, or the one which takes the right triangle and drops an altitude from the right angle and looks at the three triangles showing that they are similar. Or the other one which takes the right triangle, there you see it on one side, there you see it again, there it is again, there it is again. It fills the square except for something small in between, and I leave it to you to figure out that if you take areas and calculate them there, you again get a proof of the Pythagorean theorem. Once when I proudly told some learned audience there were two proofs of the Pythagorean theorem. One of the members of the audience came up to me with a little book, 239 proofs of the Pythagorean theorem. So, mathematical proof is single, but the proofs are multiple. So, geometry has the protean character of which I speak. It applies in more than one. The same is true even more strikingly for the calculus.

7:30 It begins with the notion of the derivative, and the derivative you can think of, the derivative of space with respect to time is your velocity, or velocity with respect to time is your acceleration, or if you're an economist, the added cost of producing one more unit is a derivative, and that's a marginal cost, or you can do it geometrically by the slope of the tangent line and the like. So that the first basic notion of the calculus is a form which has many different scientific interpretations, and the same holds for the other, second basic notion of the calculus, that of the integral, where the integral can represent a major area or volume or moment, you may remember that... There are many different types of students who have to learn integral calculus, where they have a whole sequence of different applications, pressure on the dam and the like. So the integral has many, many different meanings. Partial differential equations apply to movement of fluid, to movement of heat, to the flowing of air around an airplane, or electrical or other potentials. Optics will let the magnetic fields know that. And in between I skipped over one of the things which I put on this, that when Newton described this calculus, he said explicitly the merit of it is that it applies to many different things, to the motions of the solar system, to that of comets, to the moons around planets, to the pendulum, the tides, and the lava. So there, in the case of relativists, one has evidence for this thesis that the forms of mathematics, the structures which mathematicians use, are indeed ones with many different scientific meanings. And so that's the basis of this thesis that I have asked that mathematics is... That part of science which is not about one particular type of phenomena, but about the structure which appears in different phenomena. So another case of this is what I call unexpected connections.

10:00 So if mathematical ideas are ones which have several different meanings, And this is one of the remarkable things about mathematical ideas. You will know many examples. I cite only a few of them. I cite at the beginning here the case of tensor analysis. Use tensors. In those days, tensors were things like a letter T with two subscripts below and three subscripts above, or two subscripts below and one above, and very complicated rules about what happens when you change the coordinates. Of those same tensors... Turned up later in an abstract fashion in the sense that if I have two vector spaces, B and W, the tensor product of those vector spaces is something universal for the right of the assumptions on it. And I used to think that this was really a 20th century idea until I looked back at a famous American textbook, one of the most noted, American scientists of the 19th century was Josiah Luther Gibbs, who was a professor at Yale, originally a professor of physics without any salary, but he finally got a salary. Gibbs was a very absent-minded man, close to my family because he happened to live on the same street with my grandfather. And so my grandfather, who was a young man then, had passed Josiah Willard Gibbs many times on the street, he was never noticed him, until one day when there was a rainstorm and my grandfather took shelter in a doorway, and there we find Josiah Willard, who looked down at him and said, young man, haven't I seen you somewhere before? So Gibbs actually had the modern notion of a natural product and so what I call with my unexpected connection, the Austrian mathematician Radon studied the way in which you could build up a three-dimensional integral out of slices, out of two-dimensional slices and had an elaborate theory of it back in the 1910s I believe.

12:30 Just recently, with the so-called CAT scan, the machine that takes many pictures of the brain, the CAT scan presented the same problem. How do you reconstruct the brain from these horizontal classes? This is another case of the same mathematical idea coming in again. Well, one of the most beautiful theorems of mathematics One that my good friend Tom omitted from his list of basic theorems is the Riemann Rock theorem. The Riemann Rock theorem says if you have a Riemann surface, a surface of a torus or a plane, and you specify points where you want zeroes of a function and other points where you want so-called poles, and you tell the order of these things, the Riemann Rock theorem tells you how many you can get. Riemann got most of it. His student, Roth, got the rest of it. And that was a theorem about functions of one complex variable. It turned out to have extraordinary generalizations through a number of stages, through the Hertzsprung, Riemann-Roth theorem, through the Groten-Dietz-Riemann-Roth theorem, on the Atiyah-Singer theorem. Suddenly it raised its head and became a very active field, and immediately took over various parts of mathematics that had been perhaps languished. The most notable languisher was the so-called Lambda Calculus. Here in blue, I have the letter Lambda. Back in about 1940, Alonzo Church, the first professor of logic at an American university in the Department of Mathematics, Professor Crenshaw. Alonzo Church had thought he could get a new foundation of mathematics using what he called the Lambda Calculus. The Lambda Calculus says, if you have a formula like f, x, y, a function of two variables, then you, by applying Lambda x to it, you get the thing which is the function of x.

15:00 So you pick that. So it turns out that this same Lambda Calculus is the basis of lists. This being one of the first languages used in computer science, and my friends in the United States in mathematical logic complain sometimes that the young logicians can't get enough jobs in logic departments. They shift over to computer science and rapidly apply things there. Well, even my own favorite topic of category theory gets applied for polymorphic types. The same operation applies to different types, to numbers, to spaces, and the like. So, these are all examples of what I call unexpected connections. They illustrate the fact that a single form, say, lambda, arising in one place, applies in totally unexpected ways, and so therefore form is multiple. I have to deal with these same tensor products. So here I have, I haven't written it with a fancy, but the tensor product of just a dot. The tensor product, a times b times c, is not quite equal to a times b times c, but it's actually more. There are many cases like this. If I take sets, if a, b, and c are sets, b times c means the product of the two sets. A times B times C means we have ordered here all A comma B comma C and so on. It is isomorphic to the other. So there is the associative law that I had way back at the beginning. Now the tensor product is only associated up to isomorphism. So that if you have more complicated combinations, here I have a product of four factors. On the left, you have the four vectors, perhaps a little allegedly, A times B times C times D. And you want to rearrange it. All the parentheses come in front.

17:30 As it is at the top here, all the parentheses are in back. Well, you can get them in front in two different ways. You can first move these brackets in front, getting A times B times C times D, and then move this parentheses out in front, getting A, B, C, D. Or, you can go around this way. So, first moving the inside parenthesis front, you get A times BnC times D, and then moving this outside parenthesis front, getting A times BnC times D, and then finally, that isn't quite right, moving this parenthesis front. So, three steps this way, two steps that way, come out to be the same, or should come out to be the same. So, there was a theorem through which Stachev and I proved about 1963. If the associativity is such that this result is equal to this result, then if you take more factors, if you take 17 factors with any jammed up parentheses, any way of getting from one vertex to another will be the same just because of the validity of this pentagonal diagram. And there is also a diagram of commutativity, a times b times c is equal to a. So there was a mathematical theorem. When Stashev and I first drove it, Stashev thought it applied to technology and I thought it applied to technical spaces, and it rested in decent obscurity for something like 25 years, until a couple of young men working on string theory and theoretical physics at Princeton discovered that in working with certain group representations and the tensor products of those representation spaces, They came up with a pentagon like this one, and the other one for the commutativity was a hexagon, and they went to see the grand master of string theory, that's a professor named Edward Witten at Princeton, who claims to know all the mathematics as well as all the physics, he's a professor of physics, he was just awarded for a field medal in mathematics, they said to Witten, oh, we have found a pentagon and a hexagon, Witten remembers all of that.

20:00 He said, well, that must be verbals from Eurystia. Witten remembered in his youth reading a book by Birkhoff and MacLean on algebra and confused MacLean's collars here with Birkhoff's. But the striking thing is that this apparently strictly abstract mathematical theorem cropped up in physics. And it cropped up in connection with what are called Feynman diagrams where you have a couple of particles colliding, moving together, and separating. And the maneuvering of those diagrams turns out to be the same thing as the maneuvering of these pedagogies. So, so much then for the evidence. Parts of mathematics is much more evident than those things that I have cited at the moment. But I'd like to now shift. What consequences should one draw from a thesis that mathematics... Mathematics is about those forms which apply in many places, and so I have labeled the next slide consequences of the program, and so this means firstly because mathematics is not about this thing or about that thing, mathematicians by and large are not out collecting data or making experiments, they are out drawing conclusions, and so this accounts for the formal character of mathematics. And I've listed at the beginning of this slide a number of well-known observations about mathematics being formal, but that will be familiar to all of us that there are a lot of deductions and rules and manipulations. I draw also the next conclusion, which is more drastic. Mathematics does not need a foundation. Mathematics is not about particular things but about general things. Then any one foundation which says mathematics is about numbers or mathematics is about space mistakes the nature of mathematics and can't really be the foundation.

22:30 The nature of mathematics is necessarily appropriate. And here is the argument that I put down here. Mathematics does not deal with any one substance but with the common form which many substances have. And so, in particular, if mathematics doesn't have a foundation, I shift over to a view that my friend Bill LeVere has. Mathematics is not about set theory. The common view is that set theory provides a foundation for mathematics. If mathematics doesn't need a foundation, one doesn't need set theory for this purpose. It's handy for coding things. Instead, there are many other alternative ways of founding mathematics. The one that I mention here on the slide is the notion of an elementary, which was an idea which arose partly in geometry, in the studies of Grosendieck on algebraic geometry, and partly in the work of Le Verre. Who early on saw that you could write axioms for set theory using not set membership, but arrows. In fact, I'll relate the story about this. Many years ago, in about 1951, I was visiting Columbia University to see where my good friend Samuel Eilenberg had many students working in. And among the graduate students was F. W. LeVay. And Sammy came to me one day, Sammy of Attenborough, and said, Oh Saunders, I have this very eager language dude who says he can do set theory without membership. What? Well, Sammy said, I don't really understand it, but perhaps you should talk to him. And so Laverre came around and talked to me and explained to me how you could do set theory without having membership. Here's a set, here's a set of the functions which go with one. So we explained it to him, and Laverre went on to show that he could really do that.

25:00 Some of my friends in the sociology of science claim that ideas become popular and generally accepted just at the point when they are becoming obsolete. And to some extent, that is what happens. Just as set theory was made obsolete by Le Maire's insight, the new math came into the United States, at least, and everybody, the kids were going to kindergarten, set theory, and I don't know what, how was it in space? Did you have new math? We had a pleasure of it in that little Mary was going to kindergarten. And her parents went to the teacher and said, well, how is Mary doing? Oh, said the teacher, she's doing pretty well. She doesn't, however, quite understand set theory, because in set theory, when you want to say the set and the members are one and two, you write wiggly brackets, one, two, wiggly brackets. And the teacher said, Mary can't write the wiggly brackets. Set theory is not the foundation of mathematics. And some of the things that happen in current and set theory belong, excuse me, in what I call a never-never land, and some set theorists imagine that they will settle that famous continuum hypothesis. I've written it down just to remind you what that was. The two sets are the same cardinal number, if you can put them in one-one correspondence with each other, of those of the cardinal number of the integers, the natural numbers, the cardinal number of the reals. Is there something in between? Nobody knows. Nobody will ever know because you can have it both ways. Now, any serious attempt at a philosophy of mathematics ought to have consequences. I want to be able to tell you, roughly speaking, what is good mathematics and what is not, and that's a pretty touchy thing, because mathematicians, each mathematician thinks he knows, and all different mathematicians know differently.

27:30 But nevertheless, I've made some attempts to do this, and I hope that meanings reveal dead ends. My favorite dead end is the notion of a fuzzy set. There was an engineer at Berkeley named Lofty Zala who thought and saw quite correctly, I think, that for certain control instruments, you just simply have to have a range of controls. If you were going too slow, you would speed it up. If you were going too fast, you would slow it down. You didn't need to measure it exactly. And so he introduced the notion of a fuzzy set. What's the Spanish of it? So, here I put down the definition. I have a set, a fuzzy set. I simply know, I don't know whether something is in the set or is not in the set. I simply have a level of membership, which is here written f of x. And it can be used in some engineering. Then you have a great opportunity waiting for you. Because all mathematics is about set theory. Everywhere you have set by fuzzy set. And you have a wonderful source of many. And so the present situation with fuzzy set theory, fuzzy topology, fuzzy decision theory, and much fuzz wandering around. So I think this is only an example. I invite you to find your own. I've listened to a few others that have served me. One of them is the wonderful machinery of regression, due originally, I guess, to Gauss. So if I have a quantity a which depends on three quantities x, y, and z, and I don't know the form of functional dependence, I assume that a is equal to coefficient times x plus coefficient times y plus coefficient times y. And I use the data to estimate the coefficients. This is called linear regression. And supposedly the coefficients mean something. But if that function depended not only on x, y, and z, but also on w, and you failed to measure w, then your coefficients don't mean anything.

30:00 So there is many, many dead ends in mathematics. And I claim more seriously that once... If we can understand enough about the philosophy and nature of mathematics, then we ought to be able to tell what is good mathematics and what is not. And so I have here stuck my neck out, so to speak, by disagreeing with the overuse of regressions. You can get a program that will calculate them without any trouble, and the overuse of these things. No, another consequence is what I call interrelated networks. So mathematics consists of these forms. They are not all built up out of sets, but they are all connected with each other. So since the same form comes in different facts, then that fact will involve this form and that form and that connects the different forms. And so here I give you an example. The real continuum, whatever it is, can be either geometry or one-dimensional Euclidean space. Or it can be arithmetic, the usual days in cuts, or kind of sequences, or many other things. So that I claim that mathematics is a tightly connected network of all sorts of things. I've listed here rules, formulas, formal systems, applications, theorems, algorithms, concepts. And to some extent, I was here to... Quib, some of the ideas which Rene Combe presented this morning. He had what I would call a network of continuous mathematics and discrete mathematics. Here I've got mine, somewhat different from his. It starts with various things such as moving, measuring, shaping, combining and combing, leading to these various parts of mathematics. Applied mathematics, which is connected with calculus, which gives differential equations. Geometry, which is again connected with calculus, which leads to the differential geometry, where you've got curvature and forging of curves and the like. Combinations leads to algebra, and that again connects with geometry. Geometry and counting and numbers lead to various formulas, and that connects with algebra.

32:30 Mathematics is protean. Because the same form crops up in different places, and because these different places will use different forms, there are lively interconnections. It seems to me another consequence, again, another of interconnected pieces. This is a part of mathematics, analysis. ODE stands for ordinary differential equations. DDE stands for partial differential equations. Both of them parts of analysis. ODE flows through these various things, particle mechanics, celestial mechanics, Hamiltonian mechanics, quantum theory, because the Hamiltonian, the function that was... Vital to the Hamiltonian formulation is vital also to quantum theory, and that was connected with differential geometry. On the other side, BDE, partial differential equations, could be formulated in terms of linear operators, in terms of spaces, like Bonnard spaces and Hilbert spaces. There are many different sorts of spaces. You can't tell whether they are really geometrical or really something else. In fact, I should emphasize that that notion of what is a space turns out to have extraordinarily many different interpretations. So a space can be a Euclidean space, a three-dimensional Euclidean space, or an infinite-dimensional vector space. And I remind you of the so-called Hilbert space, the original Hilbert space. The top of the transferences ahead of time lose the novelty of things, and so there was a Hilbert space, which David Hilbert, you remember David Hilbert was that enormously powerful German mathematician, President of Berlin, and by then he was already retired, but a Hilbert space for Hilbert consisted of...

35:00 A sequence of numbers, x1, x2, x3, with the property that x1 squared plus x2 squared plus x3 squared plus and so on, converges. So, in an ordinary Euclidean space with just three coordinates, x1 squared plus x2 squared plus x3 squared, of course, comes from the Pythagorean theorem, that's just the square of the distance in the origin. So Hilbert had shown that part of certain integral equations could be managed very well using a Hilbert space in this manner. But then the story goes that von Neumann, when he was young and applying these spaces to quantum mechanics, came to Gettysburg and lectured on Hilbert space. And he began this way. A Hilbert space is a vector space with suitable properties, bilinear and complete. And Johnny von Neumann was, of course, a brilliant lecturer, but very fast on his feet, and so there was Hilbert sitting in the front row, distinguished professors at the very end sitting in the back row. Von Neumann started out with the Hilbert space and went on to apply it to quantum mechanics. Beautifully, extensively done. And at the end of the lecture, as usual, it was the responsibility of the leading professors who were sitting in the first row. I would like to really know what in some there is in Hilbert space. And he didn't recognize the axiomatic version that von Leibniz had. But then the Americans are even more responsible than that.

37:30 The famous American song, you can't get to heaven in a rocking chair, the Lord don't have no rocking chairs. There's a version of that that says, you can't get to heaven in Hilbert space, because the Lord ain't heard of any such place. No. Again, I try to draw consequences from my thesis, the nature of mathematics, that consists in finding forms that fit together. I mean forms that apply in many different places and at this point I disagree with many of my mathematical colleagues who say the most the real progress in mathematics is always solving some famous conjecture the uh the most famous of course is the Riemann data function people have never been able to prove that the zeros of the data function on a so-called critical life so I claim instead The mathematics project consists in both solving problems and in understanding the results that you get from them. So the mathematics is a two-way street. If the forms apply in different places, then precisely those different places help to the understanding of those forms. So on this particular transparency before you... I have listed a few of the problems. Fermat's last theorem, that x to the n plus y to the n equals z to the n, has no integral solutions for n greater than 2. Just recently, some progress was made proving that x to the n plus y to the n equals z to the n, or rather x. The n plus y n equals 1, the rational number of vectors y, has only a finite number of solutions, but the original, the Fairmount problem remains. This finite number is a part of the so-called Mordell conjecture. Mordell being a Canadian mathematician who shook the dust of those closer to his feet and went to live at Trinity College in Cambridge.

40:00 There is the Riemann hypothesis, there is the so-called Biberbach conjecture, which was recently settled. So, many pieces of mathematics depend on solving problems, but also on understanding. So, take the matter of vector space. So, a vector space, in the simple fashion which Gibbs first used it, a vector space consists of vectors in three dimensions shooting out from the origin. And from vectors you can add two vectors by the parallel of normal vectors, and you also have an inner product of two vectors, the length of this vector times the length of this vector times the cosine of the angle between them. But vector spaces have many different uses. In arithmetic, they solve linear equations, or they provide a way of doing three-dimensional geometry elegantly. Instead of writing down three formulas, you can write one in terms of vectors, as Gibbs saw, you can manage electrodynamic fields, you can extend them to n dimensions, or infinite dimensions, and Hilbert space and the like, and here's another problem about Bonac spaces, and there's my ditty again about the Lord having heard of Bonac spaces, but again, with vector spaces and linear transformations on them, Part of the notion is that the matter of understanding of, let me remind you, funny business about matrix products. So I have a square one that's A, I, J, and another square matrix without B, J, K. Then I can modify those matrices, those two matrices, and I get another matrix. There is a formula for the product of two matrices, those miserable texts of linear algebra which are common in the United States, they fully travel from it.

42:30 I'm kind of off to the associative. You remember my associative law. And so they take this formula and make a very complicated computation to show that that multiplication is really associative. If you haven't ever done it, I recommend that you try it. It's messy. But the real situation is that... A matrix is a way of describing a transformation which sends any vectors x, i in this space into the vector a, i, j, x, j. The second matrix is another linear transformation, and the product is simply the composition of these two. First apply t, and then apply f. And it's pretty clear that if I have three of such things, then r times s times t. Because they both mean first apply P, then apply S, then apply R. So instead of a messy computation, you have a conceptual proof. And that's, again, the sense in which mathematics depends on getting the right form and the right general concepts and understanding thereby what it really might mean. Now, progress in mathematics is hardly solving problems. I'm hardly understanding what those solutions mean or how the ideas are connected. And I'd like to cite another famous example, the so-called Galois theory. Remember, the Galois theory deals with polynomial equations like x. Everybody knows the equation. Well, let's have a polynomial equation. It's like ax squared plus bx plus c. And there's a nice formula that says x equals minus b plus or minus the square root of, what is it, b squared minus 4ac over 2a. I hope I got it right. So there's a nice formula for solving quadratic equations. Many students of algebra struggle with that formula.

45:00 There is another not-so-nice formula for solving a cube equation, but it depends on cube roots and square roots. And at first, when you look at the formula, it's a mess. It can be understood. And there is another nice formula for solving equations of fourth degrees. It depends on cube roots and fourth roots and square roots. And so, historically, a great problem in the study of algebra was to get a formula with roots for the fifth degree equation. This is roughly speaking, beginning at least, the understanding of the fact, which I think Lagrange first saw, that you can't solve a fifth-degree equation by formulas with radicals, by the groups. So there is the fifth-degree equation, there are its groups, and if you look at those five groups, I call them alpha 1 up to alpha 1.