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.

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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. 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*. I think my thought-experiment shows why this doesn't make sense--we can see that Godel's theorem doesn't prove that an uploaded brain living in a closed computer simulation S would think any different from us, just that it wouldn't be able to correctly output a theorem about arithmetic equivalent to "the simulation S will never output this statement". But this doesn't show that the uploaded mind somehow is not self-aware or that we know something it doesn't, since *we* can't correctly judge that statement to be true either! It might very well be that the simulated brain will slip up and make a mistake, giving that statement as output even though the act of doing so proves it's a false statement about arithmetic--we have no way to prove this will never happen, the only way to know is to run the program forever and see. > >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". 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. Jesse _________________________________________________________________ Need a break? Find your escape route with Live Search Maps. http://maps.live.com/default.aspx?ss=Restaurants~Hotels~Amusement%20Park&cp=33.832922~-117.915659&style=r&lvl=13&tilt=-90&dir=0&alt=-1000&scene=1118863&encType=1&FORM=MGAC01 --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---