... in real life mistakes are likely to be irrevocable. Computer simulation, however, makes it economically practical to make mistakes on purpose. If you are astute, therefore, you can learn much more than they cost. Furthermore, if you are at all discreet, no one but you need ever know you made a mistake.

’Tis a pity learned virgins ever wed
With persons of no sort of education,
Or gentlemen, who, though well born and bred,
Grow tired of scientific conversation.

“I think I’d like to learn to fly”, I told the family casually … “Not a bad idea”, said my father just as casually. “When do you start?”

“That’s another thing we’ve learned from your Nation,” said Mein Herr, “map-making. But we’ve carried it much further than you. What do you consider the largest map that would be really useful?”
“About six inches to the mile.”
“Only six inches!” exclaimed Mein Herr. “We very soon got to six yards to the mile. Then we tried a hundred yards to the mile. And then came the grandest idea of all! We actually made a map of the country, on the scale of a mile to the mile!”
“Have you used it much?” I enquired.
“It has never been spread out, yet,” said Mein Herr: “the farmers objected: they said it would cover the whole country, and shut out the sunlight! So we now use the country itself, as its own map, and I assure you it does nearly as well.”

[A man] must learn to understand the motives of human beings, their illusions, and their sufferings human beings, their illusions, and their sufferings in order to acquire a proper relationship to individual fellow-men and to the community. These precious things … primarily constitutes and preserves culture. This is what I have in mind when I recommend the “humanities” as important, not just dry specialized knowledge in the fields of history and philosophy.

[Alexander the Great] was often extremely brutal to his captives, whom he sold into slavery, tortured to death, or forced to learn Greek.

[An artist] will sooner and with more certainty, establish the character of skeletons, than the most learned anatomist, whose eye has not been accustomed to seize on every peculiarity.
Asserting his (incorrect) belief that the fossil teeth of the mastodon revealed it was a carnivorous animal.]

[For] men to whom nothing seems great but reason ... nature ... is a cosmos, so admirable, that to penetrate to its ways seems to them the only thing that makes life worth living. These are the men whom we see possessed by a passion to learn ... Those are the natural scientific men; and they are the only men that have any real success in scientific research.

[In mathematics] we behold the conscious logical activity of the human mind in its purest and most perfect form. Here we learn to realize the laborious nature of the process, the great care with which it must proceed, the accuracy which is necessary to determine the exact extent of the general propositions arrived at, the difficulty of forming and comprehending abstract concepts; but here we learn also to place confidence in the certainty, scope and fruitfulness of such intellectual activity.

[In my early youth, walking with my father,] “See that bird?” he says. “It’s a Spencer’s warbler.” (I knew he didn’t know the real name.) “Well, in Italian, it’s a Chutto Lapittida. In Portuguese, it’s a Bom da Peida. In Chinese, it’s a Chung-long-tah, and in Japanese, it’s a Katano Tekeda. You can know the name of that bird in all the languages of the world, but when you’re finished, you’ll know absolutely nothing whatever about the bird. You’ll only know about humans in different places, and what they call the bird. So let’s look at the bird and see what it’s doing—that’s what counts.” (I learned very early the difference between knowing the name of something and knowing something.)

[In my home workshop,] generally I’m mending things, which is interesting because you learn a lot about why they broke.

[L]et us not overlook the further great fact, that not only does science underlie sculpture, painting, music, poetry, but that science is itself poetic. The current opinion that science and poetry are opposed is a delusion. … On the contrary science opens up realms of poetry where to the unscientific all is a blank. Those engaged in scientific researches constantly show us that they realize not less vividly, but more vividly, than others, the poetry of their subjects. Whoever will dip into Hugh Miller’s works on geology, or read Mr. Lewes's “Seaside Studies,” will perceive that science excites poetry rather than extinguishes it. And whoever will contemplate the life of Goethe will see that the poet and the man of science can co-exist in equal activity. Is it not, indeed, an absurd and almost a sacrilegious belief that the more a man studies Nature the less he reveres it? Think you that a drop of water, which to the vulgar eye is but a drop of water, loses anything in the eye of the physicist who knows that its elements are held together by a force which, if suddenly liberated, would produce a flash of lightning? Think you that what is carelessly looked upon by the uninitiated as a mere snow-flake, does not suggest higher associations to one who has seen through a microscope the wondrously varied and elegant forms of snow-crystals? Think you that the rounded rock marked with parallel scratches calls up as much poetry in an ignorant mind as in the mind of a geologist, who knows that over this rock a glacier slid a million years ago? The truth is, that those who have never entered upon scientific pursuits know not a tithe of the poetry by which they are surrounded. Whoever has not in youth collected plants and insects, knows not half the halo of interest which lanes and hedge-rows can assume. Whoever has not sought for fossils, has little idea of the poetical associations that surround the places where imbedded treasures were found. Whoever at the seaside has not had a microscope and aquarium, has yet to learn what the highest pleasures of the seaside are. Sad, indeed, is it to see how men occupy themselves with trivialities, and are indifferent to the grandest phenomena—care not to understand the architecture of the Heavens, but are deeply interested in some contemptible controversy about the intrigues of Mary Queen of Scots!—are learnedly critical over a Greek ode, and pass by without a glance that grand epic written by the finger of God upon the strata of the Earth!

[Learning is] the actual process of broadening yourself, of knowing there’s a little extra facet of the universe you know about and can think about and can understand. It seems to me that when it’s time to die, and that will come to all of us, there’ll be a certain pleasure in thinking that you had utilized your life well, that you had learned as much as you could, gathered in as much as possible of the universe, and enjoyed it. I mean, there’s only this universe and only this one lifetime to try to grasp it. And, while it is inconceivable that anyone can grasp more than a tiny portion of it, at least do that much. What a tragedy to just pass through and get nothing out of it.

[On research] It’s got to be fun. I don't think anybody should tell you that he’s slogged his way through 25 years on a problem and there's only one reward at the end, and that's the value of the Hubble constant. That’s a bunch of hooey. The reward is learning all the wonderful properties of the things that don’t work.

[On the practical applications of particle physics research with the Large Hadron Collider.] Sometimes the public says, “What's in it for Numero Uno? Am I going to get better television reception? Am I going to get better Internet reception?” Well, in some sense, yeah. … All the wonders of quantum physics were learned basically from looking at atom-smasher technology. … But let me let you in on a secret: We physicists are not driven to do this because of better color television. … That's a spin-off. We do this because we want to understand our role and our place in the universe.

[Presently, science undergraduates] do not learn to write clearly and briefly, marshalling their points in due and aesthetically satisfying order, and eliminating inessentials. They are inept at those turns of phrase or happy analogy which throw a flying bridge across a chasm of misunderstanding and make contact between mind and mind.

[Scientists] have learned to respect nothing but evidence, and to believe that their highest duty lies in submitting to it however it may jar against their inclinations.

[Simplicio] is much puzzled and perplexed. I think I hear him say, 'To whom then should we repair for the decision of our controversies if Aristotle were removed from the choir? What other author should we follow in the schools, academies, and studies? What philosopher has written all the divisions of Natural Philosophy, and so methodically, without omitting as much as a single conclusion? Shall we then overthrow the building under which so many voyagers find shelter? Shall we destroy that sanctuary, that Prytaneum, where so many students find commodious harbour; where without exposing himself to the injuries of the air, with only the turning over of a few leaves, one may learn all the secrets of Nature.'

[Technical courage means the] physician-scientist must be brave enough to adopt new methods. It is far too easy to learn one technique and then to repeat the same experiment over and over. In this fashion one can write many papers, receive large research grants, and remain solidly rooted in the middle of a scientific field. But the true innovator has the confidence to drop one set of experimental crutches and leap to another when he or she must move forward.

[The screw machine] was on the principle of the guage or sliding lathe now in every workshop throughout the world; the perfection of which consists in that most faithful agent gravity, making the joint, and that almighty perfect number three, which is in harmony itself. I was young when I learned that principle. I had never seen my grandmother putting a chip under a three-legged milking-stool; but she always had to put a chip under a four-legged table, to keep it steady. I cut screws of all dimensions by this machine, and did them perfectly. (1846)

[The surplus of basic knowledge of the atomic nucleus was] largely used up [during the war with the atomic bomb as the dividend.] We must, without further delay restore this surplus in preparation for the important peacetime job for the nucleus - power production. ... Many of the proposed applications of atomic power - even for interplanetary rockets - seem to be within the realm of possibility provided the economic factor is ruled out completely, and the doubtful physical and chemical factors are weighted heavily on the optimistic side. ... The development of economic atomic power is not a simple extrapolation of knowledge gained during the bomb work. It is a new and difficult project to reach a satisfactory answer. Needless to say, it is vital that the atomic policy legislation now being considered by the congress recognizes the essential nature of this peacetime job, and that it not only permits but encourages the cooperative research-engineering effort of industrial, government and university laboratories for the task. ... We must learn how to generate the still higher energy particles of the cosmic rays - up to 1,000,000,000 volts, for they will unlock new domains in the nucleus.

[Two college boys on the Flambeau River in a canoe]…their first…taste of freedom … The elemental simplicities of wilderness travel were thrills not only because of their novelty, but because they represented complete freedom to make mistakes. The wilderness gave them their first taste of those rewards and penalties for wise and foolish acts which every woodsman faces daily, but against which civilization has built a thousand buffers. These boys were “on their own” in this particular sense. Perhaps every youth needs an occasional wilderness trip, in order to learn the meaning of this particular freedom.

[W]e have made a thing, a most terrible weapon, that has altered abruptly and profoundly the nature of the world. We have made a thing that, by all standards of the world we grew up in, is an evil thing. And by doing so, by our participation in making it possible to make these things, we have raised again the question of whether science is good for man, of whether it is good to learn about the world, to try to understand it, to try to control it, to help give to the world of men increased insight, increased power. Because we are scientists, we must say an unalterable yes to these questions; it is our faith and our commitment, seldom made explicit, even more seldom challenged, that knowledge is a good in itself, knowledge and such power as must come with it.

[W]hen Galileo discovered he could use the tools of mathematics and mechanics to understand the motion of celestial bodies, he felt, in the words of one imminent researcher, that he had learned the language in which God recreated the universe. Today we are learning the language in which God created life. We are gaining ever more awe for the complexity, the beauty, the wonder of God's most devine and sacred gift.

[We] do not learn for want of time,
The sciences that should become our country.

[About reading Rachel Carson's Silent Spring, age 14, in the back seat of his parents' sedan. I almost threw up. I got physically ill when I learned that ospreys and peregrine falcons weren't raising chicks because of what people were spraying on bugs at their farms and lawns. This was the first time I learned that humans could impact the environment with chemicals. [That a corporation would create a product that didn't operate as advertised] was shocking in a way we weren't inured to.

[Instead of collecting stamps, he collected dictionaries and encyclopaedias:] Because you can learn more from them.

Ac astronomye is an hard thyng,
And yvel for to knowe;
Geometrie and geomesie,
So gynful of speche,
Who so thynketh werche with tho two
Thryveth ful late,
For sorcerie is the sovereyn book
That to tho sciences bilongeth.
Now, astronomy is a difficult discipline, and the devil to learn;
And geometry and geomancy have confusing terminology:
If you wish to work in these two, you will not succeed quickly.
For sorcery is the chief study that these sciences entail.

Der bis zur Vorrede, die ihn abweist, gelangte Leser hat das Buch für baares Geld gekauft und frägt, was ihn schadlos hält? – Meine letzte Zuflucht ist jetzt, ihn zu erinnern, daß er ein Buch, auch ohne es gerade zu lesen, doch auf mancherlei Art zu benutzen weiß. Es kann, so gut wie viele andere, eine Lücke seiner Bibliothek ausfüllen, wo es sich, sauber gebunden, gewiß gut ausnehmen wird. Oder auch er kann es seiner gelehrten Freundin auf die Toilette, oder den Theetisch legen. Oder endlich er kann ja, was gewiß das Beste von Allem ist und ich besonders rathe, es recensiren.
The reader who has got as far as the preface and is put off by that, has paid money for the book, and wants to know how he is to be compensated. My last refuge now is to remind him that he knows of various ways of using a book without precisely reading it. It can, like many another, fill a gap in his library, where, neatly bound, it is sure to look well. Or he can lay it on the dressing-table or tea-table of his learned lady friend. Or finally he can review it; this is assuredly the best course of all, and the one I specially advise.

Every teacher certainly should know something of non-euclidean geometry. Thus, it forms one of the few parts of mathematics which, at least in scattered catch-words, is talked about in wide circles, so that any teacher may be asked about it at any moment. … Imagine a teacher of physics who is unable to say anything about Röntgen rays, or about radium. A teacher of mathematics who could give no answer to questions about non-euclidean geometry would not make a better impression.
On the other hand, I should like to advise emphatically against bringing non-euclidean into regular school instruction (i.e., beyond occasional suggestions, upon inquiry by interested pupils), as enthusiasts are always recommending. Let us be satisfied if the preceding advice is followed and if the pupils learn to really understand euclidean geometry. After all, it is in order for the teacher to know a little more than the average pupil.

Hoc age ['do this'] is the great rule, whether you are serious or merry; whether ... learning science or duty from a folio, or floating on the Thames. Intentions must be gathered from acts.

Les médecins les plus savans en théorie sont rarement les plus habile practiciens.
The doctors most learned in theory are seldom the most skilled practitioners.

Mikroskopisch sehen lernen.
Learn to see microscopically.

Or any science under sonne,
The sevene artz and alle,
But thei ben lerned for oure Lordes love
Lost is al the tyme.
Every science under the sun, including the Seven Arts,
Unless learned for love of Our Lord, is only time lost.

Une idée anticipée ou une hypothèse est donc le point de départ nécessaire de tout raisonnement expérimental. Sans cela on ne saurait faire aucune investigation ni s’instruire ; on ne pourrait qu’entasser des observations stériles. Si l’on expérimentait sans idée préconçue, on irait à l’aventure; mais d’un autre côté, ainsi que nous l’avons dit ailleurs, si l’on observait avec des idées préconçues, on ferait de mauvaises observations.
An anticipative idea or an hypothesis is, then, the necessary starting point for all experimental reasoning. Without it, we could not make any investigation at all nor learn anything; we could only pile up sterile observations. If we experimented without a preconceived idea, we should move at random.
[Also seen translated as:] A hypothesis is … the obligatory starting point of all experimental reasoning. Without it no investigation would be possible, and one would learn nothing: one could only pile up barren observations. To experiment without a preconceived idea is to wander aimlessly.

~~[Attributed, authorship undocumented]~~ Mathematical demonstrations are a logic of as much or more use, than that commonly learned at schools, serving to a just formation of the mind, enlarging its capacity, and strengthening it so as to render the same capable of exact reasoning, and discerning truth from falsehood in all occurrences, even in subjects not mathematical. For which reason it is said, the Egyptians, Persians, and Lacedaemonians seldom elected any new kings, but such as had some knowledge in the mathematics, imagining those, who had not, men of imperfect judgments, and unfit to rule and govern.

~~[No known source from Adams]~~ If your actions inspire others to dream more, learn more, do more and become more, you are a leader.

A celebrated author and divine has written to me that “he has gradually learnt to see that it is just as noble a conception of the Deity to believe that He created a few original forms capable of self-development into other and needful forms, as to believe that He required a fresh act of creation to supply the voids caused by the action of His laws.”

A child of the new generation
Refused to learn multiplication.
He said “Don’t conclude
That I’m stupid or rude;
I am simply without motivation.”

A collective learning machine achieves its feats by using five elements … (1) conformity enforcers; (2) diversity generators; (3) inner-judges; (4) resource shifters; and (5) intergroup tournaments.

A free soul ought not to pursue any study slavishly; for while bodily labors performed under constraint do not harm the body, nothing that is learned under compulsion stays with the mind.

A learned blockhead is a greater blockhead than an ignorant one.

A learned man is an idler who kills time with study. Beware of his false knowledge: it is more dangerous than ignorance.

A linguist would be shocked to learn that if a set is not closed this does not mean that it is open, or again that “E is dense in E” does not mean the same thing as “E is dense in itself”.

A lucky physician is better than a learned one.

A man ceases to be a beginner in any given science and becomes a master in that science when he has learned that ... he is going to be a beginner all his life.

A man's own addition to what he learns is cement to bind an otherwise loose heap of stones into a structure of unity, strength, and use.

A neurotic person can be most simply described as someone who, while he was growing up, learned ways of behaving that are self-defeating in his society.

A physicist learns more and more about less and less, until he knows everything about nothing; whereas a philosopher learns less and less about more and more, until he knows nothing about everything.

A quarter-horse jockey learns to think of a twenty-second race as if it were occurring across twenty minutes—in distinct parts, spaced in his consciousness. Each nuance of the ride comes to him as he builds his race. If you can do the opposite with deep time, living in it and thinking in it until the large numbers settle into place, you can sense how swiftly the initial earth packed itself together, how swiftly continents have assembled and come apart, how far and rapidly continents travel, how quickly mountains rise and how quickly they disintegrate and disappear.

A scientist is in a sense a learned small boy. There is something of the scientist in every small boy. Others must outgrow it. Scientists can stay that way all their lives.

A scientist reads many books in his lifetime, and knows he still has a lot to learn. A religious man barely reads one book, and thinks he knows everything.

A star is drawing on some vast reservoir of energy by means unknown to us. This reservoir can scarcely be other than the subatomic energy which, it is known exists abundantly in all matter; we sometimes dream that man will one day learn how to release it and use it for his service. The store is well nigh inexhaustible, if only it could be tapped. There is sufficient in the Sun to maintain its output of heat for 15 billion years.

A statistician is one who has learned how to get valid evidence from statistics and how (usually) to avoid being misled by irrelevant facts. It’s too bad that we apply the same name to this kind of person that we use for those who only tabulate. It’s as if we had the same name for barbers and brain surgeons because they both work on the head.

A success learns how to make hay from the grass that grows under other people's feet.

A university should be a place of light, of liberty, and of learning.

A week or so after I learned that I was to receive the Miller Award, our president, Marty Morton, phoned and asked me if I would utter a few words of scientific wisdom as a part of the ceremony. Unfortunately for me, and perhaps for you, I agreed to do so. In retrospect I fear that my response was a serious error, because I do not feel wise. I do not know whether to attribute my response to foolhardiness, to conceit, to an inordinate susceptibility to flattery, to stupidity, or to some combination of these unfortunate attributes all of which I have been told are recognizable in my personality. Personally, I tend to favor stupidity, because that is a condition over which I have little control.

A wise system of education will at least teach us how little man yet knows, how much he has still to learn.

About 85 per cent of my “thinking” time was spent getting into a position to think, to make a decision, to learn something I needed to know. Much more time went into finding or obtaining information than into digesting it. Hours went into the plotting of graphs... When the graphs were finished, the relations were obvious at once, but the plotting had to be done in order to make them so.

Acceleration of knowledge generation also emphasizes the need for lifelong education. The trained teacher, scientist or engineer can no longer regard what they have learned at the university as supplying their needs for the rest of their lives.

Access to more information isn’t enough—the information needs to be correct, timely, and presented in a manner that enables the reader to learn from it. The current network is full of inaccurate, misleading, and biased information that often crowds out the valid information. People have not learned that “popular” or “available” information is not necessarily valid.

After a tremendous task has been begun in our time, first by Copernicus and then by many very learned mathematicians, and when the assertion that the earth moves can no longer be considered something new, would it not be much better to pull the wagon to its goal by our joint efforts, now that we have got it underway, and gradually, with powerful voices, to shout down the common herd, which really does not weigh arguments very carefully?

After that cancellation [of the Superconducting Super Collider in Texas, after $2 billion had been spent on it], we physicists learned that we have to sing for our supper. ... The Cold War is over. You can't simply say “Russia!” to Congress, and they whip out their checkbook and say, “How much?” We have to tell the people why this atom-smasher is going to benefit their lives.

After the discovery of spectral analysis no one trained in physics could doubt the problem of the atom would be solved when physicists had learned to understand the language of spectra. So manifold was the enormous amount of material that has been accumulated in sixty years of spectroscopic research that it seemed at first beyond the possibility of disentanglement. An almost greater enlightenment has resulted from the seven years of Röntgen spectroscopy, inasmuch as it has attacked the problem of the atom at its very root, and illuminates the interior. What we are nowadays hearing of the language of spectra is a true 'music of the spheres' in order and harmony that becomes ever more perfect in spite of the manifold variety. The theory of spectral lines will bear the name of Bohr for all time. But yet another name will be permanently associated with it, that of Planck. All integral laws of spectral lines and of atomic theory spring originally from the quantum theory. It is the mysterious organon on which Nature plays her music of the spectra, and according to the rhythm of which she regulates the structure of the atoms and nuclei.

Alexander the king of the Macedonians, began like a wretch to learn geometry, that he might know how little the earth was, whereof he had possessed very little. Thus, I say, like a wretch for this, because he was to understand that he did bear a false surname. For who can be great in so small a thing? Those things that were delivered were subtile, and to be learned by diligent attention: not which that mad man could perceive, who sent his thoughts beyond the ocean sea. Teach me, saith he, easy things. To whom his master said: These things be the same, and alike difficult unto all. Think thou that the nature of things saith this. These things whereof thou complainest, they are the same unto all: more easy things can be given unto none; but whosoever will, shall make those things more easy unto himself. How? With uprightness of mind.

All of modern physics is governed by that magnificent and thoroughly confusing discipline called quantum mechanics ... It has survived all tests and there is no reason to believe that there is any flaw in it.... We all know how to use it and how to apply it to problems; and so we have learned to live with the fact that nobody can understand it.

All Pretences of foretelling by Astrology, are Deceits; for this manifest Reason, because the Wise and Learned, who can only judge whether there be any Truth in this Science, do all unanimously agree to laugh at and despise it; and none but the poor ignorant Vulgar give it any Credit.

Although [Charles Darwin] would patiently go on repeating experiments where there was any good to be gained, he could not endure having to repeat an experiment which ought, if complete care had been taken, to have told its story at first—and this gave him a continual anxiety that the experiment should not be wasted; he felt the experiment to be sacred, however slight a one it was. He wished to learn as much as possible from an experiment, so that he did not confine himself to observing the single point to which the experiment was directed, and his power of seeing a number of other things was wonderful. ... Any experiment done was to be of some use, and ... strongly he urged the necessity of keeping the notes of experiments which failed, and to this rule he always adhered.

Although I was four years at the University [of Wisconsin], I did not take the regular course of studies, but instead picked out what I thought would be most useful to me, particularly chemistry, which opened a new world, mathematics and physics, a little Greek and Latin, botany and and geology. I was far from satisfied with what I had learned, and should have stayed longer.
[Enrolled in Feb 1861, left in 1863 without completing a degree, and began his first botanical foot journey.]

Although the ocean’s surface seems at first to be completely homogeneous, after half a month we began to differentiate various seas and even different parts of oceans by their characteristic shades. We were astonished to discover that, during an flight, you have to learn anew not only to look, but also to see. At first the finest nuances of color elude you, but gradually your vision sharpens and your color perception becomes richer, and the planet spreads out before you with all its indescribable beauty.

Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena.

Among the multitude of animals which scamper, fly, burrow and swim around us, man is the only one who is not locked into his environment. His imagination, his reason, his emotional subtlety and toughness, make it possible for him not to accept the environment, but to change it. And that series of inventions, by which man from age to age has remade his environment, is a different kind of evolution—not biological, but cultural evolution. I call that brilliant sequence of cultural peaks The Ascent of Man. I use the word ascent with a precise meaning. Man is distinguished from other animals by his imaginative gifts. He makes plans, inventions, new discoveries, by putting different talents together; and his discoveries become more subtle and penetrating, as he learns to combine his talents in more complex and intimate ways. So the great discoveries of different ages and different cultures, in technique, in science, in the arts, express in their progression a richer and more intricate conjunction of human faculties, an ascending trellis of his gifts.

An acquaintance of mine, a notary by profession, who, by perpetual writing, began first to complain of an excessive wariness of his whole right arm which could be removed by no medicines, and which was at last succeeded by a perfect palsy of the whole arm. … He learned to write with his left hand, which was soon thereafter seized with the same disorder.

An educated person is one who has learned that information almost always turns out to be at best incomplete and very often false, misleading, fictitious, mendacious—just dead wrong.

An inventor is simply a fellow who doesn’t take his education too seriously. You see, from the time a person is six years old until he graduates form college he has to take three or four examinations a year. If he flunks once, he is out. But an inventor is almost always failing. He tries and fails maybe a thousand times. It he succeeds once then he’s in. These two things are diametrically opposite. We often say that the biggest job we have is to teach a newly hired employee how to fail intelligently. We have to train him to experiment over and over and to keep on trying and failing until he learns what will work.

And as I had my father’s kind of mind—which was also his mother’s—I learned that the mind is not sex-typed.

And I do not take my medicines from the apothecaries; their shops are but foul sculleries, from which comes nothing but foul broths. As for you, you defend your kingdom with belly-crawling and flattery. How long do you think this will last? ... let me tell you this: every little hair on my neck knows more than you and all your scribes, and my shoebuckles are more learned than your Galen and Avicenna, and my beard has more experience than all your high colleges.

And when with excellent Microscopes I discern in otherwise invisible Objects the Inimitable Subtlety of Nature’s Curious Workmanship; And when, in a word, by the help of Anatomicall Knives, and the light of Chymicall Furnaces, I study the Book of Nature, and consult the Glosses of Aristotle, Epicurus, Paracelsus, Harvey, Helmont, and other learn'd Expositors of that instructive Volumne; I find my self oftentimes reduc’d to exclaim with the Psalmist, How manifold are thy works, O Lord? In wisdom hast thou made them all.

And, in this case, science could learn an important lesson from the literati–who love contingency for the same basic reason that scientists tend to regard the theme with suspicion. Because, in contingency lies the power of each person, to make a difference in an unconstrained world bristling with possibilities, and nudgeable by the smallest of unpredictable inputs into markedly different channels spelling either vast improvement or potential disaster.

Angling may be said to be so like the mathematics, that it can never be fully learnt.

Any man who says he doesn’t like applause and recognition is either a fool or a liar. You learn from mistakes, but success gives you the courage to go on and do even more.

Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe and not make messes in the house

As a second year high school chemistry student, I still have a vivid memory of my excitement when I first saw a chart of the periodic table of elements. The order in the universe seemed miraculous, and I wanted to study and learn as much as possible about the natural sciences.

As an antiquary of a new order, I have been obliged to learn the art of deciphering and restoring these remains, of discovering and bringing together, in their primitive arrangement, the scattered and mutilated fragments of which they are composed, of reproducing in all their original proportions and characters, the animals to which these fragments formerly belonged, and then of comparing them with those animals which still live on the surface of the earth; an art which is almost unknown, and which presupposes, what had scarcely been obtained before, an acquaintance with those laws which regulate the coexistence of the forms by which the different parts of organized being are distinguished.

As an exercise of the reasoning faculty, pure mathematics is an admirable exercise, because it consists of reasoning alone, and does not encumber the student with an exercise of judgment: and it is well to begin with learning one thing at a time, and to defer a combination of mental exercises to a later period.

As geologists, we learn that it is not only the present condition of the globe that has been suited to the accommodation of myriads of living creatures, but that many former states also have been equally adapted to the organization and habits of prior races of beings. The disposition of the seas, continents, and islands, and the climates have varied; so it appears that the species have been changed, and yet they have all been so modelled, on types analogous to those of existing plants and animals, as to indicate throughout a perfect harmony of design and unity of purpose. To assume that the evidence of the beginning or end of so vast a scheme lies within the reach of our philosophical inquiries, or even of our speculations, appears to us inconsistent with a just estimate of the relations which subsist between the finite powers of man and the attributes of an Infinite and Eternal Being.

As long as museums and universities send out expeditions to bring to light new forms of living and extinct animals and new data illustrating the interrelations of organisms and their environments, as long as anatomists desire a broad comparative basis human for anatomy, as long as even a few students feel a strong curiosity to learn about the course of evolution and relationships of animals, the old problems of taxonomy, phylogeny and evolution will gradually reassert themselves even in competition with brilliant and highly fruitful laboratory studies in cytology, genetics and physiological chemistry.

As soon as the circumstances of an experiment are well known, we stop gathering statistics. … The effect will occur always without exception, because the cause of the phenomena is accurately defined. Only when a phenomenon includes conditions as yet undefined,Only when a phenomenon includes conditions as yet undefined, can we compile statistics. … we must learn therefore that we compile statistics only when we cannot possibly help it; for in my opinion, statistics can never yield scientific truth.

As the issues are greater than men ever sought to realize before, the recriminations will be fiercer and pride more desperately hurt. It may help to recall that many recognized before the bomb ever feel that the time had already come when we must learn to live in One World.

As to the need of improvement there can be no question whilst the reign of Euclid continues. My own idea of a useful course is to begin with arithmetic, and then not Euclid but algebra. Next, not Euclid, but practical geometry, solid as well as plane; not demonstration, but to make acquaintance. Then not Euclid, but elementary vectors, conjoined with algebra, and applied to geometry. Addition first; then the scalar product. Elementary calculus should go on simultaneously, and come into vector algebraic geometry after a bit. Euclid might be an extra course for learned men, like Homer. But Euclid for children is barbarous.

As we advance in life we learn the limits of our abilities.

As we discern a fine line between crank and genius, so also (and unfortunately) we must acknowledge an equally graded trajectory from crank to demagogue. When people learn no tools of judgment and merely follow their hopes, the seeds of political manipulation are sown.

As we push ever more deeply into the universe, probing its secrets, discovering its way, we must also constantly try to learn to cooperate across the frontiers that really divide earth’s surface.

Astronomy is perhaps the science whose discoveries owe least to chance, in which human understanding appears in its whole magnitude, and through which man can best learn how small he is.

At first men try with magic charms
To fertilize the earth,
To keep their flocks and herds from harm
And bring new young to birth.

Then to capricious gods they turn
To save from fire or flood;
Their smoking sacrifices burn
On altars red with blood.

Next bold philosopher and sage
A settled plan decree
And prove by thought or sacred page
What Nature ought to be.

But Nature smiles—a Sphinx-like smile
Watching their little day
She waits in patience for a while—
Their plans dissolve away.

Then come those humbler men of heart
With no completed scheme,
Content to play a modest part,
To test, observe, and dream.

Till out of chaos come in sight
Clear fragments of a Whole;
Man, learning Nature’s ways aright
Obeying, can control.

At the age of eleven, I began Euclid, with my brother as my tutor. ... I had not imagined that there was anything so delicious in the world. After I had learned the fifth proposition, my brother told me that it was generally considered difficult, but I had found no difficulty whatsoever. This was the first time it had dawned on me that I might have some intelligence.

Attaching significance to invariants is an effort to recognize what, because of its form or colour or meaning or otherwise, is important or significant in what is only trivial or ephemeral. A simple instance of failing in this is provided by the poll-man at Cambridge, who learned perfectly how to factorize a²-b² but was floored because the examiner unkindly asked for the factors of p²–q².

Babylon,
Learned and wise, hath perished utterly,
Nor leaves her speech one word to aid the sigh
That would lament her.

Because intelligence is our own most distinctive feature, we may incline to ascribe superior intelligence to the basic primate plan, or to the basic plan of the mammals in general, but this point requires some careful consideration. There is no question at all that most mammals of today are more intelligent than most reptiles of today. I am not going to try to define intelligence or to argue with those who deny thought or consciousness to any animal except man. It seems both common and scientific sense to admit that ability to learn, modification of action according to the situation, and other observable elements of behavior in animals reflect their degrees of intelligence and permit us, if only roughly, to compare these degrees. In spite of all difficulties and all the qualifications with which the expert (quite properly) hedges his conclusions, it also seems sensible to conclude that by and large an animal is likely to be more intelligent if it has a larger brain at a given body size and especially if its brain shows greater development of those areas and structures best developed in our own brains. After all, we know we are intelligent, even though we wish we were more so.

Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world could have more astonished a Greek mathematician than to learn that, under the influence of compulsory education, the whole population of Western Europe, from the highest to the lowest, could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility. … Our modern power of easy reckoning with decimal fractions is the most miraculous result of a perfect notation.

Before the printing press, the young learned by listening, watching, doing.

Berzelius' symbols are horrifying. A young student in chemistry might as soon learn Hebrew as make himself acquainted with them... They appear to me equally to perplex the adepts in science, to discourage the learner, as well as to cloud the beauty and simplicity of the atomic theory.

Better far off to leave half the ruins and nine-tenths of the churches unseen and to see well the rest; to see them not once, but again and often again; to watch them, to learn them, to live with them, to love them, till they have become a part of life and life’s recollections.

Biological evolution is a system of constant divergence without subsequent joining of branches. Lineages, once distinct, are separate forever. In human history, transmission across lineages is, perhaps, the major source of cultural change. Europeans learned about corn and potatoes from Native Americans and gave them smallpox in return.

Borel makes the amusing supposition of a million monkeys allowed to play upon the keys of a million typewriters. What is the chance that this wanton activity should reproduce exactly all of the volumes which are contained in the library of the British Museum? It certainly is not a large chance, but it may be roughly calculated, and proves in fact to be considerably larger than the chance that a mixture of oxygen and nitrogen will separate into the two pure constituents. After we have learned to estimate such minute chances, and after we have overcome our fear of numbers which are very much larger or very much smaller than those ordinarily employed, we might proceed to calculate the chance of still more extraordinary occurrences, and even have the boldness to regard the living cell as a result of random arrangement and rearrangement of its atoms. However, we cannot but feel that this would be carrying extrapolation too far. This feeling is due not merely to a recognition of the enormous complexity of living tissue but to the conviction that the whole trend of life, the whole process of building up more and more diverse and complex structures, which we call evolution, is the very opposite of that which we might expect from the laws of chance.

Botany is the school for patience, and it’s amateurs learn resignation from daily disappointments.

But Chinese civilization has the overpowering beauty of the wholly other, and only the wholly other can inspire the deepest love and the profoundest desire to learn.

But come, hear my words, for truly learning causes the mind to grow. For as I said before in declaring the ends of my words … at one time there grew to be the one alone out of many, and at another time it separated so that there were many out of the one; fire and water and earth and boundless height of air, and baneful Strife apart from these, balancing each of them, and Love among them, their equal in length and breadth.

But from the time I was in college I learned that there is nothing one could imagine which is so strange and incredible that it was not said by some philosopher; and since that time, I have recognized through my travels that all those whose views are different from our own are not necessarily, for that reason, barbarians or savages, but that many of them use their reason either as much as or even more than we do. I also considered how the same person, with the same mind, who was brought up from infancy either among the French or the Germans, becomes different from what they would have been if they had always lived among the Chinese or among the cannibals, and how, even in our clothes fashions, the very thing that we liked ten years ago, and that we may like again within the next ten years, appears extravagant and ridiculous to us today. Thus our convictions result from custom and example very much more than from any knowledge that is certain... truths will be discovered by an individual rather than a whole people.

But when great and ingenious artists behold their so inept performances, not undeservedly do they ridicule the blindness of such men; since sane judgment abhors nothing so much as a picture perpetrated with no technical knowledge, although with plenty of care and diligence. Now the sole reason why painters of this sort are not aware of their own error is that they have not learnt Geometry, without which no one can either be or become an absolute artist; but the blame for this should be laid upon their masters, who are themselves ignorant of this art.

By destroying the biological character of phenomena, the use of averages in physiology and medicine usually gives only apparent accuracy to the results. From our point of view, we may distinguish between several kinds of averages: physical averages, chemical averages and physiological and pathological averages. If, for instance, we observe the number of pulsations and the degree of blood pressure by means of the oscillations of a manometer throughout one day, and if we take the average of all our figures to get the true or average blood pressure and to learn the true or average number of pulsations, we shall simply have wrong numbers. In fact, the pulse decreases in number and intensity when we are fasting and increases during digestion or under different influences of movement and rest; all the biological characteristics of the phenomenon disappear in the average. Chemical averages are also often used. If we collect a man's urine during twenty-four hours and mix all this urine to analyze the average, we get an analysis of a urine which simply does not exist; for urine, when fasting, is different from urine during digestion. A startling instance of this kind was invented by a physiologist who took urine from a railroad station urinal where people of all nations passed, and who believed he could thus present an analysis of average European urine! Aside from physical and chemical, there are physiological averages, or what we might call average descriptions of phenomena, which are even more false. Let me assume that a physician collects a great many individual observations of a disease and that he makes an average description of symptoms observed in the individual cases; he will thus have a description that will never be matched in nature. So in physiology, we must never make average descriptions of experiments, because the true relations of phenomena disappear in the average; when dealing with complex and variable experiments, we must study their various circumstances, and then present our most perfect experiment as a type, which, however, still stands for true facts. In the cases just considered, averages must therefore be rejected, because they confuse, while aiming to unify, and distort while aiming to simplify. Averages are applicable only to reducing very slightly varying numerical data about clearly defined and absolutely simple cases.

By three methods we may learn wisdom: first, by reflection, which is noblest; second, by imitation, which is easiest; and third, by experience, which is the most bitter.

Cayley was singularly learned in the work of other men, and catholic in his range of knowledge. Yet he did not read a memoir completely through: his custom was to read only so much as would enable him to grasp the meaning of the symbols and understand its scope. The main result would then become to him a subject of investigation: he would establish it (or test it) by algebraic analysis and, not infrequently, develop it so to obtain other results. This faculty of grasping and testing rapidly the work of others, together with his great knowledge, made him an invaluable referee; his services in this capacity were used through a long series of years by a number of societies to which he was almost in the position of standing mathematical advisor.

Chemistry and physics are experimental sciences; and those who are engaged in attempting to enlarge the boundaries of science by experiment are generally unwilling to publish speculations; for they have learned, by long experience, that it is unsafe to anticipate events. It is true, they must make certain theories and hypotheses. They must form some kind of mental picture of the relations between the phenomena which they are trying to investigate, else their experiments would be made at random, and without connection.

Children are told that an apple fell on Isaac Newton’s head and he was led to state the law of gravity. This, of course, is pure foolishness. What Newton discovered was that any two particles in the universe attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. This is not learned from a falling apple, but by observing quantities of data and developing a mathematical theory that can be verified by additional data. Data gathered by Galileo on falling bodies and by Johannes Kepler on motions of the planets were invaluable aids to Newton. Unfortunately, such false impressions about science are not universally outgrown like the Santa Claus myth, and some people who don’t study much science go to their graves thinking that the human race took until the mid-seventeenth century to notice that objects fall.

Computers and rocket ships are examples of invention, not of understanding. … All that is needed to build machines is the knowledge that when one thing happens, another thing happens as a result. It’s an accumulation of simple patterns. A dog can learn patterns. There is no “why” in those examples. We don’t understand why electricity travels. We don’t know why light travels at a constant speed forever. All we can do is observe and record patterns.

Copernicus, the most learned man whom we are able to name other than Atlas and Ptolemy, even though he taught in a most learned manner the demonstrations and causes of motion based on observation, nevertheless fled from the job of constructing tables, so that if anyone computes from his tables, the computation is not even in agreement with his observations on which the foundation of the work rests. Therefore first I have compared the observations of Copernicus with those of Ptolemy and others as to which are the most accurate, but besides the bare observations, I have taken from Copernicus nothing other than traces of demonstrations. As for the tables of mean motion, and of prosthaphaereses and all the rest, I have constructed these anew, following absolutely no other reasoning than that which I have judged to be of maximum harmony.

Cosmology does, I think, affect the way that we perceive humanity’s role in nature. One thing we’ve learnt from astronomy is that the future lying ahead is more prolonged than the past. Even our sun is less than halfway through its life.

Culture can be regarded as a constitution of recipes for behavior … which are taught and learned on various levels of awareness; not all of the individuals in any society know all these recipes; and many of the recipes are alternative to one another.

Daniel Bernoulli used to tell two little adventures, which he said had given him more pleasure than all the other honours he had received. Travelling with a learned stranger, who, being pleased with his conversation, asked his name; “I am Daniel Bernoulli,” answered he with great modesty; “and I,” said the stranger (who thought he meant to laugh at him) “am Isaac Newton.” Another time, having to dine with the celebrated Koenig, the mathematician, who boasted, with some degree of self-complacency, of a difficult problem he had solved with much trouble, Bernoulli went on doing the honours of his table, and when they went to drink coffee he presented Koenig with a solution of the problem more elegant than his own.

Doubtless the reasoning faculty, the mind, is the leading and characteristic attribute of the human race. By the exercise of this, man arrives at the properties of the natural bodies. This is science, properly and emphatically so called. It is the science of pure mathematics; and in the high branches of this science lies the truly sublime of human acquisition. If any attainment deserves that epithet, it is the knowledge, which, from the mensuration of the minutest dust of the balance, proceeds on the rising scale of material bodies, everywhere weighing, everywhere measuring, everywhere detecting and explaining the laws of force and motion, penetrating into the secret principles which hold the universe of God together, and balancing worlds against worlds, and system against system. When we seek to accompany those who pursue studies at once so high, so vast, and so exact; when we arrive at the discoveries of Newton, which pour in day on the works of God, as if a second fiat had gone forth from his own mouth; when, further, we attempt to follow those who set out where Newton paused, making his goal their starting-place, and, proceeding with demonstration upon demonstration, and discovery upon discovery, bring new worlds and new systems of worlds within the limits of the known universe, failing to learn all only because all is infinite; however we may say of man, in admiration of his physical structure, that “in form and moving he is express and admirable,” it is here, and here without irreverence, we may exclaim, “In apprehension how like a god!” The study of the pure mathematics will of course not be extensively pursued in an institution, which, like this [Boston Mechanics’ Institute], has a direct practical tendency and aim. But it is still to be remembered, that pure mathematics lie at the foundation of mechanical philosophy, and that it is ignorance only which can speak or think of that sublime science as useless research or barren speculation.