The thing is with hip-hop, it has its waves and the waves crash against the beach and the new waves come in. So to stay relevant you have to roll with that.

The universe may have a purpose, but nothing we know suggests that, if so, this purpose has any similarity to ours.

The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of highest excellence, is to be found in mathematics as surely as in poetry.

The trouble with the world is that the stupid are cocksure and the intelligent full of doubt.

The time you enjoy wasting is not wasted time.

The theoretical understanding of the world, which is the aim of philosophy, is not a matter of great practical importance to animals, or to savages, or even to most civilized men.

The solution of the difficulties which formerly surrounded the mathematical infinite is probably the greatest achievement of which our age has to boast.

The significance of a fact is relative to [the general body of scientific] knowledge. To say that a fact is significant in science, is to say that it helps to establish or refute some general law; for science, though it starts from observation of the particular, is not concerned essentially with the particular, but with the general. A fact, in science, is not a mere fact, but an instance. In this the scientist differs from the artist, who, if he deigns to notice facts at all, is likely to notice them in all their particularity.

The scientific attitude of mind involves a sweeping away of all other desires in the interest of the desire to know.

The root of the matter the thing I mean is love, Christian love, or compassion. If you feel this, you have a motive for existence, a guide for action, a reason for courage, an imperative necessity for intellectual honesty.

The road to happiness and prosperity lies in an organized diminution of work.

The results of science, in the form of mechanism, poison gas, and the yellow press, bid fair to lead to the total downfall of our civilization.

The pure mathematician, like the musician, is a free creator of his world of ordered beauty.

The point of philosophy is to start with something so simple as not to seem worth stating and to end with something so paradoxical that no one will believe it

The oranges, it is true, are not all exactly of the same size, but careful machinery sorts them so that automatically all those in one box are exactly similar. They travel along with suitable things being done to them by suitable machines at suitable points until they enter a suitable refrigerator car in which they travel to a suitable market. The machine stamps the word “Sunkist” upon them, but otherwise there is nothing to suggest that nature has any part in their production.

The opinions that are held with passion are always those for which no good ground exists; indeed the passion is the measure of the holders lack of

The only thing that will redeem mankind is cooperation.

The nineteenth century, which prided itself upon the invention of steam and evolution, might have derived a more legitimate title to fame from the discovery of pure mathematics.

The nineteenth century which prides itself upon the invention of steam and evolution, might have derived a more legitimate title to fame from the discovery of pure mathematics.

The most savage controversies are those about matters as to which there is no good evidence either way. Persecution is used in theology, not in arithmetic.

The most obvious and easy things in mathematics are not those that come logically at the beginning; they are things that, from the point of view of logical deduction, come somewhere in the middle. Just as the easiest bodies to see are those that are neither very near nor very far…

The more we realize our minuteness and our impotence in the face of cosmic forces, the more amazing becomes what human beings have achieved.

The modern development of mathematical logic dates from Boole’s Laws of Thought (1854). But in him and his successors, before Peano and Frege, the only thing really achieved, apart from certain details, was the invention of a mathematical symbolism for deducing consequences from the premises which the newer methods shared with Aristotle.

The method of “postulating” what we want has many advantages; they are the same as the advantages of theft over honest toil.

The Italian Renaissance, though not medieval, is not modern; it is more akin to the best age of Greece. … No Italian of the Renaissance would have been unintelligible to Plato or Aristotle…. With the seventeenth century it is different: Plato and Aristotle, Aquinas and Occam, could not have made head or tail of Newton.

The human race may well become extinct before the end of the century. Speaking as a mathematician, I should say the odds are about three to one against survival.

The good life is one inspired by life and guided by knowledge.

The fundamental concept in social science is Power, in the same sense in which Energy is the fundamental concept in physics.

The first man who said “fire burns” was employing scientific method, at any rate if he had allowed himself to be burnt several times. This man had already passed through the two stages of observation and generalization. He had not, however, what scientific technique demands—a careful choice of significant facts on the one hand, and, on the other hand, various means of arriving at laws otherwise than my mere generalization. (1931)

The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists of the analysis of Symbolic Logic itself.

The examination system, and the fact that instruction is treated mainly as a training for a livelihood, leads the young to regard knowledge from a purely utilitarian point of view as the road to money, not as the gateway to wisdom.

The distinctive Western character begins with the Greeks, who invented the habit of deductive reasoning and the science of geometry.

The desire to understand the world and the desire to reform it are the two great engines of progress.

The demand for certainty is one which is natural to man, but is nevertheless an intellectual vice. If you take your children for a picnic on a doubtful day, they will demand a dogmatic answer as to whether it will be fine or wet, and be disappointed in you when you cannot be sure.

The degree of one’s emotions varies inversely with one’s knowledge of the facts—the less you know the hotter you get.

Symbolism is useful because it makes things difficult. Now in the beginning everything is self-evident, and it is hard to see whether one self-evident proposition follows from another or not. Obviousness is always the enemy to correctness. Hence we must invent a new and difficult symbolism in which nothing is obvious. … Thus the whole of Arithmetic and Algebra has been shown to require three indefinable notions and five indemonstrable propositions.

Sir Arthur Eddington deduces religion from the fact that atoms do not obey the laws of mathematics. Sir James Jeans deduces it from the fact that they do.

Since the world is what it is, it is clear that valid reasoning from sound principles cannot lead to error; but a principle may be so nearly true as to deserve theoretical respect, and yet may lead to practical consequences which we feel to be absurd. There is therefore a justification for common sense in philosophy, but only as showing that our theoretical principles cannot be quite correct so long as their consequences are condemned by an appeal to common sense which we feel to be irresistible.

Scientific method, although in its more refined forms it may seem complicated, is in essence remarkably simply. It consists in observing such facts as will enable the observer to discover general laws governing facts of the kind in question. The two stages, first of observation, and second of inference to a law, are both essential, and each is susceptible of almost indefinite refinement. (1931)

Science, in its ultimate ideal, consists of a set of propositions arranged in a hierarchy, the lowest level of the hierarchy being concerned with particular facts, and the highest with some general law, governing everything in the universe. The various levels in the hierarchy have a two-fold logical connection, travelling one up, one down; the upward connection proceeds by induction, the downward by deduction.

Science, by itself, cannot supply us with an ethic. It can show us how to achieve a given end, and it may show us that some ends cannot be achieved. But among ends that can be achieved our choice must be decided by other than purely scientific considerations. If a man were to say, “I hate the human race, and I think it would be a good thing if it were exterminated,” we could say, “Well, my dear sir, let us begin the process with you.” But this is hardly argument, and no amount of science could prove such a man mistaken.

Science may set limits to knowledge, but should not set limits to imagination.

Science may be a boon if war can be abolished and democracy and cultural liberty preserved. If this cannot be done, science will precipitate evils greater than any that mankind has ever experienced.

Science is what you more or less know and philosophy is what you do not know.

Science is what you know, philosophy what you don’t know.

Science is what we know, and philosophy is what we don’t know.

Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos [mathematics], where pure thought can dwell in its natural home…

Religion is something left over from the infancy of our intelligence, it will fade away as we adopt read on and science as our guidelines.

Reason is a harmonizing, controlling force rather than a creative one.

Real life is, to most men, a long second-best, a perpetual compromise between the ideal and the possible; but the world of pure reason ;knows no compromise, no practical limitations, no barrier to the creative activity.

Pure mathematics was discovered by Boole in a work which he called “The Laws of Thought” (1854).… His book was in fact concerned with formal logic, and this is the same thing as mathematics.

Pure Mathematics is the class of all propositions of the form “p implies q,” where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form. In addition to these, mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth.

Pure mathematics consists entirely of such asseverations as that, if such and such is a proposition is true of anything, then such and such another propositions is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is of which it is supposed to be true. Both these points would belong to applied mathematics. … If our hypothesis is about anything and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the the subject in which we never know what we are talking about, not whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate.

Probability is the most important concept in modern science, especially as nobody has the slightest notion of what it means.

Physics tells us much less about the physical world than we thought it did.

Physics is mathematical not because we know so much about the physical world, but because we know so little: it is only its mathematical properties that we can discover.

Physical science is thus approaching the stage when it will be complete, and therefore uninteresting. Given the laws governing the motions of electrons and protons, the rest is merely geography—a collection of particular facts.

Philosophy, though unable to tell us with certainty what is the true answer to the doubts which it raises, is able to suggest many possibilities which enlarge our thoughts and free them from the tyranny of custom.

Philosophy is that part of science which at present people chose to have opinions about, but which they have no knowledge about. Therefore every advance in knowledge robs philosophy of some problems which formerly it had …and will belong to science.

Persecution is used in theology, not in arithmetic, because in arithmetic there is knowledge, but in theology there is only opinion. So whenever you find yourself getting angry about a difference of opinion, be on your guard, you will probably find, on examination, that your belief is going beyond what the evidence warrants?

Perhaps in the times of Ahmes the multiplication table was exciting.

People are born ignorant, not stupid; they are made stupid by education.

Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say.

Only mathematics and mathematical logic can say as little as the physicist means to say. (1931)

One of the symptoms of approaching nervous breakdown is the belief that one’s work is terribly important.

One of the main purposes of scientific inference is to justify beliefs which we entertain already; but as a rule they are justified with a difference. Our pre-scientific general beliefs are hardly ever without exceptions; in science, a law with exceptions can only be tolerated as a makeshift. Scientific laws, when we have reason to think them accurate, are different in form from the common-sense rules which have exceptions: they are always, at least in physics, either differential equations, or statistical averages. It might be thought that a statistical average is not very different from a rule with exceptions, but this would be a mistake. Statistics, ideally, are accurate laws about large groups; they differ from other laws only in being about groups, not about individuals. Statistical laws are inferred by induction from particular statistics, just as other laws are inferred from particular single occurrences.

One of the chiefest triumphs of modern mathematics consists in having discovered what mathematics really is.

One must expect a war between U.S.A. and U.S.S.R. which will begin with the total destruction of London. I think the war will last 30 years, and leave a world without civilised people, from which everything will have to build afresh—a process taking (say) 500 years.
Stated just one month after the Hiroshima atomic explosion. Russell became one of the best-known antinuclear activists of his era.

One must care about a world one will not see.

Of these austerer virtues the love of truth is the chief, and in mathematics, more than elsewhere, the love of truth may find encouragement for waning faith. Every great study is not only an end in itself, but also a means of creating and sustaining a lofty habit of mind; and this purpose should be kept always in view throughout the teaching and learning of mathematics.

No man treats a motorcar as foolishly as he treats another human being. When the car will not go, he does not attribute its annoying behavior to sin; he does not say, “You are a wicked motorcar, and I shall not give you any more petrol until you go.” He attempts to find out what is wrong and to set it right. An analogous way of treating human beings is, however, considered to be contrary to the truths of our holy religion.

Next to enjoying ourselves, the next greatest pleasure consists in preventing others from enjoying themselves, or, more generally, in the acquisition of power.

Most people would rather die than think: many do.

Most people would rather die than think; many do.

Mere instruction to memorise data is empty. The attempt to enforce conventional mediocrity on the young is criminal.

Men who are unhappy, like men who sleep badly, are always proud of the fact.

Men fear thought as they fear nothing else on earth more than ruin more even than death. Thought is subversive and revolutionary, destructive and terrible, thought is merciless to privilege, established institutions, and comfortable habit. Thought looks into the pit of hell and is not afraid. Thought is great and swift and free, the light of the world, and the chief glory of man.

Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

Mathematics takes us still further from what is human, into the region of absolute necessity, to which not only the world, but every possible world, must conform.

Mathematics may be defined as the the subject in which we never know what we are talking about, not whether what we are saying is true.

Mathematics is a study which, when we start from its most familiar portions, may be pursued in either of two opposite directions. The more familiar direction is constructive, towards gradually increasing complexity: from integers to fractions, real numbers, complex numbers; from addition and multiplication to differentiation and integration, and on to higher mathematics. The other direction, which is less familiar, proceeds, by analysing, to greater and greater abstractness and logical simplicity; instead of asking what can be defined and deduced from what is assumed to begin with, we ask instead what more general ideas and principles can be found, in terms of which what was our starting-point can be defined or deduced. It is the fact of pursuing this opposite direction that characterises mathematical philosophy as opposed to ordinary mathematics.

Many people would sooner die than think. In fact they do.

Many orthodox people speak as though it were the business of sceptics to disprove received dogmas rather than of dogmatists to prove them. This is, of course, a mistake.

Man is a rational animal—so at least I have been told. … Aristotle, so far as I know, was the first man to proclaim explicitly that man is a rational animal. His reason for this view was … that some people can do sums. … It is in virtue of the intellect that man is a rational animal. The intellect is shown in various ways, but most emphatically by mastery of arithmetic. The Greek system of numerals was very bad, so that the multiplication table was quite difficult, and complicated calculations could only be made by very clever people.

Man is a part of nature, not something contrasted with nature. His thoughts and his bodily movements follow the same laws that describe the motions of stars and atoms.

Man is a credulous animal, and must believe something; in the absence of good grounds for belief, he will be satisfied with bad ones.

Machines are worshipped because they are beautiful and valued because they confer power; they are hated because they are hideous and loathed because they impose slavery.

John Locke invented common sense, and only Englishmen have had it ever since!

It seemed that animals always behave in a manner showing the rightness of the philosophy entertained by the man who observes them… . Throughout the reign of Queen Victoria all apes were virtuous monogamists, but during the dissolute twenties their morals underwent a disastrous deterioration.

It is undesirable to believe a proposition when there is no ground whatever for supposing it to be true.

It is one of the chief merits of proofs that they instil a certain scepticism as to the result proved.

It is a curious and painful fact that almost all the completely futile treatments that have been believed in during the long history of medical folly have been such as caused acute suffering to the patient. When anesthetics were discovered, pious people considered them an attempt to evade the will of God. It was pointed out, however, that when God extracted Adam’s rib He put him into a deep sleep. This proved that anesthetics are all right for men; women, however, ought to suffer, because of the curse of Eve.

Into every tidy scheme for arranging the pattern of human life, it is necessary to inject a certain dose of anarchism.

Infinity…belonged in former days to philosophy, but belongs now to mathematics.

In the streets of a modern city the night sky is invisible; in rural districts, we move in cars with bright headlights. We have blotted out the heavens, and only a few scientists remain aware of stars and planets, meteorites and comets.

In the higher walks of politics the same sort of thing occurs. The statesman who has gradually concentrated all power within himself … may have had anything but a public motive… The phrases which are customary on the platform and in the Party Press have gradually come to him to seem to express truths, and he mistakes the rhetoric of partisanship for a genuine analysis of motives… He retires from the world after the world has retired from him.

In science men have discovered an activity of the very highest value in which they are no longer, as in art, dependent for progress upon the appearance of continually greater genius, for in science the successors stand upon the shoulders of their predecessors; where one man of supreme genius has invented a method, a thousand lesser men can apply it. … In art nothing worth doing can be done without genius; in science even a very moderate capacity can contribute to a supreme achievement.

In general … science per se does not increase the destructiveness of war, since, as a rule, it strengthens the defense as much as the attack.

In attempting to understand the elements out of which mental phenomena are compounded, it is of the greatest importance to remember that from the protozoa to man there is nowhere a very wide gap either in structure or in behaviour. From this fact it is a highly probable inference that there is also nowhere a very wide mental gap.

If the matter is one that can be settled by observation, make the observation yourself. Aristotle could have avoided the mistake of thinking that women have fewer teeth than men, by the simple device of asking Mrs. Aristotle to keep her mouth open while he counted.

If I were to suggest that between the Earth and Mars there is a china teapot revolving about the sun in an elliptical orbit, nobody would be able to disprove my assertion provided I were careful to add that the teapot is too small to be revealed even by our most powerful telescopes. But if I were to go on to say that, since my assertion cannot be disproved, it is intolerable presumption on the part of human reason to doubt it, I should rightly be thought to be talking nonsense. If, however, the existence of such a teapot were affirmed in ancient books, taught as the sacred truth every Sunday, and instilled into the minds of children at school, hesitation to believe in its existence would become a mark of eccentricity and entitle the doubter to the attentions of the psychiatrist in an enlightened age or of the Inquisitor in an earlier time.

If I were a medical man, I should prescribe a holiday to any patient who considered his work important.

If I were a comet, I should consider the men of our present age a degenerate breed. In former times, the respect for comets was universal and profound.

If fifty million people say a foolish thing, it is still a foolish thing.

If any philosopher had been asked for a definition of infinity, he might have produced some unintelligible rigmarole, but he would certainly not have been able to give a definition that had any meaning at all.

I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.

I think it would be just to say the most essential characteristic of mind is memory, using this word in its broadest sense to include every influence of past experience on present reactions.

I ought to call myself an agnostic; but, for all practical purposes, I am an atheist. I do not think the existence of the Christian God any more probable than the existence of the Gods of Olympus or Valhalla. To take another illustration: nobody can prove that there is not between the Earth and Mars a china teapot revolving in an elliptical orbit, but nobody thinks this sufficiently likely to be taken into account in practice. I think the Christian God just as unlikely.

I had at one time a very bad fever of which I almost died. In my fever I had a long consistent delirium. I dreamt that I was in Hell, and that Hell is a place full of all those happenings that are improbable but not impossible. The effects of this are curious. Some of the damned, when they first arrive below, imagine that they will beguile the tedium of eternity by games of cards. But they find this impossible, because, whenever a pack is shuffled, it comes out in perfect order, beginning with the Ace of Spades and ending with the King of Hearts. There is a special department of Hell for students of probability. In this department there are many typewriters and many monkeys. Every time that a monkey walks on a typewriter, it types by chance one of Shakespeare’s sonnets. There is another place of torment for physicists. In this there are kettles and fires, but when the kettles are put on the fires, the water in them freezes. There are also stuffy rooms. But experience has taught the physicists never to open a window because, when they do, all the air rushes out and leaves the room a vacuum.

I do not believe that science per se is an adequate source of happiness, nor do I think that my own scientific outlook has contributed very greatly to my own happiness, which I attribute to defecating twice a day with unfailing regularity. Science in itself appears to me neutral, that is to say, it increases men’s power whether for good or for evil. An appreciation of the ends of life is something which must be superadded to science if it is to bring happiness, but only the kind of society to which science is apt to give rise. I am afraid you may be disappointed that I am not more of an apostle of science, but as I grow older, and no doubt—as a result of the decay of my tissues, I begin to see the good life more and more as a matter of balance and to dread all over-emphasis upon anyone ingredient.

I conclude that, while it is true that science cannot decide questions of value, that is because they cannot be intellectually decided at all, and lie outside the realm of truth and falsehood. Whatever knowledge is attainable, must be attained by scientific methods; and what science cannot discover, mankind cannot know.

I am paid by the word, so I always write the shortest words possible.

I am compelled to fear that science will be used to promote the power of dominant groups rather than to make men happy.

I am afraid that education is conceived more in terms of indoctrination by most school officials than in terms of enlightenment.

Gradually, … the aspect of science as knowledge is being thrust into the background by the aspect of science as the power of manipulating nature. It is because science gives us the power of manipulating nature that it has more social importance than art. Science as the pursuit of truth is the equal, but not the superior, of art. Science as a technique, though it may have little intrinsic value, has a practical importance to which art cannot aspire.

Frege has the merit of … finding a third assertion by recognising the world of logic which is neither mental nor physical.

Fear is the main source of superstition, and one of the main sources of cruelty. To conquer fear is the beginning of wisdom, in the pursuit of truth as in the endeavour after a worthy manner of life.

Every proposition which we can understand must be composed wholly of constituents with which we are acquainted.

Every living thing is a sort of imperialist, seeking to transform as much as possible of its environment into itself and its seed. When we compare the (present) human population of the globe with… that of former times, we see that “chemical imperialism” has been… the main end to which human intelligence has been devoted.

Even when the experts all agree, they may well be mistaken.

Even if the open windows of science at first make us shiver after the cozy indoor warmth of traditional humanizing myths, in the end the fresh air brings vigor, and the great spaces have a splendor of their own.

Education must be subversive if it is to be meaningful. By this I mean that it must challenge all the things we take for granted, examine all accepted assumptions, tamper with every sacred cow, and instil a desire to question and doubt.

Do not fear to be eccentric in opinion, for every opinion now accepted was once eccentric.

Descartes, the father of modern philosophy … would never—so he assures us—have been led to construct his philosophy if he had had only one teacher, for then he would have believed what he had been told; but, finding that his professors disagreed with each other, he was forced to conclude that no existing doctrine was certain.

Conventional people are roused to fury by departures from convention, largely because they regard such departures as a criticism of themselves.

Can a society in which thought and technique are scientific persist for a long period, as, for example, ancient Egypt persisted, or does it necessarily contain within itself forces which must bring either decay or explosion?

Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what the infinitely little might be.

Calculating machines do sums better than even the cleverest people… As arithmetic has grown easier, it has come to be less respected.

But, you might say, “none of this shakes my belief that 2 and 2 are 4.” You are quite right, except in marginal cases—and it is only in marginal cases that you are doubtful whether a certain animal is a dog or a certain length is less than a meter. Two must be two of something, and the proposition “2 and 2 are 4” is useless unless it can be applied. Two dogs and two dogs are certainly four dogs, but cases arise in which you are doubtful whether two of them are dogs. “Well, at any rate there are four animals,” you may say. But there are microorganisms concerning which it is doubtful whether they are animals or plants. “Well, then living organisms,” you say. But there are things of which it is doubtful whether they are living organisms or not. You will be driven into saying: “Two entities and two entities are four entities.” When you have told me what you mean by “entity,” we will resume the argument.

But it is just this characteristic of simplicity in the laws of nature hitherto discovered which it would be fallacious to generalize, for it is obvious that simplicity has been a part cause of their discovery, and can, therefore, give no ground for the supposition that other undiscovered laws are equally simple.

Broadly speaking, we are in the middle of a race between human skill as a means and human folly as an end.

Bad philosophers may have a certain influence; good philosophers, never.

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.

At first it seems obvious, but the more you think about it the stranger the deductions from this axiom seem to become; in the end you cease to understand what is meant by it.

Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians.

Aristotle, in spite of his reputation, is full of absurdities. He says that children should be conceived in the Winter, when the wind is in the North, and that if people marry too young the children will be female. He tells us that the blood of females is blacker then that of males; that the pig is the only animal liable to measles; that an elephant suffering from insomnia should have its shoulders rubbed with salt, olive-oil, and warm water; that women have fewer teeth than men, and so on. Nevertheless, he is considered by the great majority of philosophers a paragon of wisdom.

Aristotle maintained that women have fewer teeth than men; although he was twice married, it never occurred to him to verify this statement by examining his wives’ mouths.

An extra-terrestrial philosopher, who had watched a single youth up to the age of twenty-one and had never come across any other human being, might conclude that it is the nature of human beings to grow continually taller and wiser in an indefinite progress towards perfection; and this generalization would be just as well founded as the generalization which evolutionists base upon the previous history of this planet.

Although this may seem a paradox, all exact science is dominated by the idea of approximation. When a man tells you that he knows the exact truth about anything, you are safe in infering that he is an inexact man. Every careful measurement in science is always given with the probable error … every observer admits that he is likely wrong, and knows about how much wrong he is likely to be.

Almost everything that distinguishes the modern world from earlier centuries is attributable to science, which achieved its most spectacular triumphs in the seventeenth century.

Almost all great advances have sprung originally from disinterested motives. Scientific discoveries have been made for their own sake and not for their utilization, and a race of men without a disinterested love of knowledge would never have achieved our present scientific technique. … Faraday, Maxwell, and Hertz, so far as can be discovered, never for a moment considered the possibility of any practical application of their investigations.

All the conditions of happiness are realized in the life of the man of science.

All that passes for knowledge can be arranged in a hierarchy of degrees of certainty, with arithmetic and the facts of perception at the top.

A truer image of the world, I think, is obtained by picturing things as entering into the stream of time from an eternal world outside, than from a view which regards time as the devouring tyrant of all that is.

A sense of duty is useful in work but offensive in personal relations.

A process which led from the amoeba to man appeared to the philosophers to be obviously a progress—though whether the amoeba would agree with this opinion is not known.

A hallucination is a fact, not an error; what is erroneous is a judgment based upon it.

A habit of basing convictions upon evidence, and of giving to them only that degree or certainty which the evidence warrants, would, if it became general, cure most of the ills from which the world suffers.

A good notation has a subtlety and suggestiveness which at times make it almost seem like a live teacher. … a perfect notation would be a substitute for thought.

A fact, in science, is not a mere fact, but an instance.

A book should have either intelligibility or correctness; to combine the two is impossible, but to lack both is to be unworthy of a place as Euclid has occupied in education.