Tuesday, July 26, 2011

Mathematics in (not quite) 500 words

I'm grateful to Elisa Gabbert for the prompt for this post, esp. as the blog has been a little dead lately. (Either the internet has been letting me down or I haven't been in a mood to appreciate it.) The prompt is to describe what one studied in college in 500 words. I thought of writing an overview of physics but this was a dreary prospect -- "I, too, dislike it"; I've become anti-Science though there are of course lots of interesting specific questions in the sciences -- so I decided to write about my other college major, math, instead.

Math, even “applied math,” is stylistically different from other “mathematical” disciplines like physics: esp. in its emphasis on defining terms precisely and making all definitions and results as general as possible, so, e.g., results about circles should be extended to d-dimensional hyperspheres, and further to pi or -2 dimensions if possible… (The point is to interpret everything whenever interpretation is possible.) I have always liked this formalist/structuralist tendency, and often regret not having pursued mathematics as a career.

Branches:

Analysis. Formerly calculus, this has to do with distances and volumes in a general sense—so, e.g., it is concerned with giving a meaning to questions like “what fraction of real numbers are rational” (zero), “how likely is an arbitrary curve to be smooth,” “how close is the nearest smooth approximation to this jagged mess,” etc. I would lump “differential equations” under this rubric, at least conceptually.

Abstract algebra. Given a set of objects, you can define operations on them: e.g., for a set consisting of an apple and an orange, you could say apple “+” orange = apple, orange “+” apple = orange, etc. Abstract algebra is the theory of such relations between objects or operations: e.g., what is a reflection “times” a rotation? Etc. As the success of Galois theory attests, taking a very general view sometimes helps solve specific problems.

Geometry and topology. These are about shapes. Roughly speaking, modern geometry is chiefly about an object’s “curvature” and topology is about the number of holes in it. (These are related.) Topology in particular is a taxonomic field; the interest is in classifying all objects into groups by identifying the simplest shape you can deform them into without tearing or gluing. (E.g)

Number theory. Self-explanatory; NB “number” here almost always means natural number or integer. An example of a number-theoretic result is the Tao-Green theorem that there are arbitrarily long arithmetical progressions of prime numbers. Analysis (which, a priori, has nothing to do with natural numbers) is a surprisingly powerful tool.

Combinatorics. Questions, often practical, involving permutations etc. An important subfield is graph theory, which is the study of p points with q lines drawn between them, either at random or according to some rule. Naturally, given the propensities of mathematicians, permutations of infinitely many things are also studied.

Set theory and logic. A “set” is naively a bunch of stuff. However, for reasons related to Russell’s paradox, one needs to be quite careful about defining operations on sets, and esp. interpreting the notion of “sets of sets”; this is where formal set theory started. An idea that comes up a lot is that of cardinalities, or different sizes of infinity, and whether we are missing intermediate sizes of infinity. Logic is about the structure of proofs (and by extension of computer programs), etc. Important results include the incompleteness theorem and also (a particular favorite of mine) the Lowenheim-Skolem theorem, which says (roughly) that every system of logic has one of two problems: either (a) you cannot specify the cardinality of a set through statements about it that are “utterable” within the system, or (b) there is no strict correspondence between semantic truth and syntactic provability in the system.

Applied math. A grab-bag of stuff from other fields that can be reduced to math. Examples keep changing, but e.g., I know pattern recognition was a research problem in this area not so long ago.

(There are various permutations of these terms that are current fields of research: combinatorial set theory, algebraic geometry, analytic number theory, topological graph theory, algebraic topology, etc. Structures that are introduced for some specific purpose often have other interesting features, e.g., the sets of solutions to certain equations might have interesting geometric properties.)