Agnostics

"All articles of great expense, of vexatious and operose provision, [Diogenes] disallowed, and demonstrated their pernicious effects upon the user; yet forbade none of those bodily conveniences, which may be procured without difficulty and molestation, whether to alleviate cold or hunger or other craving appetites, but manifested by his own practice a preference to healthy situations before sickly, and to such as were more tolerable than others through all the vicissitudes of seasons. Nor was he less attentive to a plentiful supply of wholesome food, and a moderate portion of apparel: but kept himself aloof from public business, from lawsuits, from animosities, from wars and political conspiracies. The life of the Gods was the principal model of his practice: for Homer characterizes them as living at their ease with reference to the laborious and troublesome condition of mankind."

"When [Diogenes of Sinope] was sold as a slave, he endured it most nobly. For on a voyage to Aegina he was captured by pirates under the command of Scirpalus, conveyed to Crete and exposed for sale. When the auctioneer asked in what he was proficient, he replied, "In ruling men." Thereupon he pointed to a certain Corinthian with a fine purple border to his robe, the man named Xeniades above-mentioned, and said, “Sell me to this man; he needs a master.”"

"When some people urged that it is impossible for man to live like the animals owing to the tenderness of his flesh and because he is naked and unprotected, [Diogenes] would say in reply that men are so very tender because of their mode of life. ... Man’s ingenuity and his discovering and contriving so many helps to life had not been altogether advantageous to later generations, since men do not employ their cleverness to promote courage or justice, but to procure pleasure."

"As for Diogenes, Basil never ceased admiring him, the philosopher who was so set upon being content with nothing but the gifts of nature that he even threw away his drinking-cup, after he had learned from a boy how to bend over and drink from the hollow of his hands."

"When [Diogenes] observed how other men were harassed throughout their whole lives, ever plotting against one another, ever encompassed by a thousand ills and never able to enjoy a moment’s rest, nay, not even during the great festivals nor when they proclaimed a truce; and when he beheld that they did or suffered all this simply in order to keep themselves alive, and that their greatest fear was lest their so-called necessities should fail them, and how, furthermore, they planned and strove to leave great riches to their children, he marvelled that he too did not do the like, but was the only independent man in the world."

"No labor, according to Diogenes, is good but that which aims at producing courage and strength of soul rather than of body."

"Thieves and fighters do not come from eaters of barley-bread; but informers and tyrants come from meat-eaters."

"Diogenes the cynic, seeing one of the so-called freedmen pluming himself, while many heartily congratulated him, marveled at the absence of reason and discernment. “A man might as well,” he said, “proclaim that one of his servants became a grammarian, a geometrician, or musician, when he has no idea whatever of the art.” For as the proclamation cannot make them men of knowledge, so neither can it make them free."

"If I were not Alexander, I should wish to be Diogenes."

"Self-taught poverty is a help toward philosophy, for the things which philosophy attempts to teach by reasoning, poverty forces us to practice."

"Virtue cannot dwell with wealth either in a city or in a house."

"Poverty is a virtue which one can teach oneself."

"Even bronze is aged by time, but not all the ages, Diogenes, shall destroy thy fame, since you alone did show to mortals the rule of self-sufficiency and the easiest path of life."

"Diogenes compared them to fig-trees growing over precipices; for their fruit was devoured by daws and crows, not by men."

"Good approximations often lead to better ones."

"For many of the stories told the final form resulted from a sort of informal psychological experiment. I discussed the subject with several different classes... Several passages... have been suggested by answers of my students, or... modified... by the reaction of my audience."

"The volume of the cone was discovered by Democritus... He did not prove it, he guessed it... not a blind guess, rather it was reasoned conjecture. As Archimedes has remarked, great credit is due to Democritus for his conjecture since this made proof much easier. Eudoxes... a pupil of Plato, subsequently gave a rigorous proof. Surely the labor or writing limited his manuscript to a few copies; none has survived. In those days editions did not run to thousands or hundreds of thousands of copies as modern books—especially, bad books—do. However, the substance of what he wrote is nevertheless available to us. ...Euclid's great achievement was the systematization of the works of his predecessors. The Elements preserve several of Eudoxes' proofs."

"I shall often discuss mathematical discoveries... I shall try to make up a likely story how the discovery could have happened. I shall try to emphasize the motives underlying the discovery, the plausible inferences that led to it... everything that deserves imitation."

"I... present also examples of historic interest, examples of real mathematical beauty, and examples illustrating the parallelism of the procedures in other sciences, or in everyday life."

"In my presentation I... follow the genetic method. The essential idea... is that the order in which knowledge has been acquired by the human race will be a good teacher for its acquisition by the individual. The sciences came in a certain order; an order determined by human interest and inherent difficulty. Mathematics and astronomy were the first sciences really worth the name; later came mechanics, optics, and so on. At each stage of its development the human race has had a certain climate of opinion, a way of looking, conceptually, at the world. The next glimmer of fresh understanding had to grow out of what was already understood. The next move forward, halting shuffle, faltering step, or stride with some confidence, was developed upon how well the [human] race could then walk. As for the human race, so for the human child. But this is not to say that to teach science we must repeat the thousand and one errors of the past, each ill-directed shuffle. It is to say that the sequence in which the major strides forward were made is a good sequence in which to teach them. The genetic method is a guide to, not a substitute for, judgement."

"Mathematics succeeds in dealing with tangible reality by being conceptual. We cannot cope with the full physical complexity; we must idealize."

"In plausible reasoning the principal thing is to distinguish... a more reasonable guess from a less reasonable guess."

"The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference."

"The general or amateur student should also get a taste of demonstrative reasoning... he should acquire a standard with which he can compare alleged evidence of all sorts aimed at him in modern life."

"The efficient use of plausible reasoning is a practical skill and it is learned... by imitation and practice. ...what I can offer are only examples for imitation and opportunity for practice."

"Why should the typical student be interested in those wretched triangles? ...He is to be brought to see that without the knowledge of triangles there is not trigonometry; that without trigonometry we put back the clock millennia to Standard Darkness Time and antedate the Greeks."

"We wish to see... the typical attitude of the scientist who uses mathematics to understand the world around us. ...In the solution of a problem ...there are typically three phases. The first phase is entirely or almost entirely a matter of physics; the third, a matter of mathematics; and the intermediate phase, a transition from physics to mathematics. The first phase is the formulation of the physical hypothesis or conjecture; the second, its translation into equations; the third, the solution of the equations. Each phase calls for a different kind of work and demands a different attitude."

"Facing any part of the observable reality, we are never in possession of complete knowledge, nor in a state of complete ignorance, although usually much closer to the latter state."

"Diogenes the Cynic, it is related, was mighty of all people in regard to everything from self-control to endurance. He indulged in sexual lusts, not associating it with pleasure, an attractive good thing to some, but because of the harm that the retention of semen would cause if he avoided the habit of releasing it. When a prostitute who promised to visit him was delayed for some time, he rubbed his genitals with his hand, ejecting semen. After the whore arrived, he sent her away, saying: "my hand celebrated the wedding-hymn first." But it is clearly correct that, likewise, the disciplined man does not on account of pleasure indulge in lusts, but in order to relieve the hindrance acting as if this was not associated with pleasure."

"Diogenes received an invitation to dine with one whose house was splendidly furnished, in the highest order and taste, and nothing therein wanting. Diogenes, hawking, and as if about to spit, looked in all directions, and finding nothing adapted thereto, spat right in the face of the master. He, indignant, asked why he did so? "Because," Diogenes, "I saw nothing so dirty and filthy in all your house. For the walls were covered with pictures, the floors of the most precious tessellated character — and ranged with the various images of gods, and other ornamental figures.""

"Pedantry and mastery are opposite attitudes toward rules. To apply a rule to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry. … To apply a rule with natural ease, with judgment, noticing the cases where it fits, and without ever letting the words of the rule obscure the purpose of the action or the opportunities of the situation, is mastery."

"We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building."

"To write and speak correctly is certainly necessary; but it is not sufficient. A derivation correctly presented in the book or on the blackboard may be inaccessible and uninstructive, if the purpose of the successive steps is incomprehensible, if the reader or listener cannot understand how it was humanly possible to find such an argument...."

"Euclid's manner of exposition, progressing relentlessly from the data to the unknown and from the hypothesis to the conclusion, is perfect for checking the argument in detail but far from being perfect for making understandable the main line of the argument."

"Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements."

"The best of ideas is hurt by uncritical acceptance and thrives on critical examination."

"The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. … Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information."

""Groping" and "muddling through" is usually described as a solution by trial and error. ...a series of trials, each of which attempts to correct the error committed by the preceding and, on the whole, the errors diminished as we proceed and the successive trials come closer and closer to the desired final result. ...we may wish a better characterization ..."successive trials" or "successive corrections" or "successive approximations." ...You use successive approximations when ...looking for a word in the dictionary ...A mathematician may apply the term ...to a highly sophisticated procedure ...to treat some very advanced problem ...that he cannot treat otherwise. The term even applies to science as a whole; the scientific theories which succeed each other, each claiming a better explanation ...may appear as successive approximations to the truth. Therefore, the teacher should not discourage his students from using trial and error—on the contrary, he should encourage the intelligent use of the fundamental method of successive approximations. Yet he should convincingly show that for ...many ... situations, straightforward algebra is more efficient than successive approximations."

"[M]athematicians and physicists think alike; they are led, and sometimes misled, by the same patterns of ."

"Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper."

"There was a seminar for advanced students in Zürich 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."

"Demonstrative reasoning penetrates the sciences just as far as mathematics does, but it is in itself (as mathematics is in itself) incapable of yielding essentially new knowledge about the world around us. Anything new that we learn about the world involves plausible reasoning, which is the only kind of reasoning for which we care in everyday affairs."

"The noblest people are those despising wealth, learning, pleasure and life; esteeming above them poverty, ignorance, hardship and death."

"Everyone knows that mathematics offers an excellent opportunity to learn demonstrative reasoning, but I contend also that there is no other subject in the usual curricula of the schools that affords a comparable opportunity to learn plausible reasoning. ...let us learn proving, but also let us learn guessing."

"It is the relentless onward march of the texters, the SMS (Short Message Service) vandals who are doing to our language what Genghis Khan did to his neighbours 800 years ago.They are destroying it: pillaging our punctuation; savaging our sentences; raping our vocabulary. And they must be stopped."

"[the pact with Yasir Arafat had caused the deaths of sixteen hundred Israelis in terror attacks] It’s on your hands! You should be ashamed! You gave them guns and they shot at us!"

"...The ultimate test of this agreement will be a test of blood... If it becomes clear that they [the Palestinians] cannot overcome terror, this will be temporary accord and... we will have no choice but to abrogate it... And if there is no choice, the IDF will return to the places it is about to leave in the upcoming months."

"The art of music above all the other arts is the expression of the soul of a nation."

"The business of finding a nation's soul is a long and slow one at the best and a great many prophets must be slain in the course of it. Perhaps when we have slain enough prophets future generations will begin to build their tombs."

"I don't know whether I like it, but it's what I meant."