Godfrey Harold Hardy: A Mathematician’s Apology

G.H. Hardy is one of the very few mathematicians who’s been immortalized in song, by the Embarrassment no less:

Hardy’s little book, A Mathematician’s Apology (that’s apology in the sense of defense, not regret), written in 1940 near the end of his career, is an eloquent and concise statement of the mathematician’s, theoretician’s, and Platonist’s worldview. It is worth reading especially by anyone who is not a member of those clubs. It is the memoir of a person who has spent so much time discovering theorems of numbers, formulae, and equations that they have come to seem far more real than the discovery of a new species of plant or a new planet, which after all is just one more instance of a form that was already known.

I am not exaggerating on the Platonist front. Hardy states it plainly:

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. A reader who does not the philosophy can alter the language: it will make very little difference to my conclusions.

This is the key point, never to be forgotten. This mathematical reality is more real to him than the world we appear to inhabit. Hardy witnessed connection to this reality in an even stronger form in his friend Ramanujan, the great mystic mathematician, who had sent him a sample his unpolished but noetically brilliant work. Two other mathematicians had dismissed Ramanujan’s work, but on seeing the unknown Ramanujan’s work, Hardy recognized him for what he was and brought him to Cambridge. I have no doubt that Ramanujan cemented Hardy’s Platonism: Hardy rated Ramanujan as the most talented mathematician he had ever known.

Correspondingly, the prose is a mixture of plainspoken simplicity and blatant elitism, cosmic humility and human arrogance, as though Moses had come down from the mountain without desiring to convince anyone that he was right…or not even being sure that he could. His very opening suggests he has only come down from the mountain because his powers have faded and failed him:

It is a melancholy experience for a professional mathematician to find himself writing about mathematics. 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.

Yet in the human world, mathematics seems a talent of marginal utility (at least to Hardy), and he defends his mortal life by saying only that he did mathematics because he was good at it.

Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating. And that I have created is undeniable: the question is about its value.

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.

As W. J. Turner has said so truly, it is only the ‘highbrows’ (in the unpleasant sense) who do not admire the ‘real swells’.

But in terms of the greater pageant of time, the mathematician has the greatest chance at immortality. He doesn’t compare his field to the empirical sciences (though he looks down on applied mathematics), but I gather that he is more confident of mathematical achievements because their results cannot be overturned by things like as-yet-undiscovered evidence. As for language and literature, they are merely human creations and even more evanescent.

If intellectual curiosity, professional pride, and ambition are the dominant incentives to research, then assuredly no one has a fairer chance of satisfying them than a mathematician. His subject is the most curious of all—there is none in which truth plays such odd pranks. It has the most elaborate and the most fascinating technique, and gives unrivalled openings for the display of sheer professional skill. Finally, as history proves abundantly, mathematical achievement, whatever its intrinsic worth, is the most enduring of all. Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. ‘Immortality’ may be a silly word, but probably a mathematician has the best chance of whatever it may mean.

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.

Could lines be better, and could ideas be at once more trite and more false? The poverty of the ideas seems hardly to affect the beauty of the verbal pattern. A mathematician, on the other hand, has no material to work with but ideas, and so his patterns are likely to last longer, since ideas wear less with time than words.

Indeed, the entire world of the contingent, the observed, the evidentiary, seems instilled with a frailness that makes it ephemeral and far less meaningful. Any connection to the everyday nominal world is something that endangers the solid rock of eternal truths which Descartes described as the sole object of posthumous contemplation. (Our memories do not exist after death, for Descartes, so the only things our souls can contemplate are a priori truths: mathematical and logical ones.)

It is quite common, for example, for an astronomer or a physicist to claim that he has found a ‘mathematical proof’ that the physical universe must behave in a particular way. All such claim, if interpreted literally, are strictly nonsense. It cannot be possible to prove mathematically that there will be an eclipse to-morrow, because eclipses, and other physical phenomena, do not form part of the abstract world of mathematics.

We can describe, sometimes fairly accurately, sometimes very roughly, the relations which hold between some of its constituents, and compare them with the exact relations holding between constituents of some system of pure geometry. We may be able to trace a certain resemblance between the two sets of relations, and then the pure geometry will become interesting to physicists; it will give us, to that extent, a map which ‘fits the facts’ of the physical world. The geometer offers to the physicist a whole set of maps from which to choose. One map, perhaps, will fit the facts better than others, and then the geometry which provides that particular map will be the geometry most important for applied mathematics. I may add that even a pure mathematician may find his appreciation of this geometry quickened, since there is no mathematician so pure that he feels no interest at all in the physical world; but, in so far as he succumbs to this temptations, he will be abandoning his purely mathematical position.

And so applied mathematics is inferior to pure mathematics because it is hamstrung by contingent particulars. Airborne truth is brought down to earth by the accumulated weight of midges and gnats:

One rather curious conclusion emerges, that pure mathematics is one the whole distinctly more useful than applied. A pure mathematician seems to have the advantage on the practical as well as on the aesthetic side. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics. I hope that I need not say that I am trying to decry mathematical physics, a splendid subject with tremendous problems where the finest imaginations have run riot.

But is not the position of an ordinary applied mathematician in some ways a little pathetic? If he wants to be useful, he must work in a humdrum way, and he cannot give full play to his fancy even when he wishes to rise to the heights. ‘Imaginary’ universes are so much more beautiful than this stupidly constructed ‘real’ one; and most of the finest products of an applied mathematician’s fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts.

“Fancy” and “facts” being somewhat self-effacing language, since by this point it is clear that for Hardy, fancy is more enduring than fact. And for anyone who works in these fields long enough, it is hard to imagine how a mathematician could not end up a Platonist after working so dutifully with non-material, abstract entities that constantly produce new, surprising, emergent properties.

This is not a new attitude; the Pythagorean cult is only one of the oldest known manifestations of this tendency. And it exists today in hardly a different form: the “quant” of finance describes being sucked into the world of mathematical reality in a similar though less eloquent way. And the insistence with which string theorists proclaim that their equations are so perfect that they simply must describe the ultimate truth of reality is more or less just a variation on Hardy’s ideas of theoretical elegance and beauty.

C.P. Snow knew Hardy and Hardy thanks Snow in the book, but the book belies Snow’s famous generalization about the two cultures of humanities and science. To hear Hardy tell it, the real divide is not between the humanities and the sciences but between the theoreticians and the engineers, idea and praxis, rationalists and empiricists, philosophers and storytellers, gnostics and skeptics.

It is more a continuum than it is a dichotomy, but each pole is a strong attractor and tends to draw in those who already lean toward it. As someone who by temperament or talents has always tended to fall closer to the engineer’s side, I always hope for the theorists to remember that suffering is as real as any theorem. Hardy refers to the anodyne of escape provided by theory, but not only can it also be a dereliction of human duty, but it is also ultimately an unreliable respite for mere particulars such as ourselves:

There is one purpose at any rate which the real mathematics may serve in war. When the world is mad, a mathematician may find in mathematics an incomparable anodyne. For mathematics is, of all the arts and sciences, the most austere and the most remote, and a mathematician should be of all men the one who can most easily take refuge where, as Bertrand Russell says, “one at least of our nobler impulses can best escape from the dreary exile of the actual world.” It is a pity that it should be necessary to make one very serious reservation—he must not be too old. Mathematics is not a contemplative but a creative subject; no one can draw much consolation from it when he has lost the power or the desire to create; and that is apt to happen to a mathematician rather soon. It is a pity, but in that case he does not matter a great deal anyhow, and it would be silly to bother about him.



  1. Dave M
    26 December 2011

    I have tended to think that Platonism is less poisonous in the mathematical context than elsewhere, and even pretty much harmless, but you make a good point about suffering. How one thinks about mathematics, however defensible there (and even there I’m not really defending it), affects how one thinks about other things as well. Interesting that Hardy says that “Mathematics is not a contemplative but a creative subject” – that sounds more like the anti-Platonist view, at least until you see what his point is there.

  2. nostalgebraist
    26 December 2011

    This essay has always bothered me, probably just because I’m skeptical of conventional views of pure math, and it is a classic statement of those views. I can’t help but feel like Hardy hasn’t really considered the relationship between pure and applied math in any serious way. His essay seems more like a collection of emotional attitudes than a real argument; he clearly enjoys doing pure mathematics and has no (personal) interest in applications, but he mistakes these preferences for a philosophical view of the situation.

    Take this example: he states that the irrationality of sqrt(2) could have no practical importance, since any quantity important for practical purposes can be approximately arbitrarily well by a rational number. What this ignores is that the real number system itself, of which rational numbers are a part, is itself quite useful for science and engineering. The engineer who attempted to do (say) calculus solely in the rational number system would have to go to a lot of extra trouble for no added benefit. The theorems about the real numbers are just nicer than the theorems about the rational numbers; by adding a set of practically “irrelevant” entities, we complete the rational numbers to a larger entity that is much easier to talk about than the rational numbers themselves. The same goes for the next leap, from real numbers to complex numbers. Surely no practically relevant quantity actually has an “imaginary part,” yet the complex numbers (by being algebraically closed, for instance) are in some ways much easier to talk about than the real numbers, and there is no reason that applied folks shouldn’t exploit this ease while deriving results about relevant quantities.

    Hardy’s mistake here illustrates an indifference to the origin of his concepts that is typical of Platonist pure mathematicians. He loves to play with the structures he has been given, but he seems to have no interest in where those structures came from (the answer is, in many cases: applications), or whether other structures might be equally suitable for his ludic purposes. To speak of “the geometry most important for applied mathematics,” as if the very subject of geometry did not emerge from our attempts as physical creatures to make sense of shapes, lengths and angles in physical space! If we can free our minds enough to consider “geometries” that do not behave like our world, then what are we doing considering “geometries” at all? Historically, the set of structures that Hardy loves to play with is one that has emerged in a messy fashion from the continual interaction of pure and applied mathematicians, and even its “purest” bits tend to be mere generalizations of ideas that began as physical ones.

    At root, though, my issue with Hardy and pure mathematics is a different one: their unquestioned, largely unspoken aesthetic standards. Pure mathematics is not just the production of true results, but the production of “interesting” true results — which becomes clear if one reflects on the chances of getting perfectly true, probably original theorems like “1^2034928 = 1″ published in a math journal. But while the truth-criteria in pure math are crystal clear, the interestingness-criteria are vague, intuitive, and rarely discussed. In Plato’s heaven, there are many truths, perhaps so many that they fit together like a jigsaw puzzle to produce a blank and shapeless whole. How do we know that all this “structure” we feel we see there isn’t just the shape of our own, unexamined aesthetic intuitions? Once we have produced all of the “interesting” truths, we will have only a picture of what we, a corporeal creature, find interesting.

  3. James
    26 December 2011

    C.P. Snow wrote an introduction to the 1967 edition of Hardy’s Apology. Of it, T.W. Korner says:

    The second edition contains a foreword by C.P. Snow which should not be read at the same time as Hardy’s essay. First look at the artist’s work — then look at the artist’s life (if you must).

    Korner’s books, particularly his Fourier analysis text, offer something like an extended defense of applied mathematics, although never quite stated as such.

  4. NN
    28 December 2011

    I feel a bit uneasy about his thoughts on what an eternal truth can be. Why shouldn’t literature or philosophy be able to produce eternal truths on par with mathematics? Of course maybe this view simply follows from his platonism, but since neither the notation of Aeschylus or Archimedes has any guarantee of being sensible to future generations, what exactly does he mean? It seems sensible to me to think that any person who read literature might recognize its truth, if he is the same kind of thing as Aeschylus was. Might not the same apply to math? It just seems odd to us that someone might have entirely different sensibility when it comes to math, but to me it seems that the person who finds nothing to relate to in literature and the student who does no know how to continue a certain mathematical sequence (as in the example by Wittgenstein) is the same kind of incomprehensible creature.

    Meh. Maybe I am just not much of a platonist?

  5. Ray Davis
    28 December 2011

    I don’t dispute the general substance of nostalgebraist’s points, but Hardy does briefly describe his aesthetics (“a very high degree of unexpectedness, combined with inevitability and economy”), and justly protests that literary scholars tend to be just as vague on the topic.

    To me the Apology sounds like a grilled martyr shouting the omnipotence of his deity: Hardy attempts to link personal creative proprietorship to eternal wordless abstraction even while suffering their inevitable rupture. Less poignantly, during unrelated reading today I found this note sent from humanistic scholar Henry Adams to Charles Milnes Gaskell in 1910:

    “I enjoy poking fun at mathematicians, who are truly the bottom of all possible depths of imbecility. Pure thought must be absence of movement,– mere lines in nothing. I much prefer my eastern Yogis, who are at least amusing and picturesque.”

  6. David Auerbach
    30 December 2011

    Dave: I might have more properly called it neo-Platonism or gnosticism…something that entails turning away from the world, finding a salve or solution that suggests that there is permanence and good in a realm that is most certainly *not* ours, which is ruled over by a false, cruel deity. Not that this comfort isn’t very appealing at times, but it’s good to know what it is.

    The thing about mathematics is that it’s a particularly compelling sort of gnosticism, far more so than anything that you find in Lull or Bruno’s systems, though perhaps they were seeking something similar. Every time you’re surprised by some sort of result in mathematics or logic or even non-quotidian physics, that’s a thrill of a sort that does draw you out of empirical reality as it is known to most of us, and it’s a sufficient puzzle that philosophy of mathematics still can’t get to square one around it. This sort of gets back to what nostalgebraist is saying. That is, the solidity and enduring nature of the tradition and its sheer specificity makes it seem like it is far more robust than anything so vague as language.

    My own opinion, with NN, is that it is a question of degree rather than of kind: access to math/numbers is achieved through some sort of mechanism in common with language, though different valences are at work. But the fact that math + science as a package crosses cultures vastly more easily than anything else similarly sized (well, except for power and exploitation and lots of other bad things) gives its exponents a key piece of ammunition that, for example, Christian Scholastics did not have, however intricate and supposedly rigorous their logic was. So somehow Aristotelianism falls down and Platonism (or maybe Pythagoreanism) persists. Yet to return to nostalgebraist’s last point, the issue remains that standards such as elegance, simplicity, and beauty are unquantifiable despite *seeming* to possess stronger intuitive claims than many, many other things.

    Hence the two divergent trends of analytic philosophy in the 20th century: one to uncover our world as actually being the world of logic and mathematics (the Tractatus, the Aufbau, Kripke, Timothy Williamson), and one to distinguish OUR world from the world of logic and mathematics (PF Strawson, for example, and I think Quine). Perhaps it’s not so different from Leibniz vs. Newton (respectively).

    Ironically, Carnap wanted to exclude art from rigorous philosophy, which Popper claimed gave away the store to the aesthetes and left science and philosophy with almost nothing left to study.

    Hardy to me sounds like Adam after exile from the Garden. “Truly it was a paradise.”

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