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: <agi@v2.listbox.com> > 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: <agi@v2.listbox.com> >>> 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: <agi@v2.listbox.com> >>>>> 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/?member_id=8660244&id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
