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.

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.

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.

In practice this led to a sharp divergence from physics. 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.

Of course a purist attitude to mathematics goes right back to Plato. The question is now more about the roots of mathematical progress — whether they are internal and generated by problem-solving suggested by the shape of the subject itself, or external.

Quotations

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

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