A Mathematician's Apology

Most math has no practical consequences, and that which does usually has little aesthetic value. The value of a math theorem lies in its seriousness, which, in turn, depends on its “significance”—its ability to say something important within a large region of mathematical concepts and perhaps to improve that area and the sciences connected to it.

William Shakespeare is great not because of his impact on the English language but because of the beauty of his writing. That beauty includes the ways that his verse presents ideas, but this is only a part of its success and not the main cause of it.

It’s hard to find an example of a math theorem that’s elegant, used by modern mathematicians, and not too obscure to understand. A perfect exception, drawn from ancient Greece, is Euclid’s proof of the infinity of prime numbers, an achievement “as fresh and significant as when it was discovered” (92). Prime numbers are whole numbers that can’t be divided by smaller numbers except for 1. If P is “the largest” prime number, and Q equals the product of multiplying several numbers by P and then adding 1, Q is no longer divisible by any of its factors, including P (to arrive at a whole number). Thus, Q is a larger prime than P. This means that the list of prime numbers is endless. This proof is a form of reductio ad absurdum, or showing that the opposite of the thing to be proved makes no sense.

Another elegant reductio from the ancients is based on Pythagoras’s demonstrations relating to the square root of the number 2. One of these uses the Pythagorean Theorem to show that the diagonal of a square cannot be a rational number (a number cleanly divisible by another). If the side of a square = 1, then the diagonal, d, fits into the Pythagorean equation as “d^2 = 1^2+1^2 = 2” (96) and therefore d = √2, which is irrational. Any value of d must include, as a factor, the square root of 2; thus, all diagonals of squares are irrational.

Other theorems are intuitively obvious. For example, the fundamental theorem of arithmetic states that every integer larger than 1 can be expressed as the product of certain prime numbers. For example, “666 = 2 x 3 x 3 x 37, and there is no other decomposition” (96). The proof, though, requires some tedium.

Chess problems and other types of puzzles prove out in similar ways. However, they lack the seriousness of mathematical proofs.

A chess problem has no application outside the game, but the proofs of Euclid, Pythagoras, and others have a huge effect, not only in mathematics but in other realms of thought. In math, Euclid showed the fundamentals of arithmetic, but Pythagoras demonstrated that some arithmetical processes lead to answers that can’t fully be resolved, as with irrational numbers.

In practice, irrational numbers are rounded off, so Pythagoras’s discovery may have no real-life application.

The significance of a serious math theorem depends on its “generality” and “depth.” It is general if it is widely used in the proofs of other theorems. An example is the Pythagorean theorem. If a theorem has limited usefulness—like, for instance, the theorem that only four numbers other than 1 are “the sums of the cubes of their digits, viz. 153 = 1^3 + 5^3 + 3^3” (105)—then it lacks generality and therefore is insignificant, except perhaps to puzzle enthusiasts.

In the sense that math’s terms are entirely abstract, math is general. Adding things up, for example, works regardless of the things involved. This abstract quality differentiates math theorems from scientific theories of nature—for example, predicting eclipses—because their correctness depends on experimentation and cannot be derived from mathematical proofs.

Aside from the abstractness common to all math, its theorems acquire generality when they speak to many but not all other math theorems. As mathematician Alfred North Whitehead put it, “[I]t is the large generalization, limited by a happy particularity, which is the fruitful conception” (109).

To have significance, a serious theorem must be more than general; it also must have depth. A simple theorem may be important yet not deep. For example, proving the infinity of prime numbers is straightforward, but, to know how many primes exist within a given set of integers, a mathematician must go deeper and confront more difficult ideas. The precise meaning of a problem’s depth, though, “is an elusive one even for a mathematician who can recognize it” (111-12).

A good theorem should display unexpectedness, inevitability, and economy (113). As in chess, where a good new approach ought to generate many variations, in mathematics a good theorem should open the gates to more ideas. The many examples that flow from a new theorem, however, shouldn’t be used as evidence for its proof; this is a dull way to do math.

Great chess players have a reserve of logical ability, but the joy of the game isn’t in mundane calculations: Instead, it’s in the battle between highly trained minds.