A Mathematician's Apology (Canto)

Rating:
Summary: A classic of the mathematical literature
Review: Every discipline has a list of items that must be read if one is to be considered educated in that field. There is no doubt that this book should be required reading for any degree in mathematics. Most of the soul of mathematics is contained in the 91 pages of the 'Apology' (the first 58 pages consists of the foreword by Snow). Written in his later years when Hardy knew his mathematical powers were failing, this is a superb exposition by a brilliant, eccentric personality. He not only captures the grandeur of mathematical discovery, but also clearly articulates the feelings of a man who knows that his time has passed. First published in 1940, the twin messages are timeless.
Clearly distinguishing between the real mathematician and the puzzle solver, Hardy is exceptional in declaring what the real beauty of mathematics is. Among all the beautiful things that exist, the percentage of individuals that can truly appreciate an elegant theorem is among the smallest. However, anyone who can read this work and not see at least some of the poetic qualities of mathematics has a blind spot in their soul. One of the masterpieces of literature, this book can be understood and appreciated by anyone with an eye for the beautiful things that life has to offer.

Published in Journal of Recreational Mathematics, reprinted with permission.

Rating:
Summary: A unique book, should be read by everybody
Review: Hardy was a man that comes along rarely in life, and this book is an even rarer portrait of how men like him think. As a mathematician, Hardy was excellent, his collaboration produced much fruitful work, and he is perhaps most renowned for discovering the young protege Ramanujan. But this book is not really about his work, but about his views on life, and mathematics, as a whole. Considering how little people in American society know about mathematics and its practitioners, this book, which is emminently readable, will give all people a unique view of what some mathematicians think like. The book is short, but interesting from first page to last. Hardy was past his mathematical prime when he wrote this book, but this book probably is his most influencial he ever wrote.

The introduction by C.P. Snow is more like a short biography about Hardy, and it's about the same length as Hardy's actual text. It gives us insights into what one of Hardy's friends thought of him, and it also frames the life Hardy was living in as he wrote this book.

Hardy's opinions are strong, and undoubtedly every reader will disagree here and there with him. But he shows the reader some of the gems of mathematics, and perhaps the reader will be able to appreciate those even without formal mathematical training. He also talks about war and what he thinks of it. Whatever the reader thinks about Hardy's opinions, this book gives us the opportunity to glimpse into the mind of an artist - one different than the usual meaning attached to the word, but one nonetheless - and experience a part of human life not experienced by many - the wonders of mathematics.

Rating:
Summary: Motivation for anyone questioning being a Mathematician
Review: Hardy, in just a few short chapters, crafted a beautiful book that is nothing less than a devotional for mathematicians. Reading one chapter a day for the rest of my life would keep anyone in a state of mathematical bliss. In my own educational journey, Hardy gave me the feul for the fire to persist in my degree program. Every sentence of this short memoir is quotable, and I would recommend this book for anyone considering entering the field of math as well as anyone who feels stale in their choice of becoming a mathematician. In A Mathematician's Apology, Hardy clearly shows that he is the Knute Rockne of math.

Rating:
Summary: The Dark Side of ?Be All You Can Be?
Review: I confess to a weakness for the Oxbridge culture of the early 20th century, so I'm a pushover for something like this. I'm not personally familiar with Hardy's work (my studies, such as they were, were in computability, provability, and the like), but it's enchanting to think of him sitting down with Housman or Russell at lunch and disputing or joking, so much shared tacit knowledge at the table.

The long foreward by C. P. Snow makes the subsequent text richer and more sad than it would otherwise be. Still, the most important point is made by Hardy: mathematics is a serious creative art, and is worth doing for that reason. Moreover, you should move heaven and earth to develop your abilities if you have the talent, and not bother with it otherwise.

More generally, Hardy places great value on the doing of something - anything! - supremely well, and has little interest in the lot of most people, which is to muddle through in their arbitrary careers. And there's the rub. Like a great athlete, a great mathematician is finished rather early. Yet he must contrive to live long after his powers have deserted him. Athletes often go to seed when their playing days are over, where less well-endowed people might remain physically active into old age. Hardy lost his math "legs" and never got over it. And here he's weighing in on Housman's side in the perennial problem of getting old: better to slip betimes away.

Anyway, it's a darn good read, and short, at that. And we must remember that it was written in those dark days when Germany was rolling over Europe. Can we blame Hardy for taking solace in the fact that his beloved number theory seemed to have no applications to war? (These days, of course, prime factorization methods constitute a strategic advantage...)

Rating:
Summary: Interesting book for the mathematics lover.
Review: I think this is overall a worthy read for any true lover of mathematics. Firstly, its a very short book and can be covered in one or two sittings easily. The book isn't very detailed in a discussion of 'pure' mathematics, so with that said, this is a good book to compare and contrast a person's own ideas about mathematics to. It is NOT a good book to solely get reasons why it is is worthy to study pure mathematics. I also disagree with a thing or two in the book(the major one being that Hardy believes mathematics exists itself in reality and that mathematicians are only observers to this)and because of the historical time the book was written in, some of the statements in the book are completely inaccurate (ie that quantum mechanics and relativity have no 'practical use' and that pure mathematics is of no 'practical' use in wartime and is an 'innocent' or 'immaculate' science because of this). To sum it up, if you have personal ideas about 'pure' mathematics and its justification, then this is a good book to get a few ideas on, but if you have no clue what 'pure' mathematics is and want to know more about it and/or want a justification of its uses, then read a book on the history of mathematics.

Rating:
Summary: A major let-down
Review: I was really looking forward to reading this book, being familiar with Hardy and having read two books on Ramanujan; but the book was very disappointing. The foreword by Snow is almost as long as the book itself, and really doesn't add anything of note. Instead of a foreword one could use footnotes explaining the frequent cricket terms and names of people no longer recognized by the public.
What does one learn about what math is, why people become mathematicians, the artistic beauty of math, etc?
Very little. Hardy wrote the book for the public, and basically feels that anything interesting he could say about these issues would be unintelligible. All that Hardy says can be summed up as
1) I became a mathematician because it is the only thing I could do well, 2) pure mathematics is useless but harmless [he even says that relativity theory was good because it couldn't be used for war - remember the book was written before WWII]
3) doing mathematics requires creativity, and thus is done by younger people (but Hardy himself was most creative in his 40s).
Since Hardy was already in his sixties when he wrote the book he was past his prime, and says so several times.
If you are interested in why people do math and how math can be beautiful, you won't find much in this book.
Even if you are specificaly interested in Hardy and English math at the turn of the 20th century there are much better books available.

Rating:
Summary: A unique book, should be read by everybody
Review: If I could prove by logic that you would die in five minutes, I should be sorry you were going to die, but my sorrow would be very much mitigated by pleasure in the proof -G.H. Hardy to Bertrand Russell

There is something ineffably sad about this book by the mathematician G.H. Hardy, by all accounts a brilliant and a decent man, who was approaching the end of his career and realized that his contributions in the field all lay behind him. The "apology" seems in many ways to be for not being the kind of surpassing genius who revolutionizes scientific/mathematical thought--a Newton, an Einstein or a Gauss, not even the kind of natural genius that his protégé Ramanujan was. Hardy described himself as "at best, for a short time, the fifth best pure mathematician in the world." His best work came in collaboration with Ramanujan and Littlewood. But it was great work and he was one of the best mathematicians of his time and that gives the apology a kind of self-indulgent tone; who but he cares whether he was number five or number one? It's as if Jim Kelly felt compelled to write a book justifying his career in football because he was a runner up five times.

Hardy makes it clear that he sees mathematics as a unique endeavor and achievement in the field as the one guarantor of true immortality:

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.

But he has a glaring blind spot; perhaps understandably, he overemphasizes the centrality of math to human existence. C.P. Snow's Introduction to the book discusses Hardy's lifelong animus towards religion and Hardy himself is pretty dismissive of politicians, philosophers, artists and authors. But the fact is that billions of people remember the words of Moses and Christ and Mohammed and the Buddha, can quote long passages of Shakespeare, can hum a few bars of Bach and Beethoven and Elvis. But ask us what Gauss did and our eyes will glaze over. Hardy was apparently attracted to mathematics because it was a closed system (or series of systems) with set parameters and certain proofs. Within that system men can achieve certainty and can achieve immortality at least in the eyes of those who study the system, but how do the accomplishments of these men measure up against those of men like John Locke and Adam Smith and James Madison who created the liberal democratic capitalist state?

It is not surprising that Hardy had such tunnel vision, because mathematics was apparently his whole life, except for cricket. He spent his whole career in the ivory tower. He had no family. As mentioned, he had ditched any religious beliefs. What was left to him but his chosen field of study? It comes as no surprise when Snow reveals that Hardy later, with his health failing, tried to commit suicide. If math was his life and he had lost the creativity necessary to excel at it, what would be the point of continuing?

This is a very fine book, interesting, accessible and eminently quotable. But like many of the intellectuals of his generation (he was loosely affiliated with the Apostles and the Bloomsbury Group), there is a hollowness apparent at his core. In his discussion of a couple of mathematical proofs, it is obvious that Hardy revels in the achievements of the men who demonstrated that these numerical relationships exist. As I read, I was struck by a tangible sense that the beauty of these relationships implies a creator behind them. I repeat implies, not proves. And I mean creator, not Creator. Recalling Einstein's statement that "God does not play dice with the Universe", I can not believe that there is a physicist or mathematician who, in his heart of hearts, truly believes that we will never know the fundamental equations for the processes that created the Universe. The logical implication is that if it is possible eventually to understand these formulae, or even merely conceivable that they can be understood, then isn't it also conceivable, even likely, that some kind of intelligent being may have set them in motion in the first place. If these suppositions are comprehensible, isn't it also possible, even likely, that in the process of coming to understand them, we are becoming divine, that one day we will use a Grand Unified Theory to ourselves become Creators?

Suffice it to say, these are not the sorts of speculations that engaged G.H. Hardy. Instead he ended his, by his own measure, mediocre career, cloistered away in the insular world of the university, alone and compelled to produce a written justification for his life. The self portrait he has produced is one of the saddest images I've encountered of the bitter harvest that has been reaped by the hollow men of the 20th Century.

GRADE: B

Rating:
Summary: A biography ?
Review: Number Theory had been my fascinating subject since BS days.This is one of the reasons that I got attracted towards the book by Hardy.
When I started reading this book,I found myself reading a biography of Hardy and the book was supposed to convey something else as I had heard.I re-read the title again to check if I had the wrong one.The foreword by Snow covers nearly the first 20-25 pages.Rest comes the writings of Hardy.This being the negative aspect of the book,here come the +ve aspects.(bcoz u shdnt ask me "why 4 stars if u find it not to ur expectations?")

Snow is also a great mathematician of Hardy times.The foreword gives us a glimpse of the author ,his humility,his passions for sports especially cricket etc.Now coming to the topic,Hardy gives us a clear idea of who a Mathematician is ,one who generates patterns/ideas.And how do you distinguish betweeen Pure and Applied Mathematics which vary just by the very marginal fact that when u start applying ur mathematics its no more pure and the so called applied scientists cannot boast of the fact that *unlike pure mathematicians they are doing something to the society*, bcoz if there is no theory how can anything be formulated and understood to be useful?Here comes a little contradiction to the title.While Hardy himself conveys the above fact,he apologises that he has not done anything useful but just added to *knowledge*.And u and I will surely look at him wondering and say "Hey!U have added to Knowledge.Wow!Thats really Great";).Such was the humility of this Great mathematician .

More to it comes if you read the book.A really inspiring one.

Rating:
Summary: A Classic of Exposition
Review: This short book with a biographical sketch by C. P. Snow is a superbly written explanation of what mathematicians
do and how they think. Hardy was a fascinating man then and he remains so today. His prose style is marvelous
and lucid. I have reread this book many times since I first discovered it in the 1960's. His essay is somewhat
melancholic in tone because he knew as he wrote this book that his fertile creative period was essentially ended.
As he remarks 'mathematics is a young man's game.' The biographical sketch by C. P. Snow is very illuminating
also in its portrayel of Hardy as a somewhat eccentric yet brilliant indivdual.

Rating:
Summary: One should not need an apology
Review: What is discomforting to me about this book is simply that Hardy decided to write it. Hardy describes that people do what they do because they do it well (the so called standard apology), that they do it because they don't do anything else well, or they didn't have a chance to do anything else. Hardy also exclaims that pure mathematics has no utilitarian value, and does not benefit society.

But why can't mathematicians study math for the love of the subject? What is so bad about not caring about utilitarian value? Why should we have to justify our existence to others? The fact that Hardy seems so compelled to justify his existence, and all he comes up with is benefiting a pool of knowledge in the platonic realm, is almost pathetic. Why couldn't he have studied math for his own self-interest? This book would have been all the more refreshing if he stated he loved working with Ramanujan and Littlewood, and that this in itself is a justification, and not some means to some cloudy end.

Other areas of the book are equally disappointing. We hear the commonplace notion that after 40, one's mathematical abilities are pretty much over. Yet this is a self-fulfilling prophecy. Without confidence in your abilities, how do you expect to get anywhere? What about Erdos, who still actively did math up until his death? Wiles was over 40 when he finally resolved Fermat's Last theorem.

You may wonder with this criticism, why I chose 4 stars. This is because, for all its drawbacks, it is at least an interesting account of Hardy's relationship with math. It is however, disheartening how fatalistic he is. It would have been refreshing to hear something of pride, not pretentious or sneering, but pride exclaiming that nobody should ever feel the need to write an apology for their existence.