The development of mathematics was mostly driven by a need to understand, describe and predict certain phenomena in our physical world. The idea of a separate discipline of pure mathematics emerged in the nineteenth century. The generation of Gauss did not draw a clear distinction between pure and applied mathematics, but the development of formal mathematical analysis by Weierstrass (1815 – 1897) started to make such a distinction more apparent.
At the start of the twentieth century, mathematicians took up the axiomatic method strongly influenced by David Hilbert (1862 – 1943). In addition, the logical formulation of pure mathematics suggested by Bertrand Russel (1872 – 1970) seemed more and more plausible, as large parts of mathematics became axiomatised and thus subject to the simple criteria of rigorous proof. The Bourbaki group, an influential group of French mathematicians active since 1935, held the view that pure mathematics is what is proved. Pure mathematics became a recognized vocation, achievable through training.
A typical example of one of the central ideas of pure mathematics is generalisation. Pure mathematics often tends towards increased generality. Amongst the advantages of generality are:
• Generalizing theorems or mathematical structures often lead to a better understanding of the original theorems or structures
• Generality often simplify the presentation of material resulting in shorter proofs or easier arguments.
• By generalisation, duplication can be eliminated.
• Generality can facilitate connections between different branches of mathematics.
For example, the study of numbers, called algebra at the beginning undergraduate level, extends to abstract algebra at more advanced levels. The study of functions, called calculus at the first year level becomes mathematical analysis and functional analysis at more advanced levels. Each of these branches of more abstract mathematics have many sub-specialties.
Abstract mathematics is not necessarily independent of applied mathematics and there are many connections between pure mathematics and applied mathematics.
Studying mathematics at university usually entails two broad areas, namely analysis and algebra. The origins of what you study in these two areas during the first two years at university come from long before the nineteenth century, so it is mostly mathematics to be used in applications. In particular, in analysis it is calculus (differentiation, integration, differential equations) and in algebra it is linear algebra (matrices, systems of equations, determinants, etc.). Only in the third year will you really encounter proper pure mathematics, where the mathematical justification for much of what you did in calculus will be taught (real and complex analysis) and you will also have your first exposure to abstraction in mathematics with abstract algebra.