Incompleteness: The Proof and Paradox of Kurt Godel (Great Discoveries) | Date: 04 May 2011, 08:57
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KURT GODEL IS CONSIDERED the twentieth century's greatest mathematician. His monumental theorem of incompleteness overturned the prevailing conviction that the only true statements in math were those that could be proved. Inspired by Plato's philosophy of a higher reality, Godel demonstrated conclusively that there are in every formal system undeniably true statements that nevertheless cannot be proved. The result was an upheaval in mathematics. From the famous Vienna Circle and sparring with Wittgenstein to Princeton's Institute for Advanced Study, where he was Einstein's constant companion. Godel was both a towering intellect and a deeply mysterious figure, whose strange habits and ever-increasing paranoia led to his sad death by self-starvation. In this lucid and accessible study, Rebecca Goldstein, a philosopher of science and a gifted novelist whose work often focuses on science, explains the significance of Godel's theorems and the remarkable vision behind them, while bringing this eccentric, tortured genius and his world to life.
Amazon.com Review
Kurt Godel is often held up as an intellectual revolutionary whose incompleteness theorem helped tear down the notion that there was anything certain about the universe. Philosophy professor, novelist, and MacArthur Fellow Rebecca Goldstein reinterprets the evidence and restores to Godel's famous idea the meaning he claimed he intended: that there is a mathematical truth--an objective certainty--underlying everything and existing independently of human thought. Godel, Goldstein maintains, was an intellectual heir to Plato whose sense of alienation from the positivists and postmodernists of the 1940s was only ameliorated by his friendship with another intellectual giant, Albert Einstein. As Goldstein writes, "That his work, like Einstein's, has been interpreted as not only consistent with the revolt against objectivity but also as among its most compelling driving forces is ... more than a little ironic."
This and other paradoxes of Godel's life are woven throughout Incompleteness, with biographical details taking something of a back seat to the philosophical and mathematical underpinnings of his theories. As an introduction to one of the three most profound scientific insights of the 20th century (the other two being Einstein's relativity and Heisenberg's uncertainty principle), Incompleteness is accessible, yet intellectually rigorous. Goldstein succeeds admirably in retiring inaccurate interpretations of Godel's ideas. --Therese Littleton
From Publishers Weekly
Godel's Incompleteness Theorem, which proved that no formal mathematical system can demonstrate every mathematical truth, is a landmark of modern thought. It's a simple but profound statement, but the technicalities of Godel's proof are forbidding. If MacArthur Fellow and Whiting-winning novelist and philosopher Goldstein (The Mind-Body Problem) doesn't quite succeed in explaining the proof's mechanics to lay readers, she does a magnificent job of exploring its rich philosophical implications. Postmodernists have appropriated it to undermine science's claims of certainty, objectivity and rationality, but Godel insisted, to the contrary, that the theorem buttresses a Platonist conception of a transcendent mathematical reality that exists independent of human logic. Goldstein is an excellent choice for this installment of Norton's Great Discoveries series, which seeks to explain the ways of science to humanists. Her philosophical background makes her a sure guide to the underlying ideas, and she brings a novelistic depth of character and atmosphere to her account of the positivist intellectual milieu surrounding Godel (including a caustic portrait of philosopher Ludwig Wittgenstein) and to her sympathetic depiction of the logician's tortured psyche, as his relentless search for logical patterns behind life's contingencies gradually darkened into paranoia. The result is a stimulating exploration of both the power and the limitations of the human intellect. Photos.
Reviews
Summary: Brief and Engaging Book on Godel
Rating: 5
This book centers on the irony that Godel'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.
Godel 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. Godel'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. Godel 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.
Summary: A Most Important Read
Rating: 5
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 Godel 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. Godel'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. Godel 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 Godel'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.
Summary: Not What I Expected, But . . .
Rating: 5
Probably a better choice for most of us (including me) to read first, which I am glad I did. I was expecting a mathematical book about Goedel's incompleteness theorem, but this is really a biography of Kurt Goedel [Note: 'oe' is the standard substitute for an umlauted 'o' when one doesn't have the option of using the latter, which this text box doesn't provide.]
Professor Goldstein does provide a simplified explanation of Goedel's incompleteness theorems (there are 2), and a reference to Godel's Proo
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