The Incomplete Gödel

American Scientist Online ^ | September-October, 2005, issue | Gregory H. Moore

[snarks_when_bored]

The Incomplete Gödel

Gregory H. Moore

Incompleteness: The Proof and Paradox of Kurt Gödel. Rebecca Goldstein. 296 pp. W. W. Norton, 2005. $22.95.

A World Without Time: The Forgotten Legacy of Gödel and Einstein. Palle Yourgrau. x + 210 pp. Basic Books, 2005. $24.

Such eminent 20th-century physicists as Albert Einstein, Niels Bohr and Werner Heisenberg are well known to almost all scientists, whether or not they happen to be physicists. Yet most scientists are unfamiliar with eminent mathematicians from the same period, such as David Hilbert (Germany) and Oswald Veblen (United States). A rare exception is John von Neumann (Hungary and the United States), a mathematician whose contributions to quantum mechanics, the stored-program concept for computers, and the atomic bomb resonate with many physical scientists.

One mathematician who deserves to be better known, and who was highly esteemed by von Neumann, is Kurt Gödel (1906-1978). In 1951 Gödel shared the first Einstein Award with physicist Julian Schwinger (who later won the Nobel Prize). At the award ceremony, von Neumann gave a speech calling Gödel's work "a landmark which will remain visible far in space and time."

[snip]

 

	Included in this photograph taken at Albert Einstein's 70th birthday celebration in 1949 are (left to right) Eugene Wigner, Hermann Weyl, Kurt Gödel, I. I. Rabi, Einstein, Rudolf Ladenburg and J. Robert Oppenheimer.
	From A World Without Time.

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This is another review of two books noted on previous FR threads:

Gödel and Einstein: Friendship and Relativity [an excerpt from Palle Yourgrau's book]

Truth, Incompleteness and the Gödelian Way [a review of Rebecca Goldstein's book]

GÖDEL AND THE NATURE OF MATHEMATICAL TRUTH [6.8.05] - A Talk with Rebecca Goldstein

GÖDEL AND THE NATURE OF MATHEMATICAL TRUTH II [7.27.05] - A Talk with Verena Huber-Dyson

Having now read both Yourgrau's and Goldstein's books, I must concur with Moore's judgment that Yourgrau's appears to be the more accurate account of what Gödel was attempting to accomplish with his two great Incompleteness theorems, and also with his work on General Relativity during the 1940's. Goldstein has a literary sensibility, but doesn't convince her reader that she really has a grip either on the technical aspects of Gödel's work or on what Gödel's inner life must have been like. A novelist's imagination proved insufficiently powerful to see into the heart of what Gödel accomplished and who he was.

On the other hand, Verena Huber-Dyson's book, Gödel's Theorems; a workbook on Formalization, is a relentlessly detailed exploration of the formal logic and mathematics needed to really grasp Gödel's theorems. If I had a year or so of free time, I'd be tempted to work my way through the exercises. In another life, perhaps...

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Gödel ping

I've pinged everybody who showed up on any of the four earlier threads I've posted on Gödel, with the idea that a previously displayed interest might be a reliable sign of continuing interest.

[BykrBayb]

Thanks for the ping. Bumping for later reading.

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Welcome, BB...

[ThePythonicCow]

I recall when Godel died, in January of 1978. I had to give a talk the next day at Georgia Tech, and began the session by announcing his death, and asking for a moment of silence.

The great ones speak with a clarity that is delightful.

[snarks_when_bored]

I'm ashamed to say that I don't recall what I was doing the day Gödel died. I envy you your act of that day.

[D-fendr]

Thanks for the ping.

Yourgrau's appears to be the more accurate account of what Gödel was attempting to accomplish with his two great Incompleteness theorems

Would this be it? From the link:

Finally, Gödel's incompleteness theorem… set a permanent limit on our knowledge of the basic truths of mathematics: The complete set of mathematical truths will never be captured by any finite or recursive list of axioms that is fully formal. Thus, no mechanical device, no computer, will ever be able to exhaust the truths of mathematics. It follows immediately, as Gödel was quick to point out, that if we are able somehow to grasp the complete truth in this domain, then we, or our minds, are not machines or computers.

[mathluv]

Thanks for the ping.

bookmarked

[snarks_when_bored]

I find nothing to criticize in that quote from Yourgrau as far as what Gödel accomplished, but I do point out that he ends by saying this: "...if we are able somehow to grasp the complete truth in this domain, then we, or our minds, are not machines or computers." That 'if' is huge. I see nothing that leads me to believe that we're capable of grasping the complete truth in mathematics.

[PatrickHenry]

Ping acknowledged.

[Flightdeck]

In a slight change of topic, has anyone posted that article from the latest Popular Science that mentions freerepublic?

[aculeus]

Thanks for the ping.

[GOPJ]

Thanks for the ping.

[headsonpikes]

I see nothing that leads me to believe that we're capable of grasping the complete truth in mathematics.

We only have to grasp one truth uncaptured by a formal system to satisfy Goedel's criterion, not all truths.

As I read it.

[GOPJ]

Goldstein has a literary sensibility, but doesn't convince her reader that she really has a grip either on the technical aspects of Gödel's work or on what Gödel's inner life must have been like. A novelist's imagination proved insufficiently powerful to see into the heart of what Gödel accomplished and who he was.

Math, the truest language for describing reality, falls flat when describing humans. For now...

[snarks_when_bored]

I see nothing that leads me to believe that we're capable of grasping the complete truth in mathematics.

We only have to grasp one truth uncaptured by a formal system to satisfy Goedel's criterion, not all truths.

As I read it.

I was responding to this specific quote from Yourgrau (my underlines):

... The complete set of mathematical truths will never be captured by any finite or recursive list of axioms that is fully formal. Thus, no mechanical device, no computer, will ever be able to exhaust the truths of mathematics. It follows immediately, as Gödel was quick to point out, that if we are able somehow to grasp the complete truth in this domain, then we, or our minds, are not machines or computers.

So, by this argument, we'd have to grasp the complete truth of mathematics (whatever that might mean) in order to conclude that "we, or our minds, are not machines or computers." I was calling into question the likelihood of our being able to grasp the complete truth of mathematics. And, if we can't, it no longer follows (from this argument) that "we, or our minds, are not machines or computers".

Good to hear from you, bud...

[snarks_when_bored]

Math, the truest language for describing reality, falls flat when describing humans. For now...

I can't argue with that...

[headsonpikes]

I agree with your analysis of Yourgrau's assertions. I disagree that he accurately stated Goedel's view.

Surely, ONE non-provable truth would be enough to distinguish oneself from a well-progranned IBM 360(remember those?).

Thanks for posting - I'd never run across this material otherwise.

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Surely, ONE non-provable truth would be enough to distinguish oneself from a well-progranned IBM 360(remember those?).

You have to keep in mind that the truth will be non-provable within a specified formal system (of sufficient strength to express integer arithmetic with the operations of addition and multiplication). By augmenting that system with additional axioms, the formerly unprovable truth becomes provable. But (and here's where Gödel's insight really cuts) within the newly augmented system, another unprovable but true proposition will exist; and so on ad infinitum.

Hence, to make the argument that Gödel wanted to make, the argument that Yourgrau summarizes, one must suppose it possible that mathematical truth in its entirety might somehow be graspable, and so the human mind might transcend all formal systems, formal systems which are forever showing themselves to contain true but unprovable propositions.

[snarks_when_bored]

I haven't seen that article. No link for it, I guess?