It is in the nature of polymath, which is simply put ‘more of the same’, that progress on such a fundamental question is slow. Organizers and participants should simmer that at low heat. After all, it is not just a math Olympiad problem with a solution known to exist.
What have we got so far (according to my frail understanding)?
A first observation is, that the players seem to be interested in implementation and representation rather than the underlying symmetries or rules of the problem. They have in mind a cellular automaton like a sand-pile model or Conway’s game of life. This is interesting in so far as the best mathematicians tell us first to learn about the problem, state the laws governing it and then, as a last step, search for representations as known objects. Is it obvious that the problem has a solution among Turing machines? Not to me, at least. Why should we then restrict ourselves to Turing machines? One reason might be, that they are well understood. A second might be that if you believe in the Church-Turing thesis, Turing machines are all you need.
This summarizes the first observation: The players apparently presuppose the Church-Turing thesis (albeit without mentioning it).
But, what else could they do instead of looking for Turing machines?
They could go for the laws and they did. It was rapidly recognized that Gowers’s item 5 or even the one before play a central role.
4. It should have a tendency to produce identifiable macroscopic structures.
What does that mean? In his comment Tim Gowers explains:
My impression is that there are several levels of complexity and that each level provides a sort of primeval soup for the next level up, if that makes sense.
Anyhow, a sort of answer is that from any model I would look for the potential to produce objects that could serve as the basic building blocks for a higher-level model.
This last sentence seems to be a recurring motive among players. There seems to be a consensus that such levels should be part of a model. If you now start with some bottom level, then how do you proceed from one level to another? That is indeed a problem.
Some believe, for example, that if you put electrons and other elementary particles together in the right combination you will end up with a human. Like a watchmaker making a watch. The watch obeys the same laws as do their parts. In this spirit a human obeys the same laws as say an electron. That line of thinking nearly inevitably leads to the ‘free will vs. determinism‘ controversy.
Others believe that there is no evidence for the above to hold. There is essentially only one known process to ‘produce’ humans. The natural one. This process is radically different from ‘putting pieces together’ and it is not obvious that the laws governing the pieces hold for the whole. There is, let’s call it, a ‘barrier’ separating the level of humans from the level of electrons. This is hard to imagine for a reductionist.
If we take the existence of ‘levels of complexity’ as a given, then the decisive question currently seems to be:
- What are the rules to go from one ‘level of complexity’ to the next?
I am very excited on what answers they come up with …