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A Mathematician's Apology (Canto Classics) (Englisch) Taschenbuch – 29. März 2012

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'A classic tale of retrospection considering wider, powerful themes in mathematics.' Mathematics Today

'Hardy provides an amazing insight into the mind of the mathematician.' Marcus du Sautoy, The Week

'Hardy provides an amazing insight into the mind of the mathematician.' Marcus du Sautoy, Waitrose Weekend Magazine

Über das Produkt

G. H. Hardy was one of this century's finest mathematical thinkers, renowned among his contemporaries as a 'real mathematician … the purest of the pure'. This is a unique account of the fascination of mathematics and of one of its most compelling exponents in modern times.

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Format: Taschenbuch Verifizierter Kauf
G. Hardy war einer der grössten Zahlentheoretiker des letzten Jahrhunderts, im Hiblick auf seine Philosophie und seine Kompetenz gewissermassen ein Grandseigneur der Mathematik.

Sein "Klassiker" zur Rechtfertigung mathematischer Forschung versucht keineswegs, die Nützlichkeit der Mathematik hervorzukehren, im Gegenteil:Die "wertvolle" Mathematik ist in seinen Augen eine Kunst, die keinen Nutzen mit sich bringt. Die Rechtfertigung dafür, sich mit Mathematik zu befassen, besteht darin, dass man dieser Aufgabe gewachsen ist.

Offenbar lässt sich dieses Argument auf viele ungwöhnliche Aktivitäten und viele erstaunliche Leistungen verallgemeinern.
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Die hilfreichsten Kundenrezensionen auf Amazon.com (beta)

Amazon.com: 3.8 von 5 Sternen 22 Rezensionen
10 von 10 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen "A passionate lament for creative powers that used to be and that will never come again" 31. August 2013
Von Jordan Bell - Veröffentlicht auf Amazon.com
Format: Taschenbuch
I strongly disagree with Hardy's opening statement that "there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain." It is indeed easier to write uninspired but passable explanation or criticism than it is to write uninspired but passable mathematics, and excellent exposition is also easier to do than excellent mathematics. But good exposition is both more useful for the discipline of mathematics and harder to do than uninspired but passable mathematics.

Although I disagree with what Hardy thinks is important and also his belief that people who are able to be excellent in one field are unlikely to be able to be excellent in another field, he expounds his ideas clearly and beautifully and I was glad to read them. And Hardy has made a serious though brief attempt at making precise ideas mathematicians have about what important and good mathematics is, which are usually avoided by saying that one knows quality when one sees it. In section 11, Hardy says "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." In sections 15-17 he talks about "generality" and "depth" of mathematical ideas. In section 18 he presents three qualities of good mathematics: unexpectedness, inevitability, and economy.

Hardy makes a point that I don't think I've read before and with which I strongly agree: "the most 'useful' subjects are quite commonly just those which it is most useless for most of us to learn." (For example, there is huge social benefit from having good sewers, but sewer design is probably not something that should be a mandatory course to get a high school diploma.) This in fact leads to a good argument about why pure mathematics, about prime numbers and irrationals for example, is more useful to teach than applied mathematics.

Someone wanting to know about Hardy's mathematical development would need more than this book. (It's been a while since I've read it, but Kanigel's biography of Ramanujan is probably the next place to look for biographical information about Hardy.) When did Hardy first learn a proof? What material did he learn at Cranleigh and Winchester and then at Cambridge? Snow just tells us that at Winchester he was in a class of one, but not what that class covered. I would like to know what material he had worked through before reading Jordan's Cours d'analyse; once he starts to read the Cours d'analyse he can properly called a mathematician, although not yet a productive researcher. The historian who wants to know about Hardy's Cambridge studies at least is told when he took the Tripos if they want to know the specific material that he mastered before becoming a researcher.
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3.0 von 5 Sternen False dichotomy but worth reading 6. Dezember 2014
Von Ed Battistella - Veröffentlicht auf Amazon.com
Format: Taschenbuch
A friend loaned me a copy of G. H. Hardy’s “A Mathematcian;s Apology” which focuses on the creative aspect of mathematical proofs. After a 60-page biographical essay/forward by C. P. Snow, the bulk of the slim volume (page 61-151) is Hardy’s ruminations on the creative aesthetic of mathematics—an apology in the older sense of a defense of a choice (i.e., the apology of Socrates). It’s hard not to be charmed by a book that begins with the (ironic?) line that “Exposition, criticisms, appreciation is work for second-rate minds,” dissing not but readers but himself as well. Feeling past his prime and concerned about the military uses of applied math, Hardy sets out to defend pure math versus applied. It’s a false dichotomy, I think, (and makes me think of C. P. Snow’s later two cultures false dichotomy), but Hardy’s apology turns out to be a discourse on the idea of being committed to a discipline—math, linguistics, criticism, physics, anthropology, you name it (or as one of my old professors put it, the value of having a perspective and theory with which to analyze the world). There is some math (Pythagoras’s proof of the irrationality of the square root of 2, etc.), but not enough to give anyone a headache. For me though, the real value of Hardy’s essay is that the seriousness of intellectual work lies is the ideas that it connects, not in immediate applications, making it wortha read by any academic.
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5.0 von 5 Sternen Beautiful 18. Juni 2012
Von UL - Veröffentlicht auf Amazon.com
Format: Taschenbuch
This is one of the best books I have read in a while. The foreword by C.P. Snow was a perfect start. Hardy has done a marvellous job of explaning his point of view about Mathematics, the impact the subject has on every day life and his own reasons to get into the study of this very challanging subject. One could say that Hardy was a little pompous and self-congratulatory but if what he says is read in context, the reader will realize that, as Snow points out in the foreword, Hardy had a broken heart when he was writing this book. He was at the end of his creative 'boom' and was wary of the fact that he was no longer the same agile mind that was needed to perform real mathematical miracles.
Brilliant book and one that should be read by everyone ..... regardless of how they feel about mathematics because more than about math, this book is about how life is the great equalizer and how we will all one day be humbled by our aging neurons and weakening muscles .... regardless of how much prodigous talent we may possess.
1 von 1 Kunden fanden die folgende Rezension hilfreich
3.0 von 5 Sternen Three stars 15. Mai 2016
Von Prof Dr Dr - Veröffentlicht auf Amazon.com
Format: Kindle Edition Verifizierter Kauf
The book has its moments -- the chapter in which Hardy leads us through a couple of mathematical proofs is very interesting, and the autobiographical chapter at the end is actually quite moving -- but it was written decades ago and it unfortunately doesn't feel timeless.
1.0 von 5 Sternen Mathematical Arrogance 9. August 2016
Von H. J. Spencer PhD, renegade-Physicist - Veröffentlicht auf Amazon.com
Format: Taschenbuch Verifizierter Kauf
There is a deep mystery about this book – it has been widely praised and remained in print for nearly 80 years over 22 editions since its publication in 1940. The question is why? An answer will be attempted at the end of this review, especially as it is not well organized and meanders across many topics.

It has no clear structure or direction, bouncing lightly over many tough ideas while promising an uninformative autobiography. Fortunately, the esteemed novelist, C. P. Snow wrote a 60 page foreword that was added to the 4th edition in 1967. Snow wrote a sympathetic synopsis of the life of his friend Godfrey Harold, so we can see some of his better aspects of his character, like their mutual obsession with cricket and the privileged lives of Cambridge university ‘dons’. Otherwise, there would be almost no biographical details but we would be left with the impression of a very proud man (“good work is not done by humble men” [p.66]), who had few interests outside his professional research. Hardy was always a very conventional member of the English elite, defending ambition as a worthy motivation. Cambridge was dedicated to educating the best of the next generation of the ruling class, especially in science and mathematics. He wrote an “apologia” because he was obsessed with the question: “Is mathematics worth doing?” Hardy surprised himself by having to defend mathematics as he admits it is “generally recognized as profitable and praiseworthy” but acknowledges that he is really defending himself and his obsessive dedication, with only stellar astronomy and atomic physics, as sciences standing in higher esteem in the popular estimation.

Much of the book is dedicated to convincing the non-mathematician that there are (at least in Hardy’s eyes) two types of mathematics: Pure (or ‘real’) mathematics and all the rest! Only a few real mathematicians can truly appreciate (or even recognize) pure mathematics. Most educated people will only learn “trivial” or simple ‘school’ mathematics and even those whose technical careers depend on mathematics (such as engineers and physicists) will only learn ‘applied’ mathematics, such as integral and differential calculus (but these topics are both “dull and lacking in aesthetic appeal”). Although admitting that a definition is difficult, he later lists modern geometry and algebra, number theory and the theory of functions as good examples of pure mathematics. This is confirmed in his eyes by declaring them “useless” – a critical feature for him, as such topics are usually ‘harmless’ (Hardy was a notorious pacifist along with Bertrand Russell). He believes that ‘real’ mathematics is “serious” because its theorems play significant roles in other, major mathematical areas. The book contains two sections where ‘real’ mathematical theorems are discussed: these include Euclid’s proof of the infinitude of prime numbers and Pythagoras’s proof that the square root of two cannot be expressed as a rational fraction (e.g. n/m); in other words, the shattering realization that there are quantities that are not related to integers (“counting numbers”). This fascination with prime numbers (not divisible by any other number) lies at the heart of Number Theory, which leads to the ‘fundamental’ theorem of arithmetic, where every integer number can be expressed as a unique multiplication of prime numbers. Most people visualize non-prime numbers as rectangular arrangements of rows of equal numbers of objects. Hardy makes no mention of Descartes’ radical invention of “real” numbers (like 2.7134) that allowed all physical quantities to be assigned to such numbers; vital to all modern physics with its arbitrary units of measure.

Perhaps, the reason for this omission was that Hardy viewed physics as too closely linked to material reality; in fact, he makes the extreme claim that mathematicians are much more in direct contact with reality (they call it “mathematical reality”) than physicists (ignoring the key role of experimental physicists that make physics an empirical science). It is convenient for him that he counts famous physicists, such as Maxwell, Einstein, Eddington and Dirac as “real” mathematicians. He is more impressed with the properties of specific numbers, like the ‘fact’ that ‘317’ is a prime (whereas most people would respond: “So what?”). He is proud to be associated with Plato’s views of ‘deep’ reality; not surprisingly, as Plato was also a fanatic follower of the ancient religious mystic, Pythagoras. In fact, Catholic intellectuals have also pointed to mathematics as perfect examples of their own timeless world created by their God (ironically, Hardy was a harsh atheist). All these intellectual mystics deny that mathematics (and theology) were just mental constructions but lie objectively outside of all people; failing to see that artists too do not ‘discover’ their original creations but reflect the hard work of their communicable imaginations, reflecting socially evolving ideas. Most mathematicians allude to the generality of arithmetic, like 2+3=5. Once again, they fail to think deeply about their basic ideas, such that integers are the common results of the physical act of counting stable, distinguishable existents. In fact, a small number of mathematicians (the “constructivists”) believe that mathematics is just an extended intellectual analysis of abstract definitions; this certainly covers much of Euclidean geometry, which has become the classic example of what constitutes a logical proof.

The true motivation for this book was Hardy’s psychological crisis brought on by his awareness that his creative abilities had deserted him. At first, he tried to commit suicide but took too many sleeping pills; he reconciled himself with the view that “mathematics is a young man’s game” and he was still quite creative in his forties. Although Hardy recognized that pure mathematics was much like great art “in promoting a lofty habit mind”, he believed that it was superior due to its greater demonstrable ability to endure through many centuries and civilizations. Like many well-educated intellectuals, he was a snob; indeed, he admits that he is only interested in mathematics as a “creative art”, so that pure mathematics does not have to be ‘useful’ (i.e. “increase humanity’s well-being or comfort”).

So, why does this book keep finding readers? In my own case, it was to learn more about his famous relationship with the Indian mathematical genius, Ramanujan after seeing the movie “The Man who knew Infinity”. My own theory here is that generations of mathematics teachers and professors recommend it to their better students, hoping it will help commit them to this ancient profession. In fact, the abstract symbolism and definitions of mathematics have helped make it the ideal subject for secondary education across all societies. Mathematics is harmless and useful and easy to teach, and easy to mark: giving an ‘objective’ measure of the student’s intelligence (so many believe – but like most educational subjects: a good memory is important). In fact, as a manmade invention (I am an Aristotelian, not a Platonist) the finitude of mathematics actually makes it much easier (to learn and remember) than science, which is challenged by the complexity and open-ended characteristics of Nature. Indeed, the timeless nature of mathematics both preserves its structures and appeals to many intellectuals, who (like Plato) were threatened by the inevitability of their personal existence.
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