On Proofs (Unsourced)

"A proof tells us where to concentrate our doubts." (Morris Kline)

"Any good theorem should have several proofs, the more the better. For two reasons: usually, different proofs have different strengths and weaknesses, and they generalise in different directions: they are not just repetitions of each other." (Sir Michael Atiyah)

"Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs." (Paul Halmos)

"I think some intuition leaks out in every step of an induction proof." (Jim Propp)

"In science nothing capable of proof ought to be accepted without proof." (Richard Dedekind)

"It always seems to me absurd to speak of a complete proof, or of a theorem being rigorously demonstrated. An incomplete proof is no proof, and a mathematical truth not rigorously demonstrated is not demonstrated at all." (James J Sylvester)

"Just give me the insights. I can always come up with the proofs!" (Bernhard Riemann)

"[…] mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs." (Felix Klein)

"Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional." (Charles P Steinmetz)

"Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning." (John Locke)

"No human investigation can claim to be scientific if it doesn't pass the test of mathematical proof." (Leonardo Da Vinci)

"No mathematical exactness without explicit proof from assumed principles – such is the motto of the modern geometer." (George B Halsted)

"Proofs really aren't there to convince you that something is true they're there to show you why it is true." (Andrew Gleason)

"Simplification of modes of proof is not merely an indication of advance in our knowledge of a subject, but is also the surest guarantee of readiness for farther progress.” (Baron William Thomson Kelvin)

"Some facts can be seen more clearly by example than by proof." (Leonard Euler)

"The art of drawing conclusions from experiments and observations consists in evaluating probabilities and in estimating whether they are sufficiently great or numerous enough to constitute proofs." (Antoine Lavoisier)

"The essential quality of a proof is to compel belief." (Pierre de Fermat)