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.

How (not) to avoid the G-world in cosmology?

The first chapter’s title in Mary – Jane Rubinstein’s recently published book entitled Worlds Without End. The Many Lives of the Multiverse is: How to Avoid the G-word.
Namely, the God.
It is an interesting trend that not trying to avoid the G-word is an acceptable attitude for some modern cosmologist, although the Creator’s existence or non-existence hasn't play a role in modern optics or mechanics (what is more, it is not a question in the explanation
At the beginning of the 17th century Galileo argued that there were two books written by God. One of them was the the Bible and another was the book of nature. Although it was a common belief of that age that both of them used the same language, according to Galileo, the second one used the language of mathematics. [John Hedley Brooke: Science and Religion, p. 104] Supposing the existence of a Creator who created the World for humans, it is an open question why He had decided to use two different languages, and why an exotic and complicated system was introduced to describe the physical reality created by Him. In other words: why the physical reality is so complicated that it is impossible to describe it in everyday words? Was it a necessity to hide the structure of the world behind a complicated and non-evident formulism?
Obviously, Galileo’s concept based on a kind of Neoplatonic mathematical mysticism. According to it, the fundamental structure of our Universe is not only can be described by mathematics, but the nature itself is purely mathematical. But it is not an answer for the question why decided God to choose this language.
On the other hand, a group of physicists in the 17th century refused the Aristotelian concept which stated that an explanation cannot be complete without pointing out a final cause. Since Aristotle didn’t favor mathematics as a language of world description, Galileo’s approach offered a way to deny any argumentation based on final cause, and Descartes decided to focus on immediate causes of physical events. [Brooke, ibid, p. 72]
Since then it became a tradition in natural sciences to study those immediate causes and omitting either a final cause or the intervention of a Creator. But – to oversimplify the problem – when somebody studies either the “fine-tuned” nature of our Universe or the supposed existence of multiverses, there isn't an opportunity to study his or her subject from Descartes’ approach, because there is no an existing physical event to examine.
According to Bernard Carr, “If you don’t want God, you would better have a multiverse.” Namely: if you don’t want to refer to God, then you can replace Him with the idea of Multiverse – or vice versa. The “Multiverse replaces God with what is perhaps an equally baffling article of faith: the actual existence of an infinite number of worlds.” [Rubinstein, p. 29]
We have two different strategies to handle the causes. One can choose Descartes’ approach to always focus on immediate ones – and then he or she can ask another question about the solved problem’s immediate cause. We can move forward step by step – its mathematical analogy is the potential infinity where every piece of the result is finite and it can be achieved in a finite number of steps, but this process never has an endpoint.
The other approach is more problematic. If we are allowed to ask for a not immediate cause, then we could ask what caused that cause, etc. [Rubinstein, p. 21] and the result is an infinite regress without end.