 But the skeptic should not lose his nerve. Proof does not always yield knowledge. Consider a student who correctly guesses that a step in his proof is valid. The student does not know the conclusion but did prove the theorem. His instructor might have trouble getting the student to understand why his answer constitutes a valid proof. The intransigence may stem from the prover’s intelligence rather than his stupidity. L. E. J. Brouwer is best known in mathematics for his brilliant fixed point theorem. But a doubtful reading of Immanuel Kant’s philosophy of mathematics led Brouwer retract his proof. Brouwer also had philosophical doubts about the Axiom of Choice and the Law of Excluded Middle. Brouwer persuaded a minority of mathematicians and philosophers, known as intuitionists, to emulate his abstention from non-constructive proofs. This led them to develop constructive proofs of theorems that were earlier proved by less informative means. Everybody agrees that there is more to learn from a proof of an existential generalization that proceeds from a proved instance than from an unspecific reductio ad absurdum of the corresponding universal generalization. But this does not vindicate the intuitionists’s refusal to be persuaded by the reductio ad absurdum. The intuitionist, even in the eyes of the skeptic, has too high a standard of proof. An excessively high standard of proof can prevent knowledge by proof. 