ISTI-CNR (Nat. Research Council)
Registered: Jan 2005
Does this all boil down to saying that, given an ECA (or any other simple rule) and an initial condition, there are some cell configurations that will be reached, and some that will not?
I find it hard to say much more than this about an individual, deterministic ECA computation. I see an ECA computation just as a single, deterministic run that computes ... itself, not as something set up to prove or disprove theorems. I cannot even understand what 'truth' means for a single, deterministic computation (other than elementhood for a cell configuration, as mentioned above).
Of course, IF you take the rule and allow yourself to play with it, to run it on ANY initial condition you like, AND you have a way to interpret initial conditions and perhaps final conditions (they CODE some configuration of another formal system you have in mind), then you can use the rule to prove things, and to enumerate truths. But all this requires external work, human interpretation...
On the other hand, all these human brain activities, involving proofs and the idea of 'truth', are phenomena that take place in the universe itself, and that should be governed exactly by that same universal deterministic rule.
How can we then reconcile the single track nature of the universal computation with the multiway nature of proof systems and proof activities that go on in the human brain?
This reminds me of the notion of ergodicity, which I understand in tems of dolphins: take SEVERAL snapshots, at different times t1, ..., tn, of a SINGLE dolphin d jumping in and out the ocean, and then take a SINGLE snapshot, at time t, of a group of SEVERAL dolphins d1, ..., dn (you'd see just the face of one, just the tail of another, and so on): the information you get about the 'dolphin process' is the same in the two cases -- in a sense, you trade space for time.
Perhaps the universal computation is single-track, like a single dolphin, but it is 'ergodic'-like, so that it also exhibits, although displaced in time, the features of a multiway system.
But how about the 'Goedellian unproven truths'?
What can we say about the process that takes place in the human mind and leads to the conclusion that something unprovable is indeed true?
Did this process follow the Rule?
(I guess this topic must have been covered a few times in the Forum...)
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