Charles, It might be off-track here, but it would be perfectly on-track in the "agi-philosophy" list that Ben might eventually split off of this one.
But, thanks, that clarifies what you were saying greatly. --Abram On Mon, Nov 3, 2008 at 10:50 PM, Charles Hixson <[EMAIL PROTECTED]> wrote: > That's a lot stronger and more interesting that the theories that I was > referring to. Also a lot more complex. > **This is getting way off topic, so the rest should probably be ignored.** > > One of the theories that I was referring to contained only 0 and a rule that > given a number allowed you to construct the successor to that number. It > clearly couldn't prove it's own consistency, but given enough time and > effort it could clearly [from external observation] generate all finite > integers. If that were your theory, then just about all you could say about > a number was what it's successor was and possibly what it's predecessors > were...though you couldn't do that latter within the theory itself. > > My point was that numbers defined under that theory don't HAVE any other > characteristics. They may be equivalent in some sense to numbers defined > under Number Theory that were generated in an equivalent way, but they don't > have the extra characteristics that the Number Theory numbers have. And > this is because numbers aren't characteristics of the universe, but rather > particular abstractions from various characteristics of the universe. And > if I adopt a constructivist stance, then only a finite number of numbers > exist, in fact precisely those numbers that have been generated. I don't > happen to think that this is the best approach, because I think that it > arises out of an attempt to reify numbers, but it has it's value in some > cases. > > In a way this is like my argument against existentialism. I assert that a > true existentialist couldn't walk across the room, because he couldn't be > sure that the floor would exist when he took a step. Nobody is that > complete an existentialist, and there is merit in the existentialist > stance...but it's not the one that is usually described. > > The way this becomes important is that all simple representations of numbers > within a computer are inherently finite. (I'm saying simple to avoid things > like lazily evaluated functions which given a transfinite amount of time, > RAM, and energy could generate transfinite numbers...and so by existing in > an unevaluated state can be said to represent said transfinite numbers.) > I.e., for computers constructivist theories describe what's actually > possible, though typically they don't impose strict enough restrictions to > serve that function, they COULD do so. But it's not clear to me that this > is an appropriate approach, even though it models simply onto the > physicality of the situation. In a way it reminds me of the arguments that > used to be raised between the assembler programmers and the compiler > language programmers. Compiler languages use a more abstract > representation, but they require more system resources, and you can do > anything in assembler that's actually possible. But there are good reasons > why the more abstract and general choice is more commonly made. But for > some purposes there's really no choice but to actually understand things at > an assembler level. Similarly with the more abstract math and the > constructivist approach. The constructivists are correct about what we can > actually know and count and be certain of. But they aren't right when they > claim that this is generally the appropriate stance to take towards math. > Doing that is like deriving planetary motions from quantum equations. > > Sorry for going so off-track. > > Abram Demski wrote: >> >> Charles, >> >> defining "formal system" is just as difficult as defining "number"-- >> in fact, those problems are basically equivalent. Godel made use of >> that to make his proof work (or at least that's one way of looking at >> it). So, if you claim that the concept "number" is dependent on what >> formal system it is defined in, shouldn't you also say the same thing >> of "formal system"? >> >> So, for example, we might agree to interpret "number" as >> "Peano-arithmatic number" for the purpose of some discussion. But, we >> might still disagree on how to interpret Peano Arithmetic. I might >> say, "statement X isn't derivable from PA, by transfinite induction", >> and you might reply, "Hey, no, you're not allowed to use transfinite >> induction." Then we would need to settle which logic to interpret PA >> in: maybe you convince me that we've got to stick to robinson >> arithmetic. But then, I use some line of reasoning about the >> properties of Robinson arithmetic, and we've got to settle on an even >> higher system to resolve the argument... >> >> The point is, we've got to make some actual choice of default at some >> point. By arguing with anything I want to at every level, I'm using >> the classical-type default, which is essentially to use the strongest >> system available. Perhaps you would choose one of these logics that >> Godel's theorem fails for as your default. If we can figure out what >> default is normatively ideal, then in my opinion we've made important >> headway. >> >> I think I found the logics you're referring to? Looks *very* interesting. >> >> http://en.wikipedia.org/wiki/Self-verifying_theories >> >> --Abram >> >> On Fri, Oct 31, 2008 at 2:26 AM, Charles Hixson >> <[EMAIL PROTECTED]> wrote: >> >>> >>> It all depends on what definition of number you are using. If it's >>> constructive, then it must be a finite set of numbers. If it's based on >>> full Number Theory, then it's either incomplete or inconsistent. If it's >>> based on any of several subsets of Number Theory that don't allow >>> incompleteness to be proven (or even described) then the numbers are >>> precisely this which is included in that subset of the theory. >>> >>> Number Theory is the one with the largest (i.e., and infinite number) of >>> unprovable theories about numbers of the variations that I have been >>> considering. My point in the just prior post is that numbers are >>> precisely >>> that item which the theory you are using to describe them says they are, >>> since they are artifacts created for computational convenience, as >>> opposed >>> to direct sensory experiences of the universe. >>> >>> As such, it doesn't make sense to say that a subset of number theory >>> leaves >>> more facts about numbers undefined. In the subsets those aren't facts >>> about >>> numbers. >>> >>> Abram Demski wrote: >>> >>>> >>>> Charles, >>>> >>>> OK, but if you argue in that manner, then your original point is a >>>> little strange, doesn't it? Why worry about Godelian incompleteness if >>>> you think incompleteness is just fine? >>>> >>>> "Therefore, I would assert that it isn't that it leaves "*even more* >>>> about numbers left undefined", but that those characteristics aren't >>>> in such a case properties of numbers. Merely of the simplifications >>>> an abstractions made to ease computation." >>>> >>>> In this language, what I'm saying is that it is important to examine >>>> the "simplifications and abstractions", and discover how they work, so >>>> that we can "ease computation" in our implementations. >>>> >>>> --Abram >>>> >>>> On Thu, Oct 30, 2008 at 7:58 PM, Charles Hixson >>>> <[EMAIL PROTECTED]> wrote: >>>> >>>> >>>>> >>>>> If you were talking about something actual, then you would have a valid >>>>> point. Numbers, though, only exist in so far as they exist in the >>>>> theory >>>>> that you are using to define them. E.g., if I were to claim that no >>>>> number >>>>> larger than the power-set of energy states within the universe were >>>>> valid, >>>>> it would not be disprovable. That would immediately mean that only >>>>> finite >>>>> numbers were valid. >>>>> >>>>> P.S.: Just because you have a rule that could generate a particular >>>>> number >>>>> given a larger than possible number of steps doesn't mean that it is a >>>>> valid >>>>> number, as you can't actually ever generate it. I suspect that >>>>> infinity >>>>> is >>>>> primarily a computational convenience. But one shouldn't mistake the >>>>> fact >>>>> that it's very convenient for meaning that it's true. Or, given >>>>> Occam's >>>>> Razor, should one? But Occam's Razor only detects provisional truths, >>>>> not >>>>> actual ones. >>>>> >>>>> If you're going to be constructive, then you must restrict yourself to >>>>> finitely many steps, each composed of finitely complex reasoning. And >>>>> this >>>>> means that you must give up both infinite numbers and irrational >>>>> numbers. >>>>> To do otherwise means assuming that you can make infinitely precise >>>>> measurements (which would, at any rate, allow irrational numbers back >>>>> in). >>>>> >>>>> Therefore, I would assert that it isn't that it leaves "*even more* >>>>> about >>>>> numbers left undefined", but that those characteristics aren't in such >>>>> a >>>>> case properties of numbers. Merely of the simplifications an >>>>> abstractions >>>>> made to ease computation. >>>>> >>>>> > > > > ------------------------------------------- > agi > Archives: https://www.listbox.com/member/archive/303/=now > RSS Feed: https://www.listbox.com/member/archive/rss/303/ > Modify Your Subscription: > https://www.listbox.com/member/?&; > Powered by Listbox: http://www.listbox.com > ------------------------------------------- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244&id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
