Proof Theory

Abstract

Proof theory is that part of mathematics that studies the concept of proof, the structure of proof, and mathematical provability. A proof theorist is interested in the process of finding a proof, in machine-generated proof, and in finding the shortest proof of a given statement. The proof theorist is also interested in which axiom schemes might generate proofs most efficiently. Following [BUS], we note that for many purposes a very long proof that is easy to find is much more useful than a short proof that is hard to find.

Proofs are the last thing looked for by a truly religious mind which feels the imaginative fitness of its faith.—George Santayana

Faith embraces many truths which seem to contradict each other.—Blaise Pascal

Imagination is more robust in proportion as reasoning power is weak.—Giambattista Vico

But if thought is to become the possession of the many, not the privilege of the few, we must have done with fear. It is fear that holds men back—fear lest they should prove less worthy of respect than they have supposed themselves to be.—Bertrand Russell

Where all is but dream, reasoning and arguments are of no use, truth and knowledge nothing.—John Locke

All beliefs are demonstrably true. All men are demonstrably in the right. Anything can be demonstrated by logic.—Antoine de Saint-Exupéry

Logic: The art of thinking and reasoning in strict accordance with the limitations and incapacities of the human understanding.—Ambrose Bierce

The want of logic annoys. Too much logic bores. Life eludes logic, and everything that logic alone constructs remains artificial and forced.—André Gide

The seed haunted by the sun never fails to find its way between the stones in the ground. And the pure logician, if no sun draws him forth, remains entangled in his logic.—Antoine de Saint-Exupéry