Monday, January 17, 2005

Mathematical styles

Jim asked me this weekend How many types of mathematics are there, anyhow? It's a good question, and it took me a few seconds to say anything at all. I rambled on for a bit about the AMS classification of mathematical topics, and that provided us both with a few minutes of amusement, but I kept thinking about the topic afterward. I'd thought a little about the question earlier in the week, when I took a little time to put my office bookshelves in order. The divisions between branches of mathematics are rather artificial, of course, and I suspect that most of the mathematicians I admire would feel pretty strongly that at least a few divisions are spurious, historical accidents at best and divisive political nonsense at worst. Nevertheless, there are definitely different traditions and different modes of thought within mathematics, and I have some thoughts about what defines them.

I will organize my thoughts according to how I organized my bookshelf:

  1. Computer science

    I already have written that I don't think computer science is properly a scientific discipline -- at least, it isn't mostly. It also isn't entirely a mathematical discipline. Nevertheless, computer science as a discipline includes a strong mathematical component that isn't emphasized so strongly elswewhere. In particular, computer scientists pay a great deal of attention to graph theory, certain areas of combinatorics, automata theory, and both the very pragmatic and the very abstruse characteristics of algorithm performance.

    As mathematicians, computer scientists tend to be strongly combinatorial and algebraic in their thinking. Many of us have some grounding in probability and statistics as well, but it's often restricted to a discrete setting. There are computer scientists who spend time thinking about continuous problems, too: scientific computing folks (like me), graphics and vision folks, and some others. But it's not the most common strength.

    My computer science shelves are organized into books on languages and tools; on core CS topics; and on software engineering. The algorithms and data structures books, the cryptography book, and parts of the compiler book could be considered mostly mathematics.

  2. Mathematical physics

    My physics shelf includes books on general physics, classical mechanics, solid and fluid mechanics, electrodynamics, and a little quantum and statistical mechanics. It also includes a couple books that I nearly classified in geometry -- a closely related discipline.

    Mathematical physics is a frustrating misnomer. Elementary books on mathematical physics rarely include much of real physical interest; rather, they include yet another rehash of the standard linear differential equations, some special functions, Fourier analysis, and perhaps elements of the calculus of variations. They could be re-titled as texts in advanced engineering mathematics, and nobody would be the wiser -- except that most engineers write dx at the end of an integral, and many physicists like write it at the beginning. There are more advanced books that essentially follow the same line of development, but with more sophistication -- and usually without the name mathematical physics in the title. Suffice it to say that some of the mechanics books on my shelf are full of very physical insights, backed up by occasional calculations; and some of them are mathematics books that happen to have a lot of examples drawn from physics.

  3. Linear algebra

    You knew this was coming, right? Linear algebra is a curious area, partly because it's fundamental to so many disciplines: analysis, statistics, optimization, geometry, algebra, ... The influence goes both ways, too. Despite the name algebra, researchers in linear algebra often spend as much time on analytic work as they do on algebraic work.

    Numerical linear algebra is one sub-discipline of linear algebra. Matrix theory is another (as I've indicated elsewhere, matrices have structure beyond what they inherit as representatives of operators on linear spaces). To some extent, functional analysis is a branch of linear algebra, too, but I've classified it elsewhere for the purposes of shelving.

  4. Integral/differential equations and analysis

    If half the proofs start with the words for all epsilon, there exits..., I'll probably call it analysis. Analysts deal primarily in estimates, inequalities, and notions of convergence. Real and complex analysis are fields that deal primarily with the rigorous development of the real and complex fields, and with different types of real and complex functions and their basic properties. Functional analysis deals with spaces of functions, and is as much a branch of linear algebra as of analysis. The theory of ordinary and partial differential equations is hard analysis, tied closely to the field of functional analysis. I also classify my books on the numerical solution of ODEs and PDEs on this shelf, partly because even the most hard-headedly practical books on numerical solution of ODEs and PDEs will include some discussion of elementary functional analysis -- at least, they will if they're any good.

  5. Geometry and nonlinear systems

    When I speak of geometry, I'm not talking about Euclid. The geometric objects that most interest me tend to come from physics, either classical or modern. They represent the surfaces of objects, or sets of solutions to some interesting equations (e.g. sets of configurations with a given energy), or whatnot. Nonlinear systems can almost never be solved, but they can be analyzed; and investigating the geometry of their solutions, together with certain analytic properties (particularly in the neighborhood of singularities) is the clearest route to a understanding what most nonlinear systems will do.

    Poincare wrote about two basic types of mathematical minds: the geometer and the analyst, he called them, referring to those who proceed by intuition and those who proceed more methodically. At the end of the chapter where he set up this dichotomy, he admitted that it wasn't perfect, but stated that he still thought it was a real division -- and I agree. If I see a proof that involves global qualitative properties of a solution set, I'll probably call it geometry. If I see a proof that involves detailed estimation based on local quantities, I'll probably call it analysis. That's not a very good description of the difference between the fields; maybe it would be more accurate to follow Poincare's description, and say there are those who primarily launch out intuitively, and there are those who insist on the more solid ground of carefully-reasoned proofs. In Poincare's classification, I think I'm more of an analyst than a geometer.

  6. Applied mathematics and statistics

    This is a catch-all category. Some of my statistics books are really about applications; some of them are really about hard analysis. The books about applied mathematics are... eclectic. This is fine by me, of course, but it does make them hard to classify. I include in this category my books on perturbation methods.

    Most of the books in this shelf involve the analysis of problems from some discipline outside of mathematics, be it operations research, meteorology, physics, engineering, or something else entirely. When done well, a paper or book on a topic in applied mathematics should be interesting both because the problem is interesting (to someone other than a mathematician) and because the problem generates interesting mathematics.

    My books on special functions go roughly under this heading, too, though space constraints forced me to put them next to my library books instead.

  7. Nonlinear equation solving, optimization, and numerics

    This is another catch-all classification, basically containing all the numerical mathematics books I have that deal with finite-dimensional phenomena and are not devoted exclusively to numerical linear algebra. Let me mention the book Numerical Methods That Work by Forman Acton: the front cover has the word Usually embossed below the title, and the book is full of cautions which boil down to the command to think before you compute. I agree wholeheartedly.

    Numerical analysis is concerned with the design of algorithms to compute approximations (usually) which are both accurate and inexpensive. It is an art that combines the estimates of mathematical analysis, the algorithmic thinking of computer science, the trickery of applied mathematics, and a certain amount of healthy skepticism.

  8. Algebra

    Algebra involves the study of a wide variety of types of formal structures and the functions that preserve those structures (homomorphisms). Unlike analysis, most of the proofs of algebra do not involve estimation. Algebra is not my strongest subject, though I have the standard background that any math grad student might have (a graduate algebra course, a little number theory, and a fervent appreciation of whatever group-theoretic or ring-theoretic properties might help me better understand algorithmic or physical properties of systems I work with more regularly).