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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 Proposition s 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 Function s, called Calculus at the 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 , particular 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.k NOTABLE PROBLEMS IN PURE MATHEMATICS The logician Kurt Gödel , created a proof, called Gödel's Incompleteness Theorems , which proves that every logical system contains a premise which it cannot define without contradicting itself. PURISM A ''purist'' attitude to mathematics is nothing new. It 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''. QUOTES
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