Pure mathematics

Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles.

While pure mathematics has existed as an activity since at least ancient Greece, the concept was elaborated upon around the year 1900,[2] after the introduction of theories with counter-intuitive properties (such as non-Euclidean geometries and Cantor's theory of infinite sets), and the discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable, and Russell's paradox).

A famous early example is Isaac Newton's demonstration that his law of universal gravitation implied that planets move in orbits that are conic sections, geometrical curves that had been studied in antiquity by Apollonius.

Another example is the problem of factoring large integers, which is the basis of the RSA cryptosystem, widely used to secure internet communications.

[4] Ancient Greek mathematicians were among the earliest to make a distinction between pure and applied mathematics.

Plato regarded logistic (arithmetic) as appropriate for businessmen and men of war who "must learn the art of numbers or [they] will not know how to array [their] troops" and arithmetic (number theory) as appropriate for philosophers "because [they have] to arise out of the sea of change and lay hold of true being.

"[6] The Greek mathematician Apollonius of Perga was asked about the usefulness of some of his theorems in Book IV of Conics to which he proudly asserted,[7] 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 that "...seem worthy of study for their own sake.

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.

Another insightful view is offered by American mathematician Andy Magid: I've always thought that a good model here could be drawn from ring theory.