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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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I.4     The General Goals ofMathematical Research

The previous article introduced many concepts that appear throughout mathematics. This one discusses what mathematicians do with those concepts, and the sorts of questions they ask about them.

1 Solving Equations

As we have seen in earlier articles, mathematics is full of objects and structures (of a mathematical kind), but they do not simply sit there for our contemplation: we also like to do things to them. For example, given a number, there will be contexts in which we want to double it, or square it, or work out its reciprocal; given a suitable function, we may wish to differentiate it; given a geometrical shape, we may wish to transform it; and so on.

Transformations like these give rise ...

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