G. H. Hardy (1940/1969) A Mathematician’s Apology

Written by Nobel Laureate G. H. Hardy; a rare peek inside the mind of a genius mathematician.
 
The Foreword by C.P. Snow is a superbly written portraiture of Hardy by a man who knew him intimately; I wondered how someone from the sciences could write so well; only after Wikipedia-ing him could I confirm my suspicions—Snow has also a double career as a writer and novelist. Snow’s Foreword provides the necessary somber tint to the Apology: “…[A Mathematician’s Apology] is also, in an understated stoical fashion, a passionate lament for creative powers that used to be and that will never come again… it is very rare for a writer to realize, with the finality of truth, that he is absolutely finished” (p. 51). Reading the book alongside the Foreword, we realize that the Apology is the work of a man reconsidering his life’s work after the childlike joy of creative ability has left him.
 
As for the actual Apology itself, Hardy writes a rational justification for the pursuit of pure mathematics as a creative art, one capable of great beauty like poetry, although the association may feel alien to the contemporary mind. Where mathematics may differ is in its permanence: “The Greeks first spoke a language which modern mathematicians can understand… so Greek mathematics is ‘permanent’, more permanent even than Greek literature… ‘Immorality’ may be a silly word, but probably a mathematician has the best chance of whatever it may mean” (p. 81). (Although it must be noted that Bachelard in Poetics of Space writes something like “the metaphor [and perhaps poetry, by extension] is eternal”).
 
Hardy provides a large number of insights into the field, valuable due to his stature and his life’s devotion to the field. In no particular order:
 

1. The best mathematics is serious, by which he means it connects several mathematical areas, advancing the field and adjacent fields. 
2. Serious mathematics contains “significant ideas,” which are defined by the quality of generality and depth, where generality means capable of extension and connection with mathematical ideas, and depth means descending deeper into the strata, given that “mathematical ideas are arranged somehow in strata, the ideas in each stratum being linked by a complex of relations both among themselves and with those above and below” (p. 110). 
3. Mathematics as a creative art has little practical applications. Usually, the most elementary mathematics has the most practical utility. Hardy was writing when the devastation of war was still fresh: mathematics as a creative art has little potential for harm compared to trivial math. 
4. The difference between pure and applied mathematics does not have to do with their utility. Instead, the difference lies in what they try to describe: the latter is related to “physical reality”—the material world; while the former is concerned with “mathematical reality,” or the Platonic reality outside and independent of our senses. Mathematical reality, instead of being created by human beings, lies waiting to be discovered, like in Meno, one of Plato’s dialogues. Hardy uses geometry to show the difference between pure mathematics and applied mathematics: “The play is independent of the pages on which it is printed, and ‘pure geometries’ are independent of lecture rooms, or of any other detail of the physical world” (p. 126). 
5. Interestingly, Hardy asserts that, in comparison to a physicist, “the mathematician is in much more direct contact with reality” (p. 128). This is because “neither physicists nor philosophers have ever given any convincing account of what ‘physical reality’ is, or of how the physicist passes, from the confused mass of fact or sensation with which he starts, to the construction of the objects which he calls ‘real’. Thus we cannot be said to know what the subject-matter of physics is... (p. 129). On the other hand, the mathematician is working directly with their concept of reality. 

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