A Review of A mathematicians Apology

gh hardy

If a person were to read the essay A Mathematician’s Apology by Godfrey Harold Hardy and expect it to be an actual apology (as in, an expression of regret), then that person would be left sorely disappointed. More than anything, this essay is a defense of mathematics.

Before we begin, it is good to give some context as to who the writer of this essay was. Hardy was an English Mathematician. He was known for his achievements in number theory and mathematical analysis, both of which are the purer fields in mathematics. He is also known for his collaboration with English Mathematician John Edensor Littlewood, and Indian Mathematician Srinivasa Ramanujan (there was actually a film made about this: The Man Who Knew Infinity). At the time of his writing of A Mathematician’s Apology, Hardy was 63 years old.

In this essay, Hardy talked about the beauty of mathematics. To him, it is a creative field more than anything. In his words, “A mathematician, like a painter or a poet, is a maker of patterns” and I fully agree. For context, I am a Mathematics major, and Mathematics at the university level is more proving and manipulation than tedious boring computations. We seek to find patterns and prove theorems; in fact, this was apparent to me even as a year 1 student.

For Hardy, one of the reasons why Mathematics is beautiful is because of its permanence. Unlike many other disciplines, mathematical truth is eternal and non-subjective – a theorem that was proven true a thousand years ago will remain true a thousand year later. There can be no error in a theorem once its proof has been accepted by other Mathematicians (this is perhaps an advantage of working in a field where results are not empirically driven).

Hardy explains that there are really two kinds of mathematics – “real” and “trivial” mathematics. To Hardy, “real” mathematics has aesthetic value while “trivial” mathematics is dull (albeit, possibly useful). He then introduces two theorems to the readers: 1. The existence of the infinity of prime numbers by Euclid and 2. The irrationality of  by Pythagoras. Both of these theorems were proven by ancient Greek Mathematicians a long time ago. We shall take a look at the proof for the first theorem.

 

Theorem: There is an infinity of prime numbers.

Proof: First, we note that every number is either a prime or a composite. We shall proceed to prove this theorem by reductio ad absurdum (proof by contradiction).

Suppose for a contradiction that there is a finite number of primes: 2, 3, 5, 7, …, p (so p is the largest prime) and there does not exist a prime that is larger than p.

Consider then, the number .

Now, recall that  is either prime or composite. Obviously, Q is not a composite number because it leaves a remainder of 1 when divided by any of the prime numbers 2, 3, 5, 7, …, p.

Then Q has to be a prime number. But this would contradict the assumption that p is the largest prime number.

So, our initial assumption was wrong and there is an infinity of prime numbers. (QED)

 

I shall not show the proof for the irrationality of  as this is an essay review rather than an introduction to “real” mathematics. If the reader of this review is interested, he or she shall be able to find the proof with a quick google search. The point is, such proofs are not uncommon in Mathematics, and many of such theorems exists in mathematics. The beauty of such theorems lies in their simplicity, both in idea and execution; and to Hardy, these are theorems of the highest class! Good theorems are also, in my opinion, self-perpetuating.

Some natural questions to ask upon proving the infinity of prime numbers are: How is this infinity of prime numbers distributed? Given a large number , say  or , how many primes are there less than ? Such are the questions that Mathematicians grapple with.

Before I digress too much, let us return to the contents of the essay. In the essay, Hardy wrote, “There is one comforting conclusion which is easy for a real Mathematician. Real mathematics has no effects on war. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years. It is true that there are branches of applied mathematics, such as ballistics and aerodynamics, which have been developed deliberately for war and demand a quite elaborate technique: it is perhaps hard to call them ‘trivial’, but none of them has any claim to rank as ‘real’.” For context, Hardy was a pacifist and wrote this essay during the era of World War II.

I think if Hardy were to rise from his grave, he would be very disappointed to learn that his field of interest – number theory, is now used widely in everyday life (unbeknownst to everyone) for things like encryption and decryption during our internet transactions and in war to crack the Nazis’ Enigma Machine. There is a movie made on this, “The Imitation Game”. It stars Benedict Cumberbatch as another English Mathematician Alan Turing. I guess Hardy can be wrong after all. But this is to be expected… after all, his essay is not a mathematical essay 😊

Hardy also laments, in his essay, about the waning of his mathematical abilities in old age (remember he was already past 60 when he wrote the essay!). To him, mathematics is a “young man’s game” as it is a creative craft, and he was already past the age of creative endeavors. No doubt, “real” mathematics as Hardy would describe is definitely not something an aged person can masterfully do.

All in all, I find A Mathematician’s Apology an insightful read into the psyche of a professional pure Mathematician and I would definitely recommend it to anyone who is interested in learning about how a Mathematician may view the world. On a side note, if you liked the proof that I introduced in the review and are a current undergraduate in NUS, I highly recommend taking MA1100. You would be introduced to mathematical proofs like the one I showed and acquire techniques to prove or disprove future mathematical theorems!

By: Soo Qing, ReadNUS Deputy Director