On Proofs I

“There is no other scientific or analytical discipline that uses proof as readily and routinely as does mathematics. This is the device that makes theoretical mathematics special: the tightly knit chain of reasoning, following strict logical rules, that leads inexorably to a particular conclusion. It is proof that is our device for establishing the absolute and irrevocable truth of statements […].” (Steven G Krantz, “The Proof is in the Pudding”, 2007)

“There are two aspects of proof to be borne in mind. One is that it is our lingua franca. It is the mathematical mode of discourse. It is our tried-and true methodology for recording discoveries in a bullet-proof fashion that will stand the test of time. The second, and for the working mathematician the most important, aspect of proof is that the proof of a new theorem explains why the result is true. In the end what we seek is new understanding, and ’proof’ provides us with that golden nugget.” (Steven G Krantz, “The Proof is in the Pudding”, 2007)

“Rigorous proofs are the hallmark of mathematics, they are an essential part of mathematics’ contribution to general culture.” (George Polya, “Mathematical Discovery”, 1981)

“An intuitive proof allows you to understand why the theorem must be true; the logic merely provides firm grounds to show that it is true.” (Ian Stewart, “Concepts of Modern Mathematics”, 1975)

“A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.” (G H Hardy, “A Mathematician’s Apology”, 1940)

“The best proofs in mathematics are short and crisp like epigrams, and the longest have swings and rhythms that are like music.” (Scott Buchanan, “Poetry and Mathematics”, 1975)

”[…] a mathematician is more anonymous than an artist. While we may greatly admire a mathematician who discovers a beautiful proof, the human story behind the discovery eventually fades away and it is, in the end, the mathematics itself that delights us.” (Timothy Gowers, “Mathematics”, 2002)

“Some people believe that a theorem is proved when a logically correct proof is given; but some people believe a theorem is proved only when the student sees why it is inevitably true.” (Wesley R Hamming, “Coding and Information Theory”, 1980)

“Proofs knit the fabric of mathematics together, and if a single thread is weak, the entire fabric may unravel.” (Ian Stewart, “Nature’s Numbers”, 1995)

“An intuitive proof allows you to understand why the theorem must be true; the logic merely provides firm grounds to show that it is true.” (Ian Stewart, “Concepts of Modern Mathematics”, 1995)