If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it.

**George Polya**

if-you-have-to-prove-theorem-do-not-rush-first-all-understand-fully-what-theorem-says-try-to-see-clearly-what-it-means-then-check-theorem-it-could-george-polya

The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas. Thus a serious mathematical theorem, a theorem which connects significant ideas, is likely to lead to important advances in mathematics itself and even in other sciences.

**G.H. Hardy**

the-seriousness-mathematical-theorem-lies-not-in-its-practical-consequences-which-are-usually-negligible-but-in-significance-mathematical-ideas-which-it-connects-we-may-say-rough

A reply to Olbers' attempt in 1816 to entice him to work on Fermat's Theorem. I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of. []

**Carl Friedrich Gauss**

a-reply-to-olbers-attempt-in-1816-to-entice-him-to-work-on-fermats-theorem-i-confess-that-fermats-theorem-as-isolated-proposition-has-little-carl-friedrich-gauss

We re-make nature by the act of discovery, in the poem or in the theorem. And the great poem and the deep theorem are new to every reader, and yet are his own experiences, because he himself re-creates them. They are the marks of unity in variety; and in the instant when the mind seizes this for itself, in art or in science, the heart misses a beat.

**Jacob Bronowski**

we-remake-nature-by-act-discovery-in-poem-in-theorem-and-great-poem-deep-theorem-are-new-to-every-reader-yet-are-his-own-experiences-because-jacob-bronowski

In the "commentatio" (note presented to the Russian Academy) in which his theorem on polyhedra (on the number of faces, edges and vertices) was first published Euler gives no proof. In place of a proof, he offers an inductive argument: he verifies the relation in a variety of special cases. There is little doubt that he also discovered the theorem, as many of his other results, inductively.

**George Polya**

in-commentatio-note-presented-to-russian-academy-in-which-his-theorem-on-polyhedra-on-number-faces-edges-vertices-was-first-published-euler-gives-george-polya

We live in a society where we're not taught how to deal with our weaknesses and frailties as human beings. We're not taught how to speak to our difficulties and challenges. We're taught the Pythagorean theorem and chemistry and biology and history. We're not taught anger management. We're not taught dissolution of fear and how to process shame and guilt. I've never in my life ever used the Pythagorean theorem!

**Iyanla Vanzant**

we-live-in-society-where-were-not-taught-how-to-deal-with-our-weaknesses-frailties-as-human-beings-were-not-taught-how-to-speak-to-our-difficulties-iyanla-vanzant

She wanted to tell him so much, on the tarmac, the day he left. The world is run by brutal men and the surest proof is their armies. If they ask you to stand still, you should dance. If they ask you to burn the flag, wave it. If they ask you to murder, re-create. Theorem, anti-theorem, corollary, anti-corollary. Underline it twice. It's all there in the numbers. Listen to your mother. Listen to me, Joshua. Look me in the eyes. I have something to tell you.

**Colum McCann**

she-wanted-to-tell-him-much-on-tarmac-day-he-left-the-world-is-run-by-brutal-men-surest-proof-is-their-armies-if-they-ask-you-to-stand-still-you-should-dance-if-they-ask-you-to-b

A human being without the proper empathy or feeling is the same as an android built so as to lack it, either by design or mistake. We mean, basically, someone who does not care about the fate which his fellow living creatures fall victim to; he stands detached, a spectator, acting out by his indifference John Donne's theorem that 'No man is an island, ' but giving that theorem a twist: that which is a mental and a moral island is not a man.

**Philip K. Dick**

a-human-being-without-proper-empathy-feeling-is-same-as-android-built-as-to-lack-it-either-by-design-mistake-we-mean-basically-someone-who-does-not-care-about-fate-which-his-fell

Occasionally, I get a letter from someone who is in 'contact' with extraterrestrials. I am invited to 'ask them anything.' And so over the years I've prepared a little list of questions. The extraterrestrials are very advanced, remember. So I ask things like, 'Please provide a short proof of Fermat's Last Theorem.' Or the Goldbach Conjecture. And then I have to explain what these are, because extraterrestrials will not call it Fermat's Last Theorem. So I write out the simple equation with the exponents. I never get an answer. On the other hand, if I ask something like 'Should we be good?' I almost always get an answer.

**Carl Sagan**

occasionally-i-get-letter-from-someone-who-is-in-contact-with-extraterrestrials-i-am-invited-to-ask-them-anything-and-over-years-ive-prepared-little-list-questions-the-extraterre

I confess that Fermat's Theorem as an isolated proposition has very little interest for me, for a multitude of such theorems can easily be set up, which one could neither prove nor disprove. But I have been stimulated by it to bring our again several old ideas for a great extension of the theory of numbers. Of course, this theory belongs to the things where one cannot predict to what extent one will succeed in reaching obscurely hovering distant goals. A happy star must also rule, and my situation and so manifold distracting affairs of course do not permit me to pursue such meditations as in the happy years 1796-1798 when I created the principal topics of my Disquisitiones arithmeticae. But I am convinced that if good fortune should do more than I expect, and make me successful in some advances in that theory, even the Fermat theorem will appear in it only as one of the least interesting corollaries. {In reply to Olbers' attempt in 1816 to entice him to work on Fermat's Theorem. The hope Gauss expressed for his success was never realised.}

**Carl Friedrich Gauss**

i-confess-that-fermats-theorem-as-isolated-proposition-has-little-interest-for-me-for-multitude-such-theorems-can-easily-be-set-up-which-one-could-neither-prove-nor-disprove-but-

How can you shorten the subject? That stern struggle with the multiplication table, for many people not yet ended in victory, how can you make it less? Square root, as obdurate as a hardwood stump in a pasturenothing but years of effort can extract it. You can't hurry the process. Or pass from arithmetic to algebra; you can't shoulder your way past quadratic equations or ripple through the binomial theorem. Instead, the other way; your feet are impeded in the tangled growth, your pace slackens, you sink and fall somewhere near the binomial theorem with the calculus in sight on the horizon.

**Stephen Leacock**

how-can-you-shorten-subject-that-stern-struggle-with-multiplication-table-for-many-people-not-yet-ended-in-victory-how-can-you-make-it-less-square-stephen-leacock

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

it-gives-me-same-pleasure-when-someone-else-proves-good-theorem-as-when-i-do-it-myself-edmund-landau

the-greatest-unsolved-theorem-in-mathematics-is-why-some-people-are-better-at-it-than-others

i-realized-that-anything-to-do-with-fermats-last-theorem-generates-too-much-interest

what-philosophy-worthy-name-has-truly-been-able-to-avoid-link-between-poem-theorem-michel-serres

a-theorem-is-proposition-which-is-strict-logical-consequence-certain-definitions-other-propositions-anatol-rapoport

heaven-is-angered-by-my-arrogance-my-proof-fourcolor-theorem-is-also-defective-hermann-minkowski

the-axiom-conditioned-repetition-like-binomial-theorem-is-nothing-but-piece-insolence-edward-abbey

the-limbaugh-theorem-was-not-about-me-giving-me-credit-for-something-it-was-simply-sharing-with-you-when-light-went-off-rush-limbaugh

a-felicitous-but-unproved-conjecture-may-be-much-more-consequence-for-mathematics-than-proof-many-respectable-theorem-atle-selberg

the-analysis-variance-is-not-mathematical-theorem-but-rather-convenient-method-arranging-arithmetic-ronald-fisher

under-bayes-theorem-no-theory-is-perfect-rather-it-is-work-in-progress-always-subject-to-further-refinement-testing-nate-silver

to-insure-the-adoration-of-a-theorem-for-any-length-of-time-faith-is-not-enough-a-police-force-is-needed-as-well

and-i-believe-that-binomial-theorem-bach-fugue-are-in-long-run-more-important-than-all-battles-history-james-hilton

i-tend-to-regard-coase-theorem-as-stepping-stone-on-way-to-analysis-economy-with-positive-transaction-costs

The poem or the discovery exists in two moments of vision: the moment of appreciation as much as that of creation; for the appreciator must see the movement, wake to the echo which was started in the creation of the work. In the moment of appreciation we live again the moment when the creator saw and held the hidden likeness. When a simile takes us aback and persuades us together, when we find a juxtaposition in a picture both odd and intriguing, when a theory is at once fresh and convincing, we do not merely nod over someone else's work. We re-enact the creative act, and we ourselves make the discovery again... Reality is not an exhibit for man's inspection, labeled: "Do not touch." There are no appearances to be photographed, no experiences to be copied, in which we do not take part. We re-make nature by the act of discovery, in the poem or in the theorem. And the great poem and the deep theorem are new to every reader, and yet are his own experiences, because he himself re-creates them. They are the marks of unity in variety; and in the instant when the mind seizes this for itself, in art or in science, the heart misses a beat.

**Jacob Bronowski**

the-poem-discovery-exists-in-two-moments-vision-moment-appreciation-as-much-as-that-creation-for-appreciator-must-see-movement-wake-to-echo-which-was-started-in-creation-work-in-

a-proven-theorem-game-theory-states-that-every-game-with-complete-information-possesses-saddle-point-therefore-solution-richard-arnold-epstein

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

I took a break from acting for four years to get a degree in mathematics at UCLA, and during that time I had the rare opportunity to actually do research as an undergraduate. And myself and two other people co-authored a new theorem: Percolation and Gibbs States Multiplicity for Ferromagnetic Ashkin-Teller Models on Two Dimensions, or Z2.

**Danica McKellar**

i-took-break-from-acting-for-four-years-to-get-degree-in-mathematics-at-ucla-during-that-time-i-had-rare-opportunity-to-actually-do-research-as-danica-mckellar

Why does Alexander the Great never tell us about the exact location of his tomb, Fermat about his Last Theorem, John Wilkes Booth about the Lincoln assassination conspiracy, Hermann Ge¶ring about the Reichstag fire? Why don't Sophocles, Democritus, and Aristarchus dictate their lost books?

**Carl Sagan**

why-does-alexander-great-never-tell-us-about-exact-location-his-tomb-fermat-about-his-last-theorem-john-wilkes-booth-about-lincoln-assassination-conspiracy-hermann-gering-about-r

theorem-incompleteness-shows-there-is-nothing-on-this-level-existence-that-can-fully-explain-this-level-existence-pat-cadigan

Some mathematics problems look simple, and you try them for a year or so, and then you try them for a hundred years, and it turns out that they're extremely hard to solve. There's no reason why these problems shouldn't be easy, and yet they turn out to be extremely intricate. [Fermat's] Last Theorem is the most beautiful example of this.

**Andrew John Wiles**

some-mathematics-problems-look-simple-you-try-them-for-year-then-you-try-them-for-hundred-years-it-turns-out-that-theyre-extremely-hard-to-solve-theres-no-reason-why-these-proble

If market pricing is the only legitimate test of quality, why are we still bothering with proven theorems? Why don't we just have a vote on whether a theorem is true? To make it better we'll have everyone vote on it, especially the hundreds of millions of people who don't understand the math. Would that satisfy you?

**Jaron Lanier**

if-market-pricing-is-only-legitimate-test-quality-why-are-we-still-bothering-with-proven-theorems-why-dont-we-just-have-vote-on-whether-theorem-is-true-to-make-it-better-well-hav

Combinatorial analysis, in the trivial sense of manipulating binomial and multinomial coefficients, and formally expanding powers of infinite series by applications ad libitum and ad nauseamque of the multinomial theorem, represented the best that academic mathematics could do in the Germany of the late 18th century.

**Richard Askey**

combinatorial-analysis-in-trivial-sense-manipulating-binomial-multinomial-coefficients-formally-expanding-powers-infinite-series-by-applications-richard-askey

I loved doing problems in school. I'd take them home and make up new ones of my own. But the best problem I ever found, I found in my local public library. I was just browsing through the section of math books and I found this one book, which was all about one particular problem - Fermat's Last Theorem.

**Andrew Wiles**

i-loved-doing-problems-in-school-id-take-them-home-make-up-new-ones-my-own-but-best-problem-i-ever-found-i-found-in-my-local-public-library-i-was-just-browsing-through-section-ma

Can the difficulty of an exam be measured by how many bits of information a student would need to pass it? This may not be so absurd in the encyclopedic subjects but in mathematics it doesn't make any sense since things follow from each other and, in principle, whoever knows the bases knows everything. All of the results of a mathematical theorem are in the axioms of mathematics in embryonic form, aren't they?

**Alfred Renyi**

can-difficulty-exam-be-measured-by-how-many-bits-information-student-would-need-to-pass-it-this-may-not-be-absurd-in-encyclopedic-subjects-but-alfred-renyi

Carnal embrace is sexual congress, which is the insertion of the male genital organ into the female genital organ for purposes of procreation and pleasure. Fermat's last theorem, by contrast, asserts that when x, y and z are whole numbers each raised to power of n, the sum of the first two can never equal the third when n is greater than 2.

**Tom Stoppard**

carnal-embrace-is-sexual-congress-which-is-insertion-male-genital-organ-into-female-genital-organ-for-purposes-procreation-pleasure-fermats-last-tom-stoppard

I have a basic theorem as to how I do my jokes. Growing up, I knew when to cross the line and when not to cross the line. It's the same with my comedy. I know what my audience will take and how much they won't take. I can't give you a formula for it. It's my own personal formula inside my head. Somebody else's might be different.

**Larry the Cable Guy**

i-have-basic-theorem-as-to-how-i-do-my-jokes-growing-up-i-knew-when-to-cross-line-when-not-to-cross-line-its-same-with-my-comedy-i-know-what-my-larry-cable-guy

Uh-oh, " Moni sang, and nodded her head in Chantal's direction. "I think someone's a wee bit upset with us." She turned and walked a few steps backward. "Careful, " I said. "We're not out of range." "Have no fear, Super Brain is here." Moni whipped out her calculator, holding it up like a shield. "What are you going to do, daze her with denominators?" "Maybe. But first I'm going to pummel her with my Pythagorean theorem.

**Charity Tahmaseb**

uhoh-moni-sang-nodded-her-head-in-chantals-direction-i-think-someones-wee-bit-upset-with-us-she-turned-walked-few-steps-backward-careful-i-said-were-not-out-range-have-no-fear-su

There is a theorem that colloquially translates, You cannot comb the hair on a bowling ball. ... Clearly, none of these mathematicians had Afros, because to comb an Afro is to pick it straight away from the scalp. If bowling balls had Afros, then yes, they could be combed without violation of mathematical theorems.

**Neil deGrasse Tyson**

there-is-theorem-that-colloquially-translates-you-cannot-comb-hair-on-bowling-ball-clearly-none-these-mathematicians-had-afros-because-to-comb-neil-degrasse-tyson

The goal of a definition is to introduce a mathematical object. The goal of a theorem is to state some of its properties, or interrelations between various objects. The goal of a proof is to make such a statement convincing by presenting a reasoning subdivided into small steps each of which is justified as an "elementary" convincing argument.

**IU?. I. Manin**

the-goal-definition-is-to-introduce-mathematical-object-the-goal-theorem-is-to-state-some-its-properties-interrelations-between-various-objects-iu-i-manin

I think mathematics is a vast territory. The outskirts of mathematics are the outskirts of mathematical civilization. There are certain subjects that people learn about and gather together. Then there is a sort of inevitable development in those fields. You get to the point where a certain theorem is bound to be proved, independent of any particular individual, because it is just in the path of development.

**William Thurston**

i-think-mathematics-is-vast-territory-the-outskirts-mathematics-are-outskirts-mathematical-civilization-there-are-certain-subjects-that-people-william-thurston

There was a seminar for advanced students in Ze¼rich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn't say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann.

**George Polya**

there-was-seminar-for-advanced-students-in-zerich-that-i-was-teaching-von-neumann-was-in-class-i-came-to-certain-theorem-i-said-it-is-not-proved-it-may-be-difficult-von-neumann-d

Human rights are an aspect of natural law, a consequence of the way the universe works, as solid and as real as photons or the concept of pi. The idea of self- ownership is the equivalent of Pythagoras' theorem, of evolution by natural selection, of general relativity, and of quantum theory. Before humankind discovered any of these, it suffered, to varying degrees, in misery and ignorance.

**L. Neil Smith**

human-rights-are-aspect-natural-law-consequence-way-universe-works-as-solid-as-real-as-photons-concept-pi-the-idea-self-ownership-is-l-neil-smith

Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. The geometer might be replaced by the "logic piano" imagined by Stanley Jevons; or, if you choose, a machine might be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in alive and come out transformed into hams and sausages. No more than these machines need the mathematician know what he does.

**Henri Poincare**

thus-be-it-understood-to-demonstrate-theorem-it-is-neither-necessary-nor-even-advantageous-to-know-what-it-means-the-geometer-might-be-replaced-by-henri-poincare

I carried this problem around in my head basically the whole time. I would wake up with it first thing in the morning, I would be thinking about it all day, and I would be thinking about it when I went to sleep. Without distraction I would have the same thing going round and round in my mind. (Recalling the degree of focus and determination that eventually yielded the proof of Fermat's Last Theorem.)

**Andrew John Wiles**

i-carried-this-problem-around-in-my-head-basically-whole-time-i-would-wake-up-with-it-first-thing-in-morning-i-would-be-thinking-about-it-all-day-i-would-be-thinking-about-it-whe

Do people believe in human rights because such rights actually exist, like mathematical truths, sitting on a cosmic shelf next to the Pythagorean theorem just waiting to be discovered by Platonic reasoners? Or do people feel revulsion and sympathy when they read accounts of torture, and then invent a story about universal rights to help justify their feelings?

**Jonathan Haidt**

do-people-believe-in-human-rights-because-such-rights-actually-exist-like-mathematical-truths-sitting-on-cosmic-shelf-next-to-pythagorean-theorem-jonathan-haidt

... fain would I turn back the clock and devote to French or some other language the hours I spent upon algebra, geometry, and trigonometry, of which not one principle remains with me. Stay! There is one theorem painfully drummed into my head which seems to have inhabited some corner of my brain since that early time: "The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides!" There it sticks, but what of it, ye gods, what of it?

**Jessie Belle Rittenhouse**

fain-would-i-turn-back-clock-devote-to-french-some-other-language-hours-i-spent-upon-algebra-geometry-trigonometry-which-not-one-principle-jessie-belle-rittenhouse

Professor Eddington has recently remarked that 'The law that entropy always increases the second law of thermodynamics holds, I think, the supreme position among the laws of nature'. It is not a little instructive that so similar a law [the fundamental theorem of natural selection] should hold the supreme position among the biological sciences.

**Ronald Fisher**

professor-eddington-has-recently-remarked-that-the-law-that-entropy-always-increases-second-law-thermodynamics-holds-i-think-supreme-position-among-ronald-fisher

Gradually, at various points in our childhoods, we discover different forms of conviction. There's the rock-hard certainty of personal experience ("I put my finger in the fire and it hurt,"), which is probably the earliest kind we learn. Then there's the logically convincing, which we probably come to first through maths, in the context of Pythagoras's theorem or something similar, and which, if we first encounter it at exactly the right moment, bursts on our minds like sunrise with the whole universe playing a great chord of C Major.

**Philip Pullman**

gradually-at-various-points-in-our-childhoods-we-discover-different-forms-conviction-theres-rockhard-certainty-personal-experience-i-put-my-finger-philip-pullman

My favourite fellow of the Royal Society is the Reverend Thomas Bayes, an obscure 18th-century Kent clergyman and a brilliant mathematician who devised a complex equation known as the Bayes theorem, which can be used to work out probability distributions. It had no practical application in his lifetime, but today, thanks to computers, is routinely used in the modelling of climate change, astrophysics and stock-market analysis.

**Bill Bryson**

my-favourite-fellow-royal-society-is-reverend-thomas-bayes-obscure-18thcentury-kent-clergyman-brilliant-mathematician-who-devised-complex-bill-bryson

It seems that mathematical ideas are arranged somehow in strata, the ideas in each stratum being linked by a complex of relations both among themselves and with those above and below. The lower the stratum, the deeper (and in general more difficult) the idea. Thus the idea of an 'irrational' is deeper than that of an integer; and Pythagoras's theorem is, for that reason, deeper than Euclid's.

**G.H. Hardy**

it-seems-that-mathematical-ideas-are-arranged-somehow-in-strata-ideas-in-each-stratum-being-linked-by-complex-relations-both-among-themselves-with-those-above-below-the-lower-str

A human being without the proper empathy or feeling is the same as an android built so as to lack it, either by design or mistake. We mean, basically, someone who does not care about thefate which his fellow living creatures fall victim to; he stands detached, a spectator, acting out by his indifference John Donne's theorem that "No man is an island," but giving thattheorem a twist: that which is a mental and a moral island is not a man.

**Philip K. Dick**

a-human-being-without-proper-empathy-feeling-is-same-as-android-built-as-to-lack-it-either-by-design-mistake-we-mean-basically-someone-who-does-philip-k-dick

The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules... One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms.

**Kurt GÃ¶del**

the-development-mathematics-toward-greater-precision-has-led-as-is-well-known-to-formalization-large-tracts-it-that-one-can-prove-any-theorem-kurt-gdel

For what is important when we give children a theorem to use is not that they should memorize it. What matters most is that by growing up with a few very powerful theorems one comes to appreciate how certain ideas can be used as tools to think with over a lifetime. One learns to enjoy and to respect the power of powerful ideas. One learns that the most powerful idea of all is the idea of powerful ideas.

**Seymour Papert**

for-what-is-important-when-we-give-children-theorem-to-use-is-not-that-they-should-memorize-it-what-matters-most-is-that-by-growing-up-with-few-seymour-papert

The scientist has to take 95 per cent of his subject on trust. He has to because he can't possibly do all the experiments, therefore he has to take on trust the experiments all his colleagues and predecessors have done. Whereas a mathematician doesn't have to take anything on trust. Any theorem that's proved, he doesn't believe it, really, until he goes through the proof himself, and therefore he knows his whole subject from scratch. He's absolutely 100 per cent certain of it. And that gives him an extraordinary conviction of certainty, and an arrogance that scientists don't have.

**Christopher Zeeman**

the-scientist-has-to-take-95-per-cent-his-subject-on-trust-he-has-to-because-he-cant-possibly-do-all-experiments-therefore-he-has-to-take-on-trust-christopher-zeeman

...contemporary physicists come in two varieties. Type 1 physicists are bothered by EPR and Bell's Theorem. Type 2 (the majority) are not, but one has to distinguish two subvarieties. Type 2a physicists explain why they are not bothered. Their explanations tend either to miss the point entirely (like Born's to Einstein) or to contain physical assertions that can be shown to be false. Type 2b are not bothered and refuse to explain why.

**David Mermin**

contemporary-physicists-come-in-two-varieties-type-1-physicists-are-bothered-by-epr-bells-theorem-type-2-majority-are-not-but-one-has-to-distinguish-david-mermin

In 1975, ... [speaking with Shiing Shen Chern], I told him I had finally learned ... the beauty of fiber-bundle theory and the profound Chern-Weil theorem. I said I found it amazing that gauge fields are exactly connections on fiber bundles, which the mathematicians developed without reference to the physical world. I added, "this is both thrilling and puzzling, since you mathematicians dreamed up these concepts out of nowhere." He immediately protested: "No, no. These concepts were not dreamed up. They were natural and real."

**Chen-Ning Yang**

in-1975-speaking-with-shiing-shen-chern-i-told-him-i-had-finally-learned-beauty-fiberbundle-theory-profound-chernweil-theorem-i-said-i-found-it-chenning-yang

About Thomas Hobbes: He was 40 years old before he looked on geometry; which happened accidentally. Being in a gentleman's library, Euclid's Elements lay open, and "twas the 47 El. libri I" [Pythagoras' Theorem]. He read the proposition "By God", sayd he, "this is impossible:" So he reads the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps, that at last he was demonstratively convinced of that truth. This made him in love with geometry.

**John Aubrey**

about-thomas-hobbes-he-was-40-years-old-before-he-looked-on-geometry-which-happened-accidentally-being-in-gentlemans-library-euclids-elements-lay-john-aubrey

The Open Source theorem says that if you give away source code, innovation will occur. Certainly, Unix was done this way... However, the corollary states that the innovation will occur elsewhere. No matter how many people you hire. So the only way to get close to the state of the art is to give the people who are going to be doing the innovative things the means to do it. That's why we had built-in source code with Unix. Open source is tapping the energy that's out there.

**Bill Joy**

the-open-source-theorem-says-that-if-you-give-away-source-code-innovation-will-occur-certainly-unix-was-done-this-way-however-corollary-states-that-bill-joy

Mathematical knowledge is unlike any other knowledge. While our perception of the physical world can always be distorted, our perception of mathematical truths can't be. They are objective, persistent, necessary truths. A mathematical formula or theorem means the same thing to anyone anywhere - no matter what gender, religion, or skin color; it will mean the same thing to anyone a thousand years from now. And what's also amazing is that we own all of them. No one can patent a mathematical formula, it's ours to share. There is nothing in this world that is so deep and exquisite and yet so readily available to all. That such a reservoir of knowledge really exists is nearly unbelievable. It's too precious to be given away to the 'initiated few.' It belongs to all of us.

**Edward Frenkel**

mathematical-knowledge-is-unlike-any-other-knowledge-while-our-perception-physical-world-can-always-be-distorted-our-perception-mathematical-truths-cant-be-they-are-objective-per

You forget all of it anyway. First, you forget everything you learned-the dates of the Hay-Herran Treaty and Pythagorean Theorem. You especially forget everything you didn't really learn, but just memorized the night before. You forget the names of all but one or two of your teachers, and eventually you'll forget those, too. You forget your junior class schedule and where you used to sit and your best friend's home phone number and the lyrics to that song you must have played a million times. For me, it was something by Simon & Garfunkel. Who knows what it will be for you? And eventually, but slowly, oh so slowly, you forget your humiliations-even the ones that seemed indelible just fade away. You forget who was cool and who was not, who was pretty, smart, athletic, and not. Who went to a good college. Who threw the best parties Who could get you pot. You forget all of them. Even the ones you said you loved, and even the ones you actually did. They're the last to go. And then once you've forgotten enough, you love someone else.

**Gabrielle Zevin**

you-forget-all-it-anyway-first-you-forget-everything-you-learned-dates-hayherran-treaty-pythagorean-theorem-you-especially-forget-everything-you-didnt-really-learn-but-just-memor

When there were no customers, he thought about geometry. He tried to perform the Pythagorean Theorem on the light fixture above him, given his estimate of its circumference, but he failed. He wanted to be good at math. But he just wasn't. He wanted so badly for the math club to accept him, but to them he was a loser. During rush week they battered him blue with yard sticks; they tied him to a table naked and made him prove problems involving half circles before a huge swinging protractor cut him in half; they forced him to continually calculate the remaining volume of the kegs he had to drink, working it out by hand as he was held upside down. After he didn't get in the club, they had started ganging up on him every day, sticking his head in the toilet and stealing his lunch money. Business was slow at the moment, and he thought about ending his life in the kitchen appliance aisle.

**Benson Bruno**

when-there-were-no-customers-he-thought-about-geometry-he-tried-to-perform-pythagorean-theorem-on-light-fixture-above-him-given-his-estimate-its-circumference-but-he-failed-he-wa

It will be noticed that the fundamental theorem proved above bears some remarkable resemblances to the second law of thermodynamics. Both are properties of populations, or aggregates, true irrespective of the nature of the units which compose them; both are statistical laws; each requires the constant increase of a measurable quantity, in the one case the entropy of a physical system and in the other the fitness, measured by m, of a biological population. As in the physical world we can conceive the theoretical systems in which dissipative forces are wholly absent, and in which the entropy consequently remains constant, so we can conceive, though we need not expect to find, biological populations in which the genetic variance is absolutely zero, and in which fitness does not increase. Professor Eddington has recently remarked that 'The law that entropy always increases-the second law of thermodynamics-holds, I think, the supreme position among the laws of nature'. It is not a little instructive that so similar a law should hold the supreme position among the biological sciences. While it is possible that both may ultimately be absorbed by some more general principle, for the present we should note that the laws as they stand present profound differences-(1) The systems considered in thermodynamics are permanent; species on the contrary are liable to extinction, although biological improvement must be expected to occur up to the end of their existence. (2) Fitness, although measured by a uniform method, is qualitatively different for every different organism, whereas entropy, like temperature, is taken to have the same meaning for all physical systems. (3) Fitness may be increased or decreased by changes in the environment, without reacting quantitatively upon that environment. (4) Entropy changes are exceptional in the physical world in being irreversible, while irreversible evolutionary changes form no exception among biological phenomena. Finally, (5) entropy changes lead to a progressive disorganization of the physical world, at least from the human standpoint of the utilization of energy, while evolutionary changes are generally recognized as producing progressively higher organization in the organic world.

**Ronald A. Fisher**

it-will-be-noticed-that-fundamental-theorem-proved-above-bears-some-remarkable-resemblances-to-second-law-thermodynamics-both-are-properties-populations-aggregates-true-irrespect

He walked straight out of college into the waiting arms of the Navy. They gave him an intelligence test. The first question on the math part had to do with boats on a river: Port Smith is 100 miles upstream of Port Jones. The river flows at 5 miles per hour. The boat goes through water at 10 miles per hour. How long does it take to go from Port Smith to Port Jones? How long to come back? Lawrence immediately saw that it was a trick question. You would have to be some kind of idiot to make the facile assumption that the current would add or subtract 5 miles per hour to or from the speed of the boat. Clearly, 5 miles per hour was nothing more than the average speed. The current would be faster in the middle of the river and slower at the banks. More complicated variations could be expected at bends in the river. Basically it was a question of hydrodynamics, which could be tackled using certain well-known systems of differential equations. Lawrence dove into the problem, rapidly (or so he thought) covering both sides of ten sheets of paper with calculations. Along the way, he realized that one of his assumptions, in combination with the simplified Navier Stokes equations, had led him into an exploration of a particularly interesting family of partial differential equations. Before he knew it, he had proved a new theorem. If that didn't prove his intelligence, what would? Then the time bell rang and the papers were collected. Lawrence managed to hang onto his scratch paper. He took it back to his dorm, typed it up, and mailed it to one of the more approachable math professors at Princeton, who promptly arranged for it to be published in a Parisian mathematics journal. Lawrence received two free, freshly printed copies of the journal a few months later, in San Diego, California, during mail call on board a large ship called the U.S.S. Nevada. The ship had a band, and the Navy had given Lawrence the job of playing the glockenspiel in it, because their testing procedures had proven that he was not intelligent enough to do anything else.

**Neal Stephenson**

he-walked-straight-out-college-into-waiting-arms-navy-they-gave-him-intelligence-test-the-first-question-on-math-part-had-to-do-with-boats-on-river-port-smith-is-100-miles-upstre