Mark, An example of people who would argue with the meaningfulness of classical mathematics: there are some people who contest the concept of real numbers. The cite things like that the vast majority of real numbers cannot even be named or referenced in any way as individuals, since the infinity of real numbers is larger than the infinity of possible names/descriptions.
"OK. But I'm not sure where this is going . . . . I agree with all that you're saying but can't see where/how it's supposed to address/go back into my domain model ;-)" Well, you already agreed that classical mathematics is meaningful. But, you also asserted that you are a constructivist where meaning is concerned, and therefore collapse Godel's and Tarski's theorems. I do not think you can consistently assert both! If the Godelian truths are unreachable because they are undefined, then there is something *wrong* with the classical insistence that they are true or false but we just don't know which. To take a concrete example: One of these truths that suffers from Godelian incompleteness is the consistency of arithmetic. I, being of the classical persuasion, believe that arithmetic is either consistent or inconsistent. You, to the extent that you are a constructivist, should say that the matter is undecidable and therefore undefined. --Abram On Mon, Oct 27, 2008 at 12:04 PM, Mark Waser <[EMAIL PROTECTED]> wrote: > Hi, > > It's interesting (and useful) that you didn't use the word meaning until > your last paragraph. > >> I'm not sure what you mean when you say that meaning is constructed, >> yet truth is absolute. Could you clarify? > > Hmmm. What if I say that meaning is your domain model and that truth is > whether that domain model (or rather, a given preposition phrased in the > semantics of the domain model) accurately represents the empirical world? > > = = = = = = = = >>> >>> I'm a classicalist in the sense that I think classical mathematics needs >>> to be accounted for in a theory of meaning. > > Would *anyone* argue with this? Is there anyone (with a clue ;-) who isn't > a classicist in this sense? > >>> I am also a classicalist in the sense that I think that the >>> mathematically true is a proper subset of the mathematically provable, so >>> that Gödelian truths are not undefined, just unprovable. > > OK. But that is talking about a formal (and complete -- though still > infinite) system. > >>> I might be called a constructivist in the sense that I think there needs >>> to be a tight, well-defined connection between syntax and semantics... > > Agreed but you seem to be overlooking the question of "Syntax and semantics > of what?" > >>> The semantics of an AGI's internal logic needs to follow from its >>> manipulation rules. > > Absolutely. > >>> But, partly because I accept the > > implementability of super-recursive algorithms, I think there is a > chance to allow at least *some* classical mathematics into the > picture. And, since I believe in the computational nature of the mind, > I think that and classical mathematics that *can't* fit into the > picture is literally nonsense! So, since I don't feel like much of > math is nonsense, I won't be satisfied until I've fit most of it in. > > OK. But I'm not sure where this is going . . . . I agree with all that > you're saying but can't see where/how it's supposed to address/go back into > my domain model ;-) > > > > ----- Original Message ----- From: "Abram Demski" <[EMAIL PROTECTED]> > To: <[email protected]> > Sent: Monday, October 27, 2008 11:05 AM > Subject: Re: [agi] constructivist issues > > > Mark, > > I'm a classicalist in the sense that I think classical mathematics > needs to be accounted for in a theory of meaning. (Ben seems to think > that a constructivist can do this by equating classical mathematics > with axiom-systems-of-classical-mathematics, but I am unconvinced.) I > am also a classicalist in the sense that I think that the > mathematically true is a proper subset of the mathematically provable, > so that Godelian truths are not undefined, just unprovable. > > I might be called a constructivist in the sense that I think there > needs to be a tight, well-defined connection between syntax and > semantics... The semantics of an AGI's internal logic needs to follow > from its manipulation rules. But, partly because I accept the > implementability of super-recursive algorithms, I think there is a > chance to allow at least *some* classical mathematics into the > picture. And, since I believe in the computational nature of the mind, > I think that and classical mathematics that *can't* fit into the > picture is literally nonsense! So, since I don't feel like much of > math is nonsense, I won't be satisfied until I've fit most of it in. > > I'm not sure what you mean when you say that meaning is constructed, > yet truth is absolute. Could you clarify? > > --Abram > > On Mon, Oct 27, 2008 at 10:27 AM, Mark Waser <[EMAIL PROTECTED]> wrote: >> >> Hmmm. I think that some of our miscommunication might have been due to >> the >> fact that you seem to be talking about two things while I think that I'm >> talking about third . . . . >> >> I believe that *meaning* is constructed. >> I believe that truth is absolute (within a given context) and is a proper >> subset of meaning. >> I believe that proof is constructed and is a proper subset of truth (and >> therefore a proper subset of meaning as well). >> >> So, fundamentally, I *am* a constructivist as far as meaning is concerned >> and take Gödel's theorem to say that meaning is not completely defined or >> definable. >> >> Since I'm being a constructionist about meaning, it would seem that your >> statement that >>> >>> A constructivist would be justified in asserting the equivalence of >>> Gödel's incompleteness theorem and Tarski's undefinability theorem, >> >> would mean that I was "correct" (or, at least, not wrong) in using Gödel's >> theorem but probably not as clear as I could have been if I'd used Tarski >> since an additional condition/assumption (constructivism) was required. >> >>> So, interchanging the two theorems is fully justifiable in some >>> intellectual circles! Just don't do it when non-constructivists are >>> around :). >> >> I guess the question is . . . . How many people *aren't* constructivists >> when it comes to meaning? Actually, I get the impression that this >> mailing >> list is seriously split . . . . >> >> Where do you fall on the constructivism of meaning? >> >> ----- Original Message ----- From: "Abram Demski" <[EMAIL PROTECTED]> >> To: <[email protected]> >> Sent: Sunday, October 26, 2008 10:00 PM >> Subject: Re: [agi] constructivist issues >> >> >>> Mark, >>> >>> After some thought... >>> >>> A constructivist would be justified in asserting the equivalence of >>> Godel's incompleteness theorem and Tarski's undefinability theorem, >>> based on the idea that truth is constructable truth. Where classical >>> logicians take Godels theorem to prove that provability cannot equal >>> truth, constructivists can take it to show that provability is not >>> completely defined or definable (and neither is truth, since they are >>> the same). >>> >>> So, interchanging the two theorems is fully justifiable in some >>> intellectual circles! Just don't do it when non-constructivists are >>> around :). >>> >>> --Abram >>> >>> On Sat, Oct 25, 2008 at 6:18 PM, Mark Waser <[EMAIL PROTECTED]> wrote: >>>> >>>> OK. A good explanation and I stand corrected and more educated. Thank >>>> you. >>>> >>>> ----- Original Message ----- From: "Abram Demski" >>>> <[EMAIL PROTECTED]> >>>> To: <[email protected]> >>>> Sent: Saturday, October 25, 2008 6:06 PM >>>> Subject: Re: [agi] constructivist issues >>>> >>>> >>>>> Mark, >>>>> >>>>> Yes. >>>>> >>>>> I wouldn't normally be so picky, but Godel's theorem *really* gets >>>>> misused. >>>>> >>>>> Using Godel's theorem to say made it sound (to me) as if you have a >>>>> very fundamental confusion. You were using a theorem about the >>>>> incompleteness of proof to talk about the incompleteness of truth, so >>>>> it sounded like you thought "logically true" and "logically provable" >>>>> were equivalent, which is of course the *opposite* of what Godel >>>>> proved. >>>>> >>>>> Intuitively, Godel's theorem says "If a logic can talk about number >>>>> theory, it can't have a complete system of proof." Tarski's says, "If >>>>> a logic can talk about number theory, it can't talk about its own >>>>> notion of truth." Both theorems rely on the Diagonal Lemma, which >>>>> states "If a logic can talk about number theory, it can talk about its >>>>> own proof method." So, Tarski's theorem immediately implies Godel's >>>>> theorem: if a logic can talk about its own notion of proof, but not >>>>> its own notion of truth, then the two can't be equivalent! >>>>> >>>>> So, since Godel's theorem follows so closely from Tarski's (even >>>>> though Tarski's came later), it is better to invoke Tarski's by >>>>> default if you aren't sure which one applies. >>>>> >>>>> --Abram >>>>> >>>>> On Sat, Oct 25, 2008 at 4:22 PM, Mark Waser <[EMAIL PROTECTED]> >>>>> wrote: >>>>>> >>>>>> So you're saying that if I switch to using Tarski's theory (which I >>>>>> believe >>>>>> is fundamentally just a very slightly different aspect of the same >>>>>> critical >>>>>> concept -- but unfortunately much less well-known and therefore less >>>>>> powerful as an explanation) that you'll agree with me? >>>>>> >>>>>> That seems akin to picayune arguments over phrasing when trying to >>>>>> simply >>>>>> reach general broad agreement . . . . (or am I misinterpreting?) >>>>>> >>>>>> ----- Original Message ----- From: "Abram Demski" >>>>>> <[EMAIL PROTECTED]> >>>>>> To: <[email protected]> >>>>>> Sent: Friday, October 24, 2008 5:29 PM >>>>>> Subject: Re: [agi] constructivist issues >>>>>> >>>>>> >>>>> >>>>> >>>>> ------------------------------------------- >>>>> 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/?&; >>>> 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/?&; >>> 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/?&; >> 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/?&; > 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/?&; > 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
