Pure mathematics

Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. From the eighteenth century onwards, this was a recognised category of mathematical activity, sometimes characterised as speculative mathematics, and at variance with the trend towards meeting the needs of navigation, astronomy, physics, engineering, and so on.

History

19th century

The term itself is enshrined in the full title of the Sadleirian Chair, founded (as a professorship) in the mid-nineteenth century. The idea of a separate discipline of pure mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of the kind, between pure and applied. In the following years, specialisation and professionalisation (particularly in the Weierstrass approach to mathematical analysis) started to make a rift more apparent.

20th century

At the start of the twentieth century mathematicians took up the axiomatic method, strongly influenced by David Hilbert's example. The logical formulation of pure mathematics suggested by Bertrand Russell in terms of a quantifier structure of propositions seemed more and more plausible, as large parts of mathematics became axiomatised and thus subject to the simple criteria of rigorous proof.

In fact in an axiomatic setting rigorous adds nothing to the idea of proof. Pure mathematics, according to a view that continued to and through the Bourbaki group, is what is proved. Pure mathematician began to be a recognisable vocation, with access through a training.

Generality and abstraction

Geometry has expanded to accommodate topology. The study of number, called algebra at the beginning undergraduate level, extends to abstract algebra at a more advanced level; and the study of functions, called calculus at the college freshman level becomes mathematical analysis and functional analysis at a more advanced level. Each of these branches of more abstract mathematics have many sub-specialties, and there are in fact many connections between pure mathematics and applied mathematics disciplines. Undeniably, though, a steep rise in abstraction was seen mid-century.

In practice, however, these developments led to a sharp divergence from physics, particularly from 1950 to 1980. Later this was criticised, for example by Vladimir Arnold, as too much Hilbert, not enough Poincaré. The point does not yet seem to be settled (unlike the foundational controversies over set theory), in that string theory pulls one way, while discrete mathematics pulls back towards proof as central.

Purism

Mathematicians have always had differing opinions regarding the distinction between pure and applied mathematics. One of the clearest modern examples of the debate between the subfields can be found in G.H. Hardy's A Mathematician's Apology, in which the author (a number theorist) attacks applied mathematics, calling it "ugly" and "dull", while comparing pure mathematics to painting and poetry. Hardy's love of number theory's apparent uselessness would be later proven unjustified by the development of public key cryptography and other advanced cryptographic methods which relied heavily on the field.

Quotes

"There is no branch of mathematics, however abstract, which may not someday be applied to the phenomena of the real world."
- Nikolai Lobachevsky

External links

Mathematical terminology

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