Journal

The scope of Gödel's incompleteness theorem

Posted on March 11th, 2005, in the small hours

This article from Slate magazine has got me thinking again about the implications of Gödel's incompleteness theorem(s). (Not, I hasten to add, that I'm guilty of anything like the spectacular kind of short circuit the author cites: "Gödel's theorem proves the Doctrine of Original Sin"(?!), nor, I also hasten to add, that I imagine I understand everything at stake here.)

The article is in response to a recent book making high claims about the significance of Gödel's theorems, the contention of the article being that our esteem needs to be kept in perspective with the actual difference they have made to mathematics (not much, they argue) and their limited scope of relevance (they are only "direct proofs" about maths, not about truth in general).

It's the last paragraph of the article that has got me thinking. The last paragraph comments favourably on the link the book's author traces between Gödel's philosophical commitments and his mathematical results, specifically with regards the relationship between numbers (which Gödel understood as 'real things') and the axiomatic systems that are attempts to describe them and their relationships (which are human constructs).

I don't disagree with the article's admonition against a romanticised view of the implications of Gödel's mathematical proofs; it is only by analogy that we can make broader claims about human knowledge from Gödel (not by "direct proof") and we should therefore be appropriately cautious in putting too much weight on such analogy (as is not in evidence in the bold pronouncement on Original Sin). But it seems to me that (analogy though it is) the analogy actually bears more weight (for a certain kind of purpose) when the relationship between Gödel's philosophical commitments and his mathemetical results is emphasised, precisely because that relationship is so informative.