Martin Gardner The New Ambidextrous Universe: Symmetry & Asymmetry from Mirror Reflections to Superstrings
Dover, 401 pages, $18.95.
What is mathematics about? That is not so easy to explain. With biology, say, we know where we are. Biology is about living things. Linguistics is about language. Sociology is about the social aspects of human life. But it is hard to point to the objects of mathematics. Numbers? What and where are they? Even if we take the heroic (or foolhardy) Platonic option that they are inhabitants of an abstract world beyond space and time, which we access through a mysterious faculty of intuition, we are left with no understanding of what mathematics tells us about the actual world we live in.
So by default mathematics has often been considered as not about anything at all. Instead it is thought of as a toolkit of methods, formulas, and rules, or as a “theoretical juice extractor” for getting one truth from another, but not itself actually about any aspect of reality. Sadly that view, understandable enough in physicists and engineers, is reinforced by the rule-based style of teaching that fills the traditional school curriculum: “Minus times minus equals plus/ The reason for this we need not discuss.”
Neither of those views of mathematics is correct. Mathematics is about certain real properties of objects. The words “pattern,” “structure,” and “complexity” are sometimes used to describe what properties mathematics studies, but they have a certain excessive generality and vagueness to
