George Lakoff in Women, Fire, and Dangerous Things talks about categories and how we use them to divide the world into fundamental elements of thought. Lakoff convincingly shows that human categorization is messy and complex, not at all like the abstract bins of mathematical logic. He argues, therefore, that we must beware attempts (common in some areas of AI research) to reduce intelligence to axioms and theorems.
But there is a place for ideal, pure, logically precise categories ... even though people, as imperfect "meat machines", may not implement them very well in everyday life. Mathematics shines a light on our universe; it gives us a rock-solid foundation to build upon. Points, lines, and planes are utterly imaginary geometric entities --- but they hint at how we may think about actual objects. Numbers are mere abstractions, which throw away all but one aspect of real life --- but that aspect turns out to be tremendously important. Sets and subsets are the basis of a literally infinite variety of concepts, at such a deep level that we can barely talk about them.
Humans don't need to become mechanical automata --- far from it! But it can't hurt us to develop more self-awareness, in the Stoic philosophical spirit, of the roots of our thought processes and the ground in which they find purchase. Those roots, and that ground, are mathematics.
Wednesday, June 30, 1999 at 17:50:55 (EDT) = Datetag19990630