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 ...
Thanks for the ping! Will read tonight.
BUMP
mark
Thank you so much for these recommended readings and book reviews!!!
All too true, mate.
In the meantime, don't feel so bad. I once read that David Hilbert turned down an offer to work on Fermat's last theorem, on the grounds he didn't want to spare the three (!!) years it would take him to get up to speed on the background.
Cheers!
Nevertheless, there are large parts of mathematics that are catagorical. Not only are all theorems provable, there is a proof scheme for these theorems. Euclidean (and consequently, all the non-Euclidean) geometry is one such. First order logic is another as is Pressburger arithemetic.
I defer to your more comprehensive and current understanding of Goedel.
I had always read him to be making a more modest assertion.
'One-off' mathematical truths, so to speak, were possible. ;^)
I once read that David Hilbert turned down an offer to work on Fermat's last theorem, on the grounds he didn't want to spare the three (!!) years it would take him to get up to speed on the background.
That's one I hadn't heard. But, man, was he being optimistic! Wiles worked seven years to get to his proof of Fermat's Last Theorem (and at the end needed a little help from a former student to finish it)...and that was after he became convinced that a proof was possible (so he'd already worked many years acquiring the background necessary to even attempt such a proof). Of course, he proved something much more general, which is often the way of such things, but still...
On reflection, maybe my "year or so" was optimistic, too...
Nah...
Nevertheless, there are large parts of mathematics that are catagorical. Not only are all theorems provable, there is a proof scheme for these theorems. Euclidean (and consequently, all the non-Euclidean) geometry is one such. First order logic is another as is Pressburger arithemetic.
Always good to be reminded that things aren't totally hopeless.
I recommend Huber-Dyson's book, DS, even though it has a surfeit of typos. The book is one of those 'camera-ready copy' jobs, and she should've had a good proof reader go over it, but didn't.
More Godel.
[headsonpikes:] We only have to grasp one truth uncaptured by a formal system to satisfy Goedel's criterion, not all truths. As I read it.
Too many people make the mistake of concluding that if human minds can grasp more theorems than a consistent formal system like mathematics, then our minds must not be "computer-like" (i.e. not a formal system, or not deterministic).
This is a fallacy.
Here's what I wrote on that issue in my review of Robert Penrose's, "The Emperor's New Mind":
Additionally, it's not encouraging that even when arguing a point in his own field of mathematics, he miapplies it. He invokes Godel's Incompleteness Theorem in support of his speculations about mind, but makes the same mistaken made by countless metaphysicists before him -- and rebutted by countless other people already.It only demonstrates that the human mind is not a *consistent* formal system -- and even that conclusion holds only if the human mind is demonstrably "complete", which is certainly an unwarranted presumption.Godel's Theorem states that a formal system cannot be simultaneously consistent and complete. Penrose makes two mistakes when he applies this to mind:
1. He equates "formal system" with "deterministic system", but they are not synonyms. All formal systems are deterministic, but not all deterministic systems operate in the manner of formal systems. [i.e., the limitation of formal systems is not necessarily a limitation on *all* types of deterministic systems.]
2. Penrose presumes that since human minds can arrive at conclusions which formal systems can not, that therefore we are not subject to Godel's Incompleteness Theorem, and thus our minds are not formal/deterministic systems. Penrose forgets that the more obvious reason we can do so is contained in the flip-side of Godel's two opposing conditions -- the human mind is not *consistent*. That is, it is capable of reaching conclusions which are wrong and/or contradict other conclusions. Godel's Theorem is *only* a limitation on *consistent* formal systems -- that is, those which never produce contradictory results.
Needless to say, the human mind is not such a system. We're quite capable of contradicting ourselves, in a logical sense. We sacrifice logical consistency (and therefore rigid accuracy) for flexibility. This does indeed allow us to recognize truths that formal systems are "blind" to -- but at the expense of being able to make mistakes as well. Formal systems may be limited by the Godel Incompleteness Theorem -- but at the same time they're capable of producing *only* correct proofs. There are trade-offs and advantages either way.
In short, the human mind can exceed the abilities of formal systems (in *some* regards) not because the mind is nondeterministic, but because it uses "fuzzy" logic that isn't guaranteed to be always accurate. Win some, lose some.
Godel's Incompleteness Theorem is no support for the hypothesis that the human mind cannot be, at its foundation, some sort of formal or deterministic system.
It's also important to note that a deterministic computer which just mechanically enumerated all postulates would likewise be complete, albeit at the same cost -- inconsistency. So would one which enumerated all postulates, then weeded out some of the more obviously false (Godel's theorem shows that it can't weed out *all* of them). Such a computer, if self-aware, might likewise try to argue that since it can "find" more truths than a formal system, it's non-deterministic, but again that would be incorrect, for the reasons described above (and obviously so in this case, since the computer *is* acting deterministically).
Yet another book on the subject:
d) the whole idea of universal truths is a misunderstanding based on a limited philosophical system?
Again, "universal truths" is one of those "everyone knows what I mean" terms that nonetheless start to feel like nailing Jell-o to a wall when you actually begin to try to pin it down. Half the room will agree to a particular meaning, and the other half will object strenuously (*whichever* of dozens of possible ways you try to define it).
One of (hell, *THE*) best books for the layman on these types of issues (as well as many others) is "Godel, Escher, Bach: An Eternal Golden Braid", by Douglas Hofstadter. Truly one of the great books in the history of mankind. And no, I'm not exaggerating. In a step-by-step, understandable, *entertaining* manner, he walks the reader on a grand tour through (using Amazon.com's subject list for the book):
Topics Covered: J.S. Bach, M.C. Escher, Kurt Gödel: biographical information and work, artificial intelligence (AI) history and theories, strange loops and tangled hierarchies, formal and informal systems, number theory, form in mathematics, figure and ground, consistency, completeness, Euclidean and non-Euclidean geometry, recursive structures, theories of meaning, propositional calculus, typographical number theory, Zen and mathematics, levels of description and computers; theory of mind: neurons, minds and thoughts; undecidability; self-reference and self-representation; Turing test for machine intelligence.And unlike most books on any/all of these topics, the material is not "dumbed down" -- anyone who works through the book will get a *real* appreciation and working understanding of these subjects.And as the "braid" in the title promises, the author delivers on showing the intimate interconnections between these topics. In a sense, although the book is about *all* those things, it's really all about the *same* thing. And the book itself is so tightly constructed that it sometimes feels that if one sentence were accidentally removed, the whole book would unravel like a snagged tapestry. Even when it seems that the author is starting a new chapter on an entirely different topic, at some point you'll find that your brain suddenly goes, *oooooooohh*...., as you realize that he has managed to weave another "thread" into the very same subject matter that the last chapter covered.
A truly remarkable book, and I'm not the only one to think so. It won the Pulitzer Prize, an amazing achievement for a book in this genre. There are better books on the subject for a technical audience, of course, but this one's excellent for a layman who has little or no prior background. It starts from "the ground up", but climbs to amazing heights for a book of its type.
I concur! I wish the hell I could find my copy. It apparently fell into a black hole during one of my job switches.
Probably at the bottom of that Bankers Box, second from the left, on the bottom layer of the stacks of boxes, out in my garage.
Too many people make the mistake of concluding that if human minds can grasp more theorems than a consistent formal system like mathematics, then our minds must not be "computer-like" (i.e. not a formal system, or not deterministic).
This was Yourgrau's interpretation of Gödel's position. Whether that's correct as an interpretation of what Gödel really thought, I'm not certain. But, of course, 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. And that's why I wrote this in post #16 (my underlines):
"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".
As you point out, it's not at all clear that this argument holds water when viewed from a wider perspective.
Just FYI, Verena Huber-Dyson thinks Hofstadter's book is crappy and wrote a satirical review of it for the Canadian Journal of Philosophy. I'd kind of like to read that review, but haven't. I read most of the book decades ago and remember thinking it was kind of show-offy, but I'm not prepared to judge it on content without a re-read (a not-very-likely circumstance).
Thanks for the link. I somehow missed all that.
thanks
Are you familiar with Gödel's slingshot?
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