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On 29 jun, 19:10, "Jesse Mazer" <[EMAIL PROTECTED]> wrote: > LauLuna wrote: > > >On 29 jun, 02:13, "Jesse Mazer" <[EMAIL PROTECTED]> wrote: > > > LauLuna wrote: > > > > >For any Turing machine there is an equivalent axiomatic system; > > > >whether we could construct it or not, is of no significance here. > > > > But for a simulation of a mathematician's brain, the axioms wouldn't be > > > statements about arithmetic which we could inspect and judge whether > >they > > > were true or false individually, they'd just be statements about the > >initial > > > state and behavior of the simulated brain. So again, there'd be no way > >to > > > inspect the system and feel perfectly confident the system would never > > > output a false statement about arithmetic, unlike in the case of the > > > axiomatic systems used by mathematicians to prove theorems. > > >Yes, but this is not the point. For any Turing machine performing > >mathematical skills there is also an equivalent mathematical axiomatic > >system; if we are sound Turing machines, then we could never know that > >mathematical system sound, in spite that its axioms are the same we > >use. > > I agree, a simulation of a mathematician's brain (or of a giant simulated > community of mathematicians) cannot be a *knowably* sound system, because we > can't do the trick of examining each axiom and seeing they are individually > correct statements about arithmetic as with the normal axiomatic systems > used by mathematicians. But that doesn't mean it's unsound either--it may in > fact never produce a false statement about arithmetic, it's just that we > can't be sure in advance, the only way to find out is to run it forever and > check. Yes, but how can there be a logical impossibility for us to acknowledge as sound the same principles and rules we are using? > > But Penrose was not just arguing that human mathematical ability can't be > based on a knowably sound algorithm, he was arguing that it must be > *non-algorithmic*. No, he argues in Shadows of the Mind exactly what I say. He goes on arguing why a sound algorithm representing human intelligence is unlikely to be not knowably sound. > > > >And the impossibility has to be a logical impossibility, not merely a > >technical or physical one since it depends on GĂ¶del's theorem. That's > >a bit odd, isn't it? > > No, I don't see anything very odd about the idea that human mathematical > abilities can't be a knowably sound algorithm--it is no more odd than the > idea that there are some cellular automata where there is no shortcut to > knowing whether they'll reach a certain state or not other than actually > simulating them, as Wolfram suggests in "A New Kind of Science". The point is that the axioms are exactly our axioms! >In fact I'd > say it fits nicely with our feeling of "free will", that there should be no > way to be sure in advance that we won't break some rules we have been told > to obey, apart from actually "running" us and seeing what we actually end up > doing. I don't see how to reconcile free will with computationalism either. Regards --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---