What Is Mathematics, Really? by Reuben Hersh
My rating: 5 of 5 stars

Peace in Troubled Times

For many, including myself, mathematics is comforting. In an era of fake news, worldwide illness, and economic uncertainty, mathematics provides proof of another reality which is harmonious, universal, and eternal. Or so it would seem.

In fact mathematics, like all literature, is none of these things. Mathematics is, of course, a human artefact. It is a language which consists of a vocabulary, a grammar, and a community which employs these enthusiastically. Arguably, mathematics is the most refined language ever produced.

Or rather, set of languages. There are apparently some 3400 recognised branches of mathematics. Many of these have their peculiar dialects which are unintelligible to members of other mathematical communities. At least some have never been translated.

Hersh identifies two historical schools of thought which have dominated popular as well as professional discussion of mathematics: Platonists and Formalists. Platonists consider mathematics as a kind of religion. Numbers, they believe, exist independently of human thought about them. They constitute the basic fabric of the universe and determine its orderliness and predictability. For them, mathematics is reality.

Formalists dismiss this quasi-spiritual view. Their opinion is that mathematics is a game, the rules of which are entirely arbitrary. If Platonists are the religious enthusiasts of mathematics, Formalists are the agnostic clergy who have lost the certainty of belief but continue to exercise their ritualistic duties regardless.

Hersh dislikes both Platonists and Formalists. His credible claim is that mathematics developed and continues to develop because it is useful. And it’s usefulness varies so that what mathematics means and how it develops also varies continuously. There is no fixed ‘mathematical method’ by which good mathematics can be distinguished from bad. There are just mathematicians talking among themselves.

This fact - that mathematics emerges from its adherents discussing mathematics - may appear a truism. What else could be happening? But the recognition that mathematics emerges from a restricted community is an important insight. The usefulness of mathematics is not that of engineers or architects or astrophysicists or people filing tax returns.

These and other ‘users’ of mathematics eventually benefit from the products of mathematical discussions in their own work but they are not mathematicians. We may tolerate mathematicians among us because of what their work allows the rest of us to do; but mathematicians could care less. It’s not why they do mathematics.

The practical (or in their minds pedestrian) usefulness of the work of mathematicians does not concern them. Even a brief exposure to number theory, for example, is sufficient to convince most outside the mathematical community (or even outside the community of number theorists) that the things mathematicians are concerned about are essentially trivial. The strange and often captivating relationships among numbers are simply alien to practical experience. The non-mathematician can only ask ‘Why bother?’.

And the answer to this question must be the same as it is to the issue of literature in general. There is no reason for mathematics other than itself. Mathematics is a form of highly refined, esoteric poetry. Its form and subject matter is not to everyone’s taste. But neither is the Iliad, or The Wasteland, or Finnegans Wake. It takes considerable linguistic skill and aesthetic fortitude to comprehend the content of mathematical poetry. Success in such an endeavour is, as usual, its own reward.

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