A - Harmonious Foundations

Summary

Mathematics is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency.

—René Descartes (1596–1650)

Mathematical definitions appear inevitable, as if they exist independently of human thought. The appearance of inevitability prompts the question of whether mathematics is discovered or invented. We can't answer that question, but we note that someone had to think of the definitions that we now take for granted. This results from a historical process of formulating problems, looking for solutions, and creating the best mathematics for the given situations. In this appendix, we give background information on the mathematical concepts in the book. As a utilitarian fork or a chair can be beautiful, everyday mathematical constructs are also beautiful. Simple definitions can give rise to surprising phenomena. A good reference on mathematical foundations is [48].

Sets

Sets provide the building blocks for many mathematical definitions. The modern notion of sets was introduced by Georg Cantor (1845–1918). However, Cantor's set theory admitted some paradoxes, the most famous of which is Russell's paradox. It concerns the set S of all sets that are not members of themselves. If S is a member of itself, then by definition S is not a member of itself. But if S is not a member of itself, then by definition S is a member of itself. There is a contradiction either way.