Rebecca Goldstein's Incompleteness: The Proof and Paradox of Kurt Gödel is another book in the Great Discoveries series of short books by noted authors about important moments in the history of science, and the people behind them. Previous volumes include Everything and More and A Force of Nature, both of which were excellent in their own way, and Incompleteness fits right in there with them.
As the subtitle makes clear, this is a book about Kurt Gödel's famous Incompleteness Theorem, which shows that any formal logical system complex enough to describe arithmetic must allow the formation of statements that are true, but cannot be proved to be true within that system. If you've always wanted to know what all the fuss over Gödel, Escher, Bach was about, but can't make it through, this book would be an excellent alternative. Goldstein does an excellent job of explaining the meaning of the theorem, putting it in historical context, and sketching out the unique way Gödel did the proof.
Of course, like all the other books in the series, the book also reflects the author's tastes and professional inclination. Goldstein is a philosopher, and so she uses the book to make an argument about Gödel's philosophy: that contrary to the common impression of the theorem as the work of some sort of mathematical postmodernist, Gödel was in fact a passionate and committed Platonist, firmly believing in the independent reality of mathematical ideas. To him, the important part of the theorem was that the unprovable statements were true, suggesting a wider and deeper mathematical universe than the formalist program would admit. He wasn't out to destroy mathematical truth, but to confirm and in some sense ennoble it.
I'm not qualified to evaluate the accuracy of this claim, but Goldstein makes a convincing argument. She also does an excellent job of putting the theorem in historical context, sketching out both the philosophical circles of Vienna where Gödel cut his teeth, and the formalist program of David Hilbert (among others) that his famous result overthrew. And, of course, no biography of Gödel could hope to avoid his personal eccentricities, and the tragic descent into paranoia that led to his death.
This is a very well-done book, and if you have any interest in mathematics or the history and philosophy thereof, I recommend checking it out.
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I second Chad's recommendation. Gödel was "pure genius," and Goldstein does a fantastic job putting his incompleteness theorem in its historical context.
It's kind of weird that you state that you are unqualified to evaluate whether Gödel's undecidables are true having already paraphrased the theorem as "any formal logical system complex enough to describe arithmetic must allow the formation of statements that are true, but cannot be proved to be true within that system". Or did you mean that you are unqualified to evaluate whether that's how Gödel really saw it?
Anyway, the paraphrase with "true" (rather than always restricting oneself to talking about "true in a given system" and that meaning "provable") kind of makes my brain hurt. I know lots of very smart people are happy with it, but it always strikes me as taking far too much for granted about statements maintaining their meanings when moved between systems.
I meant that I'm unable to really evaluate her argument about the philosophy. I'm perfectly capable of repeating it, but I don't know enough about mathematical logic and the history thereof to say if she's presented things accurately, or if she's misrepresenting Goedel, Hilbert, Wittgenstein, or any of the rest.
That is consistent with what little I've read of the man on a personal level-- I think the basic notion was something along the lines of, "There are things in the formal program we simply cannot prove one way or the other. But for at least some of them, we recognize their basic truth. Mathematics is incomplete, but either we are not, or our minds are larger systems than the formal program."
Don't quote me, I could be garbling that-- I'm more interested in the actual math than in his interpretations of it. The thing is, even if that view is correct, it would still be the case that the larger system of the human mind is also incomplete. (I guess you could argue that the human mind is not axiomatic, but that's... a stretch, in my opinion.) I find it hard to believe that Goedel would not have considered that, but it's easy for me to say that looking backwards from nearly 80 years.
For what it's worth, I would not recommend "A Madman Dreams of Turing Machines," by Janna Levin. I found it very unsatisfying.
Yay! Thanks for reviewing this book. It is sitting on my bookshelf. I picked it up randomly, thinking it might make a good gift for my Dad, and haven't gotten to it yet. Perhaps now I will...
I don't have the citation handy, but the review of this book in the Notices of the American Mathematical Society suggests that, however professional Rebecca Goldstein is as Philosopher and as writer, she makes several mistakes in understanding and/or explaining particularly important MATHEMATICS by Gödel.
Someone with a current AMS membership might want to give the actual citation or, by now, a PDF of the review.
Isn't there a book in the past couple of years specifically about the friendship between Gödel and Einstein?
Gödel did invent a solution to Einstein's field equations which allows time travel into the past, assuming that the universe is rotating fast enough (it's not) and one goes all the way around it. Einstein agreed.
This started the modern approach to time machines, via Tipler, Hawking, Thorne, et al.
Also amusing is the anecdote about Gödel going to be naturalized as a U.S. citizen and having to be counseled DON'T tell the official about the logical flaw that you found in the U.S. Constitution to allow it to be legally changed into a dictatorship...
Reviewing Goldstein's book for the London Review of Books (February 9, 2006), Solomon Feferman was less than ecstatic. He praised her treatment of the biographical material, but complained that she made a hash of the mathematics and philosophy: "these weighty claims disintegrate under closer examination, while the book is marred by a number of conceptual and historical errors. ... As to the core of Goldstein's book, anyone familiar with Gödel's work has to flinch. ... a seriously incorrect (even impossible) formulation of Gödel's crucial lemma providing for the construction of mathematical statements that indirectly refer to themselves ... Those who are fascinated by Gödel's theorems - and the general idea of limits to what we can know - may still hunger for a more universal view of their possible significance. They should not be satisfied with Goldstein's 'vast and messy' goulash: hers is not a recipe for true understanding." Feferman is a professor of mathematics and philosophy at Stanford University, and one of the editors of Gödel's Collected Works.
First of all thanks to Chad and Bob for the recommendation; the book is ordered and I look forward to reading Ms. Goldstein's interpretation of Mr Gödel.
Matt, a comments column of a blog is not the right place to hold a seminar on meta-mathematics but what you have written couldn't be more wrong. One of the consequences of Gödel is that "provable" and "true" are not equivalent. Expressed in terms of sets Gödel's theorem states that for a given formal system of sufficient strength the set of provable statements and the set of true statements are not equal. Provability is definable and is in fact defined within a formal system. Tarski's Truth Theorem shows that "truth" for a formal system cannot be defined within the system. Smullyan demonstrated that Tarski's Theorem and Gödel's Theorem are in fact logically equivalent.
Thanks, Glenn and Thony.
"To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature ... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in."
Richard Feynman, The Character of Physical Law (1965) Ch. 2
Thony, You are probably right that this isn't the best forum to discuss this but no theorem about formal systems proves anything about the correct use of the English word "true". What constitutes "truth in mathematics" is an open discussion amongst philosophers and I don't think any of them mean things like a Tarski truth function.
As I understand it, one approach is to follow (late) Wittgenstein an say "the meaning of "truth" in mathematics is in its usage by mathematicians". Now most mathematicians are not set theorists or logicians. If we say "the closed graph theorem is true" we mean "in our fixed set theory (presumably ZFC) there is a proof of the closed graph theorem". Most (IME) would hold that the question of whether the continuum hypothesis is true has no real meaning; "it's undecidable" would be the whole answer. If asked specifically about different systems we would say "Hahn-Banach is true in ZFC" but not in "ZF (NOT C)". "True" is not in this set up a formal concept in maths itself much less a formula within a theory, but why should it be?
Thony, I've tried to explain what I mean about the use of "true" here if you want to continue your seminar, over there.:)
I'm pretty sure I'm not coming from a clueless position on this; I am a professional mathematician and others (more widely read than I) have said complementary things about my discussions of foundational issues.
Matt, I don't think that the use of the word "true" in reference to arithmetical statements needs to be mysterious. To say that "It is true that 2+2=4" means nothing more and nothing less than "2+2=4". The Godel statement for, say Peano Arithmetic, is just a statement of arithmetic, of the form "there are no integer solutions to the polynomial equation P(x,y,z) = 0". When someone says "If Peano Arithmetic is consistent, then its Godel sentence is true but unprovable", it means nothing more and nothing less than "If Peano Arithmetic is consistent, then there are no integer solutions to the polynomial equation P(x,y,z) = 0, but Peano Arithmetic cannot prove that there are no integer solutions to the polynomial equation P(x,y,z) = 0" Of course, the explicit polynomial corresponding to the Godel sentence is enormously complicated, so people just give the sentence a name, G, and say "G is true" rather than write out G.
Rebecca Goldstein gave us an endearing, if not a little sad, portrait of KG. The anecdote of the "spy hypothesis" he offered up as regarded a Princeton appointment had to be dry humor though, despite that irrational authority is not really farfetched. There are many arenas which require the suspension of the impact of the true (NOT to be confused with suspension of the true) for purposes of peace and order.