Ah, but the strawberries! That's where I had them. They laughed at me and made jokes, but I proved beyond the shadow of a doubt, and with geometric logic, that a duplicate key to the wardroom icebox did exist!

That we can't prove via reason, logic, math and/or science the things we believe to be true simply demonstrates the primacy of faith.

Typically grandiose, Orrin. Not what Goedel said at all.

From the article:

Yet, Gödel is routinely deployed by people with antirationalist agendas as a stick to whack any offending piece of science that happens by.

Though I admit, this guy

A typical recent article, "Why Evolutionary Theories Are Unbelievable," claims, "Basically, Gödel's theorems prove the Doctrine of Original Sin, the need for the sacrament of penance, and that there is a future eternity."

came out ahead of you in the full-blown fantasy department.

Eugene:

All he did was confirm what philosophers had known all along.

This Ellenberg guy is pulling a fast one. For a specialist in mathematical logic, a knowledge of Godel's work is indispensible. For most other mathematicians, Godel's stuff would be nice to know but not critically important.

Just as physicists working in relativity-related fields would need a knowledge of Einstein's research. For physicists specializing in other areas, Einstein's results would be nice to know but again, not crucial.

To be the total contrarian on this topic, Gödel's Theorems can simply be viewed as a continuation of the criticism of Tertium non Datur which began with the Liar Paradox. The reason they have so little impact on most mathematicians is that the formula Gödel produced has no external content -- it is self-contained and self-referential, isolated, with no other extension. They have had a tremendous impact however, for they produced a mathematical nihilism blocking any further attempts to prove the consistency or completeness of mathematics.

jd:

Of course it's self-contained, so is mathematics and each branch thereof.

In fact, Godel's work was far from self-contained. For instance, it formed the basis for a solution of Hilbert's first problem, the continuum hypothesis, which was proven undecidable from the standard axioms for set theory. Hilbert's tenth problem on Diophantine equations was also proven undecidable. For a non-technical account with lots of references see:

http://at.yorku.ca/t/a/i/c/52.htm

Oops, sorry, Hilebert's tenth problem (a general technique for solving any Diophantine equation) was proven unsolvable, not undecidable. I should have said that Hilbert's second problem was resolved by Godel. A list of all Hilbert's problems:

http://mathworld.wolfram.com/HilbertsProblems.html

If you could solve number nine, you'd be rich and famous.

Casey:

From that essay:

"The Second Problem asked for a proof of the consistency of the foundations of mathematics. Some of the flavor of the urgency of that problem is provided by the following passage from an article by S.G. Simpson in the same volume of JSL as the article by P. Maddy:

We must remember that in Hilbert's time, all mathematicians were excited about the foundations of mathematics. Intense controversy centered around the problem of the legitimacy of abstract objects. Weierstrass had greatly clarified the role of the infinite in calculus. Cantor's set theory promised to raise mathematics to new heights of generality, clarity and rigor. But Frege's attempt to base mathematics on a general theory of properties led to an embarrassing contradiction. Great mathematicians such as Kronecker, Poincaré, and Brouwer challenged the validity of all infinitistic reasoning. Hilbert vowed to defend the Cantorian paradise. The fires of controversy were fueled by revolutionary developments in mathematical physics. There was a stormy climate of debate and criticism. The contrast with today's climate of intellectual exhaustion and compartmentalization could not be more striking.

... Actually, Hilbert saw the issue as having supramathematical significance. Mathematics is not only the most logical and rigorous of the sciences but also the most spectacular example of the power of "unaided" human reason. If mathematics fails, then so does the human spirit. I was deeply moved by the following passage [13, pp. 370-371]: "The definitive clarification of the nature of the infinite has become necessary, not merely for the special interests of the individual sciences but for the honor of human understanding itself."

Hilbert was already aware, at the time of his 1900 lecture, of some connection between the provability of the consistency of a mathematical theory and the decidability of statements by the axioms of the theory. But it was Kurt Gödel who showed the true nature of this connection in the process of showing that Hilbert's Second Problem has a negative solution: "

The foundations are as inconsistent as we.

Resolved in the negative.

Gawd, I'm having a bad day. I meant Hilbert's eighth problem, the Riemann hypothesis. There's a million dollars waiting for a solution:

http://www.claymath.org/millennium/

jd: I ran across this in the course of looking for a readable linkable proof of the Incompleteness Theorem (this was for an earlier round of the Permanent Floating Goedel's a Big Enough Club to Beat Darwin With Thread ):

Gödel's Theorem and Information

From the abstract:

Gödel's theorem may be demonstrated using arguments having an information-theoretic flavor. In such an approach it is possible to argue that if a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be derived from the axioms. In contrast with the traditional proof based on the paradox of the liar, this new viewpoint suggests that the incompleteness phenomenon discovered by Gödel is natural and widespread rather than pathological and unusual.

I'd like to know what you think of it.

Joe,

That's funny, I just mentioned Chaitin here the other day.

His book, META MATH! is available for free online reading.

Very enjoyable, puts the fun back into math and does not neglect philosophical implications.

Recommended for almost anyone except certain weblog hosts who count even their groceries bill on their fingers. (On the other hand, Orrin does not move his lips when reading, a feat I haven't accomplished.)

Sorry, I messed up the URL for Chaitin's book.

http://xxx.lanl.gov/PS_cache/math/pdf/0404/0404335.pdf

Thanks, Eugene, I'll put it on the stack. Dontcha know it -- I thought he was joking when he couldn't figure out what 15 x 15000 was.

"They have had a tremendous impact however, for they produced a mathematical nihilism blocking any further attempts to prove the consistency or completeness of mathematics."

Maybe its the word nihilism, but I think Gödel proved that their efforts would be unavailing, so being rational men they quit. C'n'est pas? The interesting thing is that so few professional mathematicians have grappled with the implications of the proof for their own work in other areas. Although, Mathematics: The Loss of Certainty by Morris Kline (1980) is a very worthwhile investigation.

"what's most startling about Gödel's theorem ... is not how much it's changed mathematics, but how little...But most pure mathematicians can easily go through life with only a vague acquaintance with Gödel's work..."

Confession and avoidance. One reviewer of Klien's book on Amazon said:

"mathematics is not the exploration of hard edged objective reality or the discovery of universal certainties, but is more akin to music or story telling or any of a number of very human activities."

Which is one reason why, I supose, mathematicians have ignored it so serenely, except that they have, quite properly, abandoned work on foundations.

But out here in, what is laughingly called, the real world, Godel's proof triggered an avalanch. The follow-on work, in particular Alan Turing's, became the foundation of computer science. And this is one case where the theory preceded the practice. So it is true that Godel's proof has not affected pure mathematicians, but it has had an enormous practical impact on the modern world.

It is as if a coal train ran into a mountain side and produced a carload of diamonds.

Any faith is as good as any other, then. They are all equally faiths, no?

All are equally faiths but some are superior to others.