I think it is said that Gauss had ten different proofs for the law of quadratic reciprocity. 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.

**Michael Atiyah**

i-think-it-is-said-that-gauss-had-ten-different-proofs-for-law-quadratic-reciprocity-any-good-theorem-should-have-several-proofs-more-better-for-michael-atiyah

Ever since Plato most philosophers have considered it part of their business to produce 'proofs' of immortality and the existence of God. They have found fault with the proofs of their predecessors - Saint Thomas rejected Saint Anselm's proofs, and Kant rejected Descartes' - but they have supplied new ones of their own. In order to make their proofs seem valid, they have had to falsify logic, to make mathematics mystical, and to pretend that deepseated prejudices were heaven-sent intuitions.

**Bertrand Russell**

ever-since-plato-most-philosophers-have-considered-it-part-their-business-to-produce-proofs-immortality-existence-god-they-have-found-fault-with-proofs-their-predecessors-saint-t

Those who have racked their brains to discover new proofs have perhaps been induced to do so by a compulsion they could not quite explain to themselves. Instead of giving us their new proofs they should have explained to us the motivation that constrained them to search for them.

**Georg C. Lichtenberg**

those-who-have-racked-their-brains-to-discover-new-proofs-have-perhaps-been-induced-to-do-by-compulsion-they-could-not-quite-explain-to-themselves-georg-c-lichtenberg

It is an unfortunate fact that proofs can be very misleading. Proofs exist to establish once and for all, according to very high standards, that certain mathematical statements are irrefutable facts. What is unfortunate about this is that a proof, in spite of the fact that it is perfectly correct, does not in any way have to be enlightening. Thus, mathematicians, and mathematics students, are faced with two problems: the generation of proofs, and the generation of internal enlightenment. To understand a theorem requires enlightenment. If one has enlightenment, one knows in one's soul why a particular theorem must be true.

**Herbert S. Gaskill**

it-is-unfortunate-fact-that-proofs-can-be-misleading-proofs-exist-to-establish-once-for-all-according-to-high-standards-that-certain-mathematical-statements-are-irrefutable-facts

One can be enlightened about proofs as well as theorems. Without enlightenment, one is merely reduced to memorizing proofs. With enlightenment about a proof, its flow becomes clear and it can become an item of astonishing beauty. In addition, the need to memorize disappears because the proof has become part of your soul.

**Herbert S. Gaskill**

one-can-be-enlightened-about-proofs-as-well-as-theorems-without-enlightenment-one-is-merely-reduced-to-memorizing-proofs-with-enlightenment-about-proof-its-flow-becomes-clear-it-

It is a matter for considerable regret that Fermat, who cultivated the theory of numbers with so much success, did not leave us with the proofs of the theorems he discovered. In truth, Messrs Euler and Lagrange, who have not disdained this kind of research, have proved most of these theorems, and have even substituted extensive theories for the isolated propositions of Fermat. But there are several proofs which have resisted their efforts.

**Adrien-Marie Legendre**

it-is-matter-for-considerable-regret-that-fermat-who-cultivated-theory-numbers-with-much-success-did-not-leave-us-with-proofs-theorems-he-discovered-in-truth-messrs-euler-lagrang

The establishment of formal standards for proofs about programs [...] and the proposal that the semantics of a programming language may be defined independently of all processors for that language, by establishing standards of rigor for proofs about programs in the language, appears to be novel.

**Robert W. Floyd**

the-establishment-formal-standards-for-proofs-about-programs-proposal-that-semantics-programming-language-may-be-defined-independently-all-robert-w-floyd

Actually, the 'leap of faith'-to give it the memorable name that Soren Kierkegaard bestowed upon it-is an imposture. As he himself pointed out, it is not a 'leap' that can be made once and for all. It is a leap that has to go on and on being performed, in spite of mounting evidence to the contrary. This effort is actually too much for the human mind, and leads to delusions and manias. Religion understands perfectly well that the 'leap' is subject to sharply diminishing returns, which is why it often doesn't in fact rely on 'faith' at all but instead corrupts faith and insults reason by offering evidence and pointing to confected 'proofs.' This evidence and these proofs include arguments from design, revelations, punishments, and miracles. Now that religion's monopoly has been broken, it is within the compass of any human being to see these evidences and proofs as the feeble-minded inventions that they are.

**Christopher Hitchens**

actually-leap-faithto-give-it-memorable-name-that-soren-kierkegaard-bestowed-upon-itis-imposture-as-he-himself-pointed-out-it-is-not-leap-that-can-be-made-once-for-all-it-is-leap

It is the invariable habit of bureaucracies, at all times and everywhere, to assume...that every citizen is a criminal. Their one apparent purpose, pursued with a relentless and furious diligence, is to convert the assumption into a fact. They hunt endlessly for proofs, and, when proofs are lacking, for mere suspicions. The moment they become aware of a definite citizen, John Doe, seeking what is his right under the law, they begin searching feverishly for an excuse for withholding it from him.

**H. L. Mencken**

it-is-invariable-habit-bureaucracies-at-all-times-everywhere-to-assumethat-every-citizen-is-criminal-their-one-apparent-purpose-pursued-with-h-l-mencken

The first and most important field of philosophy is the application of principles such as 'Do not lie.' Next come the proofs, such as why we should not lie. The third field supports and articulates the proofs, by asking, for example, 'How does this prove it? What exactly is a proof, what is logical inference, what is contradiction, what is truth, what is falsehood?' Thus, the third field is necessary because of the second, and the second because of the first. The most important, though, the one that should occupy most of our time, is the first. But we do just the opposite. We are preoccupied with the third field and give that all our attention, passing the first by altogether. The result is that we lie - but have no difficulty proving why we shouldn't.

**Epictetus**

the-first-most-important-field-philosophy-is-application-principles-such-as-do-not-lie-next-come-proofs-such-as-why-we-should-not-lie-the-third-field-supports-articulates-proofs-

How could it occur to anyone to demonstrate that God exists unless one has already allowed Himself to ignore Him? A king's existence is demonstrated by way of subjection and submissiveness. Do you want to try and demonstrate that the king exists? Will you do so by offering a string of proofs, a series of arguments? No. If you are serious, you will demonstrate the king's existence by your submission, by the way you live. And so it is with demonstrating God's existence. It is accomplished not by proofs but by worship. Any other way is but a thinker's pious bungling.

**Soren Kierkegaard**

how-could-it-occur-to-anyone-to-demonstrate-that-god-exists-unless-one-has-already-allowed-himself-to-ignore-him-a-kings-existence-is-demonstrated-by-soren-kierkegaard

he-said-no-go-both-you-with-our-proofs-we-will-be-with-you-listening-ashshuara-15

if-only-i-had-theorems-then-i-should-find-proofs-easily-enough-bernhard-riemann

but-if-you-slip-after-proofs-have-come-to-you-know-that-allah-is-powerful-wise-albaqarah-209

tricks-treachery-are-merely-proofs-lack-skill-francois-de-la-rochefoucauld

an-administration-without-police-executive-is-powerless-there-were-many-proofs-this

and-do-not-be-like-those-who-separated-disputed-after-clear-proofs-came-to-them-for-them-is-great-punishment-ali-imran-105

what-they-ask-you-for-is-actions-proofs-works-all-you-can-produce-are-transformed-tears-emil-cioran

thus-in-sense-mathematics-has-been-most-advanced-by-those-who-distinguished-themselves-by-intuition-rather-than-by-rigorous-proofs

now-all-orators-effect-their-demonstrative-proofs-by-allegation-either-enthymems-examples-besides-these-in-no-other-way-whatever-aristotle

through-our-expectations-we-violently-expand-try-to-fill-world-with-our-proofs-worth-bryant-mcgill

moses-came-to-you-with-clear-proofs-yet-you-adopted-calf-in-his-absence-you-were-in-wrong-albaqarah-92

thats-problem-with-false-proofs-true-theorems-its-not-easy-to-produce-counterexample-jeffrey-shallit

to-speak-evil-any-one-unless-there-is-unequivocal-proofs-their-deserving-it-is-injury-for-which-there-is-no-adequate-reparation-george-washington

after-turbulence-death-moral-principles-even-religious-proofs-are-called-into-question

god-has-big-book-beautiful-proofs-mathematical-theorems-are-listed-here-paul-erdos

if-i-shall-be-condemned-upon-surmises-all-proofs-sleeping-else-but-what-your-jealousies-awake-i-tell-you-tis-rigor-not-law-william-shakespeare

trifles-light-as-air-are-to-jealous-confirmations-strong-as-proofs-holy-writ-william-shakespeare

i-dont-have-lot-paper-in-my-immediate-work-environment-except-when-im-doing-things-like-checking-godforsaken-proofs-jonathan-lethem

proofs-are-last-thing-looked-for-by-truly-religious-mind-which-feels-imaginary-fitness-its-faith-george-santayana

then-after-them-we-sent-moses-aaron-with-our-proofs-to-pharaoh-his-dignitaries-but-they-acted-arrogantly-they-were-sinful-people-yunus-75

for-strangely-his-officers-looked-upon-jack-aubrey-as-moral-figure-in-spite-all-proofs-contrary-patrick-obrian

the-proofs-existence-god-are-to-such-extent-fallen-into-discredit-that-they-pass-for-something-antiquated-belonging-to-days-gone-by-georg-wilhelm-friedrich-hegel

to-be-capable-of-steady-friendship-or-lasting-love-are-the-two-greatest-proofs-not-only-of-goodness-of-heart-but-of-strength-of-mind

long-may-louis-de-broglie-continue-to-inspire-those-who-suspect-that-what-is-proved-by-impossibility-proofs-is-lack-imagination-john-stewart-bell

it-is-evidently-equally-foolish-to-accept-probable-reasoning-from-mathematician-to-demand-from-rhetorician-demonstrative-proofs-aristotle

i-gave-you-three-proofs-witchcraft-a-cat-that-drinks-blood-a-horse-that-talks-and-man-who-propagates-poodles-richard-curtis

to-be-capable-steady-friendship-lasting-love-are-two-greatest-proofs-not-only-goodness-heart-but-strength-mind-william-hazlitt

one-unerring-mark-of-the-love-of-truth-is-not-entertaining-any-proposition-with-greater-assurance-than-the-proofs-it-is-built-upon-will-warrant

Those who said, "Allah has made a covenant with us, that we shall not believe in any messenger unless he brings us an offering to be consumed by fire." Say, "Messengers have come to you before me with proofs, and with what you asked for; so why did you assassinate them, if you are truthful?"

**Ali 'Imran 183**

those-who-said-allah-has-made-covenant-with-us-that-we-shall-not-believe-in-any-messenger-unless-he-brings-us-offering-to-be-consumed-by-fire-say-messengers-have-come-to-you-befo

Has not the story reached you, of those before you, the people of Noah, and Aad, and Thamood-and those after them? None knows them except Allah. Their messengers came to them with the clear proofs, but they tried to silence them, and said, "We reject what you are sent with, and we are in serious doubt regarding what you are calling us to."

**Ibrahim 9**

has-not-story-reached-you-those-before-you-people-noah-aad-thamood-those-after-them-none-knows-them-except-allah-their-messengers-came-to-them-with-clear-proofs-but-they-tried-to

if-they-accuse-you-lying-messengers-before-you-were-accused-lying-they-came-with-proofs-psalms-illuminating-scripture-ali-imran-184

computer-assisted-proofs-are-getting-better-better-computers-will-play-bigger-bigger-role-in-future-enrico-bombieri

those-who-claim-to-discover-everything-but-produce-no-proofs-same-may-be-confuted-as-having-actually-pretended-to-discover-impossible-archimedes

a-poet-should-leave-traces-his-passage-not-proofs-traces-alone-engender-dreams-rene-char

one-proofs-immortality-soul-is-that-myriads-have-believed-it-they-also-believed-world-was-flat-mark-twain

No body wishes more than I do to see such proofs as you exhibit, that nature has given to our black brethren, talents equal to those of the other colors of men, and that the appearance of a want of them is owing merely to the degraded condition of their existence, both in Africa and America.

**Thomas Jefferson**

no-body-wishes-more-than-i-do-to-see-such-proofs-as-you-exhibit-that-nature-has-given-to-our-black-brethren-talents-equal-to-those-other-colors-men-that-appearance-want-them-is-o

No body wishes more than I do to see such proofs as you exhibit, that nature has given to our black brethren, talents equal to those of the other colors of men, and that the appearance of a want of them is owing merely to the degraded condition of their existence, both in Africa & America.

**Thomas Jefferson**

no-body-wishes-more-than-i-do-to-see-such-proofs-as-you-exhibit-that-nature-has-given-to-our-black-brethren-talents-equal-to-those-other-colors-men-thomas-jefferson