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Mathematics and Occam's Razor: a Brief Exchange Via Facebook Mathematics and Occam’s razor: a brief exchange via Facebook Minhyong Kim (University of Oxford and Seoul 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.
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