July 14, 2006 7:10 AM   Subscribe

I just finished Rebecca Goldstein's (excellent) book Incompleteness. If I'm understanding this correctly, Godel's theorems were instrumental in ending logical postitivism, by showing that the movement could never produce a mathematical system which defines arithmetic while being complete and self-consistent. So what about Intuitionism? Isn't that nominalist as well, and subject to the same logical inconsistencies? How do they answer this? (Also, any relevant book suggestions would be appreciated)
posted by rottytooth to Religion & Philosophy (18 answers total) 2 users marked this as a favorite
Godel is wrong. Just going on the record.

You might have read 5 Golden Rules, there's a sequel out. I really like the stuff on topography and how it can be used for problems that aren't spacial at all...
posted by ewkpates at 7:39 AM on July 14, 2006

I'm a bit out of my depth here, but as I remember it, Intuitionism doesn't accept the idea of a "completed" infinity - ie, you can talk about arbitrarily large numbers, but not "the natural numbers" as an entirety.

Now thinking about the axioms for arithmetic in first order logic, you need an axiom schema (ie an infinite list) of axioms of the form (if P is a first order definable property then (if P0 and Pn => Pn+1 then ForAll n in N Pn)). I don't think that schema will stand up in a strict intuitionist logic so you can't even get far enough to talk about Godel's theorem.

Someone who knows more than me will probably show me up in a minute.
posted by crocomancer at 8:01 AM on July 14, 2006

Do you mean instrumental in ending logicism or logical positivism?
posted by ontic at 8:14 AM on July 14, 2006

Best answer: I just did a quick search on this, and the most interesting result I found was here. Scroll down to 2. Godel's '31, Constructively.

The money quote is In summary, Brouwer and the Constructivists will assert to [a Godel sentence] as a
syntactical result, but will not proceed further because of a dispute on
effective meaning. The Logistic Intuitionists can be persuaded through
reductio ad absurdum to an undecidable result but will not agree to
truth without proof.

posted by crocomancer at 8:15 AM on July 14, 2006

godel is wrong? care to elaborate?

godel's result only applies to first order logic. there may be some other logic system that is both complete and consistent, just like there may be a formal model of computation where the halting problem is decidable.
posted by paradroid at 8:21 AM on July 14, 2006

Godel is wrong. Just going on the record.

As a prediction, or as a statement? I've never heard anything about it losing favor.
posted by Hildago at 8:27 AM on July 14, 2006

Paradroid - sorry to be picky. First order arithmetic is incomplete. First order logic is (provably) complete and consistent.
posted by crocomancer at 8:37 AM on July 14, 2006

Best answer: ewkpates is merely advertising his ignorance.

Working within the formalization effected by Heyting, Gödel was able to demonstrate that the intuitionistic arithmetic contained the whole of classical arithmetic under an interpretation, differing from the usual one, but nevertheless adequate for the realization that the two arithmetics are equiconsistent. Thus Gödel showed that the intuitionistic arithmetic [31] was not "narrower" than classical arithmetic, and was not safer either. Whereas consistency was not the primary concern of the intuitionists,[32] it might have been believed that because the intuitimistic arithmetic seemed narrower, it was less likely to be contradictory.[33]
posted by gleuschk at 8:39 AM on July 14, 2006

godel's incompleteness result is exactly the statement that first order logic is either incomplete or inconsistent. it is often expressed in the easier to understand form that any (first order) formal system which is sufficiently complex as to include arithmetic is incomplete.

and first order arithmetic is obviously a set of axioms in the first order logic.
posted by paradroid at 8:46 AM on July 14, 2006

I think "a dispute on effective meaning" sums it up.

If I define a set of everything, call it set A, and somebody, say gleuschk, comes along and says, "Nu-uh, because I define set B, those things not contained in set A" that doesn't make him right and me wrong.

Saying something doesn't make it 1) real; 2) rational; or 3) meaningful.

"This sentence is false." Cunning conundrum? I say no, just meaningless noise.
posted by ewkpates at 9:32 AM on July 14, 2006

MathWorld, Wikipedia on Goedel's Incompleteness Theorem.

Ewkpates, I spent a whole semester in grad school in a logic course learning the foundation to express and understand Goedel's Incompleteness Theorem. If you're dismissing it as sophistry, I gotta go with gleuschk: you're just advertising your ignorance.

I beg the OP's forgiveness in prolonging a derail. As penance, here's a a link to some of Goedel's own work on intuitionist logic including proof of the equiconsistency of intuitionist and classical logic.
posted by Zed_Lopez at 10:15 AM on July 14, 2006

Ignore ewkpates. Gleuschk has the good citation for this -- Goedel's Dialectica interpretation of formalism is a good window to come at this from. Check out chapter 5 in the 1998 edition of the Handbook of Proof Theory (ed. Samuel Buss) for some better detail. Also Brouwer's essay "Intuitionism and Formalism" (in a lot of collections -- I know offhand it's in Benacerraf and Putnam) is a good cite for info on the relative merit of consistency for the intuitionists.
posted by j.edwards at 10:21 AM on July 14, 2006

s/formalism/intuitionism. What a mistake.
posted by j.edwards at 10:22 AM on July 14, 2006

Anyone who doesn't have at least two problems with this paragraph should probably just wait for the Rapture. It's due any day now.

Philosophically, intuitionism differs from logicism by treating logic as a part of mathematics rather than as the foundation of mathematics; from finitism by allowing (constructive) reasoning about infinite collections; and from platonism by viewing mathematical objects as mental constructs with no independent ideal existence.

Newton knew we had problems but couldn't fix them... that doesn't mean that all your crazy talk about infinities and "that's just semantics" gets you a gold star...
posted by ewkpates at 11:18 AM on July 14, 2006

Seriously though, there are some real unresolved problems in mathematics. These problem may be (probably are) corrupting science and physics, and creating fractured systems of logic.

Anyone who tells you different is selling something... (maybe it's a semester of graduate level logic).
posted by ewkpates at 11:35 AM on July 14, 2006

I am one of those who refuse to be expelled "from the Paradise Cantor has created for us," but I do sometimes find myself sniffing the air of sanctity hanging about Goedel thinking to detect just the faintest odor of an unburied corpse.

Here is the most persuasive and readable polemic against set theory and the idea of completed infinity I have run into so far.
posted by jamjam at 12:11 PM on July 14, 2006 [1 favorite]

godel's incompleteness result is exactly the statement that first order logic is either incomplete or inconsistent. it is often expressed in the easier to understand form that any (first order) formal system which is sufficiently complex as to include arithmetic is incomplete.

Sorry, no. Those are two different claims, and the first of them is false. In fact, Gödel's Completeness Theorem shows that first order logic is complete (that is, any valid first-order statement is provable). Gödel's Incompleteness Theorem shows that any first-order theory that is strong enough to capture arithmetic is either incomplete (i.e. some statements of arithmetic are neither provable nor disprovable in that theory) or inconsistent.

The completeness theorem is about first-order logic. The incompleteness theorem is about the axiomatization of arithmetic (in first-order logic).
posted by klausness at 11:23 AM on July 15, 2006

Response by poster: Anyone who's interested in this topic: It looks like the current issue of Philosophia Mathematica is all Godel, including a piece on his interpretation of Intuitionism.
posted by rottytooth at 6:27 PM on July 16, 2006

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