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Incompleteness: The Proof and Paradox of Kurt Gödel

Rebecca Goldstein
W.W. Norton
Publication Date: 
Number of Pages: 
Great Discoveries
[Reviewed by
John Martin
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The impossible dilemma of writing popular mathematics: How many equations to use? Too few and the language is insufficient to the subtlety of the ideas. Too many and the interested non-specialist reader puts your book back on the shelf having failed the flip-through test for readability.

I agree with Rebecca Goldstein: the importance of Gödel’s work on incompleteness of formal systems reverberates so broadly beyond the narrow confines of academic mathematics that someone’s got to risk blurring the ideas a little by styling a narrative that welcomes the non-specialist reader with a great story. And Rebecca Goldstein is a great storyteller. Incompleteness spins an engaging yarn introducing a few of the main players who’ve struggled with the philosophical implications of Gödel’s work over the past seventy-five years

In Incompleteness we meet her good guy, Kurt Gödel, with his good idea, mathematical Platonism. Then, because every good guy needs a bad guy, we meet Ludwig Wittgenstein with his bad idea, linguistic relativism. This bad guy has a gang, the logical positivists, who aren’t really his gang but Wittgenstein “fulminates,” “banishes,” “devastates,” “rages,” and “bewitches” them into delusional discipleship. Gödel, the lone-gun genius standing in quiet but principled opposition to the scourge of Wittgenstein and his gang, nevertheless lives a life too quirky, timid, and tragic to support his elevation to the status of icon-genius where he belongs. So, Gödel needs a buddy, Albert Einstein, whose icon-genius stature is so enormous that just walking and talking with Einstein every afternoon at Princeton strengthens Gödel’s genius platform considerably.

Cool in his white hat, Gödel measures up the positivists at their Vienna Circle hideout. A man dazzled with an idea, Gödel has experienced “loves ecstatic transfiguration,” (p.59) in a youthful encounter with Mathematical Platonism according to which mathematics, while non-empirical, is nevertheless “descriptive of objective reality.”

Hunkered down in the Circle listening to the gang’s subjectivist ideas which Goldstein characterizes early on as “subversively hostile to the enterprises of rationality, objectivity, and truth,” (p.23) Gödel sees he’s badly outnumbered. He needs a powerful trump ready to hand if he’s going to clean up this outlaw bunch.

By age 23, he’s got one, his first incompleteness result. Trouble though: Gödel’s laconic, self-effacing let-the-mathematics-do-the-talking style badly misfires. First, no mathematician other than John von Neumann even understands its significance. Then, when the mathematicians finally start to get it, the “intellectual gurus” (a shadowy group of postmodern intellectuals lurking around the fringes of Incompleteness) entirely misunderstand – reading the incompleteness theorems as support for Wittgenstein and the Logical positivists!

Goldstein is out to set the record straight. Gödel, author(!) of the incompleteness results, understood them as vindicating his first love, Mathematical Platonism: The incompleteness results drive a wedge between the narrow idea of proof within a formal system and the broader idea of truth. Platonic intuitions (the faculty by which Platonists non-empirically access objective mathematical truth) necessarily flourish on the margin of truth that exceeds the reach of mathematical formalism. Direct intuition of mathematical truth will forever be necessary to remedy formalism’s shortfall.

A correct account of Gödel’s own philosophical interpretation of his work is of course central to understanding the ongoing struggle to pin down the broader implications of his theorems. However, in Goldstein’s version, neither Wittgenstein nor ideas opposed to mathematical Platonism get the kind of sympathetic hearing accorded Gödel. Perhaps the slanted telling is explained on p. 63 where, generalizing the kind of heady experience that can engender “loves ecstatic transfiguration,” Goldstein ends with the parenthetic remark “(I remember my own.)” While the honest disclosure is admirable, the dramatic styling of good Gödel/bad Wittgenstein nevertheless crosses a line from presentation to boosterism.

And crossing this particular line delivers the reader a nasty ironic jolt: Gödel champions the objective basis of mathematical truth; Goldstein defends Gödel’s position with a broad array of subjective supports. Yes, one ought to pay careful attention to what Gödel himself thought his theorem meant philosophically. However, because his position defends objective truth, one expects an objective exposition of the evidence supporting his philosophical belief. Instead, Incompleteness delivers: a quasi-Freudian explanation of the roots of Gödel’s Platonism that carefully emphasizes its embrace antedating and spurring proof of the incompleteness theorems; a lengthy comparison of the personalities of the champions of the opposing positions; and that hagiographic elevation of Gödel’s genius by association with Einstein. How can any of this (undoubtedly useful and interesting information) bear on the correctness of Gödel’s philosophical interpretation of his results – unless, of course, one embraces a more subjectivist explanation of mathematical belief?

Thus Incompleteness suffers an odd stylistic inconsistency. Its overt message, “Believe Gödel: his result shows that mathematics is objectively true,” is viciously undercut not only by Goldstein’s believe-as–I-say–not–as–I-do style but also by a more natural reading that her subjective observations about Gödel’s belief structures suggest (to me at least). Specifically this: If, as Goldstein says, Gödel believed passionately, unshakably in Platonism prior to concocting his incompleteness results and then set out specifically to find mathematical results that support Platonism, isn’t this evidence of the very kind of personal bias, finding what you set out to find prejudice, that the postmodern intellectual gurus charge?

Which lesson should the reader learn from Incompleteness – the objectivism it overtly champions or the subjectivism it both employs and suggests?

None of this is to say you won’t enjoy Goldstein’s story. It’s a juicy personal tale about a number of interesting lives that poses (perhaps unintentionally) a difficult conundrum about objectivity and subjectivity that you might like to decide for yourself. If you do venture into Incompleteness, four small warnings in order from quibble to serious reservation about blurrings in the technical details:

As other reviewers have remarked, renaming first order logic “limpid logic” (p.150) serves no recognizable purpose and jars the senses if you are accustomed to the usual name.

Cantor’s Continuum Hypothesis (CH) concerns “cardinality” not “ordinality” as Incompleteness repeatedly asserts (p.139). Surely this is a typographical rather than conceptual confusion.

On the same page (p.139), the Continuum Hypothesis is characterized as an ignoramibus. The sentence immediately preceding this characterization is appropriately restrained: “Gödel, together with Paul Cohen, proved that the continuum hypothesis can be proved neither true nor false within current set theory.” However, the ignoramibus label with its clarifying clause “– a claim that can neither be confirmed nor discredited, a claim about which we remain ignorant.” misleadingly universalizes the unproveability of the CH. That CH is independent of ZFC (Zermelo-Fraenkel+Choice which can be taken to be “current set theory”) does not entail that CH is outside the range of all formal systems. Indeed, one understands much of the current work in set theory as searching for stronger axioms which will settle the continuum hypothesis in some “natural” way. Gödel’s incompleteness results doom any hope of finding a single universal formal system in which to settle all mathematical questions but not the hope of finding an interesting formal system that will settle a particular question like the continuum hypothesis.

Incompleteness repeats the “disaster-theory” of the set theoretic paradoxes both historically – “Russell’s discovery of his paradox had grievous consequences in the foundations of mathematics, and for one man in particular, Gottlob Frege” (p.91) – and logically – “So an inconsistent system is worthless as a tool of proof.” (p.92). On the historical: Though Russell’s paradox did temporarily interrupt Frege’s publication schedule, it has always seemed odd to hear Russell’s Paradox – which is so clever and so stimulatingly embedded in the long sorting out of foundational concerns that leads to Gödel’s breathtaking result – described with adjectives like “grievous.” Far from disasters, putative inconsistencies are historically some of the most stimulating developments in mathematics. It’s time we stopped denigrating them as “disaster”, “crisis”, “grievous consequence,” etc. On the logical: It seems pertinent to bear in mind that we have no proof within a provably consistent theory that Peano Arithmetic (PA), for instance, is consistent. Wouldn’t the modest Platonist – one not entirely certain that his/her intuition is 100% reliable (existence assumed) – therefore concede the possibility that PA may really be inconsistent. If so, we’ve all spent a lot of time doing some very interesting and insightful work in an inconsistent theory. Will we really abandon all that should we tomorrow discover proof that PA really is inconsistent? Or will we just go back and patch it up like always? Discarding an inconsistent theory as “worthless” is a little like tossing aside an otherwise interesting book just because its style is inconsistent with its content.

John Martin is a Senior Instructor in the Farrand Academic Program at the University of Colorado, Boulder.


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