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.