I'd better bite on Jerry's comments on maths, despite time commitments
elsewhere. Jerry posed:
What about the "new math", e.g. chaos theory? Although chaos theory has
the potential of being used to analyze non-linear dynamics in a more
suitable and realistic manner than matrix algebra, are there not
limitations of that form of mathematics as well? If so, what are they?
Also, there are different forms of chaotic analysis (something that Steve
K knows well). Which forms are best used for the study of particular
non-linear topics and why?
<Among other snipped things>
Very quickly, one lesson maths itself has gradually taken to heart is
that there are many things which, while they can be expressed in
mathematical form, can't be solved analytically: polynomials of
degree higher than 4, most differential equations of any order,
coupled differential equation systems of dimension 3 or above,... the
list is endless. So maths has limits.
On dialectics v mathematics, I see the relationship as being
dialectics--> mathematics. To me, the dialectical mode of thinking
(and I have my own very peculiar take on this re marx, which I'll
eventually submit to OPE) suggests the dynamic relationships that
we should, where possible, consider in mathematical form. But there
will be some issues that dialectics suggests which can't be
adequately posed mathematically.
One such example is the issue OPE has been discussing on and off for
months now--the evolution of new technology. While a colleague of
mine has devised a mathematical analysis of evolution, it can only
be considered in a computer simulation (from which he derives
eigenvalues and then makes mathematical observations about the process
of evolution, etc.). Such things will become the norm as economics
evolves: many issues that we'd like to pose mathematically defeat
standard maths, and if we're to analyse them, have to be
simulated.
On this point, take the usual practice of analysing capitalism using
matrix maths. This presumes an industry size of n (n sectors). But
technical change implies that new industries evolve out of the old
ones (and some disappear because of new technology). so much for
a fixed number of sectors. Instead, we have a "countable infinity",
a limitation to the maths of infinite vector spaces, and a much less
powerful--but more appropriate--set of tools.
There will be times, however, that maths--> dialectics. This can
be when a supposedly "dialectical" statement can't be made to
work in whatis a proper translation of it into mathematics. Despite
my sympathy for the TSS approach, this is how I see its (and every
other school of marxism's) problems over the Sraffian critique. In
that case, I see the problem as being poor dialectics. But that's
another can of worms.
Cheers,
Steve Keen