### First thoughts on Rebecca Goldstein’s, Incompleteness: The proof and paradox of Kurt Gödel

The power of the book doesn’t come from its treatment of the theorem itself (she does an adequate job, but others have done better. See for example Nagel and Newman’s classic,

*Gödel’s Proof*for a fine non-technical treatment); rather the books achievement is that it puts Gödel’s work in context. Goldstein successfully (and finally) gives Gödel’s theorems the philosophical interpretation that he himself would have intended.

Before reading

*Incompleteness*I often wondered why Gödel, an avowed Platonist, did most of his work in Mathematical Logic, the most formalist of all mathematical fields. Also, why did he join the Logical Positivists of Vienna who in their way were the most extreme kind of Formalists; and lastly why did Gödel associate himself with a group who revered the teachings of Wittgenstein – the very same Wittgenstein who essentially claimed that all of a mathematics was a mere tautology (a claim that was almost surely quite repulsive to Gödel, and to almost every other mathematician).

Goldstein answered all of this (and more). She gets her answers not from the mathematics, but from the story of Gödel’s life and the philosophical battles that drove him.

In brief, the story is that the Logical Positivists essentially believed that truth lived in the precise, meaning-aware use of language. According tho them, it is only possible to identify a statement as being true or false by proving or disproving it by experience. Logic and mathematics was excluded from this rule; they claimed that mathematics was a branch of logic and was for all intents and and purposes a mere tautology.

Gödel on the other hand was a Platonist; he believed that mathematicians uncovered truths about the universe, and mathematical concepts were merely communicated by—but not contained within—its equations and symbols. Yet, confusingly, Gödel belonged to a Positivist group. He largely stayed silent through their meetings, neither objecting nor agreeing, for that was not his way.

But the internal storm of disagreement that welled within him did lead him to prove that Positivists were wrong. He proved that the structural manipulation of mathematical symbols could not yield all statements that we know to be true. He demonstrated a ‘true’ statement that was not provable—which should have banished Logical Positivism for ever.

Yet it didn’t; for Godel, before Goldstein’s book, was never well understood.

I’ll have a lot more to say about all this in the coming weeks.