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On Jul 5, 10:14 pm, "Jesse Mazer" <[EMAIL PROTECTED]> wrote: > LauLuna wrote: > > >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? > > The axioms in a simulation of a brain would have nothing to do with the > high-level conceptual "principles and rules" we use when thinking about > mathematics, they would be axioms concerning the most basic physical laws > and microscopic initial conditions of the simulated brain and its simulated > environment, like the details of which brain cells are connected by which > synapses or how one cell will respond to a particular electrochemical signal > from another cell. Just because I think my high-level reasoning is quite > reliable in general, that's no reason for me to believe a detailed > simulation of my brain would be "sound" in the sense that I'm 100% certain > that this precise arrangement of nerve cells in this particular simulated > environment, when allowed to evolve indefinitely according to some > well-defined deterministic rules, would *never* make a mistake in reasoning > and output an incorrect statement about arithmetic (or even that it would > never choose to intentionally output a statement it believed to be false > just to be contrary). But again, for any set of such 'physiological' axioms there is a corresponding equivalent set of 'conceptual' axioms. There is all the same a logical impossibility for us to know the second set is sound. No consistent (and strong enough) system S can prove the soundness of any system S' equivalent to S: otherwise S' would prove its own soundness and would be inconsistent. And this is just what is odd. > > > 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. > > He does argue that as a first step, but then he goes on to conclude what I > said he did, that human intelligence cannot be algorithmic. For example, on > p. 40 he makes quite clear that his arguments throughout the rest of the > book are intended to show that there must be something non-computational in > human mental processes: > > "I shall primarily be concerned, in Part I of this book, with the issue of > what it is possible to achieve by use of the mental quality of > 'understanding.' Though I do not attempt to define what this word means, I > hope that its meaning will indeed be clear enough that the reader will be > persuaded that this quality--whatever it is--must indeed be an essentail > part of that mental activity needed for an acceptance of the arguments of > 2.5. I propose to show that the appresiation of these arguments must involve > something non-computational." > > Later, on p. 54: > > "Why do I claim that this 'awareness', whatever it is, must be something > non-computational, so that no robot, controlled by a computer, based merely > on the standard logical ideas of a Turing machine (or equivalent)--whether > top-down or bottom-up--can achieve or even simulate it? It is here that the > Godelian argument plays its crucial role." Yes, he ultimately argues for that. > His whole Godelian argument is based on the idea that for any computational > theorem-proving machine, by examining its construction we can use this > "understanding" to find a mathematical statement which *we* know must be > true, but which the machine can never output--that we understand something > it doesn't. I'd say this is rather Lucas's argument. Penrose's is like this: 1. Mathematicians are not using a knowably sound algorithm to do math. 2. If they were using any algorithm whatsoever, they would be using a knowably sound one. 3. Ergo, they are not using any algorithm at all. >But I think my argument shows that if you were really to build a > simulated mathematician or community of mathematicians in a computer, the > Godel statement for this system would only be true *if* they never made a > mistake in reasoning or chose to output a false statement to be perverse, > and that therefore there is no way for us on the outside to have any more > confidence about whether they will ever output this statement than they do > (and thus neither of us can know whether the statement is actually a true or > false theorem of arithmetic). > > It's true that on p. 76, Penrose does restrict his conclusions about "The > Godelian Case" to the following statement (which he denotes 'G'): > > "Human mathematicians are not using a knowably sound algorithm in order to > ascertain mathematical truth." > > I have no objection to this proposition on its own, but then in Chapter 3, > "The case for non-computability in mathematical thought" he does go on to > argue (as the chapter title suggest) that this proposition G justifies the > claim that human reasoning must be non-computable. In discussing objections > to this argument, he dismisses the possibility that G might be correct but > that humans are using an unknowable algorithm, or an unsound algorithm, but > as far as I can see he never discusses the possibility I have been > suggesting, that an algorithm that faithfully simulated the reasoning of a > human mathematician (or community of mathematicians) might be both knowable > (in the sense that the beings in the simulation are free to examine their > own algorithm) and sound (meaning that if the simulation is run forever, > they never output a false statement about arithmetic), but just not knowably > sound (meaning that neither they nor us can find a *proof* that will tell us > in advance that the simulation will never output a false statement, the only > way to check is to run it 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". > > >The point is that the axioms are exactly our axioms! > > Again, the "axioms" would be detailed statements about the initial > conditions and behavior of the most basic elements of the simulation--the > initial position and velocity of each simulated molecule along with rules > for the molecules' behavior, perhaps--not the sort of high-level conceptual > axioms we use in our minds when thinking about mathematics. If we can't even > predict whether some very simple cellular automata will ever reach a given > state, I don't see why it should be surprising that we can't predict whether > some very complex physical simulation of an immortal brain and its > environment will ever reach a given state (the state in which it decides to > output the system's Godel statement, whether because of incorrect reasoning > or just out of contrariness). > > > > > >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. > > I am only talking about the feeling of free will which is perfectly > compatible with ultimate determinism > (seehttp://en.wikipedia.org/wiki/Compatibilism), not the philosophical idea of > "libertarian free will" > (seehttp://en.wikipedia.org/wiki/Libertarianism_(metaphysics) ) which requires > determinism to be false. If we had some unerring procedure for predicting > whether other people or even ourselves would make a certain decision in the > future, it's hard to see how we could still have the same subjective sense > of making choices whose outcomes aren't certain until we actually make them. > > Jesse > > _________________________________________________________________http://im.live.com/messenger/im/home/?source=hmtextlinkjuly07- > Hide quoted text - > > - Show quoted text - --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---