Incompleteness: The Proof and Paradox of Kurt Gödel by Rebecca Goldstein
My rating: 4 of 5 stars

Gödel’s Riposte to Augustine

I find an unexpected comfort in Gödel’s Proof of Incompleteness in mathematics - essentially that we have no good reason to believe that even arithmetic has a solid logical foundation. To me the implication is that no matter how much we learn, we will still be wrong. Not because we don’t know everything, but because what we do know is fundamentally uncertain. We are not unsure only about mathematics. Physics for example will always exhibit paradoxes like those of quantum theory. People unaccountably will always do things which are bad for them. And my socks will continue to disappear in the dryer. There is, in other words, a fundamental continuity, a necessary humility, in life that will never be interrupted by the latest technology from Apple or Trump’s most recent tweet.

Goldstein appreciates the cultural import of Gödel’s Proof. In an age rocked by the counter-intuitive implications of things like Relativity Theory and Quantum Mechanics, which present paradoxes that seem resolvable by further thought, the Incompleteness Theorem is even more of a scandal. It exists in logic not in observation. It will remain in force no matter what else we learn about the world. As Goldstein says, “Gödel’s theorems, then, appear to be that rarest of rare creatures: mathematical truths that also address themselves—however ambiguously and controversially—to the central question of the humanities: what is involved in our being human?”

Incompleteness is a leveller. It applies to the deepest thinker, the wealthiest entrepreneur, the most powerful politician, as well as to any random cog in the modern economic machine and to those who have been rejected by it. It is the modern form of the ancient Christian doctrine of Original Sin, formulated so forcefully by the great saint, Augustine of Hippo.

Just as Original Sin, Incompleteness affects us all. We inherit it, not through our genes, but through our memes. Incompleteness comes packaged in language itself. To engage the world through language is to enter the domain of Incompleteness, and therefore of profound doubt. And just as Augustine said in his religious idiom, Godel has restated the situation in his: There is no escape. The user of language is trapped and is incapable of extricating himself from an existence of rational error - about himself as well as the world around him.

But unlike Augustine, Gödel doesn’t presume he has a solution. Augustine withholds his assent to radical doubt, which he neutralises through ‘faith.’ Clearly Augustine cannot stomach the intellectual humiliation of not having a way forward, of not mitigating the debilitating effects of the human condition. Like many before him and since, Augustine fills the intellectual vacuum with the magic of a divine saviour, the guarantor of the ultimate rationality of human and other life on Earth. For Augustine Christ is the deus ex machina who is capable of correcting, literally remaking, flawed human nature into something reliable. And if other people don’t ‘get it,’ he feels entirely justified in literally throwing them to the lions.

Augustine, of course, merely demonstrates the extent to which the basic human flaw can make us crazy. That his need for and presumption of a beneficent saviour is part of his Original Sin is something which doesn’t occur to him. His solution is actually the greatest delusion produced by his fundamental insight. He neurotically invents in order to avoid his own logic, and then projects his neurosis onto the world as a defect which must be eliminated. He is the first Christian terrorist.

Gödel has no such delusion and therefore accepts the bleakness of our prospects. What Godel allows us to see, however, is that mathematics is a genre of poetry with its own arbitrary, but still rather satisfying, conventions. It is something to be done for its own sake, not because it leads anywhere else (the Princeton Institute for Advanced Studies of which Godel, along with Albert Einstein, was a founding member was created on the principle of “the usefulness of useless knowledge”).

Prospects, bleak or not, - whether spiritual or material - have nothing to do with the matter, therefore. Mathematics, like the rest of poetry, is important in the continuous present. It doesn’t save us but it passes the time rather pleasantly. Gödel was no materialist or relativist, however. For my money he was more spiritual than Augustine, as well as more committed to the idea of truth. He knew there was something permanently beyond human reach. As a committed Platonist, he considered this to be the abstract realm of numbers, which exist quite independently of human thought about them.

Numbers for Gödel are eternal and impassive, that is, there is nothing we can do to affect their existence. They call to us from elsewhere, much like Augustine’s God. The principle difference however is that numbers make no absolute demands and pass no judgments. They exist for our comfort and edification not for remaking us as something we’re not. And very few have felt compelled to use violence to defend number theory.

Goldstein makes an apposite observation: “Paranoia isn’t the abandonment of rationality. Rather, it is rationality run amuck, the inventive search for explanations turned relentless.” Augustine is an example of rationality run amok. This certainly is the heart of Gödel’s riposte to Augustine’s and all other religious arrogance.

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