Inspiration, aesthetics and pure and applied mathematics

Mathematics arises wherever there are difficult problems that merit careful mental investigation. At first these were found in commerce, land measurement and later astronomy. Nowadays, mathematics derives much inspiration from the natural sciences and it is not uncommon for new mathematics to be pioneered by physicists, although it may need to be recast into more rigorous language. Some notable examples of this happening are Newton inventing calculus and Feynman inventing his Feynman path integral, but it also happens with results from string theory. The mathematics arising from this immediately has relevance for the subject which inspired it and can be applied to solve problems in that subject. Mathematics which can be so used is called applied mathematics as opposed to pure mathematics. In this way applied mathematics is an indispensable tool. With the increase in our mathematical knowledge, mathematics itself has become a source of inspiration. Mathematics is inspiring to mathematicians because it has some intrinsic aesthetics or inner beauty, which is hard to explain. Mathematicians value especially simplicity and generality and when these seemingly incompatable properties combine in a piece of mathematics, as in a unifying generalization for several subfields, or in a helpful tool for common calculations, often that piece of mathematics is called beautiful. Since the result of mathematics inspired by mathematics is often pure mathematics and thus has no applications outside of mathematics yet, the only value it has is in its aesthetics. Surprisingly often, it has happened that pure mathematics, which was considered only of interest to mathematics, has become applied mathematics because of some new insight, as if it anticipated later needs.

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