Tuesday, 17 July 2012

Mathematics and Theory

There can be tendency to see mathematics and theory as two sides of the same coin. General relativity, for example, is a theory with mathematics that supports it. This mathematics gives the theory utility in enabling calculations in accordance with the theory. Moreover, the presence of mathematics in a theory, particularly if the mathematics affords calculation, can enhance the fiduciary quality of the theory: we say "ah, there's maths, and it seems to work - the theory must be correct". Economics (econometrics) suffers from this conflation particularly badly. And in econometrics, what happens is that all-too-frequently, the maths is not very good, and certainly doesn't 'work' in any real sense (it has little predictive and no explanatory power). Yet, its presence is a barrier to having a sensible discussion as to why or why not it doesn't work, or to explaining economic phenomena in a sensible way.

It is a commonplace to remark that maths is like music. But in a very deep way, this can be seen to be true. In fact, maths is like most arts: it's fundamental concern is with form, not calculation. What is revealed by mathematicians are the peculiar forms that our logical faculties - those essential quality of human beings - can take. There are, of course, many strange things that occur. Louis Kauffman showed me this mathematical 'joke' for example:

Euler's identity tells us that:
That means that:
this is where the funny thing happens.. because i appears to re-enter the equation in this way...

which produces this infinite regress:

This is the joke... But I have to confess I didn't really get it when I first saw it, thinking that the joke was in the infinite regress. I confidently taught this back to someone else (puzzling why I wasn't really laughing), only half understanding it (but in the hope that in explaining it I might understand it better...), but Louis pointed out what the real joke was: how can an imaginary number equal a real number?

What's relevant here though is the way that mathematicians think. This is a revealing of logical form. It has a kind of 'cadential' structure (to use a musical analogy): it is satisfying and beautiful. 

Theory on the other hand, is not like this. As I explained yesterday, I think theory is the result of an attempt to reveal this kind of beauty which effectively fails. Theory is a way of accounting for the failure. [this is my theory!]. 

Of course, mathematics has 'theory'. Louis uses category theory to work out his thinking about eigenform and time (which he also explained to me). Is such a theory (like category theory) a theory in the same sense as Luhmann's theory of communication, or Baudrillard's theory of symbolic exchange and hyper-reality? I think they are different, and an artistic metaphor can help expose how they are different.

I wonder if a mathematical theory is like some base material upon which the sculptor works to reveal a form. Category theory is like the 'material' basis whose logical structure can then be exposed through the work of a mathematician. Although most social theories aspire towards this same foundational goal, they tend to be ill-defined, and hence a slippery foundation for making any defensible statements (Baudrillard is particularly bad with this). It may be that the failure of such foundations feeds back into the need to prop-up bad theories with more bad theories. 

It is here that I wonder if listening carefully to the mathematicians and the artists might be a more sensible thing to do, rather than to get increasingly bogged-down with theories...

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