Is mathematics useful?

This morning I picked up two small books from the sofa, to take upstairs to the bookshelves. They were G. H. Hardy’s A Mathematician’s Apology, and Kenneth Falconer’s Fractals: A Very Short Introduction.

These two authors take very different views about the usefulness of mathematics. Hardy says,

There are then two mathematics. There is the real mathematics of the real mathematicians, and there is what I will call “trivial” mathematics, for want of a better word.

Hardy’s “trivial” mathematics is the mathematics taught at school, and maybe in the first year of a University course. It is the mathematics which is useful in engineering, medicine, and other areas of undoubted benefit to humanity. Real mathematics is what he worked on with Littlewood and Ramanujan, and he argues that it is of no use to anyone.

In the closing pages of Falconer’s book, he talks about multifractals. He explains clearly what these are, but without attempting a precise definition; this is a fairly advanced topic in fractal geometry which I make no claim to understand, although some of my colleagues work on it. But it appears to be useful: among its uses, Falconer lists identifying different crops in satellite photographs, and automated analysis of mammograms to identify cancerous tissue.

Interestingly, there is a mathematician who features in both books: Besicovich, who was a colleague of Hardy in Cambridge and also a pioneer in fractal geometry (before the subject was named as such by Mandelbrot).