Review: A Mathematician's Apology by G.H.Hardy

I recently had the pleasure of reading A Mathematician's Apology by G.H.Hardy. The book, as the title suggests, is Hardy giving a defence of mathematics. He explains the reason why mathematics is important and why mathematics is beautiful. I will give a short biography of the author, G.H.Hardy, before I continue with my review because I feel the background to the book and why it was written helps us understand why it is such an important book.

G.H.Hardy was one of the greatest pure mathematicians of the early 20th century. Hardy spent the early and late part of his career at Trinity College Cambridge, prior to being at New College Oxford. However, his most famous work was done in collaboration with John Littlewood and his Indian prodigy Srinivasa Ramanujan after returning to Cambridge. To anyone that doesn’t know the story of Hardy’s discovery of Ramanujan, I suggest reading the The Man who Knew Infinity. Hardy had two loves in life mathematics and cricket. However, at the time the book is written Hardy is aged 63 and no longer, as he puts it, has the ‘creativity, patience or inspiration to be a successful mathematician.’ It is clear Hardy struggles to deal with the loss of his academic abilities which creates a sombre tone throughout the book. In the book Hardy gives not just a defence of mathematics, but a defence of his life as a mathematician.

Hardy starts his defence by explaining what is meant by mathematics. He does this by exploring the question of whether we invented mathematics or whether if always existed and we discovered it. Hardy is a firm believer that we discovered mathematics and every breakthrough he ever made was a discovery not an invention. This belief famously led him to say ‘317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is, because mathematical reality is built that way.’

Hardy argues that many weak defences of mathematics suggest that mathematics is important because it is important in real life situations. For example, mathematics helps us to build bridges or model variables. However, Hardy is a pure mathematician and he states that nothing he has ever achieved has any practical application. He believes that the mathematics used in engineering or other fields that utilises mathematics is not real mathematics. He believes that only pure mathematics is real. This raises the question of, if we then only take the purest mathematics as our definition of maths, can something with no applications be important? Hardy argues the beauty of mathematics is what makes it so important. He argues that a painting has no purpose other than its aesthetic beauty and the emotions it creates and mathematics is the same.

Hardy also defines mathematical beauty in the book, although even he admits he struggles to define it. He thinks that the beauty of a result or proof is when it manages to combine together a great many different unrelated ideas. Hardy believes there are only 2 examples of mathematical beauty that non mathematicians can understand. The fact there are so few, is why he believes there are misconceptions surrounding mathematical beauty. His two examples are Euclid’s proof of infinitely many primes by reductio ad absurdum and pythagoras’ proof that the square root of 2 is irrational, which I am sure many of you will be fond of also.

A final question he ponders is whether mathematics can cause harm. This is particularly important to Hardy because he was a pacifist. He saw that some mathematics was being used in a destructive way to support war such as its uses in ballistics. This causes him to discuss whether this makes mathematics a subject we should not study, because it causes harm.  He concludes that the mathematics he calls "real mathematics" can not  be used for war. He says "No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years".

I would highly recommend reading this book even if you are not a mathematician because it provides such a good insight into the life and thoughts of a mathematician. As Hardy’s chosen field was number theory, my next review will be of one of the greatest books ever written about this subject, which is The Higher Arithmetic by H. Davenport. That review will be online soon. Thank you for reading.