Topics Mathematics

Question Often, people who claim to be psychic also claim that they have an ability to perceive or to have verification of the existence of things that other people do not have (but could possibly obtain). How is this different from the mathematician who discovers, or claims to have discovered, a proof? Certainly, many mathematical proofs are not for the layman to approve of or dispute, because of a certain lack of ability. Both claim to see or to have proof (although different types of proof) of something, but in both cases one can imagine at least one person that cannot verify that the proof really is (a) proof.

Accepted:
June 7, 2006

Comments

If you claim to have gotten evidence for something using a method that I cannot employ, should I rely on your testimony? If I assume that you are sincere, then the key question is whether I should believe that you really do have a reliable evidence-producing method. I must rely on experience. Here's one way: if I can establish that your method led you to true conclusions in the past, I can justify trusting it now. Even if I cannot do that, I might be able to rely on more general considerations: perhaps you are so clearly a scrupulous evidence-conoisseur with respect to all the kinds of evidence whose reliability I can judge that it is reasonable for me to believe that you are not mistaken in taking this alien sort of evidence to be reliable.

That much seems true equally of the psychic and the mathematician. A difference may be that mathematicians do not regard the evidence embodied in proofs to be accessible only to people with some special capacity. Certainly producing a proof can involve unusual insight, genius, and even luck; but verifying that it is a proof is supposed to require nothing beyond very simple rational capacities (perhaps applied over and over, in verifying many simple steps).

In actual mathematical practice, however, proofs are not given in this idealized way. Instead of being broken down into a long sequence of simple steps, the proof of a theorem is compressed in various ways according to disciplinary standards of explicitness, rendering the proof opaque to non-mathematicians. Maybe this is not a big deal: maybe the real justification for the theorem resides in a fully explicit, simple-steps proof that has been compressed for easy expression. But to the extent that undoing the "compression" is not trivial, the mathematician requires non-trivial justification for thinking that underlying the compressed proof she has produced there is a fully explicit, simple-steps proof. So the mathematician does indeed have a special evidence-recognizing skill that is not simply a matter of applying simple steps.