A Mathematician’s Apology


In ‘A Mathematician’s Apology’, the renowned English mathematician Godfrey Harold Hardy presents an eloquent and poignant apology for a mathematician’s life. Written in 1940, towards the end of his triumphant career as a pure mathematician, Hardy makes a convincing case for the beauty, seriousness, and permanence of mathematical achievement. When I first read Hardy’s book, I was an aspiring researcher yet to begin graduate studies. Reading it again, now as a fourth-year PhD student who works on theoretical aspects of machine learning, I sense the influence this book had on me, in nudging me to pursue math-heavy theoretical research as opposed to empirical work. I must say that my interests have evolved considerably from those early days, and are still evolving.


Hardy begins his essay with two questions a person needs to answer to justify his existence and his activities:

  1. Why do you do THIS?

  2. Why do YOU do this?

Hardy remarks that the second question is easier to answer for most people, and the answer is either “I do this because THIS is the only thing I can do well,” or a humbler variant: “I do this because I am not particularly good at doing anything, and THIS came my way.” The first question is much harder, and Hardy spends the rest of the essay answering it for his own self, where THIS is pure mathematics.

At first thought, it may seem obvious that mathematics does not need an apology, as it is essential to the pursuit of science and technology which drives the economy and adds to the comfort and well-being of people. However, Hardy’s apology concerns “real” mathematics, which is more of an art form, as opposed to “trivial” mathematics that finds utility in science and technology.

Real mathematics must be justified as art if it can be justified at all.

To Hardy, the utility in pure mathematics is not in its consequences or in enhancing the well-being of the people, but rather in its beauty. Hardy considers ‘real’ mathematics as a creative art, much like poetry or music. He defends mathematics on the grounds of the joy it brings to the artist i.e., the mathematician, on the significance of the ideas it connects, and on the quality of permanence that is inherent to the pursuit of discovering hidden truths of the universe.

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

To me, the best parts of the book are Hardy’s descriptions of the subtle qualities of mathematical results like generality, depth, seriousness, and aesthetics. They offer a peek into the mind of a mathematical artist. Hardy’s artistic view of mathematics is in sharp contrast to a prevailing view of the common man, of math as a dry and boring subject involving tedious calculations.

The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.

If there is one thing that I do not like about the book, it is the subtle but persistent judgemental tone of Hardy. He casually remarks that most people cannot do anything well and so it matters little what career they choose. To Hardy, exposition, criticism, appreciation, is work for “second-rate” minds. He does not see value in an average person learning about the sciences, as they live either by the rule of thumb or rely on other people's professional knowledge. Particularly poignant is his view that mathematics is a young man’s game, as he says that his own life is finished ever since he got older. I cannot help but trace his overly critical attitude to his early days growing up in a competitive environment: he says of his school days, “I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively.”

Hardy’s book is my first recommendation to anyone who wants to know what it is like to work in theoretical disciplines, especially the ones that involve higher-level mathematics. And to those who already work in theory, I think this is must-read. Like me, you might just rediscover why you do what you do, the way you do it!


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