A Mathematician's Apology is a 1940 essay by British mathematician G. H. Hardy. It concerns the aesthetics of mathematics with some personal content, and gives the layman an insight into the mind of a working mathematician. It is, however, a very individual view, and Hardy's opinions are not universally held by mathematicians. He is reguarded as one of the greatest mathematicians of his time G. H. Hardy Professor Godfrey Harold Hardy FRS (February 7, 1877 – December 1, 1947) was a prominent English mathematician, known for his achievements in number theory and mathematical analysis. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ...

Summary

In the book's title, Hardy uses the word "apology" in the sense of a formal justification or defense (as in Plato's Apology of Socrates), not in the sense of a plea for forgiveness. There were two main reasons why Hardy felt the need to justify his life's work in mathematics at this time. Firstly, at age 62, Hardy felt the approach of old age (he had survived a heart attack in 1939) and the decline of his mathematical creativity. He wanted to explain his mathematical philosophy to the next generation of mathematicians. Secondly, at the start of the Second World War, Hardy, who was 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. As Hardy was an atheist, he makes his justification not to a god but to his fellow man. This article or section does not cite any references or sources. ... PLATO was one of the first generalized Computer assisted instruction systems, originally built by the University of Illinois (U of I) and later taken over by Control Data Corporation (CDC), who provided the machines it ran on. ... (The) Apology (of Socrates) is Platos version of the speech given by Socrates as he defends himself against the charges of being a man who corrupted the young, did not believe in the gods, and created new deities. Apology here has its earlier meaning (now usually expressed by the... Mushroom cloud from the nuclear explosion over Nagasaki rising 18 km into the air. ... Pacifist may mean: an advocate of pacifism. ... For information about the band, see Atheist (band). ...

One of the main themes of the book is the beauty of mathematics, which Hardy compares to painting and poetry. For Hardy, the most beautiful mathematics was that which had no applications in the outside world, by which he meant pure mathematics, and, in particular, his own special field of number theory. He justifies the pursuit of pure mathematics with the argument that its very "uselessness" meant that it could not be misused to cause harm. On the other hand, Hardy denigrates applied mathematics, describing it as "ugly", "trivial" and "dull". Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. ...

These characterizations concerning applied mathematics mean that it is not the fact that they are applied that makes them ugly, trivial or dull but it is because more often the most ugly, trivial and dull mathematics are usually finding application. These characterizations are attributed or not to certain branches of mathematics in accordance to the originality, depth and beauty of the underlying concepts that constitute the foundations of these branches as defined by G. H. Hardy. This is further stressed by Hardy in his comments about a phrase attributed to C. F. Gauss that "Mathematics is the queen of the sciences and number theory is the queen of mathematics". Some people believe that it is the extreme non-applicability of number theory that led Gauss to the above statement about number theory; however, Hardy points out that this is certainly not the reason. If an application of number theory were to be found then certainly no one would try to dethrone the queen of mathematics because of that. What C. F. 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. Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. ... G. H. Hardy Professor Godfrey Harold Hardy FRS (February 7, 1877 – December 1, 1947) was a prominent English mathematician, known for his achievements in number theory and mathematical analysis. ... Johann Carl Friedrich Gauss Johann Carl Friedrich Gauss (Gauß) (April 30, 1777 _ February 23, 1855) was a legendary German mathematician, astronomer and physicist with a very wide range of contributions; he is considered to be one of the greatest mathematicians of all time. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ...

Another theme is that mathematics is a "young man's game", so 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 wane of his own mathematical powers. For Hardy, real mathematics was essentially a creative activity, rather than an explanatory or expository one.

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". Charles Percy Snow, Baron Snow, CBE (15 October 1905–1 July 1980) was a scientist and novelist. ...

Critiques

Hardy's opinions were heavily influenced by the academic culture of the universities of Cambridge and Oxford between World War I and World War II. His assumption that only the very best original work in any field has any lasting value can sound elitist to the modern ear. Academia is a collective term for the scientific and cultural community engaged in higher education and research, taken as a whole. ... Geography Status City (1951) Region East of England Admin. ... Oxford is a city and local government district in Oxfordshire, England, with a population of 134,248 (2001 census). ... “The Great War ” redirects here. ... Combatants Allied powers: China France Great Britain Soviet Union United States and others Axis powers: Germany Italy Japan and others Commanders Chiang Kai-shek Charles de Gaulle Winston Churchill Joseph Stalin Franklin Roosevelt Adolf Hitler Benito Mussolini Hideki Tōjō Casualties Military dead: 17,000,000 Civilian dead: 33,000... Elitism is a belief or attitude that an elite — a selected group of persons whose personal abilities, specialized training or other attributes place them at the top of any field (see below) — are the people whose views on a matter are to be taken most seriously, or who are alone...

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, the application of relativity was part of the development of nuclear weapons, while number theory figures prominently in public-key cryptography.^[1] However, Hardy's more prominent examples of elegant mathematical discoveries with no use (proofs of the infinity of primes and the irrationality of the square root of two) still hold up. The mushroom cloud of the atomic bombing of Nagasaki, Japan, 1945, rose some 18 km (11 mi) above the epicenter. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... A big random number is used to make a public-key pair. ...

It has to be noted though that the applicability of a mathematical concept is not the reason that Hardy considered applied mathematics somehow inferior to pure mathematics; it is the simplicity and prosiness that belongs to applied mathematics that led him to describe them as he did. Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. ...

He considers that Rolle's theorem for example, though it is of some importance for calculus, cannot be compared to the elegance and preeminence of the mathematics produced by Leonhard Euler or Évariste Galois and other pure mathematicians. In calculus, Rolles theorem states that if a function f is continuous on a closed interval and differentiable on the open interval , and then there is some number c in the open interval such that . Intuitively, this means that if a smooth curve is equal at two points then... Calculus (from Latin, pebble or little stone) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... Galois at the age of fifteen from the pencil of a classmate. ...

Notes

1. ^ Experimental mathemetician Jonathan Borwein's comments on the Apology

References

• G. H. Hardy, A Mathematician's Apology, Cambridge University Press (1940). 153 pages. ISBN 0-521-42706-1.

G. H. Hardy Professor Godfrey Harold Hardy FRS (February 7, 1877 – December 1, 1947) was a prominent English mathematician, known for his achievements in number theory and mathematical analysis. ...

External links

• Full text of 'A Mathematician's Apology', in the public domain in Canada, courtesy of the University of Alberta Mathematical Science Society.