Tuesday, March 07, 2006

the two and the many.

consider cases of 'twos'.

base 2 arithmatic or its infinite limit, the real numbers. everything is just a line composed of 0's and 1's.

a 2 state ising model---spin up and down---in one dimension. again this is simply a line with 0's and 1's (or 1's and -1's).

a 1-dimensional cellular automota (CA). (again a line of 0's and 1's). (one can also think of a 'turing tape' (the input string for a turing machine) which can be thought of as the state of a computer).

a 2 person zero game. (any game can be turned into a zero sum two person game).

a two variable equation, or a one variable equation of degree two (quadratic).

then consider manys.

a spin glass (n state ising model) or n state CA.

base n arithmatic.

higher order polynomials, or high degree polynomials. (see Hilbert's 10th problem).

a complex valued ising model, or cellular automata. (similar to quantum computers). (one can also compare laplace or similar transforms with fourier transforms.)

n person games.

higher dimensional automota or ising models and spin glasses. (one can also compare knots with higher dimensional manifolds; see also Poincare's conjectures.)

one can also consider cases with long range or other complex interactions (or 'actions at a distance') versus nearest neighbor interactions.

one can always ask when is some high dimensional or many state system reducible to a lower order one.

the standard example is elementary quantum mechanics. this involves a complex valued wave function, which can be replaced by two real valued wave functions. one question is why one doesn't simply choose the real representation, and the answer is because then one is required to choose as the 'real physical' solution an average of the two, which represent the same motion except reversed in time.

some argue that an ising model in one dim, like the real numbers can represent everything. a checker board can be rewritten as a real number, as can a cube or higher dimensional object. similarily, any algorithm or equation for constructing some object can be written as a real number, via Godel numbering. (hence even two dimensional objects, like a one dimensional ising model or CA, can be viewed as single numbers. These numbers then represent 'spaces' in a sense. From the godelian view, the coordinates are labeled by primes. In the real number case, the space is a vector in high dimensions.
Indeed dynamics of a CA or ising model can be viewed as operations of a group acting to rotate a vector.
Other translations between numbers and spaces exist, such the work by chalmers on polytopes. )