6. Abstraction in Mathematics
Mathematicians do not study objects, but the relations between objects;to them it is a matter of indifference if these objects are replaced by others,provided that the relations do not change. Matter does notengage their attention, they are interested in form alone.Poincaré, Science and Hypothesis
The history of mathematics is filled with discoveries of new abstractions: finding ways to solve a more general problem. For example, we saw in Chapter 5 how Euler generalized Fermat’s Little Theorem so it would work with composite numbers as well as primes. Eventually, mathematicians realized that they could generalize beyond numbers, and derive results about abstract entities called algebraic structures—collections of ...
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