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A Mathematician's Apology ()

Modern Library Top 100 Non-Fiction Books of the 20th Century (87)

    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 (see Orrin's review) of the 20th Century.


Wingnut's review:

In G.H. Hardy's book Mathematician's Apology, he distinguishes between the "real" math of mathematicians and applicable math. Real mathematics is a rarefied field in which few humans care about and even fewer can comprehend. It deals with sets of imaginary numbers, huge primes and limits as series approach infinity. Heavy stuff:

The mass of mathematical truth is obvious and imposing; its practical applications, the bridges and steam engines and dynamos, obtrude themselves on the dullest imagination. The public does not need to be convinced that there is something in mathematics.  All this is in its way very comforting to mathematicians, but it is hardly possible for a genuine mathematician to be content with it. Any genuine mathematician must feel that it is not on these crude achievements that the real case for mathematics rests, that the popular reputation of mathematics is based largely on ignorance and confusion.

When I say to people that I majored in math at Colgate, I get looks of ignorance and confusion, also. Why? I didn't do particularly well, probably hit a wall of knowledge somewhere along the line. The math that I studied only touched on the "real" math that Hardy describes as his life work. Why are people so scared of it? Because it is the fear of the unknown, possibly combined with the fear of failure. I would like to defend his career, though. I think that real math is justified in that there is always the possibility that solutions could lead to a great discovery in other fields, something as yet undiscovered in the realms of real math will benefit us, the ignorant masses. In Hardy's bio (listed as a Brother's Judd link:,5716,40055+1,00.html) it states that one of his solutions (the Hardy-Weinberg Law) helped in genetic research. So why the need for an apology?

This is not so much an apology as an attempt to justify his own existence. It is stated in the over-long pre-ramble that he is an atheist and it is implied not so subtly that he is a homosexual, thus facing the prospect of no descendants. He feels the need to leave something of lasting value. He feels the need to give some meaning to his existence (could it also be a suicide note?) Hardy tries to justify math's "beauty" through comparisons to chess and poetry. He also tries to convince us of its seriousness as a discipline. OK, I'm buying. Mathematics is a worthwhile endeavor. But what's the point. In his conclusion he finally gets to the point:

    The case for my life, then, or for that of any one else who has been a mathematician in the same
    sense in which I have been one, it this: that I have added something to knowledge, and helped
    others to add more; and that these somethings have a value which differs in degree only, and not in
    kind, from that of the creations of the great mathematicians, or of any of the other artists, great or
    small, who have left some kind of memorial behind them.

The end of the book reveals the true nature of his apology, he is trying to justify his existence by
saying that he added to the general core of "knowledge". Well big woop. After swimming through all
the whining and name dropping, I'd much rather hear about his achievements and inspirations (both
creative and external) than what a failure he thought he was. For a much better account of creativity
in mathematics, historical intrigue and the heights people can achieve in a certain field, I highly
recommend Fermat's Enigma.


Grade: (C+)


Book-related and General Links:
    -ENCYCLOPÆDIA BRITANNICA: Hardy, Godfrey Harold
    -BIO: Godfrey Harold Hardy (born February 7, 1877; died December 1, 1947)
    -DISCUSSION GROUP: G.H. Hardy Lecture Hall (Moby Dick)
    -ESSAY: Can Science Be Ethical? (FREEMAN DYSON, NY Review of Books)
    -REVIEW: Ian Stewart: In The Jungle of the Infinite, NY Review of Books
        The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel
        Acquaintances by Arnold J. Toynbee
        Variety of Men by C.P. Snow
    -REVIEW: Rudolf Peierls: One Culture, NY Review of Books
        The Physicists by C.P. Snow
    -SPEECH: The Perils of Popularizing Science Writing Presented by Robert Kanigel (Alfred and Julia Hill Lecture on Science, Society and the Media The University of Tennessee, Knoxville on April 7, 1999)
    -REVIEW: of ZERO The Biography of a Dangerous Idea By Charles Seife (Christopher Lehmann-Haupt, NY Times)
    -REVIEW: of FERMAT'S ENIGMA The Quest to Solve the World's Greatest Mathematical Problem. By Simon Singh (Roger Penrose, NY Times Book Review)
    -REVIEW: of ARCHIMEDES' REVENGE The Joys and Perils of Mathematics. By Paul Hoffman (Jim Holt, NY Times Book Review)
    -REVIEW: of A BEAUTIFUL MIND By Sylvia Nasar (David Goodstein, NY Times Book Review)
    -REVIEW: of A MATHEMATICIAN READS THE NEWSPAPER By John Allen Paulos (Richard Bernstein, NY Times)
    -REVIEW: of THE MAN WHO LOVED ONLY NUMBERS The Story of Paul Erdos and the Search for Mathematical Truth. By Paul Hoffman (James Alexander, NY Times Book Review)
    -ESSAY: PHYSICS AND FICTION: ORDER FROM CHAOS (John Banville, NY Times Book Review)
    -ESSAY: Key concepts: the science of secrecy (London Review of Books, Brian Rotman)


I think that this is kinda interesting but at the same time gay.

- samantha

- Dec-08-2003, 09:29


I think that this is kinda interesting but at the same time gay.

- samantha

- Dec-08-2003, 09:28