How Far Can Theory Take Us?

Yesterday I wrote a note (below) with a method. The Mexican peasant I was thinking of, does take her people far. If our western civilization collapses globally, those Mexican Indians may be able to keep going. Modern Science has taken us this far. Both examples answer the question of this note affirmatively. Theory can take us far.

Nevertheless not any theory can take us far. Theory that reflects observations, like the Mexican peasant constant comparison of this year´s corn head size with last year´s; is bound to be successful. On the other hand my selection method (note below) may be worthless.

Can we know beforehand which case is which?

I have to use the method and report results. At this point all I can do, is a little more theory.

Given a situation requiring a selection, we take one. That is our life in a nutshell. Here I write an analogy with Natural Selection, to strengthen my proposal.

Selection as I have been referring to in these recent notes, is human selection. Natural Selection is different. Unless I want to give Nature human qualities as the people from some places now do, and all places long ago did. Then what I write here is useless. I assume though, that even if Nature surprises us often, humans are different. I will introduce an idea due to Stephen Wolfram in this Natural Selection. It is not Darwin´s version then, maybe I can call it Darwin-Wolfram Natural Selection.

Principle of Computational Equivalence.

My understanding of this Principle is that above a complexity threshold; complex systems are computationally equivalent. A simple cellular automaton rule, was proven by Wolfram, to be a universal computer.

One lemma for me then, is that human selection and Natural Selection, are not all that different.

With this perspective a statement of Natural Selection, may be useful for my Generalized Selection Principle.

Natural Selection: A system produces more copies of one type than other types.

The fitness property is used to distinguish permanent parts from transitory ones. We say that those permanent parts are more fit for that system. The system is contained in other systems. As scientists we may be aware of the other systems or not, still the principle is applied locally.

I suspect that the system is fractal. As the Poet put it:

So, naturalists observe, a flea
    Has smaller fleas that on him prey;
     And these have smaller still to bite 'em,
      And so proceed ad infinitum.
       Thus every poet in his kind
        Is bit by him that comes behind.
- Jonathan Swift, Poetry--A Rhapsody

We only know a few fleas, but that is enough, and that is the power of thought.

Darwin only had to see finches, what prey on finches? That was not necessary. Gauss could tell a flea if its world was spherical, without the flea leaving the surface of the sphere. That insect only had to go around, holding a flag leaving its twin behind, if the flag pointed to the same direction as the flea left behind, there was no sphere, there was curvature otherwise. Locally Gauss tells us, we can leave the curved surface, at least in our mind´s eye.

Kenneth Wilson taught us to use the phase transition to know about the system almost without knowing much about its Hamiltonian, or dynamical generator. Whole classes of Hamiltonians, universality, led to the same type of phase transition. That is good and it is bad. At the point of transition we do not need to know much about the system, by the same token at the transition point, we won´t know much about the system.

I believe that these different thinkers, Smolin, Wilson, Wolfram, and Darwin, are telling us something very deep.

I want to encompass all that in the Generalized Selection Principle.

In conclusion:

A simple method is needed to know which step to take in all cases. It is known from the Travelling Salesman Problem, that the trivial solution is not a solution. It is easy to convince oneself, that this person cannot just go to the nearest client and expect to get the shortest path when all the clients are visited, you can try it with three points judiciously located, it is easy to construct a counterexample. Some hindsight is needed, the next move needs some depth of vision in the game to be a good move. I feel that there is no universal strategy to always win, my Principle, at most is heuristic, as the great George Polya understood so well, so many years ago.

With Wolfram I believe that I have to respect and negociate even with a sufficiently complicated cellular automata. There are some around, for instance that virus that put you down with a cold the other day.

The Principle then is only a guide to take informed decisions.

In any case as Smolin still has to convince me that something of value will come out by changing parameters to produce universes worse than ours. I still have to convince myself that my theoretical game is worth anything. At least Smolin has a way to test, I still have not given a test for my Principle. Remembering Renè Descartes, I feel that maybe I will be the only one using this Method. I hope I am at least as successful with this kind of game as he was.

Conclusion

Natural selection is different than human selection. Given Wolfram´s result, now I believe they are not different, therefore for a first try on my Generalized Principle, I will assume they are the same. In a future revison of the Principle I may introduce the differences I feel may be necessary.