Theorems Quotes

God has the Big Book, the beautiful proofs of mathematical theorems are listed here. ~ Paul Erdos

To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples ... ~ John B. Conway

One of the most important lessons, perhaps, is the fact that SOFTWARE IS HARD. From now on I shall have significantly greater respect for every successful software tool that I encounter. During the past decade I was surprised to learn that the writing of programs for TeX and Metafont proved to be much more difficult than all the other things I had done (like proving theorems or writing books). The creation of good software demand a significiantly higher standard of accuracy than those other things do, and it requires a longer attention span than other intellectual tasks. ~ Donald Knuth

For even sheep do not vomit up their grass and show to the shepherds how much they have eaten; but when they have internally digested the pasture, they produce externally wool and milk. Do you also show not your theorems to the uninstructed, but show the acts which come from their digestion. ~ Epictetus

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 Godel

A felicitous but unproved conjecture may be of much more consequence for mathematics than the proof of many a respectable theorem. ~ Atle Selberg

The Limbaugh Theorem was not about me giving me credit for something. It was simply sharing with you when the light went off. ~ Rush Limbaugh

It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds. ~ G.H. Hardy

Any good theorem should have several proofs, the more the better ~ Michael Atiyah

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

The most probable of all my theorems, is that life is ordered by the principles of some religion so peculiar and obscure it has no followers, and none may fathom it, nor know the rituals by which to court its favour. ~ Alan Moore

There are infinitely many variations of the initial situation and therefore no doubt indefinitely many theorems of moral geometry. ~ John Rawls

Less depends upon the choice of words than upon this, that their introduction shall be justified by pregnant theorems. ~ Carl Friedrich Gauss

It is my dearest hope that one day I shall be the one to discover the GUT-the Grand Unified Tale, the one which will bind together all our Theorems and Laws, leaving out not one Orphan Girl or Youngest Son or Cup of Life and Death. Not one Descent or Ascent, not one Riddle or Puzzle or Trick. One perfect golden map that can guide any soul to its desire and back again. I will be the one to do it, I know it. I hope I know it. I know I hope it. ~ Catherynne M Valente

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

Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems - general and specific statements - can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences. ~ Christopher Zeeman

I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it. ~ G.H. Hardy

A mathematician is a device for turning coffee into theorems ~ Paul Erdos

Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means ... ~ Henri Poincare

Our offense is like the pythagorean theorem: There is no answer! ~ Shaquille O'Neal

No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it also gives the impression of being beautiful. ~ George Boole

Credit expansion results in the recurrence of economic crisis and periods of depression. Inflation makes the prices of all commodities and services soar. The attempts to enforce wage rates higher than those the unhampered market would have determined produce mass unemployment prolonged year after year. Price ceilings result in a drop in the supply of commodities affected. The economists have proved these theorems in an irrefutable way. No ~ Ludwig Von Mises

The paraphrase of Gödel's Theorem says that for any record player, there are records which it cannot play because they will cause its indirect self-destruction. ~ Douglas R. Hofstadter

One cannot really argue with a mathematical theorem. ~ Stephen Hawking

The missing piece in his stomach hurt so much-and eventually he stopped thinking about the Theorem and wondered only how something that isn't there can hurt you. ~ John Green

Phyllis explained to him, trying to give of her deeper self, 'Don't you find it so beautiful, math? Like an endless sheet of gold chains, each link locked into the one before it, the theorems and functions, one thing making the next inevitable. It's music, hanging there in the middle of space, meaning nothing but itself, and so moving ... ' ~ John Updike

I have found a very great number of exceedingly beautiful theorems. ~ Pierre De Fermat

There is no answer to the Pythagorean theorem. Well, there is an answer, but by the time you figure it out, I got 40 points, 10 rebounds and then we're planning for the parade. ~ Shaquille O'Neal

The primes are the raw material out of which we have to build arithmetic, and Euclid's theorem assures us that we have plenty of material for the task. ~ G.H. Hardy

Any effect, constant, theorem or equation named after Professor X was first discovered by Professor Y , for some value of Y not equal to X. ~ John C. Baez

To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well. ~ Albert Camus

In science, every question answered leads to 10 more. I love that science can never, ever be finished. From a young age, people think, 'Science is hard and boring.' We don't tell children, 'Yes, you have to learn these formulae and theorems, but then you go on to learn about nuclear reactions and stars.' ~ Elise Andrew

Q7. The total output of all the mathematicians who have ever lived, together with the output of all the human mathematicians of the next (say) thousand years is finite and could be contained in the memory banks of an appropriate computer. Surely this particular computer could, therefore, simulate this output and thus behave (externally) in the same way as a human mathematician-whatever the Godel argument might appear to tell us to the contrary?

While this is presumably true, it ignores the essential issue, which is how we (or computers) know which mathematical statements are true and which are false. (In any case, the mere storage of mathematical statements is something that could be achieved by a system much less sophisticated than a general purpose computer, e.g. photographically.) The way that the computer is being employed in Q7 totally ignores the critical issue of truth judgment. One could equally well envisage computers that contain nothing but lists of totally false mathematical 'theorems', or lists containing random jumbles of truths and falsehoods. How are we to tell which computer to trust? The arguments that I am trying to make here do not say that an effective simulation of the output of conscious human activity (here mathematics) is impossible, since purely by chance the computer might 'happen' to get it right-even without any understanding whatsoever. But the odds against this are absurdly enormous, and the issues that are being addressed here, namel ~ Roger Penrose

Never call yourself a philosopher, nor talk a great deal among the unlearned about theorems, but act conformably to them. Thus, at an entertainment, don't talk how persons ought to eat, but
eat as you ought. For remember that in this manner Socrates also universally avoided all ostentation. ~ Epictetus

If all sentient beings in the universe disappeared, there would remain a sense in which mathematical objects and theorems would continue to exist even though there would be no one around to write or talk about them. Huge prime numbers would continue to be prime, even if no one had proved them prime. ~ Martin Gardner

Those who have more power are liable to sin more; no theorem in geometry is more certain than this. ~ Lord Acton

I was a capable boy, intelligent, well-liked, but powerfully afraid. And I felt, vaguely, wordlessly, that for a child to be marked off for such a life, to be forced to live in fear was a great injustice. And what was the source of this fear? What was hiding behind the smoke screen of streets and schools? And what did it mean that number 2 pencils, conjugations without context, Pythagorean theorems, handshakes, and head nods were the difference between life and death, were the curtains drawing down between the world and me? ~ Ta-Nehisi Coates

Math does come easily to me, but I was always much more interested in what theorems imply about the world than in proving them. ~ Antony Garrett Lisi

The analysis of variance is not a mathematical theorem, but rather a convenient method of arranging the arithmetic. ~ Ronald Fisher

I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems while the root drives into the depths. ~ Gottlob Frege

A dozen more questions occurred to me. Not to mention twenty-two possible solutions to each one, sixteen resulting hypotheses and counter-theorems, eight abstract speculations, a quadrilateral equation, two axioms, and a limerick. That's raw intelligence for you. ~ Jonathan Stroud

I am persuaded that this method [for calculating the volume of a sphere] will be of no little service to mathematics. For I foresee that once it is understood and established, it will be used to discover other theorems which have not yet occurred to me, by other mathematicians, now living or yet unborn. ~ Archimedes

The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them. ~ George Polya

The axiom of conditioned repetition, like the binomial theorem, is nothing but a piece of insolence. ~ Edward Abbey

Humans like to think of themselves as unusual. We've got big brains that make it possible for us to think, and we think that we have free will and that our behavior can't be described by some mechanistic set of theorems or ideas. But even in terms of much of our behavior, we really aren't very different from other animals. ~ Mark Pagel

A mathematician," he liked to say, "is a machine for turning coffee into theorems. ~ Mason Currey

Mathematicians can and do fill in gaps, correct errors, and supply more detail and more careful scholarship when they are called on or motivated to do so. Our system is quite good at producing reliable theorems that can be solidly backed up. It's just that the reliability does not primarily come from mathematicians formally checking formal arguments; it comes from mathematicians thinking carefully and critically about mathematical ideas. ~ William Thurston

The combination of Bayes and Markov Chain Monte Carlo has been called "arguably the most powerful mechanism ever created for processing data and knowledge."
Almost instantaneously MCMC and Gibbs sampling changed statisticians' entire method of attacking problems. In the words of Thomas Kuhn, it was a paradigm shift. MCMC solved real problems, used computer algorithms instead of theorems, and led statisticians and scientists into a worked where "exact" meant "simulated" and repetitive computer operations replaced mathematical equations. It was a quantum leap in statistics. ~ Sharon Bertsch McGrayne

Did chemistry theorems exist? No: therefore you had to go further, not be satisfied with the quia, go back to the origins, to mathematics and physics. The origins of chemistry were ignoble, or at least equivocal: the dens of the alchemists, their abominable hodgepodge of ideas and language, their confessed interest in gold, their Levantine swindles typical of charlatans and magicians; instead, at the origin of physics lay the strenuous clarity of the West-Archimedes and Euclid. ~ Primo Levi

Imagine a life-form whose brainpower is to ours as ours is to a chimpanzee's. To such a species, our highest mental achievements would be trivial. Their toddlers, instead of learning their ABCs on Sesame Street, would learn multivariable calculus on Boolean Boulevard. Our most complex theorems, our deepest philosophies, the cherished works of our most creative artists, would be projects their schoolkids bring home for Mom and Dad to display on the refrigerator door. ~ Neil DeGrasse Tyson

Paul Erdos has a theory that God has a book containing all the theorems of mathematics with their absolutely most beautiful proofs, and when he wants to express particular appreciation of a proof he exclaims, "This is from the book!" ~ Ross Honsberger

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

There's no answer for my offense, just like the polythagorean theorem. ~ Shaquille O'Neal

A peculiarity of the higher arithmetic is the great difficulty which has often been experienced in proving simple general theorems which had been suggested quite naturally by numerical evidence. ~ Harold Davenport

If only I had the Theorems! Then I should find the proofs easily enough. ~ Bernhard Riemann

Economists often like startling theorems, results which seem to run counter to conventional wisdom. ~ Joseph Stiglitz

Murphy's Law, that brash proletarian restatement of Godel's Theorem ... ~ Thomas Pynchon

Finance is concerned with the relations between the values of securities and their risk, and with the behavior of those values. It aspires to be a practical, like physics or chemistry or electrical engineering. As John Maynard Keynes once remarked about economics, "If economists could manage to get themselves thought of as humble, competent people on a level with dentists, that would be splendid." Dentists rely on science, engineering, empirical knowledge, and heuristics, and there are no theorems in dentistry. Similarly, one would hope that nance would be concerned with laws rather than theorems, with behavior rather than assumptions. One doesn't seriously describe the behavior of a market with theorems. ~ Emanuel Derman

He knew by heart every last minute crack on its surface. He had made maps of the ceiling and gone exploring on them; rivers, islands, and continents. He had made guessing games of it and discovered hidden objects; faces, birds, and fishes. He made mathematical calculations of it and rediscovered his childhood; theorems, angles, and triangles. There was practically nothing else he could do but look at it. He hated the sight of it. ~ Josephine Tey

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

An axiomatic system comprises axioms and theorems and requires a certain amount of hand-eye coordination before it works. A formal system comprises an explicit list of symbols, an explicit set of rules governing their cohabitation, an explicit list of axioms, and, above all, an explicit list of rules explicitly governing the steps that the mathematician may take in going from assumptions to conclusions. No appeal to meaning nor to intuition. Symbols lose their referential powers; inferences become mechanical. ~ David Berlinski

Which is to say that culture is not a reflex of political economy, but
that society is now a reflex of key shifts in music theory and practice ...
[Sampladelia is] the sound made by those early-twentieth-century discoveries
in particle physics and relativiity theory, the projection of the minds of
Einstein, Heisenbery, and Bohr, their fateful explorations of liquid time,
curving space, uncertainty fields and relativity theorems, into densely
configured and fully ambivalent android music tracks ~ Arthur Kroker

There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. Everywhere one finds this miserable way of concluding from the special to the general and it is extremely peculiar that such a procedure has led to so few of the so-called paradoxes. ~ Niels Henrik Abel

But my shift to the serious study of economics gradually weakened my belief in Major Douglas's A+B theorem, which was replaced in my thought by the expression MV = PT. ~ James Meade

I do not deny the power and the beauty of reductionist science, as exemplified in the axioms and theorems of abstract algebra....But I assert the equal power and beauty of constructive science as exemplified in Godel's construction of an undecidable proposition.... ~ Freeman Dyson

Isolated, so-called "pretty theorems" have even less value in the eyes of a modern mathematician than the discovery of a new "pretty flower" has to the scientific botanist, though the layman finds in these the chief charm of the respective sciences. ~ Hermann Hankel

Before beginning [to try to prove Fermat's Last Theorem] I should have to put in three years of intensive study, and I haven't that much time to squander on a probable failure. ~ David Hilbert

The Mean Value Theorem is the midwife of calculus - not very important or glamorous by itself, but often helping to deliver other theorems that are of major significance. ~ Edward Mills Purcell

In the Principia Mathematica, Bertrand Russell and Alfred Whitehead attempted to give a rigorous foundation to mathematics using formal logic as their basis. They began with what they considered to be axioms, and used those to derive theorems of increasing complexity. By page 362, they had established enough to prove 1 + 1 = 2. ~ Ted Chiang

MacPherson told me that my theorem can be viewed as blah blah blah Grothendieck blah blah blah, which makes it much more respectable. ~ Jim Propp

Obviously, the final goal of scientists and mathematicians is not simply the accumulation of facts and lists of formulas, but rather they seek to understand the patterns, organizing principles, and relationships between these facts to form theorems and entirely new branches of human thought. For me, mathematics cultivates a perpetual state of wonder about the nature of mind, the limits of thoughts, and our place in this vast cosmos. ~ Clifford A. Pickover

The big question in cosmology in the early 1960s was did the universe have a beginning? Many scientists were instinctively opposed to the idea, because they felt that a point of creation would be a place where science broke down. One would have to appeal to religion and the hand of God to determine how the universe would start off. This was clearly a fundamental question, and it was just what I needed to complete my PhD thesis.
Roger Penrose had shown that once a dying star had contracted to a certain radius, there would inevitably be a singularity, that is a point where space and time came to an end. Surely, I thought, we already knew that nothing could prevent a massive cold star from collapsing under its own gravity until it reached a singularity of infinite density. I realised that similar arguments could be applied to the expansion of the universe. In this case, I could prove there were singularities where space–time had a beginning.
A eureka moment came in 1970, a few days after the birth of my daughter, Lucy. While getting into bed one evening, which my disability made a slow process, I realised that I could apply to black holes the casual structure theory I had developed for singularity theorems. If general relativity is correct and the energy density is positive, the surface area of the event horizon - the boundary of a black hole - has the property that it always increases when additional matter or radiation falls into it. Moreover, if two black holes coll ~ Stephen Hawking

Now we can see what makes mathematics unique. Only in mathematics is there no significant correction - only extension. Once the Greeks had developed the deductive method, they were correct in what they did, correct for all time. Euclid was incomplete and his work has been extended enormously, but it has not had to be corrected. His theorems are, every one of them, valid to this day. ~ Carl B. Boyer

to be grave without affectation: to observe carefully the several dispositions of my friends, not to be offended with idiots, nor unseasonably to set upon those that are carried with the vulgar opinions, with the theorems, and tenets of philosophers: his conversation being an example how a man might accommodate himself to all men and companies; so that though his company were sweeter and more pleasing than any flatterer's cogging and fawning; yet was it at the same time most respected and reverenced: ~ Marcus Aurelius

We may consequently state the fundamental theorem of Natural Selection in the form: The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time. ~ Ronald Fisher

Mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It's the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation. ~ Ivars Peterson

My dad once told me that Winstone Churchill said that Russia was riddle, wrapped in a mystery, inside an enigma. According to my dad, Churchill had been talking about my mother. This was before the divorce, and he said it half-bitterly, half-respectfully. Because even when he hated her, he admired her.
I think he would have stayed with her forever, trying to figure out the mystery. He was a puzzle solver, the kind of person who likes theorems, theories. X always had to equal something. It couldn't just be X.
To me, my mother wasn't that mysterious. She was my mother. Always reasonable, always sure of herself. To me, she was about as mysterious as a glass fo water. She knew what she wanted; she knew what she didn't want. And that was to be married to my father. I wasn't sure if it was that she fell our of love or if it was that she just never was. in love, I mean. ~ Jenny Han

One may characterize physics as the doctrine of the repeatable, be it a succession in time or the co-existence in space. The validity of physical theorems is founded on this repeatability. ~ Friedrich Hund

What philosophy worthy of the name has truly been able to avoid the link between poem and theorem? ~ Michel Serres

There are three signs of senility. The first sign is that a man forgets his theorems. The second sign is that he forgets to zip up. The third sign is that he forgets to zip down. ~ Paul Erdos

The theory of numbers, more than any other branch of mathematics, began by being an experimental science. Its most famous theorems have all been conjectured, sometimes a hundred years or more before they were proved; and they have been suggested by the evidence of a mass of computations. ~ G.H. Hardy

they propound mathematical theorems in beleaguered cities, conduct metaphysical arguments in condemned cells, make jokes on scaffolds. ~ C.S. Lewis

And, if anyone tells you that you know nothing, and you are not nettled at it, then you be sure that you have begun your business. For sheep don't throw up the grass to show the shepherds how much they have eaten; but, inwardly digesting their food, they outwardly produce wool and milk. Thus, therefore, do you likewise not show theorems to the unlearned, but the actions produced by them after they have been digested. ~ Epictetus

The pursuit of pretty formulas and neat theorems can no doubt quickly degenerate into a silly vice, but so can the quest for austere generalities which are so very general indeed that they are incapable of application to any particular. ~ E. T. Bell

Humanism ... is not a single hypothesis or theorem, and it dwells on no new facts. It is rather a slow shifting in the philosophic perspective, making things appear as from a new centre of interest or point of sight. ~ William James

Without computers we will be stuck only proving theorems that have short proofs. ~ Kenneth Appel

The Three Theorems of Psychohistorical Quantitivity:

The population under scrutiny is oblivious to the existence of the science of Psychohistory.
The time periods dealt with are in the region of 3 generations.
The population must be in the billions (±75 billions) for a statistical probability to have a psychohistorical validity. ~ Isaac Asimov

Fact: She was alone with Quint in his bedchamber.
Fact: He was talking of theorems, secret messages, and mathematical probability.
Fact: She was incredibly, distractingly aroused. ~ Joanna Shupe

How many theorems in geometry which have seemed at first impracticable are in time successfully worked out! ~ Archimedes

I am not qualified to say whether or not God exists. I kind of doubt He does. Nevertheless I'm always saying that the SF( The SF is the supreme Fascist, the Number-One guy up there) has this transfinite book-transfinite being a concept in mathematics that is larger than infinite-that contains the best proofs of all mathematical theorems, proofs that are elegant and perfect. ~ Paul Erdos

Wars and chaoses and paradoxes ago, two mathematicians between them ended an age d began another for our hosts, our ghosts called Man. One was Einstein, who with his Theory of Relativity defined the limits of man's perception by expressing mathematically just how far the condition of the observer influences the thing he perceives.
...
The other was Goedel, a contemporary of Eintstein, who was the first to bring back a mathematically precise statement about the vaster realm beyond the limits Einstein had defined: In any closed mathematical system--you may read 'the real world with its immutable laws of logic'--there are an infinite number of true theorems--you may read 'perceivable, measurable phenomena'--which, though contained in the original system, can not be deduced from it--read 'proven with ordinary or extraordinary logic.' Which is to say, there are more things in heaven and Earth than are dreamed of in your philosophy, Horatio. There are an infinite number of true things in the world with no way of ascertaining their truth. Einstein defined the extent of the rational. Goedel stuck a pin into the irrational and fixed it to the wall of the universe so that it held still long enough for people to know it was there.
...
The visible effects of Einstein's theory leaped up on a convex curve, its production huge in the first century after its discovery, then leveling off. The production of Goedel's law crept up on a concave curve, microscopic at first, th ~ Samuel R. Delany

Insight, then. Wisdom. The quest for knowledge, the derivation of theorems, science and technology and all those exclusively human pursuits that must surely rest on a conscious foundation. Maybe that's what sentience would be for - if scientific breakthroughs didn't spring fully-formed from the subconscious mind, manifest themselves in dreams, as full-blown insights after a deep night's sleep. It's the most basic rule of the stymied researcher: stop thinking about the problem. Do something else. It will come to you if you just stop being conscious of it...
Don't even try to talk about the learning curve. Don't bother citing the months of deliberate practice that precede the unconscious performance, or the years of study and experiment leading up to the gift-wrapped Eureka moment. So what if your lessons are all learned consciously? Do you think that proves there's no other way? Heuristic software's been learning from experience for over a hundred years. Machines master chess, cars learn to drive themselves, statistical programs face problems and design the experiments to solve them and you think that the only path to learning leads through sentience? You're Stone-age nomads, eking out some marginal existence on the veldt - denying even the possibility of agriculture, because hunting and gathering was good enough for your parents.
Do you want to know what consciousness is for? Do you want to know the only real purpose it serves? Training wheels. You can't see both a ~ Peter Watts

Theorems are fun especially when you are the prover, but then the pleasure fades. What keeps us going are the unsolved problems. ~ Carl Pomerance

Papers should include more side remarks, open questions, and such. Very often, these are more interesting than the theorems actually proved. Alas, most people are afraid to admit that they don't know the answer to some question, and as a consequence they refrain from mentioning the question, even if it is a very natural one. What a pity! As for myself, I enjoy saying 'I do not know'. ~ Jean-Pierre Serre

This skipping is another important point. It should be done whenever a proof seems too hard or whenever a theorem or a whole paragraph does not appeal to the reader. In most cases he will be able to go on and later he may return to the parts which he skipped. ~ Emil Artin

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