FOM: Arbitrary objects, intuition, and rigor

[JoeShipman@aol.com]

Avron writes:

>>Both logic and experience have taught me that whenever there is a clash between intuition and rigour it means that something is wrong with the intuition. Rigour simply cant be wrong.<<

I disagree.  When logical rigor and intuition clash it doesn't mean that intuition is necessarily wrong -- a conclusion rigorously arrived at can certainly be wrong if the assumptions it was rigorously derived from are wrong.  The right thing to do when confronted with such a clash is not simply to dismiss one's intuition; rather one should seek examine both the reasons for the intuition AND the assumptions behind the logic.

In the issue under discussion, I am inclined to agree that most of the confusion comes from taking a particular natural language (English in this case) too seriously.  Everyone here understands how to formalize a proof using "arbitrary objects" in the predicate calculus, and how to reason correctly using this manner of speech (and it is a manner of speech, "arbitrary" is not a predicate).  

If Fine contends that some of the intuition behind this manner of speech has not been fully captured by the formalism and offers something new, this could be significant, as Davis pointed out by analogy to Robinson's Nonstandard Analysis.  I haven't been able to find Fine's book yet, and I don't quite understand from the discussion here exactly what new ideas he has brought to the subject, though I look forward to finding out.

-- Joe Shipman