Pure mathematics
Pure mathematics is broadly speaking mathematics motivated entirely for reasons other than applications. 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 obvious.
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 quantifier structure of propositions could seem more and more plausible, as large parts of mathematics became axiomatised and so subject to 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 has been criticised, for example by Arnol'd, as too much Hilbert, not enough PoincarÃÂé. The point seems not yet 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.
See also: applied mathematics