> I'm not sure that I'm responding to your intended meaning, but: all > computers are in reality finite-state machines, including the brain > (granted we don't think the real-number calculations on the cellular > level are fundamental to intelligence). However, the finite state > machines we call PCs are so large that it is convenient to pretend > they have infinite memory; and when we do this, we get a machine that > is equivalent in power to a Turing machine. But a turing machine has > an infinite tape, so it cannot really exist (the real computer > eventually runs out of memory). Similarly, I'm arguing that the human > brain is so large in particular ways that it is convenient to treat it > as an even more powerful machine (perhaps an infinite-time turing > machine), despite the fact that such a machine cannot exist (we only > have a finite amount of time to think). Thus a "spurious infinity" is > not so spurious.
Abrahm, Thanks for responding. You know, i might be in a bit over my head with some of the terminology in your paper, so to apologize in advance, but just to clarify: "spurious infinity" according to Hegel is the sleight of hand the happens when quantity transitions surreptiously into a quality. At some point counting up, we are simply not talking about any number at all, but about a quality of being REALLY SUPER BIG as we make kind of a leap. According to him when we talk about infinity we are talking about some idea of a huge number (in this case of calculations) and to use a phrase he liked: "imaginary being." So since I am kind of a Hegelian of sorts when I scanned the paper it looked like it argued that it is not possible to compute something that I had become convinced was imaginary anyway. That would be true if you bought into Hegel's definition of infinity and I realize there aren't a log of hegelians around. But, tomorrow I will read further. Mike > > On Mon, Jun 16, 2008 at 9:19 PM, Mike Archbold <[EMAIL PROTECTED]> wrote: >>> I previously posted here claiming that the human mind (and therefore an >> ideal AGI) entertains uncomputable models, counter to the >>> AIXI/Solomonoff model. There was little enthusiasm about this idea. :) >> Anyway, I hope I'm not being too annoying if I try to argue the point >> once again. This paper also argues the point: >>> >>> http://www.osl.iu.edu/~kyross/pub/new-godelian.pdf >>> >> >> It looks like the paper hinges on: >> "None of this prior work takes account of G¨odel intuition, repeatedly >> communicated >> to Hao Wang, that human minds "converge to infinity" in their power, and >> for this reason >> surpass the reach of ordinary Turing machines." >> >> The thing to watch out for here is what Hegel described as the "spurious >> infinity" which is just the imagination thinking some imaginary quantity >> really big, but no matter how big, you always can envision "+1", but the >> result is always just another imaginary big number, to which you can add >> another "+1"... the point being that infinity is a idealistic quality, >> not >> a computable numeric quantity at all, ie., not numerical, we are talking >> about thought as such. >> >> I didn't read the whole paper, but the point I wanted to make was that >> Hegel takes up the issue of infinity in his Science of Logic, which I >> think is a good ontology in general because it mixes up a lot of issues >> AI >> struggles with, like the ideal nature of quality and quantity, and also >> infinity. >> >> Mike Archbold >> Seattle >> >> >>> The paper includes a study of the uncomputable "busy beaver" function >>> up >> to x=6. The authors claim that their success at computing busy beaver >> strongly suggests that humans can hypercompute. >>> >>> I believe the authors take this to imply that AGI cannot succeed on >> current hardware; I am not suggesting this. Instead, I offer a fairly >> concrete way to make deductions using a restricted class of >>> uncomputable models, as an illustration of the idea (and as weak >> evidence that the general case can be embodied on computers). >>> >>> The method is essentially nonmonotonic logic. Computable predicates can >> be represented in any normal way (1st-order logic, lambda >>> calculus, a standard programming language...). Computably enumerable >> predicates (such as the halting problem) are represented by a default >> assumption of "false", plus the computable method of enumerating true >> cases. To reason about such a predicate, the system allocates however >> much time it can spare to trying to prove a case true; if at the end of >> that time it has not found a proof by the enumeration method, it >> considers it false. (Of course it could come back later and try >>> harder, too.) Co-enumerable predicates are similarly assumed true until >> a counterexample is found. >>> >>> Similar methodology can extend the class of uncomputables we can handle >> somewhat farther. Consider the predicate "all turing machines of class N >> halt", where N is a computably enumerable class. Neither the true cases >> nor the false cases of this predicate are computably enumerable. >> Nonetheless, we can characterize the predicate by assuming it is true >> until a counterexample is "found": a turing machine that doesn't seem to >> halt when run as long as we can afford to run it. If our best efforts >> (within time constraints) fail to find such a >>> machine, then we stick with the default assumption "true". (A >>> simplistic nonmonotonic logic can't quite handle this: at any stage of >> the search, we would have many turing machines still at their default >> status of "nonhalting", which would make the predicate seem >>> always-false; we need to only admit assumptions that have been >>> "hardened" by trying to disprove them for some amount of time.) >>> >>> This may sound "so easy that a Turing machine could do it". And it is, >> for set cutoffs. But the point is that an AGI that only considered >> computable models, such as AIXI, would never converge to the correct >> model in a world that contained anything uncomputable, whereas it seems >> a human could. (AIXI would find turing machines that were >>> ever-closer to the right model, but could never see that there was a >> simple pattern behind these ever-larger machines.) >>> >>> I hope that this makes my claim sound less extreme. Uncomputable models >> of the world are not really so hard to reason about, if we're >> comfortable with a logic that makes probably-true conclusions rather >> than definitely-true. >>> >>> >>> ------------------------------------------- >>> agi >>> Archives: http://www.listbox.com/member/archive/303/=now >>> RSS Feed: http://www.listbox.com/member/archive/rss/303/ >>> Modify Your Subscription: >>> http://www.listbox.com/member/?&; >> Powered by Listbox: http://www.listbox.com >>> >> >> >> >> >> >> >> ------------------------------------------- >> agi >> Archives: http://www.listbox.com/member/archive/303/=now >> RSS Feed: http://www.listbox.com/member/archive/rss/303/ >> Modify Your Subscription: http://www.listbox.com/member/?&; >> Powered by Listbox: http://www.listbox.com >> > > > ------------------------------------------- > agi > Archives: http://www.listbox.com/member/archive/303/=now > RSS Feed: http://www.listbox.com/member/archive/rss/303/ > Modify Your Subscription: > http://www.listbox.com/member/?&; > Powered by Listbox: http://www.listbox.com > ------------------------------------------- agi Archives: http://www.listbox.com/member/archive/303/=now RSS Feed: http://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: http://www.listbox.com/member/?member_id=8660244&id_secret=106510220-47b225 Powered by Listbox: http://www.listbox.com
