24 June 2015

Mathematics as cellular automation

According to a popular belief, mathematics is nothing more than a big tautology, since it is a deductive system and a new result reachable through a process of steps specified by certain rules. The two sides of the equation means the same: To give an example: 2+2=4 (and it is held that the relations between the sets of axioms and the result of proof is the same).
Of course, the logic of mathematics makes possible to reach a result (e.g. a mathematical proof), but even if you accept the axioms as a given and unchangeable base, the possibility doesn’t means the necessity. Even a few elements can result “hyper astronomically” huge number of combinations (to borrow Quine’s term). Thus searching for an answer (e.g. examining whether a theorem is true) can be interpretable as (a random) walk in the phase space of mathematics. Perhaps we are convinced that there is a certain mountain pike in this virtual landscape, but we do not know the path to it; or we even don’t know whether the hill exists at all. Obviously, this image is more or less misleading, since there aren’t existing routes originally, and we have to build them–and in some cases this activity constructs the target itself. Furthermore, not only the phase space of a certain mathematics is enormously huge, but the phase space of mathematics based on different sets of axioms and considerations are similarly large as well.
It is an interesting question that to what extent overlaps different mathematics (=mathematics based different set of axioms and rules) each other. And since the “phase space” describes mathematics as an n-dimensional landscape where every point is defined by certain parameters, we can try to define them using another method. Ad analogiam: Remember Descartes’s idea to mathematize geometry.
Arithmetic is compressible: if you know the adding rules, then you can calculate the result of 2+2 directly, omitting the steps of 1+1=2; 2+1=3; 3+1=4. Simply speaking, there aren’t intermediate steps.
Opposite to it, in case of a mathematical theorem you cannot omit any part of the proofing jumping from the starting point directly to the end, thus it is history dependent and the first step is essential to reach the second one, etc. This uncompressible method is strongly resembles for the cellular automation’s operation mode.
Cellular automata (CA) is based on simple rules that determine the status of a certain cell (=a certain point of the landscape or cellular space, if you prefer) taking into consideration certain cells’ intermediate states. It is an uncompressible process: You cannot tell the result without executing the process itself.
It seems to be plausible that we can build a CA to generate point by point the path to any certain proof: After all, we can adapt the rules which guides the work of cellular automata to the objective to be achieved. What is more, probably we could construct a “universal mathematical CA” (UCA) to execute the whole mathematics. Or, we could build other UCAs to examine their ways of work. Perhaps it would result totally different mathematics.

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