Comments on: To "have the privilege of walking home with Gödel."

To "have the privilege of walking home with Gödel."

thatwhichfalls — Sat, 19 Mar 2005 13:12:55 -0800

"Gödel put logic on the mathematical map."
An excellent interview with Rebecca Goldstein, biographer of Kurt Godel

By: thatwhichfalls

thatwhichfalls — Sat, 19 Mar 2005 13:14:19 -0800

"Gödel believed that truth - most paradigmatically, mathematical truth - subsists independently of any human point of view. If ever there was a man committed to the objectivity of truth, and to objective standards of rationality, it was Gödel. And so the usurpation of his theorems by postmodernists is ironic. Jean Cocteau wrote in 1926 that "The worst tragedy for a poet is to be admired through being misunderstood." For a logician, especially one with Gödel's delicate psychology, the tragedy is perhaps even greater."

By: NickDouglas

NickDouglas — Sat, 19 Mar 2005 13:46:55 -0800

A review of the biography on Slate, downplaying Godel's importance: "Does Gödel Matter?"

By: nofundy

nofundy — Sat, 19 Mar 2005 13:47:53 -0800

This rocks! Thanks! I had to read The Eternal Golden Braid to get an understanding of Godel. (Sorry, I'm ignorant of how to do the umlaut thing.)

By: Mean Mr. Bucket

Mean Mr. Bucket — Sat, 19 Mar 2005 14:08:27 -0800

Nice post.

By: loquacious

loquacious — Sat, 19 Mar 2005 14:23:02 -0800

Excellent post, thank you.

By: koeselitz

koeselitz — Sat, 19 Mar 2005 14:23:54 -0800

This is interesting. Thanks. (By the way, nofundy, you don't need to know how to do the umlaut thing, handily enough, at least in German. The same grammatical effect is achieved by placing an 'e' after the letter which would have recieved an umlaut. So, for example, 'koeselitz' could also be spelled without the first 'e' and with an umlaut over the 'o;' and the fellow's name is spelled 'Goedel.' I too, however, wish I knew how to make an umlaut, as it would've been a cooler screen name.)

By: thatwhichfalls

thatwhichfalls — Sat, 19 Mar 2005 14:38:45 -0800

NickDouglas - from what I've read, Goedel himself wouldn't have disagreed with anything in that review. Goldstein would also seem to agree that the significance of the incompleteness theorems is considerably more limited than some would have us think. koeselitz - thanks for the tip. I was just cutting and pasting the 'o' with umlaut (although I see I missed a couple earlier). My favorite Einstein and Goedel story:

So naïve and otherworldly was the great logician that Einstein felt obliged to help look after the practical aspects of his life. One much retailed story concerns Gödel's decision after the war to become an American citizen. The character witnesses at his hearing were to be Einstein and Oskar Morgenstern, one of the founders of game theory. Gödel took the matter of citizenship with great solemnity, preparing for the exam by making a close study of the United States Constitution. On the eve of the hearing, he called Morgenstern in an agitated state, saying he had found an "inconsistency" in the Constitution, one that could allow a dictatorship to arise. Morgenstern was amused, but he realized that Gödel was serious and urged him not to mention it to the judge, fearing that it would jeopardize Gödel's citizenship bid. On the short drive to Trenton the next day, with Morgenstern serving as chauffeur, Einstein tried to distract Gödel with jokes. When they arrived at the courthouse, the judge was impressed by Gödel's eminent witnesses, and he invited the trio into his chambers. After some small talk, he said to Gödel, "Up to now you have held German citizenship." No, Gödel corrected, Austrian. "In any case, it was under an evil dictatorship," the judge continued. "Fortunately that's not possible in America." "On the contrary, I can prove it is possible!" Gödel exclaimed, and he began describing the constitutional loophole he had descried. But the judge told the examinee that "he needn't go into that," and Einstein and Morgenstern succeeded in quieting him down. A few months later, Gödel took his oath of citizenship.

From here

By: timb

timb — Sat, 19 Mar 2005 14:58:51 -0800

ö → ö

By: andrew cooke

andrew cooke — Sat, 19 Mar 2005 15:16:07 -0800

there's a good, free online text that introduces godel's work in a fairly technical manner.

By: loquacious

loquacious — Sat, 19 Mar 2005 15:31:50 -0800

That's an awesome story and link, thatwhichfalls, thanks.

By: thatwhichfalls

thatwhichfalls — Sat, 19 Mar 2005 15:56:48 -0800

Thanks for the link andrew cooke - downloaded for later.

By: 3.2.3

3.2.3 — Sat, 19 Mar 2005 17:48:37 -0800

it's a gross overstatement to say gödel's theorems are misappropriated because of what gödel believed existed beyond symbolic logic. it's more interesting to observe gödel ironically persevered with positivism despite what his theorems tenuously implied. and this has been observed, but probably not biographically enough, by the communities goldstein charges with obliviousness to gödel's intent. another more interesting observation, although unoriginal and usually made by the opponents of those communities, is that self-reference, whether with russell's set categories or using mathematics to comment on the "human activity mathematicians do," is itself suspect.

By: andrew cooke

andrew cooke — Sat, 19 Mar 2005 18:00:28 -0800

you don't have to ironically persevere if you are faced only with a tenuous implication. in such cases you simply continue to believe.

By: thatwhichfalls

thatwhichfalls — Sat, 19 Mar 2005 20:04:09 -0800

3.2.3, I'm not too sure what you are saying. Actually I had to cut and paste your comment clause by clause into notepad to get a feeling for what you were saying.

...self-reference, whether with russell's set categories or using mathematics to comment on the "human activity mathematicians do," is itself suspect.

The inherent self-reference in that statement is both obvious and calculated. So I'll just say that ... I was building myself up for a rant, but, on re-reading your comment, I see that I would be fighting roughly on the same side as you - I've been a long time out of academia (thank god) and have lost the knack for decoding this kind of stuff. It would help if you could tag it in some way. "RED = academic irony - or maybe it doesn't?". Something like that. tenuously implied"? No - he proved what he proved. Other people extended it, sometimes unfairly, but Goedel never claimed anything beyond what he had proved.

By: vacapinta

vacapinta — Sat, 19 Mar 2005 20:35:46 -0800

We also need to distinguish what Goedel showed mathematically versus what he personally believed. In one sense, the latter is irrelevant. His work should stand on its own. Its interesting to note that Cohen, Goedels most famous student, who extended much of what Goedel had done, was himself more of a formalist (vs. a Platonist)

By: soyjoy

soyjoy — Sat, 19 Mar 2005 21:57:42 -0800

Thanks for that Einstein & Goedel story, thatwhichfalls - it seems to be ultimately sourced to the book Pi in the Sky, which sounds interesting, but I'd never heard of. (Anybody?) Is it just me, or does it sound just too perfect to be true? "Mr. Incompleteness Theorem" finding the single loophole that negates the whole purpose of the document in which it exists - and which if carried out would make the whole document itself invalid? I mean, was Goedel just constantly going everywhere and noticing systems that can be made to self-destruct? Did he arrange his alphabet soup to spell "poison?" Did he walk into the Deli and say "Your pastrami is its own worst advertisement!"? ...If only he'd lived to see Pete Townsend smash his guitar, eh? I'm halfway through GEB: EGB for the second time, and for the first time in 25 years. I missed a lot of the more technical stuff the first time through as my eyes would glaze over and I would go into Evelyn Wood mode to get through the pages till the next appearance of Achilles and the Tortoise or whatever. Now I'm taking it nice and slow, with breaks here and there, like between "Books." It's very pleasant, as I found the first time through that for all his great ideas of how to draw connections between things, Hofstadter's initially scintillating voice can get a little tinny after three or four hundred pages.

By: painquale

painquale — Sat, 19 Mar 2005 23:08:06 -0800

self-reference, whether with russell's set categories or using mathematics to comment on the "human activity mathematicians do," is itself suspect. It depends what kind of self-reference and what kind of system you're talking about. The self-referencing we use in the vernacular gets us into all sorts of paradoxes ("this sentence is false"), and many philosophers want to get rid of those paradoxes when translating natural languages into formal logic. The set theory in the Principia Mathematica originally led to Russell's Paradox, which was felt to be intolerable and had to be closed up. But as far as I know, no mathematicians or philosophers object to the Goedel numbering or the types of self-reference that Goedel numbering allows. Am I wrong about this?

By: DirtyCreature

DirtyCreature — Sun, 20 Mar 2005 03:15:43 -0800

But as far as I know, no mathematicians or philosophers object to the Goedel numbering or the types of self-reference that Goedel numbering allows. Am I wrong about this? It's not really Godel numbering itself that is the source of the self-reference. Fundamentally the culprit is universal quantification (i.e. for all x, phi(x) ) within formal logic. This gives rise to recursion and the scope for self-reference. If you have issues with formal logic, that's a whole other story.... Russell's paradox arose out of a collection of axioms defined within formal logic for a theory of sets which turned out to be internally inconsistent. The paradox can be avoided by defining an axiomatic system for set theory which does not permit sets to be members of themselves. For example, Zermelo Frankel. As an interesting aside, Godel himself died from starvation while trying to avoid food poisoning! An ironic example of the inherent problems with trying to assert your own consistency!

By: mek

mek — Sun, 20 Mar 2005 03:55:40 -0800

I've been studying Einstein a great deal this year, and this article in the New Yorker is an interesting look at the friendship that he and Goedel shared. Goedel was a very tragic figure indeed, as DC alludes to - he became extremely paranoid in his later years, and eventually caused his own death.

By: painquale

painquale — Sun, 20 Mar 2005 10:08:39 -0800

Fundamentally the culprit is universal quantification (i.e. for all x, phi(x) ) within formal logic. This gives rise to recursion and the scope for self-reference. Hmmm, I don't think this is right. Universal quantification in itself doesn't lead to self-reference. It depends what sort of domain you're quantifying over. Being able to quantify over sentences in the same language (i.e. including the metalanguage within the object language) is obviously going to lead to paradoxes like the liar paradox, which is why most logicians advocate translating the vernacular into Tarskian model theory. If you don't include sentences within your domain, then you won't quantify over them. To talk about the sentences themselves, you have to kick things up a level into the metalanguage. Also, Goedel numbering doesn't have anything to do with universal quantification. With Goedel numbering, you can say things like "343: '343 is false'." That's direct self-reference.

By: DirtyCreature

DirtyCreature — Sun, 20 Mar 2005 10:34:36 -0800

Also, Goedel numbering doesn't have anything to do with universal quantification. Sigh. Ok. Your original question appeared like you were willing to learn. But now your stance is you know everything anyway. Enjoy.

By: painquale

painquale — Sun, 20 Mar 2005 11:43:20 -0800

Ha! Sorry to pull the ol' bait and switch like that. No, I do want to learn... I'm genuinely curious if there are any (respectable) people who think that there's something foully amiss with Goedel numbering because they hold a principled stance against self-reference. I haven't heard of anyone like that, but it wouldn't surprise me if they existed.

By: DirtyCreature

DirtyCreature — Sun, 20 Mar 2005 13:09:23 -0800

Self-reference is nothing to be afraid of. You just need to take care with what circumstances you permit its use. Functional programming is an excellent example of the usefulness, simplicity and power of self-reference. Standard arithmetic and modern set theory both involve self-reference. Remember, Godel numbering is really just a sneaky way of encoding formal logic statements into statements about the natural numbers. You can't really "outlaw" this without vastly reducing the power of logic system you want to reason to begin with. I just wrote a much longer explanation but re-reading these things always sound like a bunch of hand-waving opening yourself up for attack when you're talking in an informal language. Yes there has been a lot of attacks on Godel's results, formal logic, self-reference, axiomatic method since Godel published his work. It's true that attacks on universal quantification are just one of MANY areas that these attacks have focused on. It really sent the mathematical community into a tailspin from which it has never convincingly recovered in my opinion. I'm going to lame out and direct you to Wiki's spiel on Godel's incompleteness theorems. Its a great place to whet your appetite and begin a longer exploration.