Hi Jonathan,
I think that the - 2 in the Mandelbrot calculation is a mathematical artifact resulting from the properties of the 2 dimensional complex plane in which the object is constructed.
String theory takes 2 dimensions (the plane or "brane") as the fundamental space in which strings propagate. Witten discards the idea of spacetime and replaces it with a 2-dimensional field theory. That all of the physical world can be described in 2 dimensions eventually comes down to the holographic principle -- the information we can abstract from the surface of a black hole. Because black hole energy increases to the negative inverse (- 1) of temperature, and entropy simultaneously increases to the - 2 power, entropy increases with area (the surface of the black hole) which is r^2, radius squared.
So if we're speaking of a fundamentally geometric theory that applies arithmetic functions, as string theory does, we need all those mathematical artifacts that originate in the complex plane C, as a closed algebra, imaginary roots and all. Otherwise, we would be (mathematically) stuck in the infinitely open plane and could not escape incompleteness. String theory, although based in quantum field theory, is -- like special and general relativity -- mathematically complete.
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I hope you and the group will forgive me for being self-serving at this point -- I have invested a lot of time and thought into constructing an alternative basis for encoding information in n-finite dimensions of space while nevertheless preserving a time metric. For this discussion, we only refer to 1.3 -- 1.10 in the draft paper linked. The table explains correspondence of the plane to the integer - 2.
All best,
Tom