Mathematics and Occam’s razor: a brief exchange via Facebook Minhyong Kim ( and National University), Peter B. Shalen (University of Illinois, Chicago), and Deane Yang (NYU Polytechnic School of Engineering)

DY: I’ve always found descriptions of what mathematics is to be too vague and broad. I wanted to distill the essence of mathematics into one precise sentence. So this is what I came up with: Mathematics is the art and craft of deriving new knowledge from old using deductive logic and abstraction. I’m interested in reactions and improvements. For example, someone might ask why this definition would not apply to all the sciences. What is unique, in the realm of the sciences, about mathematics? I have two responses to that. One is that the sciences cannot and do not restrict themselves to deductive logic for deriving new knowledge. Second, the sciences rarely use abstraction. Mathematics is, of course, a powerful tool used by the sciences, as well as in engineering and other areas of endeavor. I would add that I use the word “craft”, because it makes clear that mathematics is something you do and not just “understand” or “appreciate”. If we want to teach mathematics honestly to anyone, we should train them in the craft, just like one would teach carpentry or basketball. I believe the appreciation of the sublime beauty of mathematics will arise from learning the craft, rather than vice versa. PS: When my daughter Eve was three, Ravi Kulkarni asked her what mathematics is, and she said “It’s what mathematicians do.” He said he thought it was a better answer than most mathematicians could give. DY: Peter, cute but, I think, harmful. PS: Any science, any discipline, any human activity makes sense only as an integrated whole. Anyone embarking on the study of music or a foreign language or automotive engineering knows that he is going to be initiated into a whole system of mysteries, no part of which can be understood except in the context of the whole system. That’s at least as true of mathematics as of any other activity, and I think that’s the kernel of truth in the three-year-old Eve’s definition of mathematics. Suppose someone asks “What is French?” I suppose the standard answer would be that it’s the language spoken in France, but that of course says nothing about the internal structure of the language, only where it is situated from the outside. Similarly, if we say mathematics uses only deduction, we may be saying something about where it is situated relative to the other sciences, but we’re not saying much of anything about what it is. DY: Peter, I definitely didn’t explain my purpose in trying to come up with a concise but precise description of mathematics. I’m not trying to get a sentence that captures anything close to the full range of mathematics. Such a description, as you point out, is essentially philosophical in nature and

1 therefore it’s difficult to do better better than what your daughter said. But I think we owe our students, as well as other non-mathematicians who are curious about mathematics, a more down-to-earth explanation of what it means to do and use mathematics. Something similar to what we would say if we were to try to explain what it means to play basketball or make furniture. The explanation would describe both the actions involved as well as what the goal of those actions is. I also think we owe ourselves a much more clear vision of what it is exactly what we want our students to get out of our courses. This is something I’ve spent a lot of time thinking about, and by now I do have what I think is a clear view of what I want my students to learn. I want my students to have learned at least some of the *craft* of mathematics, which means they have learned how to do or use mathematics in a way that is meaningful and useful to them. They have learned to use deductive logic in settings beyond a few artificial ones seen in class, and they are comfortable learning about and working with abstractions of concrete examples. I believe we do this very well for students who like and are good at mathematics, but we do it very poorly for almost everyone else. And I hear too many of my colleagues blame the students for the poor outcomes. I want to do better than that. The sentence I wrote above is for me a clear and precise description of what it is that we do, and for that reason, I believe it is a good starting point when talking to a non-mathematician about what exactly is mathematics and how is it used, not just by mathematicians, but also by others. It is in fact how I start all of my math courses now, because I don’t believe the students have acquired a proper understanding of what it really means to do or use mathematics. Then, as the course goes on, I can point out how what we’re doing fits the description. And what the goal is (what’s the old knowledge we already have and what’s the new knowledge we’re seeking?). Anyway, this is far too long of a response already. I’ve been thinking of writing a rant (polemic?) and submitting it to the Notices. But for the moment that remains a fantasy. MK: Deane, I lack the ability to capture mathematics in any concise form, but maybe it will be helpful if I express one concern that leapt to mind when I read your version: It reads a bit too much like a philosopher’s caricature of mathematics. (I emphasise again, anything I try to come up with is likely to be far worse.) Possibly, my reaction is due to something I heard from a friend years go, which I quote essentially verbatim: ’Mathematicians are people who deduce consequences of their assumptions.’ I know your sentence is different, but it looked close enough to make me worry a bit. DY: Minhyong, you are very right to be concerned, because I agree 100 % with your friend! My description matches hers exactly. Of course, if you start and end with that sentence, you do mathematics a disservice. But to me that sentence opens the door to a more honest introduction to mathematics that leads to people being able to appreciate real mathematics instead of the artificial and for me unrecognizable subject that I see in museums and other popularizations of mathematics.

2 PS: What about saying that mathematics is the part of (scientific) knowledge that can be derived by reasoning alone, and that because it involves reasoning and not observation, it is by its nature abstract? MK: But most interesting mathematics I know of makes heavy use of observation! Take a classical example like The Euler number of polyhedron is 2. This was almost certainly experimentally observed long before people came up with a theory explaining it (topology, essentially), which of course was necessary even for the correct formulation (definition of homeomorphism, etc.). Number theory is replete with important observations that breed conjectures, definitions, and theories, perhaps the most famous being those surrounding the conjecture of Birch and Swinnerton-Dyer (BSD). I think of BSD as being strongly analogous to a *law* of physics, like Newton’s law of gravitation. It was formulated on the basis of a collection of experiments (guided by strong intuition), and then subsequently checked to agree with a far greater range of experiments. Eventually, we’re all convinced that it’s true (at least most of it). I suppose the difference is we attach great importance to the eventual proof, that is, explaining the phenomena in terms of fundamental principles. But this is actually important in physics as well, explaining an old ’law’ in terms of a new and more basic theory. (Consider Einstein ⇒ Newton.) Maybe the real difference is that some of us believe we’ve already discovered the single foundation, *pure reason*, to which we can reduce all truth in mathematics, like the physicist’s theory of everything. Even were that the case, I really can’t see either of the two examples mentioned above as arriving at knowledge by ‘reasoning alone.’ I’m sure you can think of many more examples. PS: Of course when I said “observation” I meant observation of the external world. The observation that one does in mathematics (and does all the time, obviously) is observation of one’s own thoughts: introspection, if you will. At least that’s the way I think about it. A case could be made that one’s own thoughts are part of the world, and therefore that it’s not a well-defined distinction. MK: Peter, perhaps we can agree that the observations we’re all used to are of the *mathematical world*. Whether or not this world belongs entirely to pure thought, I would be reluctant to say. Certainly thought is a tool for observing and studying it, as they are for studying the physical universe. Do you think the observation of Euler numbers I mentioned earlier is pure (or even mainly) introspection? PS: Perhaps we should say that a statement, or a proof, does not become mathematics until it is disengaged from observation. Or, since in practice this is not always something absolute, it becomes mathematics to the extent that it is disengaged from observation. The Egyptians knew about 3-4-5 and 5-12-13 right triangles and a few others. Pythagoras presumably knew about the theorem that bears his name, but it was all tied up with mysticism. Eudoxus,

3 or whoever it was that did the work appearing in Euclid’s Elements, gave a proof of the Pythagorean theorem that was in principle supposed to build on axioms using pure logic. Of course some of the axioms and deductions seem pretty fuzzy to us, but the fact that they even attempted it means that they had the idea of separating deduction from observation. They are generally credited with founding what is called mathematics today, as distinguished from the Egyptian or Pythagorean version where everything was tangled up with everything else. This is like isolating the science of astronomy from the mishmash of astrology, which contained much good astronomy but did not separate it from other stuff. I believe the first fascicule of Bourbaki begins with something like “Depuis les Grecs, qui dit mathématique dit démonstration.” MK: Peter, I doubt that the conjecture of Birch and Swinnerton-Dyer will ever become disengaged from observation in any reasonable sense. So I guess it will never be mathematics. It’s plausible that you could draw a coherent picture delimiting mathematics in the manner you describe. But I don’t quite see why we should. Why not just accept that the (wonderful) reality of mathematics has many different aspects of which deductive proof is one? Other crucial ingredients include observation, conjecture, definition, theory-building, experiment, etc. And then, there’s obviously the ceaseless interaction with the material universe and technological innovation. I hope you’re not claiming that all this activity is on a par with astrology! Perhaps I should state the characterisation I favour these days: Mathematics is the natural science of the mathematical universe. Obviously, such a statement is useless by itself, but Deane was also assuming that an opening line would lead to further conversation and elaboration. PS: I’ve always thought of observation, conjecture, theory-building, experiment, and interaction with the material universe and technological innovation as being things that one uses for the purpose of giving proofs. At least that’s the point of view of “formal” mathematics. From this point of view, definitions are important for the sake of language: they are of enormous value in reducing verbiage. Now, of course, doing “formal” mathematics is not a self-contained activity, because, if nothing else, we need our judgment to decide which mathematical statements are interesting. And it’s also true that most of the things you’ve mentioned are involved in that decision, as well as being tools for finding proofs. But I would argue that while conjectures are, and should be, suggested by theories or observations or experiments, the distinguishing feature of mathematics is still that it involves giving purely deductive proofs of these conjectures. By the way, when you say the Birch-Swinnerton-Dyer conjecture will never be disengaged from observation, do you mean it will never be proved? Or do you mean something else by that? MK: Peter, I have nothing against attaching great importance to proofs! The weight one gives to the other things is also somewhat a matter of taste. (I think Yuri Manin has an essay in which he claims that definitions are the most important things in mathematics, or something of that sort.) But I just don’t

4 see the value (intrinsic or practical) of proclaiming that the rest is not mathematics. Regarding BSD, perhaps I misunderstood what you meant by ‘disengaged.’ A key interest of BSD is that it enables us to compute things. For example, most of the existing algorithms for computing ranks of elliptic curves rely on assuming BSD for their termination. (Fortunately, they do always terminate, because BSD is clearly true :=)) I certainly hope it will be proved sooner rather than later. But given the way it’s used already, I don’t see how a proof will lead to ‘disengagement’ in any sense that I understand. PS: That’s an interesting point. Once you’ve seen that the algorithm terminates, you know you have the answer without quoting the conjecture. The conjecture, if it becomes a theorem, will be a theoretical, a priori way of justifying the algorithm. This is certainly one way in which theorems are used, and it’s hard to fit that into a purely deductive framework. I still maintain–and this is a retreat from my original position–that the fact that central principle have complete deductive proofs is the feature of mathematics that distinguishes it from other sciences. MK: I’m afraid I’m not too confident about a good notion of a ‘complete’ deductive proof. PS: I realize the idea of a “complete deductive proof” brings up philosophical issues as well, but it seems OK for the level of Deane’s beginning students. MK: Well yes. But for the level of Deane’s students, physicists also give plenty of deductive proofs. Consider, say, ’t Hooft, whose Nobel prize was awarded for proving the renormalizability of Yang-Mills theories. Of course we are a good deal fussier about what counts as a proof, regardless of whether or not we believe in ultimate completeness. But I’m not sure the procedures are intrinsically different. I guess my vague impression is that scientific methodology is more or less universal, and it’s the (overlapping) subject matter that distinguishes fields. Because of its engagement with various levels of abstraction, mathematics of course ends up interacting with many disciplines. So does physics, by the way. DY: OK. I think (but I’m not sure) I’m ready to respond to everything that was said. I confess that I haven’t read it all carefully, so please don’t hesitate to object if I’m responding to my imagination instead of what you really said. Also, I don’t have much time these days, so I’m going to try to be as brief as I can (which probably won’t be so brief). There have been some remarks about math not being just abstraction and deduction. That a lot of mathematics is about observation, experiment, and testing hypotheses, similar to what is done in science. And conversely that scientists also use deductive logic. So what’s the difference? My response is that what’s done in math is *totally* different from what is done in science. In science, there is an external phenomenon you are trying to understand. If you are trying to describe it using a mathematical model, you

5 test the model against the external phenomenon. And this is what for example t’Hooft was doing. And, as we all know, physicists are allowed to and do do things that are not completely logically consistent. Their only test of success is whether their calculations (whatever that means) are able to predict observed phenomenon. On the other hand, doing math is different. There is no external phenomenon. There are only assumptions (sometimes called axioms) and their logical consequences. An observation is always just one or the other. An experiment is an effort to test whether one set of assumptions is logically consistent with another set (usually the set of all known mathematics). We are also allowed to define whatever we want and study it. And that’s exactly what happens. And there are endless disputes about which definitions are more “important” or“deeper” than others. But there is no objective way to decide this (unlike science). In the end it’s a judgement each of us makes. And let me emphasize that I am not taking sides in the Platonist versus (the other side, I forget the name) debate. I’m actually a Platonist, but that doesn’t contradict anything I said. It simply means that I believe that there *are* external reasons why certain mathematical ideas are more important than others. But that belief is really just a religious or philosophical belief. I have no way to ever test that belief. For example, when you do “experiments” with the Birch-Swinnerton-Dyer conjecture (which I cannot state, never mind “understand”), you are simply making it an assumption and determining logical consequences to see if any of them contradict what is already “known” (which means a lot of assumptions that we all believe in and their logical consequences). And, remarkably, I don’t believe I am contradicting Thurston’s assertion that proofs are just “social constructs”. All that means is that although we all strive for bulletproof logically rigorous proofs, there is no way to be 100 % sure that our proofs are. As long as we don’t encounter a contradiction, we are allowed to assume that all of our assumptions and theorems are logically consistent. And if we do encounter a contradiction, we are always try to isolate it (i.e., discovering the error in the proof) or ignore it (because for one reason or another we just don’t believe the proof). None of this contradicts the fact that the fundamental two tools of the practice of mathematics are abstraction and logical deduction. By the way, by “abstraction” I mean creating definitions and assumptions (properties of the objects defined) that either generalize older definitions or model observed external phenomenon (this is where mathematics meets science). In any case, all Thurston’s assertion means is that we can never be 100% sure that we are using these tools correctly. We are only human. Now let me return to the original point of this post, which I don’t believe I ever stated clearly. Here’s the scenario: You are talking to another human being (could be a student, could be anyone) who knows little mathematics and asks you to explain what the big deal is. Almost all, if not all, of the discussion above is indeed about very interesting but subtle aspects of mathematics. The problem is that to understand the discussion at all, you must be not just a practicing mathematician but a practicing *pure*

6 mathematician or a sophisticated amateur. Let me present an analogy. Golf is an extremely difficult sport, and there are all kinds of subtleties about how to hit the ball just right, how to plan one’s shots, how to maintain the right mental state, etc., etc., etc. Now suppose you run into someone who has never seen or heard of golf and want to explain it to them. You certainly would not want to start with the subtleties. You start with “You try to get the little white ball into the hole by hitting it as few times as possible.” Then you build on that start one step at a time. And if the person has not lost all interest, you continue until you really can start discussing the subtleties. And one more thing is, I think, obvious. That a person is going to be a lot more interested and better able to understand the subtleties, if that person has tried to play golf instead of just hearing about it. I think there’s no difference with mathematics. I believe that when we teach mathematics we need to teach them how to work with abstract definitions (it’s too early to ask them to create such things) and make rigorous logical deductions as early as possible. I also think most mathematicians misinterpret what this means. It does *not* mean starting with the Peano axioms or anything like that. When we teach, we can start with *any* axioms we like. In freshman calculus, an extremely useful assumption is that the exponential function exists and has certain well-known properties. When I see mathematicians debate whether this is a good idea or not (the alternative being waiting until the integral is defined, using it to define the logarithm function, and then defining the exponential function as the inverse function), I always ask, “But what about radians?” In any case, mathematics is about starting with a reasonable set of assumptions and reasoning rigorously from there. And it should not be hard to show a student how to do it and how it leads to useful or cool consequences. MK: Deane, your pedagogical points are reasonable. But I still don’t see why they should force us to restrict our conception of mathematics so drastically, insisting it’s *not* this or that. Furthermore, while I agree it’s possible to believe in the existence of a definite dividing line between external reality and some sort of internal one (whatever that means), my own impression is it’s pretty hard to make that belief consistent. DY: Minhyong, in fact I am willing to concede that the boundary between mathematics and other fields can indeed be fuzzy, just as they are between the different sciences. But I also believe that each field has a core (usually labeled “pure”) that distinguishes it from the others. I believe that if we are to create a clear image of what mathematics is and what mathematicians do, we need to explain from the start what the core of mathematics is and why it is worth doing. I also believe that when we talk about “experiments” and “observations,” they are metaphors but extremely useful ones. On the other hand, I believe that if we use such terms when first introducing mathematics, we create a confusion about what mathematics is that is hard to dispel later.

7 But am I wrong about this? Could you give an example of what you mean you say that it’s hard to make this belief consistent? MK: I agree that each discipline has a core domain of study. I think we also agree that mathematics is defined by the study of mathematical objects. But then, one needs to come to some understanding of what mathematical objects are. I believe you proposed that whatever they are, they are not part of external reality. This was the bit I felt uncertain about. That is, the boundary I find fuzzy is not just that between different disciplines, but also between external reality and some other kind. Belief in a clear boundary there is what I thought difficult to make consistent. As for a concrete example illustrating this, would you say a quark belongs to external reality or not? I would say essentially yes, but this is not a straightforward position to defend. Regarding the pedagogical concern, obviously you’ve thought about teaching much more seriously than I have, so I feel quite unqualified to comment. But I feel like I’ve seen excellent teachers with a range of temperaments start from a variety of viewpoints, and manage to convey something authentic and exciting in each case. For some of them the starting point was an experiment, for others intuition, metaphysics, examples, ... and of course for some, a rigorous dedication to deductive reasoning. I’m not good at any of these things myself! DY: Minhyong, many thanks for your continuing remarks. I really like your example of a quark. I hadn’t thought about the use of abstraction in other fields, but I now see that in some sense everybody in all fields use abstractions of what they believe are real objects and try to reason with them logically. Given that, it does indeed seem very hard to say what exactly makes mathematical objects more abstract (if that’s the difference) than the other ones. I guess, once we are convinced that the abstraction is itself interesting and beautiful, we focus on the abstraction itself and pay little or no attention to the real objects that the abstraction was meant to represent. MK: My feeling is that in England, many of the students tend to be bogged down by a quite formal view of mathematics (theorem-proof-theorem-proof-...). Anyways, thanks for raising this question. I was having a parallel discussion of these things over lunch and dinner with my colleagues at Merton, including physicists and philosophers. By the way, the example of a quark was somewhat random, but there are much worse examples, I think, like virtual particles. Or even quite fundamental ones: It’s widely agreed that ‘symmetry’ is very important to modern physics. You’ll probably see it used in physics papers far more than in mathematical ones. But what is it? Is it purely abstract, or is it part of external reality? How about a representation of SU(3)? I think at some point in history, there were also debates among philosophers about whether or not ‘fields’ were real. Of course these were physical ones under discussion, as in electromagnetism or gravity. With the ascent of information theory, maybe the arithmetic ones (say finite fields) became even more real than the gluon field!

8 PS: Minhyong, you wrote “My feeling is that in England, many of the students tend to be bogged down by a quite formal view of mathematics (theorem-proof-theorem-proof-...).” I found this interesting, because I think I myself have a quite formal view of mathematics, and the very first math(s) book I ever read was Hardy’s “Course of Pure Mathematics.” MK: Or maybe because you work primarily in an intuitively appealing subject like three-manifold topology, you feel a greater responsibility to discipline yourself with formal rigour. I’ll conclude my contribution with an appeal to Occam’s razor. The philosopher Simon Saunders asked me a while ago where I thought mathematical objects came from. Well, by the time I was born, most of them had been around for a long time, so it’s rather hard for me to say. We work with a wide range of mathematical objects having pretty consistent and reliable properties that come up in ordinary life all the time (for example, small natural numbers or groups). Some mathematical objects are still developing (such as ‘motives’ in ), while others have a rather unstable interaction with the physical universe. (Here, I won’t pick on any example, for fear of revealing my own ignorance.) But my key point is that axiomatic systems or set theory can be thought of as occasionally convenient models of mathematical reality.

But I don’t see why we should hold that reality hostage to them. At least, no more than the reality of quarks should depend on the reality of groups (and hence, of axioms, if we follow that line of reasoning). If we are forced to be economical about our description of what really exists, I think it’s pretty clear that it’s the axioms that have to go, not the natural mathematical structures. Simon tells me my position is known to learned people as ‘Quine’s refutation of conventionalism.’ I’d prefer to think of myself as a humble follower of Occam.

Acknowledgements: MK is very grateful to PS and DY for agreeing to have this exchange published. He is also grateful to Ralf Bader, Alan Barr, Simon Saunders, and Alex Schekochihin for numerous tutorials on the nature of mathematics.

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