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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 Arnol'd, 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.
See also: applied mathematics
External links
- What is Pure Mathematics? by Lis D'Alessio, University of
Waterloo
- What is Pure Mathematics? by Professor P.J. Giblin The University of
Liverpool
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