### Imperfections, Ambiguities and Physics

#### Posted by David Corfield

I mentioned Lautman’s association of, on the one hand, Descartes’ argument to the existence of a perfect being (God) from an awareness of his own imperfections with, on the other, a mathematical argument to the existence of an algebraically closed field from the inability to factor polynomials in a given field, or to the existence of a simply connected space from the inability to contract all loops in a given space.

This perfection/imperfection ‘dialectic’ involves our realising from an imperfect state that there is a perfect state, and also from the nature of the imperfections what are the attributes of the perfect state.

Now does this thought have any resonance in physics? Are there specifically physical manifestations of this phenomenon? Or to the extent that we find these, such as Doron Gepner’s Galois Groups in Rational Conformal Field Theory, should we say that everything Galoisian ‘factors’ through the mathematics?

On the other hand, if we wanted a natural language description of the mathematical phenomena, is the perfection/imperfection pairing the best way, or is Galois’ own ambiguity theory not more accurate? In which case, although ambiguity might be thought an imperfection, the commonality with the cartesian situation is lessened.

Finally, are physical manifestations better described as ambiguities rather than imperfections? Don’t I remember something on ‘defects’ in nematic crystals in one of those mathematics of gauge theory textbooks? Ah yes, Topology and Geometry for Physicists. So, does the existence of a crystal defect tell us about a perfected crystal, even if physically unrealisable? Similarly, how could we describe the Aharonov-Bohm effect?

Posted at August 30, 2008 9:40 AM UTC
## Re: Imperfections, Ambiguities and Physics

While I’m asking, how about the same set of questions for duality/reciprocity. Denis was extremely helpful in showing me what was at stake mathematically.

But where do find duality/reciprocity richly exemplified outside of mathematics? In physics, e.g., electric-magnetic duality? Again is this ‘factorable’ through mathematical duality?