 Gentle introduction to Godel
This book sets out to understand the incompleteness theorem through Godel's nature and intellectual relationships. In doing so, we get some excellent philosophical insights situating Godel's theorem with Einstein's relativity theory, Hilbert's formalism, Wittgenstein's philosophy of language and Turing's decidability theorems. The proof itself takes one chapter midway through the book. It's a general self-contained introduction that just gives a concise overview. The last chapter is pretty light and concerns the times of Godel in Princeton. Overall, this book is quite a tour de force to tie up all the loose ends to understand this very important theorem.
 Incompleteness: The Proof and Paradox of Kurt Gödel
Incompleteness: The Proof and Paradox of Kurt Godel (Great Discoveries) by Rebecca Goldstein tells the story of Kurt Gödel, one of the greatest mathematicians and logicians of modern times. Gödel's theorem suggesting that all mathematical truths or, for that matter, logic, cannot be known, defined or proved, rocked the scientific community of the 20 th century and the theorem significantly challenged all modern thought.
The author of Incompleteness, Rebecca Goldstein, has taught philosophy at various top U.S. universities. She is also the author of eight books. Incompleteness is a nonfiction account of Gödel's life and accomplishments. The book portrays a man of great, albeit tortured, genius.
Goldstein opens the book not by focusing on Gödel's revolutionary theorems but by exploring his intriguing friendship with Albert Einstein. On the first page we are shown an image of two men walking serenely together, ".hands clasped behind their backs, quietly speaking". As we read on we see that these two men on their daily walk are Albert Einstein and Kurt Gödel. These gentlemen used to take daily walks together at Princeton University . Einstein, the physicist, and Gödel, the mathematician, were in many respects, very different but they understood and respected each other, establishing a warm and close friendship.
The author relates the relationship between the two men to illustrate how they both saw themselves as exiles from their native Austria . Their intellect has greatly impacted human thought and at the same time, this same intellect isolated them from others. With the presentation of the Einstein/Gödel friendship, Goldstein sets up her thesis that those we consider to be "geniuses" have internal struggles with their intellect and may very well be alone and isolated.
The book then takes us through Gödel's life, from his entrance into the University of Vienna at the age of eighteen to his fascination with Platonism, the invitation to join a group of distinguished philosophers known as Vienna Circle and development of his incompleteness theorems. We are shown how Gödel's passions not only led him to his groundbreaking work, but also brought him to the brink of madness, which, in turn led to his tragic end.
Incompleteness-The Proof and Paradox of Kurt Gödel may be a daunting read for some but the strength of the work lies in the compelling look into the revolutionary mathematician's personal side of Gödel and a clear explanation of Gödel's "incompleteness" theorems.
The general reader may not be able to grasp the concepts of Incompleteness but those knowledgeable in the field of mathematics and philosophy should find this work very informative and enjoyable.
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Brief and Engaging Book on Gödel
This book centers on the irony that Gödel's own philosophical interpretation of his work (which indeed may have driven his efforts to begin with) was in complete opposition to how it was most commonly interpreted by others.
Gödel was a Platonist, believing that the mind was able to make contact with absolute mathematical reality. Given that he was an attending member of the Vienna circle in the 1920's, which was the locus of logical positivism, many assumed he was of like mind, believing there was no truth beyond what man could empirically discover. Gödel's extreme reluctance to speak or write on his views helped make this misunderstanding possible. Indeed, the incompleteness theorems have often been co-opted by sloppy post-modernists (along with relativity theory and the uncertainty principle) in making the case for truth relativism. They would focus on the conclusion that we can't construct formal systems (large enough to at least encompass arithmetic) which are both complete and provably consistent and treat this as revealing a limitation in our ability to reach absolute truth. Gödel believed the actual lesson was that the human mind can and does perceive truth beyond the capability of formal systems (equivalently, algorithmic computing machines).
This book does a nice job in the treatment of the ideas as well as the biography.
A Most Important Read
Goldstein, does a masterful job describing the life and the work of the greatest logician to ever live. Ironically the genius and logical perfection exuded by Gödel is in the end matched by the equilibrium of the universe- he becomes completely illogical and insane.
Goldstein writes with a piercing passion and pointed savvy that I envy. He deep appreciation for the mind of the great logician bleeds all the way through the entire read. Gödel's incompleteness theorem took formalistic logic and arithmetic in a time when it was getting ready to announce its supreme dominance and perfection to the world and turned it on its head. Gödel proved that logic and arithmetic will forever be incomplete within themselves. In other words, logic and arithmetic will never take the place of human reasoning or mathematical truth. Man is not machine.
This all started with Russell's paradox which is the proposition
This sentence is false.
Known as the liar's paradox, Russell's paradox has a very strange quality about it. The "false" part applies to the whole sentence and its subject simultaneously. Thus if you seek to give the sentence a true or false value we run into immediate problems.
Is the proposition is false then it cant be false within itself and so it isn't false it must be true. This means that it is self contradictory.
But then again if the proposition is true then it isn't' false; another contradiction. Russell's paradox wins no matter what. There is something very special about negations indeed.
This book is monumental not simply because Goldstein can write like a demon on a mission but because Gödel's life and accomplishment is timeless. His theorem is crystal clear and logically flawless-- one of it's, if not "the" strangest and most ironically paradoxical qualities.
If you have any interest in philosophy at all- read this book. Its a must. Not.
Excellent
Among the interesting byproducts of feminism and the admission, commencing in 1970, of women to places like Princeton are overall more interesting and "cultured" readings of analytic philosophy and mathematics, before that male ghettos.
Goldstein, who studied logic and philosophy at Princeton (and who used vignettes from her experience in "The Mind-Body Problem", a novel) met Goedel, and understands the technical details of his work thoroughly. She does a better job, in fact, than Ernest Nagel did in 1968 because she makes emotional connections that exist in mathematical work but which mathematicians often don't like to talk about.
Nagel did say something about Goedel's "intellectual symphony", but Goldstein, unlike Goedel, did deeper research into Goedel's biography, snooping for example around the Mercer County courthouse for records of his US citizenship application.
She reveals the plight of the hyper-intelligent and why we have tenure, since guys like Kurt Goedel and John "A Beautiful Mind" Nash are snuffed out in the so-called "real world": once Einstein passed on, Goedel, we learn, had nobody to talk to.
Interestingly, we get no Pop-feminist nonsense and boo-hoo-ing about Goedel's wife and her loneliness, having married a truly weird individual. Mature women know today what my Mom knew: you make your bed and you lie in it, and any marriage is a unique contract. Gretel Karplus, Adorno's wife, was far more intelligent than Mrs. Goedel but she buried the possibility of being an Arendt or a Weil in service to Teddy and was shattered by his unexpected death. Likewise, Goedel's wife seems to have gotten what she wanted and what many women would kill for: a quiet husband and a house on Linden Lane.
Goldstein's "philosophy of mathematics" is nuanced. Unlike some feminist philosophers she makes no attempt to reduce the subject-matter to some sort of Freudianism. At the same time, she knows that "what we think about when we think about math" comes as do other inputs: by way of meat.
This is an *aufhebung* worthy in its own workyday way of an Aristotle or an Aquinas, because a sharper bifurcation and reification renders lifeless the terms on either side of the cut. Just as Aristotle realized that there are Forms but always instantiated, and just as Aquinas applied this insight to religion, Goldstein manages to hold together the apparently opposing thoughts, that mathematical realities are independent of our thought...but have no existence *that we know of* outside our embodied thought. They are the closest thing we have to noumena manifesting as phenomena.
As a dialectical thinker, Rebecca Goldstein knows how negation works in embodied space. By trying to make themselves over into things, "thinking machines", the Positivists transformed themselves, as she shows, from a sought objectivity into its reverse; this was also C. S. Lewis' insight, in his novel That Hideous Strength, in which the Logical Positivists of Belbury turn out to be merely Satanists, of a sort, in a word, chumps who bow down to wood and stone, having emptied themselves of the capacity for thought through a nihilistic metaphysics.
The problem with this gesture is that (as Adorno pointed out), the categories themselves are in motion so that at the end all we "know" is that:
(1) Logical Positivism imprisoned the scientific subject within a barrage of sense-data, without explaining how sense data organizes itself.
(2) Formalism in mathematics simply denies that anything exists outside a formal system in a relationship of containing. Fearful of either benign or else vicious circles, it refuses to do mathematical philosophy.
(2) First rate minds (Goedel and Wittgenstein) wanted no part of this malarkey.
As the Austrian philosopher Gustav Bergmann pointed out, Logical Postivism's denial was a perverse sort of metaphysics. In the middle of its denial, Goedel upped the ante by discovering that the paradox of the Liar has a metaphysical implication as regards the capacities of formal systems, versus that of human beings. Goedel stood outside the machine (the formal system) and derived an indirect existence proof of truths unprovable within the machine, such that if they were incorporated as axioms, new unprovable truths would appear, and this is why today we almost never anthropomorphise computers: whereas the pronoun for a ship was she, the pronoun for computer is it (and, the adjectives are not printable).
Parenthetically, I was glad to see Goldstein mention Gustav Bergmann, a relatively minor member of the Vienna Circle, since he'd self-marginalized by moving to the Midwest, that black hole, and teaching at the University of Iowa. Bergmann gave a talk at my university in which he pronounced a Goedelian commitment to the continued existence of ontology and its truth, saying he'd die in a ditch to defend it. At this time, in 1970, Goedel was invisible and people were unaware that he felt and thought pretty much the same as Bergmann.
Does Goedel's proof have metaphysical import? Goldstein rejects what she calls the postmodern interpretation, which she re-presents as the argument that (1) mathematics is undecidable ergo (or, as First Gravedigger says in Hamlet, argal) (2) there is no "truth", only "stories".
Of course, neither Derrida nor my fat pal Adorno make this argument. Indeed, there's quite a lot of metaphysical speculation and conviction in Derrida; for example, arche-writing is an ontological analysis of meaning which, ontologically and Kantian-metaphysically rejects doing ontology with received categories of writing and speech. Derrida was merely unconvinced that the only reine vernunft on tap is mathematically expressible as opposed to using natural language.
But this is a minor aporia on Goldstein's part, caused I think by the fact that during her studies at Princeton, "deconstruction" was fashionable and usable in a sloppy way unlike mathematics.
There are many popular books on mathematics that overstress fascinating and sexy details about the biological mathematicians. While the current rage for this, sparked by the movie A Beautiful Mind, might help to get math geeks laid, a mathematical biography should balance the math and the meat, and even more than Sylvia Nasar's book eponymous to the movie, Incompleteness does this.
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