On 6/11/08, Stephen Gray <stevebg@roadrunner.com> wrote:
... The most complex proofs must be accepted provisionally, since all advanced science and math is accepted on the basis of approval by experts. As much as I dislike it, even mathematics is therefore subject to authority. ...
[A touch of subject-line creep poking its head up here, I fear ...] It's not even that simple (is it ever?). Without having to think about it, I can immediately recall three separate occasions on which I have identified clear, serious logical fallacies in a purported chain of reasoning, only to find my objections dismissed out-of-hand by other (competent, experienced) mathematicians. This phenomenon is by no means uncommon, and occurs --- occasionaly with serious repercussions for the refuter --- in many other apparently objective, scientific disciplines --- but it is disconcerting to discover how often it arises in a subject lacking any obvious reliance on experimental interpretation. The first conclusion I draw from this phenomenon is that, while the fallible human cannot always rely on his own critical powers, neither should he rely on current consensus, often driven as much by emotional and social factors as by intellectual consideration. The second is that psychology plays a larger part in our acceptance of a proof than we might like to think. In particular, as soon as a generally accepted notion is challenged, cognitive dissonance kicks in; the mind --- composure threatened by a looming contradiction which may not even have been recognised at a conscious level --- starts casting around frantically for an "explanation" of the observation which does not involve having to accept that one's current comfortable mental model of the world may have to be abandoned. To put it another way, once you accept that a proof is correct, you invest emotional security in it, which inevitably compromises your future capacity to review it critically. There really seems no way around this: if you keep on chipping away at the foundations, your tower will never reach the first floor. At some stage you have to make a qualitative decision to trust the current construction and proceed farther. But watch your back --- or maybe it should be your feet! Fred Lunnon