weemba@sagi.wistar.upenn.edu (Matthew P. Wiener) says...
>>...Goedel's incompleteness theorem >>doesn't even *apply* unless you *already* know that the drift is >>bounded for *Darwinian* reasons. > >Exactly. I do split my conjectured proof into two parts--a 0-1 law which >summarizes the effects of evolution and a Goedelian part afterwards that >exploits this. Somewhere logic does have to enter.
I don't think so. If you assume that the original TM is completely adequate for survival in the world, then the odds are that eventually any trait giving it improved reasoning ability would be lost due to genetic drift and mutation. Goedel's theorem doesn't help with the conclusion that the limit theory is no more powerful than the initial theory.
>>Why? Your argument about limit theories, etc. assumes that the TM >>mind develops the capability for physically relevant mathematics >>*before* it develops the capability for reasoning about sets. Why >>should you assume that? > >Since my argument also suggests that a TM mind beyond PM would *devolve* >down to PM.
Not if a machine capable of PM (physical mathematics) and no more were *more* complicated than one capable of beyond-PM reasoning. I think that is very likely to be the case.
>>Obviously, we *did* evolve to be able to do higher-power mathematics, >>and just as obviously, there was no direct evolutionary pressure for >>doing so. I don't understand how you can argue that our powerful >>brains were selected for in a way that doesn't *also* argue for >>similar TM minds being so selected. > >This is trivial. If our minds are not computable, then Goedelian arguments >don't apply.
Goedelian arguments *don't* apply, as far as I can see, in either the TM or non-TM case. What is relevant is this: was there a plausible alternative design for human brains (TM or otherwise) that would have resulted in completely satisfactory survival abilities but would not have allowed the contemplation of ZFC? Or, is the simplest, most plausible brain-design the one we've got? I don't see how the mechanism for reasoning (TM or not) is relevant.
>>As I said, I don't really think either 0-1 laws or Goedel's theorems >>are relevant; the only thing that is relevant is whether there exists >>a TM mind that is (A) capable of doing all physically relevant >>reasoning as well as we can, (B) incapable of doing set theory, (C) >>more likely to evolve than any TM mind that is fully as capable as we >>are. > >Eh? That is what my argument claims, more or less.
But you didn't offer a plausible alternative TM that would be less powerful than our brains. You suggested Peano Arithmetic, but that isn't really plausible; Peano arithmetic is both too powerful for survival purposes and not powerful enough. It is not likely that a TM would be directly equipped with *any* mathematical theory; instead it would be a program for recognizing patterns, making plans, and reacting to situations. What sort of reasoning is necessary for this? I don't know, for sure, but mathematics is not a direct concern. It is possible (even quite likely) that mathematical ability would be a side-effect of whatever program allows our TMs to survive, but the selected-for TM would not be principally a mathematician. The relationship between the TM's survival abilities and the amount of math it would be capable of is likely to be very indirect.
