On seeing a theorem

Sun 25 Dec 2005 05:08 PM

A stray thought that didn't make it into the induction paper: In John Worrall's 2000 BJPS article, he writes:

Recognising that some proposition is indeed a theorem of some axiomatic system is clearly an outstandingly creative act... But what else can a great mathematician be doing when recognizing that proposition P is a theorem, but somehow-- and clearly in large part subconsciously-- going through some mental process that amounts to the construction of a sketch-proof for P? [fn. 13]
Is there anything that indicates the shift from argument to bold assertion more clearly than a rhetorical question?

A mathematician, in situ, might arrive at a conclusion in any number of ways. The public defense of that conclusion requires that it meet the muster of public standards. It is important not to get confused and think that the private process must already mirror the public debate that follows.

Pattern recognition-- as a psychological matter-- is perceptual rather than inferential. Mathematicians are trained to recognize theorems. Good ones can recognize that something is a theorem on sight, without even thinking out the sketch of a proof. For most theorems identified in this way, they can provide a proof-- but that is a separate matter. It seems natural enough to think that great mathematicians might recognize in the same intuitive way that some novel, thrilling P is a theorem even when they are unable to give a proof of P. There doesn't need to be a sub-conscious sketch proof lurking in the recesses of their brain.

(This is some support for the discovery/justification distinction, even though it is now fashionable to diss on that distinction.)