Posted on 09/19/2005 1:51:42 AM PDT by snarks_when_bored
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."
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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. |
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From A World Without Time. |
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(Excerpt) Read more at americanscientist.org ...
Are you familiar with Gödel's slingshot?
No, I wasn't. But I've now googled a bit about it. I see that it has something to do with modal logic. I'll have to do some reading later. In the meantime,
I don't necessarily believe that it's possible for modal logic to be given an effective implementation.
(rimshot)
Goldstein's philosophy classes might be interesting, but like many who teach philosophy she is no Hegel [that is a hard act to follow]. What it comes down to is that the student has to develop his own understanding. The teacher can only point the way, or several ways or possible points of departure, and maybe one of them will be a sufficient spark. If Chomsky can do it, there is hope that all of us can do it.
Here's the link:
Kurt Gödel special edition of the Notices of American Mathematical Society
Enjoy!
I was at a talk with a half-dozen prominent Hungarians when Erdos died.
Thanks for the link.
Thanks for that link. Looks like they have a good collection of articles there. It should keep me busy for a while.
Thanks for that link. Looks like they have a good collection of articles there. It should keep me busy for a while.
Me, too, BB! And you're welcome...
Thanks!
thanks snarky.. great stuff.
I was only responding to D-fendr's citation of the passage, pointing out that the hypothesis, i.e., that the human mind is capable of grasping all mathematical truths, doesn't appear at all likely
How about a human mind capable of coming up with the Incompleteness Theorem?
What I'm pointing to is the ability to see the greater context of, transcend, the program and see a meta-program, and on up...
Thanks for the ping.
Thanks for the ping. Bookmark for later.
Which violates Gödel's Incompleteness Theorems, as there no way to know if a Gödel statment is true or not without supplemental axioms, which creates a new set of unproveably true Gödel statements.... and so on, as you pointed out.
In effect, the argument being made is a case of Begging the Question, because it requires you to assume that which contradicts Gödel's Theorems (that we can can somehow prove true a Gödel statement with out creating more Gödel statements) order to arrive a conclusion (that we can grasp Mathematical truth in its entirety) that violates Gödel's Theorems.
The other thing to keep in mind is that the limitation imposed by Gödel's Incompleteness theorems has little if any practical effect on most Mathematicians. Since the only sort of theorem used by Mathematicians to prove other theorems are theorems which have already been proven, Gödel's Theorems never come into play. IOW, that which we can prove, we can prove.
It's only the meta-Mathematicians, the Russells, the Whiteheads, et al., whose lofty goals were trashed forever by Gödel's results. Life for mortal Mathematicians goes on as before, they hardly even feel a bump when the pass over Gödel's Theorems.
Thanks!
Just found this link today and signed on. As a long time Gödel fan I’m pleased to see so much interest and thanks to Snarks for all the links. I came here looking for more facts on Kurt’s life. I highly recommend “A Madman Dreams of Turing Machines” also for insight into his personality. I read it only recently, after starting my own story on a website dedicated to the ABC show LOST. My goal is to introduce the magic of numbers to an audience not really very familiar with them. It’s an experiment.
You don’t need to ever have see LOST to follow this thread. Those interested in Turing, Gödel, and Riemann might find it amusing. Remember its a fiction oriented toward those with little background in math but who enjoy puzzles. Any suggestions on ways to improve it would be most welcome.
http://www.4815162342.com/forum/viewtopic.php?t=31267
I look forward to participating in this forum when I’ve had time to read more. Hard to believe I didn’t come across this site before.
I'm guessing it wasn't "Hey, yo, how 'bout them Packers, baby?"...
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