Tuesday, August 22, 2006

Foundations of mathematics, evolution, and castles in the air

I am no Platonist Realist when it comes to mathematics. Mathematics does not exist in some unseen Platonic universe. Our basic mathematical concepts arise from our ordinary experience of the world. From these basic concepts we build ever more complex mathematical concepts via analogy and metaphor.

In fact, our most basic mathematical concepts reside in our unconscious minds. It takes clever psychological experiments to tease them from their hiding places. Thus, mathematics is embedded in our evolved mental systems.

Since our most basic mathematical concepts come from our mental ability to negotiate the world, the usefulness of mathematics in explaining the world is apparent at the same time. Our sophisticated mathematical constructions depend upon the foundations in our basic biology and our creativity.

If at some point mathematics is not useful, then it becomes pure art. Pure art has its place in our lives too. Practicality is not derailed by invention nor invention derailed by practicality.

Let us take one famous episode in the history of mathematics. Until the beginning of the Nineteen Century, Euclid’s fifth postulate about parallel lines was considered basic to the structure of the world. In fact, many people tried to prove the parallel postulate from Euclid’s other postulates.

Consider a form of the postulate called Playfair’s Postulate: given a line and a point not on the line, one can draw one and only one line through the point that is parallel to the given line.

Gauss, Bolyai, and Lobachevsky showed that you can negate the parallel postulate and arrive at a geometry that is as consistent as the original Euclidean geometry. (Actually, a full proof of consistency remained to be proved later.) Negating Playfair’s Postulate gives two separate parallel postulates. One case is that more than one parallel line can be drawn through the given point (hyperbolic geometry). The other case is that no parallel line can be drawn through the point (elliptic geometry).

If one considers mathematics as a series of formal systems with undefined terms, unproven axioms, plus logic, then there seems to be no limit as to the kinds of mathematical systems one can create. Of course, their consistency must be maintained or they are nonsense because any proposition can proved from a contradiction. However, these systems may not be useful in explaining the world. That is not to say they might not be useful some day, or that the practical does not motivate invention and art.

It is at the frontiers of the foundations of mathematics that one must make a commitment to evolution or not. Our creative mathematical ability must be explained somehow. Searching for mathematics in some ethereal castle in the air that cannot be physically seen seems like searching for Zeus on top of Mount Olympus.

You can have myths about mathematics or you can have mathematics, but you cannot them both at the same time and remain consistent in your philosophy or your science.

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