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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 propositionIn modern logic, a proposition or ansatz is what is asserted as the result of uttering a sentence. In other words, it is the meaning of the sentence, rather than the sentence itself. Different sentences can express the same proposition, if they have the ss 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 ArnoldVladimir Igorevich Arnold ( , born June 12, 1937 in Odessa, USSR) is one of the world's most prolific mathematicians in the field of mechanics. His career dates from Stalinist USSR, and he is still active at the turn of the 21st century. One of his earlie, as too much Hilbert, not enough PoincaréJules Henri Poincar ( April 29, 1854 July 17, 1912) was one of France's greatest mathematicians, theoretical scientists and a philosopher of science. Poincare is often described as the last "universalist" capable of understanding and contributing in virtu. The point does not yet seem to be settled (unlike the foundational controversies over set theorySet theory is the mathematical theory of sets, which represent collections of abstract objects. It has a central role in modern mathematical theory, providing the basic language in which most of mathematics is expressed. For more information on 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
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