# Critical Overview

In his review of A Mathematician’s Apology in the *Spectator*, British
author Graham Greene asserts that Hardy’s philosophy is akin to the philosophy
of an artist. ‘‘The real mathematician,’’ according to Greene, ‘‘must justify
himself as an artist.’’ Indeed, Hardy’s work is a very successful justification
of the mathematician as artist, much in the literary tradition that includes
*The Autobiography of Benvenuto Cellini*; Vincent Van Gogh’s letters to
his brother Theo; and, as Greene points out, the work of Henry James. Greene
writes, ‘‘I know no writing—except perhaps Henry James’s introductory
essays—which conveys so clearly and with such an absence of fuss the excitement
of the creative artist.’’

While the ‘‘uninitiated’’—that is, non-mathematicians such as Greene—were
apt to focus on the work as an artist’s memoir, those with more rigorous
mathematical training focused on the rift within the field of mathematics that
A Mathematician’s Apology brought to the fore. As the anonymous reviewer in the
*Times Literary Supplement* observes, ‘‘‘Real’ mathematics deals only with
the ultimate abstractions of number, and, if not in itself incapable of being
put to ‘use,’ at least becomes only occasionally and accidentally useful.’’
‘‘Applied’’ mathematics, on the other hand, deals with numbers as useful
scientific tools, which helps bring about innovation. Its definition implies
utility, or usefulness, and is the opposite of the ‘‘math-as-art philosophy’’
Hardy espouses throughout the book. And true to Hardy’s lifelong reputation for
his candid opinions, Hardy did not hold back the scorn and derision he felt for
the functional uses of mathematics. He refers to chess problems, for instance,
as ‘‘trivial,’’ regardless of their relative degrees of difficulty, and he
similarly belittles applied mathematicians and their work throughout the
book.

Hardy sums up this attitude at the beginning of chapter 28:

There are then two mathematics. There is the real mathematics of the real mathematicians, and there is what I call the ‘‘trivial’’ mathematics, for want of a better word. The trivial mathematics may be justified by arguments that would appeal to [Lancelot] Hogben, or other writers of his school, but there is no such defense for the real mathematics, which must be justified as art if it can be justified at all.

Ironically, Hogben, the mathematician for whom Hardy reserved the word
‘‘trivial,’’ appears to have been unaffected by the criticism. In fact, a late
edition of Hogben’s book, *Mathematics for the Million*, was reviewed in
tandem with the reprint of A Mathematician’s Apology in 1967 in the *Times
Literary Supplement*, as a vivid illustration of the disagreement between
the two views. As the reviewer notes, ‘‘For [Hardy] Hogben is ‘admittedly not a
mathematician’ and ‘real’ mathematics is to Hogben ‘merely an object of
contemptuous pity.’’’ Despite the profound differences between the two works,
the reviewer writes that they both ‘‘deserve the immortality they appear to
have achieved.’’

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