A Mathematician's Apology in Mathematics Students

Sep 23, 2011 Yasiru rated it it was amazing

Shelves: mathematics, favourites

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I had the good fortune to come across this title just as I was finally beginning to see the glimmers of beauty in mathematics thanks to the efforts of some wonderful instructors on the subject during my later school years. It called upon me for a deeper reflection on my chosen pursuit, which at that point appealed to me for its fundamental importance to the other sciences and for the simple pleasure that can be gleaned of it. I was The text may be found at

I had the good fortune to come across this title just as I was finally beginning to see the glimmers of beauty in mathematics thanks to the efforts of some wonderful instructors on the subject during my later school years. It called upon me for a deeper reflection on my chosen pursuit, which at that point appealed to me for its fundamental importance to the other sciences and for the simple pleasure that can be gleaned of it. I was confronted with a fleeting, tricksome nature even in mathematical truth, which had till then been sacrosanct and taken for granted. It became apparent that there was more artistic excitement in mathematics than the safety of empirical sciences and it has been my great love ever since, however modestly ability dresses me.

Hardy's is an essay of paradoxes and duals, with examples littered through the text. These may seem contradictory at a glance, but close scrutiny reveals them to be mutually reliant points that hint to the complexities of Hardy's pursuit. One instance is in his claim that mathematical patterns are more enduring than any artist's or poet's. While this may leave many an artist or poet bristling, this endurance is manifest in the progression of a mathematician's significant ideas, even though these are all in a language of ideas that does not rely on being evocative (so unlike the artist or poet). This is at odds however, with the aesthetic appreciation to be had in nontrivial mathematics that Hardy keeps returning to. The prospect of leaving some mark so enduring is certainly catering to ambition (which Hardy encourages), but this is incompatible with failing to capture beauty, and so it is that one cannot stand without the other. It has been said that this is a tragic account, but here is a thread of hope- that one should remain steadfast in the area of one's talent and atop that cultivate ambitions and seek too what is beautiful. There is a terrible, indubitable rationale here which could strike some as arrogant (though even if it were taken to be so, not something diminishing from the work), but is ultimately a course of efficiency (something Hardy valued in his mathematical aesthetic) devoid of any lingering melancholy.

Another instance is what he has to say about the worthwhile mathematics and trivial. Frequently, Hardy is misrepresented as disdaining applied mathematics upon the haughty perch of the pure. This is unfortunate since special mention is made that mathematicians do not tend to revel in the uselessness of their work. Hardy is no fool to try untangling the mathematics that will come to some use and what stands detached and austere. He draws the crucial distinction instead based on aesthetic value once again, setting aside those abundant trifles of mathematics so often connected with the mundane world of everyday and advocating the deep framework (of rather few significant contributions) of what might be dubbed the 'true mathematics'. Hardy is rather half-hearted about elaborating on his standards of beauty, but there is no real surprise in this for anyone acquainted with higher mathematics, because it really is a difficult thing to define. There is efficiency and sweeping elegance in a great result, but too a contribution to that progression of ideas which makes mathematics so enduring. Whether these criteria capture beauty is contestable and so I make no claims. But this difficulty in explaining a thing like beauty highlights the crucial problem in Hardy's endeavour to explain to the public what a mathematician does and why. He justifies the pursuit of the true mathematics exactly because it is so removed from mundane being and so comes of purer intentions than much else that can be directed to harm as well as good, but it is a difficult thing to impart the joy of actually doing mathematics to the uninitiated, which is perhaps why so many reviews have picked upon the thread of personal tragedy here (partly owing to the foreword I suppose, thought Hardy's circumstances are the least of the work's significance, I thought, both on my first reading and now) instead of the 'apology' itself. There is nowadays a great scuffle to get some manner of emotional fix upon an author one is reading. This sort of thing is seldom the author's intent (else they would simply save themselves the trouble of writing a reasoned and coherent account with objective value and instead send their diaries to a publisher), especially in a work as this where the author is attempting to convey a position and defend it. The arguments do not require lenses coloured with the author's circumstances. I imagine Hardy the pure mathematician would rather have avoided this, as he remarks the following in the present work (something of a spoiler warning is due here)-

"Dr Snow had also made an interesting point about 搂8. Even if we grant that 鈥楢rchimedes will be remembered when Aeschylus is forgotten鈥�, is not mathematical fame a little too 鈥榓nonymous鈥� to be wholly satisfying? We could form a fairly coherent picture of the personality of Aeschylus (still more, of course, of Shakespeare or Tolstoi) from their works alone, while Archimedes and Eudoxus would remain mere names. Mr J M Lomas put this point more picturesquely when we were passing the Nelson column in Trafalgar square. If I had a statue on a column in London, would I prefer the columns to be so high that the statue was invisible, or low enough for the features to be recognizable? I would choose the first alternative, Dr Snow, presumably, the second."
(Dr Snow here is C P Snow, who wrote the foreword to this edition. It's worth reading, but perhaps after Hardy's own words, since its biographical tint somewhat along the line I've discussed above may detract from the potent general ideas Hardy is forwarding here.)

Few other works have attempted the exposition of what creative enterprise demands and entails quite so well as this one, and this from a master who has seen ability and passions come and go. I recommend it highly. ...more

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Jan 12, 2010 Fraser Kinnear rated it really liked it

Shelves: math

The C.P. Snow introduction, which was nearly as long as the Apology itself, was a fantastic inclusion, as it provided context which I felt really brought me closer to Hardy. Without it, I think, I would have found Hardy to have been a bit too full of himself, as he feels dismissive of perspectives other than his own. Having said that, Hardy is very sensible in explaining the limits of his ability to explain his thoughts, and admits that he is unqualified to approach topics like the aesthetics of The C.P. Snow introduction, which was nearly as long as the Apology itself, was a fantastic inclusion, as it provided context which I felt really brought me closer to Hardy. Without it, I think, I would have found Hardy to have been a bit too full of himself, as he feels dismissive of perspectives other than his own. Having said that, Hardy is very sensible in explaining the limits of his ability to explain his thoughts, and admits that he is unqualified to approach topics like the aesthetics of mathematics philosophically.

I've come away really wishing that I had this book as part of a dialog, rather than a soliloquy which. I think that there is a lot of fertile ground here, especially in comparing his Platonism with other schools of thought at the time, which he must have been aware of, like Wittgenstein.

Hardy's main point (I think) is that there are two types of mathematics: Real (academic, pure) and trivial (basic, simple, everyday). Trivial mathematics is applied to our lives and serves its purpose (and so is commonly known as applied mathematics). But real mathematics has almost no connection to our lives (outside of philosophy and some of the high sciences), even if it is describing a reality that Hardy, as a Platonist, believes is greater than the one we observe. Real mathematics, to Hardy, can only "be justified as art if it can be justified at all."

This may be seen as a sad book, certainly. But I see Hardy's opinion of his own mortality (he committed suicide) as starkly inconsistent with his opinion of pure mathematics. Near the close of the book, Hardy admits "It is plain now that my life, for what it is worth, is finished, and that nothing I can do can perceptibly increase or diminish its value." What a shame that he considers his life finished because he can't increase its value. Hasn't he spent an entire book justifying (or at least enjoying) something which he admits to be useless? If Hardy saw his life the way he saw his mathematics, he would have spent the remainder of it enjoying himself, simply for his own sake. ...more

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May 24, 2014 Elizabeth rated it really liked it

1. I take issue with Hardy's assumption that ambition is a noble value.
2. I love chess. Hardy says chess problems are unimportant. He makes a good case.
3. Mathematics is beautiful and Hardy's words do this idea justice.
4. This was written 75 years before I read it. It's interesting that the whole "apology" would likely not have been written today given that number theory does improve lives today.

...more

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