Yeah! And while we're at it, can we declare a moratorium on the misuse of the Many Worlds Hypothesis in science fiction stories?

I will say, though, that while Goedel's theorem doesn't say "there are true things that can't be proved" any more than Newton or Darwin's theories say "there is no personal God of history", by spelling the end of a certain kind of mathematical ambition for formal completeness (as characterized by Whitehead and Russell), Goedel does have a checking effect on philosophical claims of logical positivism. I don't think it's unfair to say that Goedel's theorem suggests that mathematics is a story we tell -- more rigorous and perfect than language, but not perfectly rigorous and perfect. A certain dream, dating from Euclid, did die with Goedel, and even if the practical consequences of this are minor, the aesthetic consequences matter.

And aesthetics matter fundamentally in science. We went from the Ptolemaian to the Copernican model, from Lorenz hacks on Newton to special relativity to general relativity, because of beauty. After Newton and Darwin, you can still legitimately believe in a personal creator-God of history... but it's clunkier somehow -- you can't really claim it should be obvious to everyone. After Goedel, you can still believe science one day will explain everything important -- but the notion has lost a little of its luster.

Even though the findings in question don't actually disprove the notions they partially supplant, it's not foolish to allow your worldview to be influenced by intuition and aesthetics, as well as logic.

Actually, I’m fine with abuse of the many-worlds hypothesis so long as the authors concerned are clear that what they’re writing is allegory. But then I think all the classics of old-school hard SF are allegorical, too.

But yeah, I agree that aesthetics matter. Gödel changed the way the math and philosophy fields look at themselves, whether they realize it or not.

Goedel does have a checking effect on philosophical claims of logical positivism

I don't really see this. Logical positivism had problems all its own, completely apart from Godel's work.

I don't think it's unfair to say that Goedel's theorem suggests that mathematics is a story we tell

I'm not sure how that follows. Do you mean that if mathematics weren't a story we tell, it would be able to prove its own consistency? Why should that be the case?

--> Ben: Goedel does have a checking effect on philosophical claims of logical positivism

--> Ted: I don't really see this. Logical positivism had problems all its own, completely apart from Godel's work.

Recall that I am not talking about a logical checking effect, but a social checking effect. (I am also probably using the term "logical positivism" loosely or incorrectly).

The fact that math couldn't be a self-proving system isn't really a problem for math as math. In some ways it's even more beautiful and mysterious than the competing ideal, from Euclid to Russell and Whitehead, of math as a tower, starting with simple axioms and expanding outward by irrefutable logic to prove everything which is strictly true and can be said in the language of math about the things of math.

But the end of that old idea -- along with Planck and company suggesting that God does play dice with the universe, and Freud, and jazz and Picasso -- had the effect, not of discrediting *logically* the idea of a world about which all essential things were knowable through experiment and unbiased observation, but of making it less cool.

--> Ben: I don't think it's unfair to say that Goedel's theorem suggests that mathematics is a story we tell

--> Ted: I'm not sure how that follows. Do you mean that if mathematics weren't a story we tell, it would be able to prove its own consistency? Why should that be the case?

I guess it depends on what else, other than a story, you think math might be.

If math were a machine for generating all math-phrasable truths, and the consistency of math was a math-phrasable truth, then it follows that math could prove its own consistency.

If math is either a machine for generating some-but-not-all math-phrasable truths, or, if the consistency of math is not a math-phrasable truth... well, I'd say that is a little closer to a story than the machine for generating all math-phrasable truths.

It is a property of stories that they leave things out, they pick and choose among truths. It is another property of stories that the truth they incorporate is not an entirely rigorous and precise set of well-defined, generic signs upon which operations can be performed -- that its truths are not math-phrasable. (For instance, how they are expressed changes them).

Do *you* think math is a story we tell? A creation of our own minds? Or do you think it is out there, preexisting, waiting to be discovered?

I'm not actually decided on this point myself, and I think the question, though illuminating, may ultimately be wrong-headed. I think if we could say with any precision just why it is wrong-headed, though, we would have come a long way.

(I am also probably using the term "logical positivism" loosely or incorrectly).

Okay. I use "logical positivism" to refer to the philosophical movement that put forth the verifiability theory of meaning. It is this movement that had its own problems, completely apart from Godel.

But the end of that old idea -- along with Planck and company suggesting that God does play dice with the universe, and Freud, and jazz and Picasso -- had the effect, not of discrediting *logically* the idea of a world about which all essential things were knowable through experiment and unbiased observation, but of making it less cool.

Making it less cool? You mean Godel changed people's idea of what was fashionable? Maybe; I don't know how widely popularized Godel's work was at the time. But in terms of fashion, misinterpretations of Godel's work are far more important than the work itself; it wouldn't really matter what Godel proved at all. Similarly, the idea that "Einstein proved that everything is relative" would be what's important, rather than Einstein's actual theory of relativity.

If math is either a machine for generating some-but-not-all math-phrasable truths, or, if the consistency of math is not a math-phrasable truth... well, I'd say that is a little closer to a story than the machine for generating all math-phrasable truths.

Hmmm. Propositional calculus ("if p and (if p then q), then q") has been shown to be complete and consistent. So does that mean that propositional calculus is less like a story than arithmetic?

It is a property of stories that they leave things out, they pick and choose among truths.

Beavers can hide things in the lodges within their dams, while ducks live in the open. But I wouldn't say arithmetic is more like a beaver and propositional calculus is more like a duck, unless we're speaking in the most metaphorical context.

[In all honesty, by the way, I should admit that I can't really follow the math of Goedel's proof (or I never have, anyway), though I can follow the Turing's proof about the Halting Problem (or did once, anyway), and I hear that they're isomorphic.]

But in terms of fashion, misinterpretations of Godel's work are far more important than the work itself; it wouldn't really matter what Godel proved at all.

An excellent point, but there are different levels to fashion. There are fashions within science and math. Some of the fashions that affect the people doing work that will win them a Nobel prize or a Fields medal are different from, though they interact with, what's fashionable to chat about in the cafes. (The influence runs both ways, but the realms are separate enough that neither world wholly dominates the other.)

So I think both the misunderstandings outside of math, and the actual end of Hilbert's program within math, had an effect.

So does that mean that propositional calculus is less like a story than arithmetic?

Hmm....

and David writes:

Salon’s running another review of Incompleteness, and the way they tell it is almost the opposite of the way you do, Ben: That if mathematics were just a story we tell (or, in Hilbert’s language, a game we play), then it should be complete and consistent.

Very interesting article. Now I really want to read the book! And I begin to wonder to what extent I have indeed been making the error it describes, maybe in a subtler way that I thought we were talking about.

(I have a feeling the article oversimplifies a lot, though, about at least Einstein's philosophy of science; I understand him to be highly influenced by Mach's idea of science as a human artifact. It's clear that Einstein was passionate about science's aesthetic, and that he felt that it spoke directly to a real physical reality and its real nature; but I suspect [on the basis of very scanty evidence, I admit] that he also believed, in a Machian sense, that no human theory could actually really capture the way the world is. One can "get closer to the Old One", but one cannot -- pace Weinberg -- get to Him. Consider how he refers to his entire contribution to physics as "my castle in the air". But maybe I'm projecting).

Okay, so...I think we suffer here from a mixed and abused metaphor. The question is, what do we mean by "story"?

If by story you mean "human artifact" -- if you mean "story versus world" -- then propositional logic is more like a story (simple human creation) than arithmetic, and Goedel's proof suggests that math is not just a story (formal language game).

But what I meant was actually "story versus game" -- story as messy, complex, intuitive human artifact, as opposed to simple, clean, formalistic, sterile human artifact. As I said, math is "more rigorous and perfect than language, but not perfectly rigorous and perfect".

Goedel -- if you believe Salon -- regarded his proof as vanquishing Hilbert's program's threat to the role of intuition (for which I adore Goldstein's phrase: "the urgent cogency that compels belief") in mathematics. That's what I was trying to get at with "story" -- and propositional logic (or the rules of chess, as the article suggests) was actually exactly what I meant to contrast with "story". Does propositional logic require any intuition?

Stories are messy, requiring of intuition. They are not the product of a clean, closed system in which everything is well-defined, complete, and correct, in which truths can be enumerated algorithmically.

Propositional logic is sterile, reliable, and perfect. It is not messy. Hilbert wanted to make all math like propositional logic. Goedel proved (and this, too, I know, is probably an oversimplification) that it was messy -- gloriously messy.

Unlike (according to Salon) Goedel, I am not a Platonist. In fact, not only do I believe math is a human artifact, I'm a Machian and, IIRC his postition, a Kuhnian -- I believe science is a human artifact.

(If that makes me a postmodernist, O Salon, so be it, though my epistemology is actually that of Gadamer, [wherein we can know the world, but only circuitously and always through the screen of our biases], rather than that of, say, Derrida, [where -- though it's always perilous to summarize Derrida -- the notion of a world exterior to language is problematized?].

And why is there no Wikipedia article on Gadamer? Anyway...)

So I don't take Goedel's proof as establishing that math is a discovery of real trans-empirical objects, rather than a way of telling a story about them. But I do take it to mean that the story that math is, is a rich and messy thing, requiring all of our capacities, irreduceable to something clean and simple. That's, in fact, what *I* meant by "story".

(That's what I want my stories to have: "the urgent cogency that compels belief". Not belief as a trivial consequence of predefined operations -- belief as a response of the heart, compelled by an urgent cogency!)

(I do mean "story" here as a metaphor, obviously -- there are still many properties post-Goedel mathematical theorems have that stories in English do not have, of course).

It's a fast and loose kind of thinking, I know, but I can't help seeing this drama of mathematics from a feminist perspective -- as being about what's under control, and what's wild. And I think it's in that sense that I broadly (and, now that I look into it, egregiously) misused the term "logical positivism" to refer to the whole program of bringing the theory of the world -- of the empirically observable world as well as the world of the mind -- under control. There's a way in which the Hilbert program would have brought math under control. There's a way in which Goedel's theorem keeps it wild.

(Chaitin is interesting in this context, too, by the way, arguing as he does that

Information theory suggests that the Gödel phenomenon is natural and widespread, not pathological and unusual. Strangely enough, it does this via counting arguments, and without exhibiting individual assertions which are true but unprovable!

)

[In all honesty, by the way, I should admit that I can't really follow the math of Goedel's proof (or I never have, anyway), though I can follow the Turing's proof about the Halting Problem (or did once, anyway), and I hear that they're isomorphic.]

My understanding is that they aren't so much isomorphic as that given the latter, the former can be easily proven. In a way, Turing's proof is a more powerful result.

There are fashions within science and math.

True. But I don't know that Godel's work made the pursuit of Hilbert's program *unfashionable*. It proved pretty conclusively that Hilbert's program was unachievable. I think that constitutes a pretty strong reason to stop pursuing it, fashion aside. So in that context, Godel's work had a logical checking effect, rather than a social checking effect.

But what I meant was actually "story versus game" -- story as messy, complex, intuitive human artifact, as opposed to simple, clean, formalistic, sterile human artifact.

Can you explain what you mean by "game"? Are all games simple, clean, formalistic, and sterile? Are we talking about Monopoly, chess, Go, poker, iterated Prisoner's Dilemma? Do none of them require intuition?

The "story vs game" distinction you appear to be using is one I've never seen before. In the past I've seen a "story vs game" distinction made in the context of computer game design, but I'm fairly sure that's not what you mean.

I get the feeling that you're speaking in highly metaphorical terms, and that's fine. But given that the original context of this discussion was about people confusing the metaphorical for the literal, I guess I would have preferred it if you had actually said you were being metaphorical.

From the Salon review:

Goldstein uses the very helpful metaphor of chess to illuminate this distinction. The game of chess has a complex and unified set of rules, but it isn't about anything but itself. As magnificently as they function, the rules of chess are a set of entirely man-made ideas and don't describe anything in the real, material world. Likewise, according to the positivists (and, in this instance, Wittgenstein), much of what philosophy had traditionally concerned itself with -- abstract questions about the nature of being, reality and God -- were "meaningless" outside the closed, man-made system of language (including the language of mathematics)....

...Gödel's faith in mathematical intuition put him in opposition to the mathematical equivalent of positivism, a movement called formalism. Formalism, led by the mathematician David Hilbert, believed that mathematics was, in Hilbert's words, "a game played according to certain simple rules with meaningless marks on paper." Or, as Goldstein describes it, "mathematicians, according to formalism, are not in the business of discovering descriptive truths, whether of the real world of things in physical space or the trans-empirical world of numbers and sets ... They are simply in the business of manipulating the mechanical rules of self-enclosed formal systems."

I think the distinction Ben's drawing between "stories" and "games" could as easily be drawn between "open games" and "closed games", or something of the sort. Iain Banks' protagonist Jernau Gurgeh, game-player extraordinaire, in The Player of Games, dismisses as lacking interest "closed games" like chess in which there's no element of randomness and it's theoretically possible to mechanistically play a perfect game . . . though another character suggests to Gurgeh that at heart he's not a game-player but a gambler.

Huh.

I find it very fruitful debating with you, Ted, because you demand so much rigor of me.

I guess game is the wrong word, too, since lots of games do require intuition.

What I was trying to say -- and I'm no longer even sure if it's true -- was something like this: Goedel proved math was messy -- that you couldn't do a thing (one very particular and specific thing) with it that would have made it feel tame and neat and under control. This may have had a broader effect than just its specific logical effect of ending Hilbert's program. It may have caused people to look at math, and by extension the whole edifice of systematic reason (from logic, to systemic philosophy, to natural science), in a different way.

The article argued that uneducated people who couldn't follow Goedel's proof, got the feeling "Math is dead! The world is chaos! Long live relativism and emotion!". The author of the article was appalled at this distortion.

But I think people who actually could follow the proof may have legitimately gotten the feeling -- the *feeling*, mind you, the emotional resonance, not the logical certainty -- "math is more mysterious, more uncertain, than we thought; math (and by extension, reason) cannot be put into a box".

That's all I was saying; I was defending Goedel's proof's relationship with the Zeitgeist. It's not a matter of a purely ironic, ignorant distortion of Goedel's work leading to the postmodern vibe. Even a correct understanding of Goedel may tend in that (emotional) direction.

I don't think I'm being any more metaphorical than necessary, given that what I'm trying to describe is an emotion. Is there a literal way to talk about emotions?

I would like a better word than "game" for the way that Hilbert and company felt about math, if you have one. But to speak literally, it might be sufficient to say "Goedel showed that math was less like propositional logic, and more like natural language, in a certain regard".

But I think people who actually could follow the proof may have legitimately gotten the feeling -- the *feeling*, mind you, the emotional resonance, not the logical certainty -- "math is more mysterious, more uncertain, than we thought; math (and by extension, reason) cannot be put into a box".

Sure. No disagreement.

But to speak literally, it might be sufficient to say "Goedel showed that math was less like propositional logic, and more like natural language, in a certain regard".

That sounds fine to me. It's just that it would never have occurred to me to compare math to a story (or a game) as a way of expressing that.