On “trivial” in mathematics

One aspect of mathematicians’ vocabulary that non-mathematicians often find non-intuitive is the word “trivial”. Mathematicians seem to call a great many things “trivial”, most of which are anything but. Here’s a joke, pointed out to me by Mohan:

Two mathematicians are discussing a theorem. The first mathematician says that the theorem is “trivial”. In response to the other’s request for an explanation, he then proceeds with two hours of exposition. At the end of the explanation, the second mathematician agrees that the theorem is trivial.

Like many jokes, this is not far from the truth. This tendency has led others to say, for example, that

In mathematics, there are only two kinds of proofs: Trivial ones, and undiscovered ones.

Or as Feynman liked to say, “mathematicians can prove only trivial theorems, because every theorem that’s proved is trivial”. (The mathematicians to whom Feynman said this do not seem to have liked the statement.)

A little examination, however, shows that all this is not far from reality, and indeed not far from the ideal of mathematics. Consider these excerpts from Gian-Carlo Rota‘s wonderful Indiscrete Thoughts:

“eventually every mathematical problem is proved trivial. The quest for ultimate triviality is characteristic of the mathematical enterprise.” (p.93)

“Every mathematical theorem is eventually proved trivial. The mathematician’s ideal of truth is triviality, and the community of mathematicians will not cease its beaver-like work on a newly discovered result until it has shown to everyone’s satisfaction that all difficulties in the early proofs were spurious, and only an analytic triviality is to be found at the end of the road.” (p. 118, in The Phenomenology of Mathematical Truth)

Are there definitive proofs?
It is an article of faith among mathematicians that after a new theorem is discovered, other simpler proofs of it will be given until a definitive one is found. A cursory inspection of the history of mathematics seems to confirm the mathematician’s faith. The first proof of a great many theorems is needlessly complicated. “Nobody blames a mathematician if the first proof of a new theorem is clumsy”, said Paul Erdős. It takes a long time, from a few decades to centuries, before the facts that are hidden in the first proof are understood, as mathematicians informally say. This gradual bringing out of the significance of a new discovery takes the appearance of a succession of proofs, each one simpler than the preceding. New and simpler versions of a theorem will stop appearing when the facts are finally understood. (p.146, in The Phenomenology of Mathematical Proof, here/here).

For more context, the section titled “Truth and Triviality” is especially worth reading, where he gives the example of proofs of the prime number theorem, starting with Hadamard and de la Vallée Poussin, through Wiener, Erdős and Selberg, and Levinson. Also p. 119, where he feels this is not unique to mathematics:

Any law of physics, when finally ensconced in a proper mathematical setting, turns into a mathematical triviality. The search for a universal law of matter, a search in which physics has been engaged throughout this century, is actually the search for a trivializing principle, for a universal “‘nothing but.” […] The ideal of all science, not only of mathematics, is to do away with any kind of synthetic a posteriori statement and to leave only analytic trivialities in its wake. Science may be defined as the transformation of synthetic facts of nature into analytic statements of reason.

So to correct the earlier quote, there are three kinds of proofs from the mathematician’s perspective: those that are trivial, those that have not been discovered, and those niggling ones that are not yet trivial.