Excerpts from A Mathematician's Apology

july 11, 2026

the justification for mathematics

Hardy opens with the melancholy of writing about mathematics instead of doing mathematics, and a criticism of those who expound.

The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done.

Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.

He then moves on the question of usefulness. How can a lawyer, a journalist, or a mathematician justify his career? There are two questions one must answer: 1) the intrinsic value of the job, and 2) why one is uniquely suited for it.

There are two answers to these questions, the second of which is a humbler version of the first. Firstly, it is because one’s talent lies in his profession: he possesses a unique ability to bat in a cricket game, calculate chess moves, or orate in a courtroom:

… perhaps five or even ten percent of men can do something rather well. It is a tiny minority who can do something really well, and the number of men who can do two things well is negligible. If a man has any genuine talent he should be ready to make almost any sacrifice in order to cultivate it to the full.

Moreover, it is extremely unlikely that a man has genuine talent in two or more fields. In the case of mathematics especially, mathematical talent is one of the most specialized; it is not known for its general ability nor its versatility.

The second reason why one practices his profession is because there is nothing one can do particularly well, and his profession simply came his way. Hardy accepts this explanation, while noting that none of his readers would be content with it.

He then moves on to his 1920 talk at Oxford, an earlier rendition of this apology. The question he seeks to answer is if mathematics worth doing at all. He outlines three main reasons, all of which will be subject to elaboration:

Firstly, mathematics is harmless as a discipline. He acknowledges the question of the effect of science on war, which he will contemplate later, but stands firmly by this point.
Secondly, assuming mathematics is useless, the scope of mathematicians wasting their time is greatly limited: it is a bunch of university ‘dons’, unlikely to have great talents elsewhere, who are wasting their time. This is negligible.
Thirdly, while he notes his reliance on rhetoric here, mathematics has a character of permanence:

What we do may be small, but it has a certain character of permanence; and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men.

In these days of conflict between ancient and modern studies, there must surely be something to be said for a study which did not begin with Pythagoras, and will not end with Einstein, but is the oldest and the youngest of all.

ambition

In this next section Hardy changes his focus:

I shall assume that I am writing for readers who are full, or have in the past been full, of a proper spirit of ambition. A man’s first duty, a young man’s at any rate, is to be ambitious. Ambition is a noble passion which may legitimately take many forms; there was something noble in the ambitions of Attila or Napoleon; but the noblest ambition is that of leaving behind something of permanent value.

Here he argues for the virtue of ambition, and says that the noblest form of it is to leave behind something permanent. Moreover, those who make great discoveries are often motivated by ambition more than anything else. Taking the example of physiology, an “obviously beneficial study”:

We must guard against a fallacy common among apologist of science, the fallacy of supposing that the men whose work most benefits humanity are thinking much of that while they do it, that physiologists, for example, have particularly noble souls. A physiologist may indeed be glad to remember that his work will benefit mankind, but the motives which provide the force and the inspiration for it 11 are indistinguishable form those of a classical scholar or a mathematician

He then says there are three motives for any man pursuing research. The first is intellectual curiosity. The second is the shame of failure: the pride a craftsman has when his work is unworthy of his talent. The third is ambition, the desire for money, fame, or power. It is fine, in the completion of your work, to find out that it benefited humanity, but he places great doubt on anyone claiming that the driving force in their work is altruistic.

beauty

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. And here I must deal with a misconception which is still widespread (though probably much less so now than it was twenty years ago), what Whitehead has called the ‘literary superstition’ that love of an aesthetic appreciation of mathematics is ‘a monomania confined to a few eccentrics in each generation’.

He notes that while it is difficult to concretize a description of mathematical beauty, it is easy to recognize it. And while many take pride in acclaiming their mathematical inability, the beauty of mathematics is quite well-recognized. This is because people recognize mathematical beauty in its lower forms:

There are masses of chess-players in every civilized country—in Russia, almost the whole educated population; and every chess-player can recognize and appreciate a ‘beautiful’ game or problem. Yet a chess problem is simply an exercise in pure mathematics (a game not entirely, since psychology also plays a part), and everyone who calls a problem ‘beautiful’ is applauding mathematical beauty, even if it is a beauty of a comparatively lowly kind. Chess problems are the hymn-tunes of mathematics.

The practical value (in the crude sense of increasing the happiness and comfort of the common man) of chess and mathematics may both be negligible. So what is the difference between the two forms of mathematics? It is the seriousness of the best mathematics, defined here:

The ‘seriousness’ of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is ‘significant’ if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas.

Moreover, seriousness is a precondition for beauty:

The beauty of a mathematical theorem depends a great deal on its seriousness, as even in poetry the beauty of a line may depend to some extent on the significance of the ideas which it contains. I quoted two lines of Shakespeare as an example of the sheer beauty of a verbal pattern, but After life’s fitful fever he sleeps well seems still more beautiful. The pattern is just as fine, and in this case the ideas have significance and the thesis is sound, so that our emotions are stirred much more deeply.

a mathematical digression

To demonstrate to the reader he means by beauty in mathematics in a way accessible to those without a strong background, he gives two simple yet beautiful proofs: Euclid’s proof of the existence of an infinity of prime numbers and Pythagoras’s proof of the irrationality of $\sqrt{2}$ . They are reproduced here:

There are infinitely many prime numbers

Prime numbers are integers that cannot be factored into smaller integers. The numbers 2, 3, 5, 7, 13, 17, 19 are all prime.

Proof: Assume this is false, i.e. there are a finite number of prime numbers. Let this finite sequence of prime numbers be $(2, 3, 5, ..., P)$ where $P$ is the largest prime number. Then let the number $Q = (2 \cdot 3 \cdot 5 \cdot ... \cdot P) + 1$ . It can be shown that this number is not divisible by any of the primes, because the remainder mod $p$ , where $p$ is the prime, is always 1. This contradicts our assumption that there is no prime greater than $P$ , therefore our hypothesis is false.

\sqrt{2}

is irrational

Rationality means that a number $q$ can be represented as $\frac{a}{b}$ where $a,b$ are integers that share no common divisor. If $\sqrt{2}$ is rational, then $\sqrt{2} = \frac{a}{b} \implies (\sqrt{2})^2 = \frac{a^2}{b^2}$ .

Proof: Assume $\sqrt{2}$ is rational, so we get that $a^2 = 2b^2$ for some integers $a,b$ with no common divisor. Because the right hand side is even, $a$ must be even as well, i.e. $a = 2c$ for some integer $c$ . Therefore we get that $2b^2 = a^2 = (2c)^2 = 4c^2$ . This reduces to $b^2 = 2c^2$ , where by a similar argument we get that $b$ is even. However, both $a$ and $b$ can’t be even, as we assumed they share no comon divisors. Hence we get a contradiction, and $\sqrt{2}$ is irrational.

Here he makes a wonderful comment on proofs by contradiction:

The proof is by reductio ad absurdum, and reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

defining seriousness

A mathematical theory is serious in great part due to the significance of its ideas. In this section he makes two defining criteria for the significance of a mathematical idea: its generality and its depth.

On generality:

The idea should be one which is a constituent in many mathematical constructs, which is used in the proof of theorems of many different kinds. The theorem should be one which, even if stated originally (like Pythagoras’s theorem) in a quite special form, is capable of considerable extension and is typical of a whole class of theorems of its kind. The relations revealed by the proof should be such as to connect many different mathematical ideas.

On difficulty:

It has something to do with difficulty; the ‘deeper’ ideas are usually the harder to grasp: but it is not at all the same. The ideas underlying Pythagoras’s theorem and its generalization are quite deep, but no mathematicians now would find them difficult. On the other hand a theorem may be essentially superficial and yet quite difficult to prove (as are many ‘Diophantine’ theorems, i.e. theorems about the solution of equations in integers).

It seems 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. The lower the stratum, the deeper (and in general more difficult) the idea. Thus the idea of an ‘irrational’ is deeper than that of an integer; and Pythagoras’s theorem is, for that reason, deeper than Euclid’s.

He gives an example: Euclid’s theorem is important, but not that deep (you don’t need anything more than divisibility to prove it). But the questions that come after it are deeper: given that there are an infinity of primes, how many are less than $10^{80}$ ? What about $10^{{10}^{10}}$ ? The theorem that explains this, the Prime Number Theorem, is much deeper, and make use of very powerful methods in analytic number theory.

more thoughts on chess

He makes one further distinction on the difference between ‘real mathematics’ and chess:

In both [Euclid’s and Pythagorus’s] theorems … there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions. There are no complications of detail—one line of attack is enough in each case; and this is true too of the proofs of many much more difficult theorems, the full appreciation of which demands quite a high degree of technical proficiency. We do not want many ‘variations’ in the proof of a mathematical theorem: ‘enumeration of cases’, indeed, is one of the duller forms of mathematical argument. A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way

While chess can have unexpectedness, as in a surprising move, and a level of economy, from each piece playing its part, the effect is more ‘cumulative.’ But for a chess move to be beautiful, the enemy must also have a variety of moves to respond with. This ‘enumeration of cases’ is what a real mathematician despises.

returning to the question of utility

Here he notes that, at this point, it is probably clear he is interested in mathematics as purely a creative art. In terms of crude utility (the comfort and welfare of the common man), mathematics, like many other subjects, has almost no usefulness at all:

It is useful to be tolerably quick at common arithmetic (and that, of course, is pure mathematics). It is useful to know a little French or German, a little history and geography, perhaps even a little economics. But a little chemistry, physics, or physiology has no value at all in ordinary life. We know that the gas will burn without knowing its constitution; when our cars break down we take them to a garage; when our stomach is out of order, we go to a doctor or a drugstore. We live either by rule of thumb or on other people’s professional knowledge.

And while some mathematics may be useful, the implication is more often “some experts knowing mathematics could increase the welfare of the majority,” than “everyone should know a great deal of mathematics.”

It is clear, then, that some elementary mathematics (such as differential and integral calculus) are useful to some laypeople, but that the majority of ‘real mathematics’ is wholly useless. How can we justify mathematics in terms of its utility, then? It is through its neutrality:

But science works for evil as well as for good (and particularly, of course, in time of war); and both Gauss and less mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean.

pure vs applied mathematics

Hardy notes that while it may seem natural to suppose there is a difference in the utility of pure and applied mathematics, he says that the distinction between them creates no difference in utility. To preface further discussion, he expresses his opinions on the nature of mathematical reality:

I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations’, are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it.

The distinction between the pure and applied mathematics is clearest in geometry:

There is the science of pure geometry, in which there are many geometries, projective geometry, Euclidean geometry, non-Euclidean geometry, and so forth. Each of these geometries is a model, a pattern of ideas, and is to be judged by the interest and beauty of its particular pattern. It is a map or picture, the joint product of many hands, a partial and imperfect copy (yet exact so far as it extends) of a section of mathematical reality.

To make this distinction clearer, he gives two examples:

Let us suppose that I am giving a lecture on some system of geometry, such as ordinary Euclidean geometry, and that I draw figures on the blackboard to stimulate the imagination of my audience, rough drawings of straight lines or circles or ellipses. It is plain, first, that the truth of the theorems which I prove is in no way affected by the quality of my drawings.

And then suppose a massive gravitating body appears in the lecture room, changing its geometry. Does that cause the theorems proved to be false? It does not:

It would be like supposing that a play of Shakespeare is changed when a reader spills his tea over a page. 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.

This map of mathematical reality is the symbol of a pure mathematician. While pure mathematicians may not be able to map this world perfectly, they can say meaningful things about it, or about relations between objects in it, and at some point the map may become useful to physicists. The geometer offers a set of maps to choose from, and the physicist picks whichever ‘fits the facts’ of the physical world best.

age

I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game. To take a simple illustration at a comparatively humble level, the average age of election to the Royal Society is lowest in mathematics. We can naturally find much more striking illustrations. We may consider, for example, the career of a man who was certainly one of the world's three greatest mathematicians. Newton gave up mathematics at fifty, and had lost his enthusiasm long before; he had recognized no doubt by the time he was forty that his greatest creative days were over. His greatest idea of all, fluxions and the law of gravitation, came to him about 1666 , when he was twentyfour—'in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time sine'. He made big discoveries until he was nearly forty (the 'elliptic orbit' at thirty-seven), but after that he did little but polish and perfect. Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work a good deal later; Gauss's great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself.