What is Pure Mathematics?
Pure mathematics differs from science, in which I include applied mathematics, in its method of reasoning. The scientific method uses inductive reasoning, i.e., inferring or inducing a general result by observation of specific examples. The researcher makes an educated guess, or hypothesis, then makes observations and, if all these fit the hypothesis, he/she may infer that the educated guess is correct or “proven”. This is most assuredly not the same thing that a mathematician means by “proof”. Further observations, especially after, e.g., the development of new technology, may refute the hypothesis, in which case the scientist must either scrap or modify the explanation. Pure mathematics uses deductive reasoning. It is very much like playing a game such as Chess. There are the game pieces and a set of rules. All the moves in the game have to be consistent with the rules and involve only the game pieces. Analogously, in deductive reasoning, the rules of the game are the axioms. These a
