Comprehensively paradoxical Gödel – Rebecca Goldstein

I mentioned noting that one of Rebecca Goldstein’s earlier works was Incompleteness – the Proof and Paradox of Kurt Gödel. Since Gödel is an existing interest of mine, and Goldstein’s writing has never let me down yet, it was a no-brainer to obtain a copy. (I’ve since also obtained a copy of her fiction The Mind-Body Problem – effectively I’m now working my way through her entire back catalogue. *)

Gödel and friends

The fact that Gödel had associations with Einstein, Wittgenstein, Von Neumann, Turing, the Vienna Circle and more, means that as well as official biographical material there is a wealth of anecdotal and indirect material to work with. And despite Gödel’s own meager introverted publishing record, the meticulous unpublished material he left behind (nachlass) proves to be another fascinating source.

Too many touch points with my own agenda to mention them all, suffice to say both Gödel and Wittgenstein were at odds with, and misunderstood by, logical positivism as well as each other. A fascinating love-hate triangle so well suited to the Goldstein treatment I’ve come to love.

Mach and more

Interesting to again find Mach as an original influence at several distinct points, so many topics lead back to Mach (and hence for me to Boscovich before him **). Only four (long) chapters, the third of which is a semi-formal description of what Gödel’s theorems actually did formally prove and how they did it. A necessary task given how misunderstood and misquoted out of context Gödel can be. A considerable amount of detail on Wittgenstein; unsurprising given the relationship to Gödel. A name-droppers list of players, but everyone of them alive on the page. Great writing, some turns of phrase I found significant to my work:

Describing paradoxes as part of philosophy since the beginning – Epimenides’ Liar’s Paradox, Russell’s own set theory version and more, she goes on to say:

(p91) Paradoxes have often been found lurking about in the deepest places of thought. Their presence is often the signal (like the canary dying?) that we have managed, sometimes unwittingly to stumble on a deep and problematic place, a fissure in the foundations.

The cracks are how the light gets in? The foundations of mathematics in this case.


One influence on Wittgenstein:

(p95) Otto Weininger, a quintessentially Viennese figure who had argued that the only way for a man to justify his life (for a woman there is no way) is by acquiring and cultivating genius.

As mentioned, there follows an extended section on Wittgenstein, his Viennese origins and influences, and those influenced and (failed to be influenced) by his work. On The Vienna Circle (p97) “reading Tractatus exegetically like the Torah” despite Wittgenstein disowning them and going to great lengths to offend them with his in-your-face displays of mystical readings, and:

(p103) Wittgenstein’s attitude toward the inherent contradiction of the Tractatus is perhaps more Zen than positivist. His insouciance in the face of paradox was an aspect of his thinking that was all but impossible for the very un-Zen-like members of the Vienna Circle to understand.

There’s lots been said about Wittgenstein’s motives towards Russell in writing Tractatus, and personally I consider it a long joke at Russell’s expense, using logic to put logic in its place, Goldstein writes:

(p103) Whereas the early Wittgenstein had laboured hard with Russell on problems of logic, the later Wittgenstein came to regard the entire field as a “curse” (while Russell, disheartened by his earlier labours with Wittgenstein – his inability to understand him – withdrew from the field and wrote best-sellers).

Talking of the Vienna Circle, particularly Waissman and leader Schlick, idolising Wittgenstein and his Tractatus, despite his rejection of both the Circle’s logical positivism and their failure to understand his position, she quotes contemporaries:

(p105) “[Schlick] returned in an ecstatic state … He adored him and so did Waissman … like others of Wittgenstein’s disciples, the Circle members came to imitate his gestures and manner of speech [and dress].”

Continuing, she describes his famous concluding aphorism as both a succinct summary of his actual position, and an expression of his exasperation at the failure of the Circle to grasp the limits to (their) logic he was exposing. “That whereof we cannot speak we must remain silent” was interpreted by the positivists as saying that there was nothing beyond the bounds of [logical] language that was worth saying,  whereas of course he was saying the opposite. Having said all we can say with logic, we can say nothing more with logic and, since that [formal] language is largely tautological and/or trivial, what we have said is of little significance and all the really important stuff is beyond such language.

(p106) “For Wittgenstein there really was ‘that whereof we cannot speak’. The ethical or – what amounts to the same thing for him – the mystical is both real and inexpressible. He believed that he had explained all that can be said [formally] in the Tractatus,  but as he told one potential publisher, what he had not said in the Tractatus – because it could not be said – was more important than what he had said:

“I once wanted to give a few words in the foreword which now actually are not in it, which however I’ll write to you now because the might be a key for you: my work consists of two parts: of the one which is here, and of everything I have not written. And precisely this second part is the important one. For the ethical is delimited from within, as it were, by my book; and I’m convinced that strictly speaking it can ONLY be delimited that way. In brief, I think: All of that which many are babbling today, I have defined in my book by being silent about it.”

He took himself to have demonstrated how little one has actually said after one has finished saying all that can be [formally] said.”

Back to Gödel

Anyway, fascinating though the interpretations of Wittgenstein remain to this day, the point is the parallel with Gödel and his incompleteness, and the wider point that we’ve inherited from Aristotle’s mis-directed systematisation of what he thought he’d learned from Socrates and Plato. Given my own experience arguing with information modellers on the place of “First Oder Logic” or “Predicate Calculus” in real-world industrial applications, the passages starting p150 where they are poetically re-branded Limpid Logic by Goldstein, appealed to my sense of irony. Logically true, even tautologous, but not of direct practical value. Or, as an early (engineering) mentor of mine was fond of saying “10 out of 10, useless” whenever one of us juniors presented him with some design or calculation.

In the same way as Hilbert had been trying to turn mathematics into a closed and comprehensive (and necessarily consistent) set of axioms, Gödel had the same aims in the “pure” logic of mathematics before discovering his proof of its impossibility. (Russell had been trying to do the same by adding additional practical rules to the set of axioms.) And, in the same way as Wittgenstein’s own disciples failed to recognise his message that all the important real world stuff lay outside their closed systems of logic, Gödel, despite announcing his result at the 7th October 1930 conference in Königsberg, actually failed to have any effect on the mainstream, until Von-Neumann took his message back to Princeton and afterwards helped to spread its significance. In the same way as Wittgenstein had turned logic on itself in Tractatus, Gödel Numbering had turned numbers back on themselves to prove their inadequacy.

Ultimately Gödel lived out a life of paranoia in Princeton, where only Einstein was his mental equal, discoursing during their mutually cherished daily walks. As a contemporary young student at Princeton’s Institute of Advanced Studies, Goldstein was clearly inspired by their stories, not only to understand the technicalities of their deeply philosophical topics, but to turn their living narratives into literary form.

I side with all of Einstein, Gödel and Wittgenstein; they were right even when they knew they were wrong, their genius was accepted in practice, and their contributions to their respective fields considered pivotal, yet despite their mythological public personae, the mainstream of life continues to this day as if they’d never existed. We still worship the (rational & scientific) value of logic and objectivity, and give them priority over, merely tolerating, even to the exclusion of, the (irrational & unscientific, human) values of the mystical and ethical. As Schumpeter would suggest, and Kuhn after him, a single human lifetime is never enough to cause a global paradigm shift in society at large – in politics and economics. It takes three, baby.


[Aside – on p124 is that an error in use of the word “acute” in the description of sketch 2? Page references based on the W W Norton “Great Discoveries” hardback 2005 edition.]

[Post Note (*) – one thing to note about Goldstein is that her writing alternates between fictional literature and historical biographical works, both with science and philosophy themes. Loved, but never did explicitly review, all of them – Googleplex, 36-Arguments, Spinoza, Gödel, Properties of Light, Mazel, Mind-Body Problem and Strange Attractors:

I was taken by a recent remark from Lisa Jardine as to how good writing of history based on documented records always requires creative fiction, and it comes to mind as I read Goldstein. Jardine is a historian by profession, but had the writing of her very personal biography of Jacob Bronowski in mind when when she said it.]

[PPS (**) – when is someone of this stature going to research and write on Roger Boscovich (1711 – 1787) I wonder.]

[PPPS – 2017 update from Andy Martin. Without hypocrisy all is tautology. Add Brunsson to the list.]