| Pure Mathematics |
Article Index for Pure |
Website Links For Pure |
Information AboutPure Mathematics |
| CATEGORIES ABOUT PURE MATHEMATICS | |
| mathematical terminology | |
|
HISTORY Ancient Greece Ancient Greek mathematicians were among the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between "arithmetic", now called Number Theory and "logistic", now called Arithmetic . Plato regarded logistic as appropriate for business men and men of war who "must learn the art of numbers or he will not know how to array his troops," while arithmetic was appropriate for philosophers "because he has to arise out of the sea of change and lay hold of true being."1 Euclid Of Alexandria , when asked by one of his students of what use was the study of geometry, asked his slave to give the student threepence, "since he must needs make gain of what he learns."2 The Greek mathematician Apollonius Of Perga was asked about the usefulness of some of his theorem in Book IV of ''Conics'' to which he proudly asserted,3 They are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason. And since many of his results were not applicable to the science or engineering of his day, Apollonius further argued in the preface of the fifth book of ''Conics'' that "the subject is one of those which seems worthy of study for their own sake." 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 can be ascribed to the Bourbaki group, is what is proved. Pure mathematician became a recognized vocation, to be achieved through training. GENERALITY AND ABSTRACTION Geometry has expanded to accommodate Topology . The study of Number s, 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 college 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 20th century. In practice, however, these developments led to a sharp divergence from Physics , particularly 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. PURISM Mathematicians have always had differing opinions regarding the distinction between pure and applied mathematics. One of the most famous (but perhaps misunderstood) modern examples of this debate can be found in G.H. Hardy 's '' A Mathematician's Apology ''. It is widely believed that Hardy considered applied mathematics to be ugly and dull. Although it is true that Hardy preferred pure mathematics, which he often compared to Painting and Poetry , Hardy saw the distinction between pure and applied mathematics to be simply: that applied mathematics sought to express ''physical'' truth in a mathematical framework, whereas pure mathematics expressed truths that were independent of the physical world. Hardy made a separate distinction in mathematics between what he called "real" mathematics, "which has permanent aesthetic value", and "the dull and elementary parts of mathematics" that have practical use. Hardy considered some physicists, such as Einstein and Dirac , to be among the "real" mathematicians, but at the time that he was writing the ''Apology'' he also considered General Relativity and Quantum Mechanics to be "useless", which allowed him to hold the opinion that only "dull" mathematics was useful. Moreover, Hardy briefly admitted that--just as the application of matrix theory and group theory to physics had come unexpectedly--the time may come where some kinds of beautiful, "real" mathematics may be useful as well. SUBFIELDS IN PURE MATHEMATICS Analysis is concerned with the properties of functions. It deals with concepts such as Continuity , Limits , Differentiation and Integration , thus providing a rigorous foundation for the calculus of infinitesimals introduced by Newton and Leibniz in the 17th century. Real Analysis studies functions of real numbers, while Complex Analysis extends the aforementioned concepts to functions of complex numbers. on a set which contains an identity element and inverses for each member of the set, the set and operation is considered to be a Group . Other structures include Rings , Fields and Vector Spaces . Geometry is the study of shapes and space, in particular, groups of transformations that act on spaces. For example, Projective Geometry is about the group of projective transformations that act on the real projective plane, whereas Inversive Geometry is concerned with the group of inversive transformations acting on the extended complex plane. Geometry has been extended to Topology , which deals with objects known as topological spaces and continuous maps between them. Topology is more concerned with the way in which a space is connected than precise distances and angles. is easily stated (but is yet to be proved or disproved). In other ways it is the least accessible discipline; for example, Wiles' proof that Fermat's equation has no nontrivial solutions requires understanding automorphic forms, which though intrinsic to nature have not found a place in Physics or in public discourse. QUOTES
NOTES SEE ALSO EXTERNAL LINKS
|
|
|