Sunday, April 26, 2015

195. The mystery of mathematics

Mathematics is mysterious to most people, including philosophers. The Platonic temptation of seeing knowledge as contemplation of absolute, i.e. universal and timeless truths, is nowhere as strong as there.

The distinction has been made between synthetic truth, about the empirical world, and analytical truth that follows purely logically from axioms or definitions. The separation between the two has been contested.

For one thing, what was once invented purely by mathematical imagination later turns out to describe, with astonishing accuracy, what we observe in the physical world. How can that be? Surely, there is progression from mutual interaction between theory and observation. Yet the explanatory efficacy of prior mathematical insight remains mysterious. How can that be?

My conjecture, based on an evolutionary argument, is as follows. Survival in evolution selected for ways of cognition that are adequate to a conceptualization of the world that is conducive to survival. In other words, we survived because our thought evolved to reflect reality in some sense.

We survived because we adapted to conceptualize the physical world adequately, inspired by running towards prey and away from predators and enemies, throwing a spear, rolling a wheel, catching fish, building a wall. The claim that we conceptualize the world adequately, in some sense and up to some point, is supported by the fact that we survived. Perhaps this aptitude moves to higher levels of abstraction, inspired by success and failure in predicting nature to invent yet ‘higher level’ mathematical theories.

From Euclidean geometry in planes to geometry on curved surfaces, onwards to a wider theory of shapes and deformations (topology).  

This success of mathematics concerning nature is in stark contrast with our relative inefficacy in developing mathematical insight into how human society works.

In the Enlightenment there was a dream (e.g. Condorcet) that social phenomena could be equally well be accounted for by the mathematics that was so successful in nature. This dream has largely not been realized.

There are mathematical economic theories but they hardly yield an adequate account of society. There are other social theories but not with much effective mathematics in them (except for the statistics behind empirical testing).

Why is that? Along the line of the evolutionary argument the explanation would be that the maths required for adequate understanding of society is of a different order, and our aptitude for it has not been equally nursed by evolution. Adequate social understanding is only now becoming a condition for survival of our species. Mental evolution may be to slow to catch up in time. Let us hope that in time some mathematics will be found that gives a better grasp of society. 

How does mathematics fit in with imperfection on the move, the central theme of this blog?  Isn’t math the paragon of perfection?

Mathematics is not in fact universal and timeless. With the formulation of basic assumptions, called axioms, a mathematical system shields itself, immunizes itself from imperfection. It rules out all contingencies and subtleties that invalidate it, since it looks only at what follows from the axioms. Perfection is built in. Its imperfection lies in the limits imposed by the axioms, and even mathematics is imperfection on the move, in moving on to other, new axioms, in an ongoing search of abstract patterns, correspondences, (a)symmetries, and transformations.

We need a mathematics of meaning and meaning change, metaphor, identity, freedom, learning, and motivation, to name a few.