A Mathematician's Apology: Difference between revisions

'''''A Mathematician's Apology''''' is a 1940 essay by British mathematician [[G. H. Hardy]] which defends the pursuit of mathematics for its own sake. Central Itto concernsHardy'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 its applications. Hardy located this value in what he called the [[Mathematical beauty|aestheticsbeauty of mathematics]] and withgave some personalexamples content,of and givescriteria thefor laymanmathematical anbeauty. The book also includes a brief autobiography which gives insight into the mind of a working [[mathematician]].

==SummaryBackground==

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.

"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}}

In Hardy's words, "Exposition, criticism, appreciation, is work for second-rate minds. [...] It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done."<ref name="Apology">{{cite book|last=Hardy |first=G. H. |title=A Mathematician's Apology |publisher=[[Cambridge University Press]] |year=1940}}</ref>{{rp|§1}}

Secondly, at the start of the [[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 wantedbegan towriting writeon athis booksubject in whichwhen he wouldwas explaininvited histo mathematicalcontribute philosophyan article to the''Eureka'',<ref nextname="Apology">{{cite generationbook|last=Hardy of|first=G. mathematicians;H. that|title=A wouldMathematician's defendApology mathematics|publisher=[[Cambridge byUniversity elaboratingPress]] on|year=1940}}</ref>{{rp|Preface}} the meritsjournal of pure[[The mathematicsArchimedeans]] solely,(the withoutCambridge havingUniversity tostudent resortmathematical tosociety). the attainmentsOne of appliedthe mathematicstopics inthe ordereditor tosuggested justifywas the"something overallabout importancemathematics ofand mathematics;the war", and thatthe wouldresult inspirewas the upcomingarticle generations"Mathematics of purein mathematicianswar-time".<ref name="Wartime">{{cite journal |last1=Hardy was|first1=G. anH. [[atheist]],|date=January and1940 makes|title=Mathematics hisin justificationwar-time not|journal=Eureka to|volume=1 [[God]]|issue=3 but|pages=5–8}}</ref> toHardy hislater fellowincorporated manthis article into ''A Mathematician's Apology''.<ref name="Apology" />{{rp|Preface}}

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.

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[[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|publisher=}}</ref>; Howeverfurthermore, Hardy'sthe more prominent examplesinter-convertability of elegantmass mathematicaland discoveriesenergy withpredicted no use (proofs ofby [[Euclid's theorem|the infinitude ofspecial primesrelativity]] andforms of [[Square root of 2|the irrationalityphysical ofbasis thefor squarenuclear root of two]]) still hold up {{Citation needed|date=February 2019}}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]].

* {{cite book |last=Hardy |first=G. H. |authorlink=G. H. Hardy |title=A Mathematician's Apology |publisherothers=With a foreword by [[C. P. Snow]]|publisher=Cambridge University Press]] |location=Cambridge |year=20042012 |origyearorig-year=1st pub. 1940, with foreword 1967 |isbn=978-0-521-42706-79781107295599 |url=http}} [https://www.cambridgearchive.org/cataloguedetails/catalogue.asp?isbn=9780521427067}}hardy_annotated/ Full text]

* {{cite journal |last1=Mordell, |first1=L. J. (1970)|author-link=Louis J. Mordell |title=Hardy's "A Mathematician's Apology". ''|journal=[[American Mathematical Monthly]]'', |date=1970 |volume=77, October, |issue=8, 831-836|pages=831–836 |doi=10.2307/2317017|jstor=2317017 }}

* {{cite journal |last1=Broad, |first1=C. D. (1941)|author-link1=C. D. Broad |title=Review of ''A Mathematician's Apology. '' |journal=[[Philosophy (journal)|Philosophy]]'', |date=1941 |volume=16, |issue=63, 323-326|pages=323–326 |doi=10.1017/s0031819100002655|s2cid=170901567 }}

* Full text of ''[httphttps://wwwarchive.math.ualberta.ca/~mssorg/miscdetails/A%20Mathematician's%20Apology.pdfhardy_annotated AAn Annotated Mathematician's Apology]'', courtesyan ofannotated edition including Hardy's essay ‘Mathematics in war-time’, with commentary on the [[Universitycontext ofand Alberta]]legacy Mathematicalof Sciencethe Society''Apology''.

[[Category:1940 booksessays]]

[[Category:Aesthetics literaturebooks]]