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| illustrator =
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| country = United Kingdom
| language = English
| series =
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| subjects = [[philosophyPhilosophy of mathematics]], [[mathematical beauty]]
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| isbn = 9781107295599
| isbn_note = (2012 reprinting)
| oclc = 488849413
| dewey = 510
| congress = QA7.H3
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'''''A Mathematician's Apology''''' is a 1940 essay by British mathematician [[G. H. Hardy]], which offersdefends athe defencepursuit of themathematics pursuitfor ofits mathematicsown sake. Central to Hardy's "[[Apologetics|apology]]" – in the sense of a formal justification or defence (as in [[Plato]]'s ''[[Apology (Plato)|Apology of Socrates]]'') – is an argument that mathematics has value independent of possibleits applications. Hardy located this value in what he called the [[Mathematical beauty|beauty of mathematics]], and gave some examples of and criteria for mathematical beauty. The book also includes a brief autobiography, andwhich gives the layman an insight into the mind of a working [[mathematician]].
 
==Background==
[[File:Ghhardy@72.jpg|thumb|left|In ''A Mathematician's Apology'', [[G. H. Hardy]] defined a set of criteria for mathematical beauty.]]
 
Hardy felt the needwished to justify his life's work in mathematics at this time mainly for two reasons. Firstly, athaving agesurvived 62,a Hardyheart feltattack theand approachbeing ofat oldthe age (heof had62, survivedHardy aknew heartthat attackhe inwas 1939)approaching andold theage declineand ofthat his mathematical creativity and skills were declining.
By devoting time to writing the Apology, Hardy was admitting that his own time as a creative mathematician was finished. In his foreword to the 1967 edition of the book, [[C. P. Snow]] describes the Apology as
"a passionate lament for creative powers that used to be and that will never come again".<ref name="snow67">{{cite book|last=Hardy |first=G. H. |title=A Mathematician's Apology |publisher=[[Cambridge University Press]] |year=1967 |contributor-last=Snow |contributor-first=C. P. |contribution=Foreword }}</ref>{{rp|51}}
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Secondly, at the start of [[World War II]], Hardy, a committed [[pacifist]], wanted to justify his belief that mathematics should be pursued for its own sake rather than for the sake of its applications. He began writing on this subject when he was invited to contribute an article to ''Eureka'',<ref name="Apology">{{cite book|last=Hardy |first=G. H. |title=A Mathematician's Apology |publisher=[[Cambridge University Press]] |year=1940}}</ref>{{rp|Preface}} the journal of [[The Archimedeans]] (the Cambridge University student mathematical society). One of the topics the editor suggested was "something about mathematics and the war", and the result was the article "Mathematics in war-time".<ref name="Wartime">{{cite journal |last1=Hardy |first1=G. H. |date=January 1940 |title=Mathematics in war-time |journal=Eureka |volume=1 |issue=3 |pages=5–8}}</ref> Hardy later incorporated this article into ''A Mathematician's Apology''.<ref name="Apology" />{{rp|Preface}}
 
HeHardy wanted to write a book in which he would explain his mathematical philosophy to the next generation of mathematicians;. thatHe wouldhoped defendthat mathematicsin bythis elaboratingbook onhe thecould meritsinspire offuture puregenerations mathematics solely, without having to resort toabout the attainments of applied mathematics in order to justify the overall importance of mathematics; andwithout that would inspire the upcoming generations of pure mathematicians. Hardy was an [[atheist]], and makes his justification notappealing to [[God]]its but to his fellowapplied manuses.
 
Hardy initially submitted ''A Mathematician's Apology'' to [[Cambridge University Press]] with the intention of personally paying for its printing, but the Press decided to fund publication with an initial run of four thousand copies.<ref name="hardy-annotated-legacy">{{cite book |last=Hardy |first=G. H. |title=An Annotated Mathematician's Apology |year=2019 |url=https://archive.org/details/hardy_annotated |contributor-last=Cain |contributor-first=A. J. |contribution=Context of the ''Apology''}}</ref>{{rp|97}} For the 1940 1st edition, Hardy sent postcards to the publisher requesting that presentation copies be sent to his sister Gertrude Emily Hardy (1878–1963), [[C. D. Broad]], [[John Edensor Littlewood]], Sir [[Arthur Eddington]], [[C. P. Snow]], the cricketer [[John Lomas (cricketer)|John Lomas]] (to whom G. H. Hardy dedicated the book), and others.<ref>{{cite book|editor=Pitici, Mircea|chapter=''In defense of pure mathematics'' by Daniel S. Silver|title=The Best Writing on Mathematics 2016|pages=17–26|year=2017|publisher=Princeton University Press|chapter-url=https://books.google.com/books?id=RXGYDwAAQBAJ&pg=PA18}} (See page 18.)</ref>
 
==Summary==
 
One of the main themes of the book is the beauty that mathematics possesses, which Hardy compares to painting and poetry.<ref>{{cite book |last1=King |first1=Jerry P. |title=The Art of Mathematics |date=1992 |publisher=Fawcett Columbine |isbn=0-449-90835-6 |pages=135–139}}</ref> For Hardy, the most beautiful mathematics was that which had no practical applications in the outside world ([[pure mathematics]]) and, in particular, his own special field of [[number theory]], Hardy's own field. Hardy contends that if useful knowledge is defined as knowledge which is likely to contribute to the material comfort ofwithout mankindrespect in the near future (if not right now), so thatto mere intellectual satisfaction is irrelevant, then the great bulkmost of higher mathematics is useless. He justifies the pursuit of pure mathematics with the argument that its very "uselessness" on the whole meantmeans that it could notcannot be misused to cause harm. On the other hand, Hardy denigrates much of the [[applied mathematics]] as either being "trivial", "ugly", or "dull", and contrasts it with "real mathematics", which is how he ranks the higher,describes pure mathematics.
 
Hardy expounds by commentingcomments about a phrase attributed to [[Carl Friedrich Gauss]] that: "Mathematics is the queen of the sciences and number theory is the queen of mathematics." SomeOne peoplemay believe that it is the extremerelative non-applicabilitysparseness of number theory in applied mathematics that led Gauss to the above statement about number theory; however, Hardy points out that this is certainly not the reasoncase. If an application of number theory were to be found, then certainly no one would try to dethrone the "queen of mathematics" becauseby of thatit. What [[Carl Friedrich Gauss|Gauss]] meant, according to Hardy, is that the underlying concepts that constitute number theory are deeper and more elegant compared to those of any other branch of mathematics.
 
Another theme is that mathematics is a "young man's game",. soHardy believed that anyone with a talent for mathematics should develop and use that talent while they are young, before their ability to create original mathematics starts to decline in middle age. This view reflects Hardy's increasing depression at the wanewaning of his own mathematical powersskill. For Hardy, real mathematics was essentially a creative activity, rather than an explanatory or expository one.
 
==Critiques==
Hardy's opinions were heavily influenced by the [[academia|academic]] culture of the universities of [[Cambridge]] and [[Oxford]] between [[World War I]] and [[World War II]].
 
Some of Hardy's examples seem unfortunate in retrospect. For example, he writes, "No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years." Since then number theory was used to crack German [[Enigma machine|Enigma codes]], and much later, figurefigured prominently in [[public-key cryptography]].<ref>{{cite web|url=http://wayback.cecm.sfu.ca/personal/jborwein/apaper.pdf|title=Experimental mathematician Jonathan Borwein's comments on the Apology}}</ref>; furthermore, the inter-convertability of mass and energy predicted by [[special relativity]] forms the physical basis for nuclear weapons.
 
TheApplicability applicability of a mathematical conceptitself is not the reason that Hardy considered applied mathematics somehow inferior to pure mathematics, though; it is the simplicity and prosinessvulgarity that belong to applied mathematics that led him to describe themit as he did. He considersconsidered that [[Rolle's theorem]], for example, cannot be compared to the elegance and preeminence of the mathematics produced by [[Évariste Galois]] and other pure mathematicians, although it is of some importance for [[calculus]].
 
==Notes==
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{{DEFAULTSORT:Mathematicians Apology, A}}
[[Category:1940 essays]]
[[Category:1940 non-fiction books]]
[[Category:Biographies and autobiographies of mathematicians]]
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