(1) I have told you more than I know about osteoporosis. (2) What I have told you is subject to change without notice. (3) I hope I raised more questions than I have given answers. (4) In any case, as usual, a lot more work is necessary.
Conclusion of one of his papers.

... one of the main functions of an analogy or model is to suggest extensions of the theory by considering extensions of the analogy, since more is known about the analogy than is known about the subject matter of the theory itself … A collection of observable concepts in a purely formal hypothesis suggesting no analogy with anything would consequently not suggest either any directions for its own development.

...[T]he natural history of the rat is tragically similar to that of man ... some of the more obvious qualities in which rats resemble men — ferocity, omnivorousness, and adaptability to all climates ... the irresponsible fecundity with which both species breed at all seasons of the year with a heedlessness of consequences, which subjects them to wholesale disaster on the inevitable, occasional failure of the food supply.... [G]radually, these two have spread across the earth, keeping pace with each other and unable to destroy each other, though continually hostile. They have wandered from East to West, driven by their physical needs, and — unlike any other species of living things — have made war upon their own kind. The gradual, relentless, progressive extermination of the black rat by the brown has no parallel in nature so close as that of the similar extermination of one race of man by another...

Dicere enim bene nemo potest, nisi qui prudenter intelligit.
No one can speak well, unless he thoroughly understands his subject.

Ein Fachmann ist ein Mann, der einige der gröbsten Fehler kennt, die man in dem betreffenden Fach machen kann, und der sie deshalb zu vermeiden versteht.
An expert is someone who knows some of the worst mistakes that can be made in his subject, and how to avoid them.

Les causes primordiales ne nous sont point connues; mais elles sont assujetties à des lois simples et constantes, que l’on peut découvrir par l’observation, et dont l’étude est l’objet de la philosophie naturelle.
Primary causes are unknown to us; but are subject to simple and constant laws, which may be discovered by observation, the study of them being the object of natural philosophy.

Newsreader: A huge asteroid could destroy Earth! And by coincidence, that's the subject of tonight's miniseries.
Dogbert: In science, researchers proved that this simple device can keep idiots off your television screen. [TV remote control] Click.

A common fallacy in much of the adverse criticism to which science is subjected today is that it claims certainty, infallibility and complete emotional objectivity. It would be more nearly true to say that it is based upon wonder, adventure and hope.

A good title should aim at making what follows as far as possible superfluous to those who know anything of the subject.

A great department of thought must have its own inner life, however transcendent may be the importance of its relations to the outside. No department of science, least of all one requiring so high a degree of mental concentration as Mathematics, can be developed entirely, or even mainly, with a view to applications outside its own range. The increased complexity and specialisation of all branches of knowledge makes it true in the present, however it may have been in former times, that important advances in such a department as Mathematics can be expected only from men who are interested in the subject for its own sake, and who, whilst keeping an open mind for suggestions from outside, allow their thought to range freely in those lines of advance which are indicated by the present state of their subject, untrammelled by any preoccupation as to applications to other departments of science. Even with a view to applications, if Mathematics is to be adequately equipped for the purpose of coping with the intricate problems which will be presented to it in the future by Physics, Chemistry and other branches of physical science, many of these problems probably of a character which we cannot at present forecast, it is essential that Mathematics should be allowed to develop freely on its own lines.

A hundred years ago … an engineer, Herbert Spencer, was willing to expound every aspect of life, with an effect on his admiring readers which has not worn off today.
Things do not happen quite in this way nowadays. This, we are told, is an age of specialists. The pursuit of knowledge has become a profession. The time when a man could master several sciences is past. He must now, they say, put all his efforts into one subject. And presumably, he must get all his ideas from this one subject. The world, to be sure, needs men who will follow such a rule with enthusiasm. It needs the greatest numbers of the ablest technicians. But apart from them it also needs men who will converse and think and even work in more than one science and know how to combine or connect them. Such men, I believe, are still to be found today. They are still as glad to exchange ideas as they have been in the past. But we cannot say that our way of life is well-fitted to help them. Why is this?

A multitude of words doth rather obscure than illustrate, they being a burden to the memory, and the first apt to be forgotten, before we come to the last. So that he that uses many words for the explaining of any subject, doth, like the cuttle-fish, hide himself, for the most part, in his own ink.

A patient pursuit of facts, and cautious combination and comparison of them, is the drudgery to which man is subjected by his Maker, if he wishes to attain sure knowledge.

A person by study must try to disengage the subject from useless matter, and to seize on points capable of improvement. ... When subjects are viewed through the mists of prejudice, useful truths may escape.

A physician’s subject of study is necessarily the patient, and his first field for observation is the hospital. But if clinical observation teaches him to know the form and course of diseases, it cannot suffice to make him understand their nature; to this end he must penetrate into the body to find which of the internal parts are injured in their functions. That is why dissection of cadavers and microscopic study of diseases were soon added to clinical observation. But to-day these various methods no longer suffice; we must push investigation further and, in analyzing the elementary phenomena of organic bodies, must compare normal with abnormal states. We showed elsewhere how incapable is anatomy alone to take account of vital phenenoma, and we saw that we must add study of all physico-chemical conditions which contribute necessary elements to normal or pathological manifestations of life. This simple suggestion already makes us feel that the laboratory of a physiologist-physician must be the most complicated of all laboratories, because he has to experiment with phenomena of life which are the most complex of all natural phenomena.

A professor is one who can speak on any subject—for precisely fifty minutes.

A professor … may be to produce a perfect mathematical work of art, having every axiom stated, every conclusion drawn with flawless logic, the whole syllabus covered. This sounds excellent, but in practice the result is often that the class does not have the faintest idea of what is going on. … The framework is lacking; students do not know where the subject fits in, and this has a paralyzing effect on the mind.

A rock or stone is not a subject that, of itself, may interest a philosopher to study; but, when he comes to see the necessity of those hard bodies, in the constitution of this earth, or for the permanency of the land on which we dwell, and when he finds that there are means wisely provided for the renovation of this necessary decaying part, as well as that of every other, he then, with pleasure, contemplates this manifestation of design, and thus connects the mineral system of this earth with that by which the heavenly bodies are made to move perpetually in their orbits.

A scientist may exhaust himself; he frequently exhausts his colleagues, always exhausts his money, but never exhausts his subject.

A superficial knowledge of mathematics may lead to the belief that this subject can be taught incidentally, and that exercises akin to counting the petals of flowers or the legs of a grasshopper are mathematical. Such work ignores the fundamental idea out of which quantitative reasoning grows—the equality of magnitudes. It leaves the pupil unaware of that relativity which is the essence of mathematical science. Numerical statements are frequently required in the study of natural history, but to repeat these as a drill upon numbers will scarcely lend charm to these studies, and certainly will not result in mathematical knowledge.

A superficial knowledge of mathematics may lead to the belief that this subject can be taught incidentally, and that exercises akin to counting the petals of flowers or the legs of a grasshopper are mathematical. Such work ignores the fundamental idea out of which quantitative reasoning grows—the equality of magnitudes. It leaves the pupil unaware of that relativity which is the essence of mathematical science. Numerical statements are frequently required in the study of natural history, but to repeat these as a drill upon numbers will scarcely lend charm to these studies, and certainly will not result in mathematical knowledge.

Accordingly, we find Euler and D'Alembert devoting their talent and their patience to the establishment of the laws of rotation of the solid bodies. Lagrange has incorporated his own analysis of the problem with his general treatment of mechanics, and since his time M. Poinsôt has brought the subject under the power of a more searching analysis than that of the calculus, in which ideas take the place of symbols, and intelligent propositions supersede equations.

After five years' work I allowed myself to speculate on the subject, and drew up some short notes; these I enlarged in 1844 into a sketch of the conclusions, which then seemed to me probable: from that period to the present day I have steadily pursued the same object. I hope that I may be excused for entering on these personal details, as I give them to show that I have not been hasty in coming to a decision.

All scientists must focus closely on limited targets. Whether or not one’s findings on a limited subject will have wide applicability depends to some extent on chance, but biologists of superior ability repeatedly focus on questions the answers to which either have wide ramifications or lead to new areas of investigation. One procedure that can be effective is to attempt both reduction and synthesis; that is, direct a question at a phenomenon on one integrative level, identify its mechanism at a simpler level, then extrapolate its consequences to a more complex level of integration.

All that can be said upon the number and nature of elements is, in my opinion, confined to discussions entirely of a metaphysical nature. The subject only furnishes us with indefinite problems, which may be solved in a thousand different ways, not one of which, in all probability, is consistent with nature. I shall therefore only add upon this subject, that if, by the term elements, we mean to express those simple and indivisible atoms of which matter is composed, it is extremely probable we know nothing at all about them; but, if we apply the term elements, or principles of bodies, to express our idea of the last point which analysis is capable of reaching, we must admit, as elements, all the substances into which we are capable, by any means, to reduce bodies by decomposition.

All true science must aim at objective truth, and that means that the human observer must never allow himself to get emotionally mixed up with his subject-matter. His concern is to understand the universe, not to improve it. Detachment is obligatory.

Although the way ahead [for immunology] is full of pitfalls and difficulties, this is indeed an exhilarating prospect. There is no danger of a shortage of forthcoming excitement in the subject. Yet, as always, the highlights of tomorrow are the unpredictabilities of today.

Amid all the revolutions of the globe, the economy of Nature has been uniform, ... and her laws are the only things that have resisted the general movement. The rivers and the rocks, the seas and the continents, have been changed in all their parts; but the laws which direct those changes, and the rules to which they are subject, have remained invariably the same.

An engineer, a physicist and a mathematician find themselves in an anecdote, indeed an anecdote quite similar to many that you have no doubt already heard.
After some observations and rough calculations the engineer realizes the situation and starts laughing.
A few minutes later the physicist understands too and chuckles to himself happily, as he now has enough experimental evidence to publish a paper.
This leaves the mathematician somewhat perplexed, as he had observed right away that he was the subject of an anecdote, and deduced quite rapidly the presence of humor from similar anecdotes, but considers this anecdote to be too trivial a corollary to be significant, let alone funny.

An enthusiastic philosopher, of whose name we are not informed, had constructed a very satisfactory theory on some subject or other, and was not a little proud of it. “But the facts, my dear fellow,” said his friend, “the facts do not agree with your theory.”—“Don't they?” replied the philosopher, shrugging his shoulders, “then, tant pis pour les faits;”—so much the worse for the facts!

And I believe there are many Species in Nature, which were never yet taken notice of by Man, and consequently of no use to him, which yet we are not to think were created in vain; but it’s likely … to partake of the overflowing Goodness of the Creator, and enjoy their own Beings. But though in this sense it be not true, that all things were made for Man; yet thus far it is, that all the Creatures in the World may be some way or other useful to us, at least to exercise our Wits and Understandings, in considering and contemplating of them, and so afford us Subject of Admiring and Glorifying their and our Maker. Seeing them, we do believe and assert that all things were in some sense made for us, we are thereby obliged to make use of them for those purposes for which they serve us, else we frustrate this End of their Creation.

And new philosophy calls all in doubt,
The Element of fire is quite put out;
The Sun is lost, and th’earth, and no mans wit
Can well direct him where to look for it.
And freely men confesse that this world’s spent,
When in the Planets, and the Firmament
They seeke so many new; and then see that this
Is crumbled out againe to his Atomies.
’Tis all in pieces, all cohaerence gone;
All just supply, and all Relation;
Prince, Subject, Father, Sonne, are things forgot,
For every man alone thinkes he hath got
To be a phoenix, and that then can bee
None of that kinde, of which he is, but hee.

Another diversity of Methods is according to the subject or matter which is handled; for there is a great difference in delivery of the Mathematics, which are the most abstracted of knowledges, and Policy, which is the most immersed ... , yet we see how that opinion, besides the weakness of it, hath been of ill desert towards learning, as that which taketh the way to reduce learning to certain empty and barren generalities; being but the very husks and shells of sciences, all the kernel being forced out and expulsed with the torture and press of the method.

Any chemist reading this book can see, in some detail, how I have spent most of my mature life. They can become familiar with the quality of my mind and imagination. They can make judgements about my research abilities. They can tell how well I have documented my claims of experimental results. Any scientist can redo my experiments to see if they still work—and this has happened! I know of no other field in which contributions to world culture are so clearly on exhibit, so cumulative, and so subject to verification.

Archimedes possessed so high a spirit, so profound a soul, and such treasures of highly scientific knowledge, that though these inventions [used to defend Syracuse against the Romans] had now obtained him the renown of more than human sagacity, he yet would not deign to leave behind him any commentary or writing on such subjects; but, repudiating as sordid and ignoble the whole trade of engineering, and every sort of art that lends itself to mere use and profit, he placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life; studies, the superiority of which to all others is unquestioned, and in which the only doubt can be whether the beauty and grandeur of the subjects examined, or the precision and cogency of the methods and means of proof, most deserve our admiration.

Art and religion first; then philosophy; lastly science. That is the order of the great subjects of life, that’s their order of importance.

Art arises in those strange complexities of action that are called human beings. It is a kind of human behavior. As such it is not magic, except as human beings are magical. Nor is it concerned in absolutes, eternities, “forms,” beyond those that may reside in the context of the human being and be subject to his vicissitudes. Art is not an inner state of consciousness, whatever that may mean. Neither is it essentially a supreme form of communication. Art is human behavior, and its values are contained in human behavior.

Art is an expression of the world order and is, therefore, orderly, organic, subject to mathematical law, and susceptible to mathematical analysis.

Art, science, discovery and invention, startle and bewilder us at every turn, by their rapid, vast and wonderful achievements. These forces have made men lords where they were vassals, masters where they were slaves, and kings where they were subjects. They have abolished the limitations of time and space and have brought the ends of the earth together.

As few subjects are more interesting to society, so few have been more frequently written upon than the education of youth.

As for me ... I would much rather be a perfected ape than a degraded Adam. Yes, if it is shown to me that my humble ancestors were quadrupedal animals, arboreal herbivores, brothers or cousins of those who were also the ancestors of monkeys and apes, far from blushing in shame for my species because of its genealogy and parentage, I will be proud of all that evolution has accomplished, of the continuous improvement which takes us up to the highest order, of the successive triumphs that have made us superior to all of the other species ... the splendid work of progress.
I will conclude in saying: the fixity of species is almost impossible, it contradicts the mode of succession and of the distribution of species in the sequence of extant and extinct creatures. It is therefore extremely likely that species are variable and are subject to evolution. But the causes, the mechanisms of this evolution are still unknown.

As science is more and more subject to grave misuse as well as to use for human benefit it has also become the scientist's responsibility to become aware of the social relations and applications of his subject, and to exert his influence in such a direction as will result in the best applications of the findings in his own and related fields. Thus he must help in educating the public, in the broad sense, and this means first educating himself, not only in science but in regard to the great issues confronting mankind today.

As to Science, she has never sought to ally herself to civil power. She has never attempted to throw odium or inflict social ruin on any human being. She has never subjected anyone to mental torment, physical torture, least of all to death, for the purpose of upholding or promoting her ideas. She presents herself unstained by cruelties and crimes. But in the Vatican—we have only to recall the Inquisition—the hands that are now raised in appeals to the Most Merciful are crimsoned. They have been steeped in blood!

At no period of [Michael Faraday’s] unmatched career was he interested in utility. He was absorbed in disentangling the riddles of the universe, at first chemical riddles, in later periods, physical riddles. As far as he cared, the question of utility was never raised. Any suspicion of utility would have restricted his restless curiosity. In the end, utility resulted, but it was never a criterion to which his ceaseless experimentation could be subjected.

At present good work in science pays less well very often than mediocrity in other subjects. This, as was pointed out by Sir Lyon Playfair in his Presidential Address to the British Association in 1885 helps to arrest progress in science teaching.

At that point, my sense of dissatisfaction was so strong that I firmly resolved to start thinking until I should find a purely arithmetic and absolutely rigorous foundation of the principles of infinitesimal analysis. … I achieved this goal on November 24th, 1858, … but I could not really decide upon a proper publication, because, firstly, the subject is not easy to present, and, secondly, the material is not very fruitful.

Bacteria are highly adaptable. They frequently change both morphologically and functionally. Their virulence is also an essentially fluctuating property, that increases or diminishes according to the conditions to which the pathogenic organism is subjected.

Beasts have not the high advantages which we possess; but they have some which we have not. They have not our hopes, but then they have not our fears; they are subject like us to death, but it is without being aware of it; most of them are better able to preserve themselves than we are, and make a less bad use of their passions.

Before delivering your lectures, the manuscript should be in such a perfect form that, if need be, it could be set in type. Whether you follow the manuscript during the delivery of the lecture is purely incidental. The essential point is that you are thus master of the subject matter.

Before I came here I was confused about this subject. Having listened to your lecture I am still confused. But on a higher level.

Bertrand Russell had given a talk on the then new quantum mechanics, of whose wonders he was most appreciative. He spoke hard and earnestly in the New Lecture Hall. And when he was done, Professor Whitehead, who presided, thanked him for his efforts, and not least for “leaving the vast darkness of the subject unobscured.”

Bertrand, Darboux, and Glaisher have compared Cayley to Euler, alike for his range, his analytical power, and, not least, for his prolific production of new views and fertile theories. There is hardly a subject in the whole of pure mathematics at which he has not worked.

Biology can be divided into the study of proximate causes, the study of the physiological sciences (broadly conceived), and into the study of ultimate (evolutionary) causes, the subject of natural history.

Biology cannot go far in its subject without being met by mind.

But as no two (theoreticians) agree on this (skin friction) or any other subject, some not agreeing today with what they wrote a year ago, I think we might put down all their results, add them together, and then divide by the number of mathematicians, and thus find the average coefficient of error. (1908)

But for the persistence of a student of this university in urging upon me his desire to study with me the modern algebra I should never have been led into this investigation; and the new facts and principles which I have discovered in regard to it (important facts, I believe), would, so far as I am concerned, have remained still hidden in the womb of time. In vain I represented to this inquisitive student that he would do better to take up some other subject lying less off the beaten track of study, such as the higher parts of the calculus or elliptic functions, or the theory of substitutions, or I wot not what besides. He stuck with perfect respectfulness, but with invincible pertinacity, to his point. He would have the new algebra (Heaven knows where he had heard about it, for it is almost unknown in this continent), that or nothing. I was obliged to yield, and what was the consequence? In trying to throw light upon an obscure explanation in our text-book, my brain took fire, I plunged with re-quickened zeal into a subject which I had for years abandoned, and found food for thoughts which have engaged my attention for a considerable time past, and will probably occupy all my powers of contemplation advantageously for several months to come.

But nothing of a nature foreign to the duties of my profession [clergyman] engaged my attention while I was at Leeds so much as the, prosecution of my experiments relating to electricity, and especially the doctrine of air. The last I was led into a consequence of inhabiting a house adjoining to a public brewery, where first amused myself with making experiments on fixed air [carbon dioxide] which found ready made in the process of fermentation. When I removed from that house, I was under the necessity making the fixed air for myself; and one experiment leading to another, as I have distinctly and faithfully noted in my various publications on the subject, I by degrees contrived a convenient apparatus for the purpose, but of the cheapest kind. When I began these experiments I knew very little of chemistry, and had in a manner no idea on the subject before I attended a course of chymical lectures delivered in the Academy at Warrington by Dr. Turner of Liverpool. But I have often thought that upon the whole, this circumstance was no disadvantage to me; as in this situation I was led to devise an apparatus and processes of my own, adapted to my peculiar views. Whereas, if I had been previously accustomed to the usual chemical processes, I should not have so easily thought of any other; and without new modes of operation I should hardly have discovered anything materially new.

But the fact is that when wine is taken in moderation, it gives rise to a large amount of breath, whose character is balanced, and whose luminosity is strong and brilliant. Hence wine disposes greatly to gladness, and the person is subject to quite trivial exciting agents. The breath now takes up the impression of agents belonging to the present time more easily than it does those which relate to the future; it responds to agents conducive to delight rather than those conducive to a sense of beauty.

By science, then, I understand the consideration of all subjects, whether of a pure or mixed nature, capable of being reduced to measurement and calculation. All things comprehended under the categories of space, time and number properly belong to our investigations; and all phenomena capable of being brought under the semblance of a law are legitimate objects of our inquiries.

Calculation touches, at most, certain phenomena of organic destruction. Organic creation, on the contrary, the evolutionary phenomena which properly constitute life, we cannot in any way subject to a mathematical treatment.

Can we separate object and subject? Myself is nothing but a part of my body, my body is nothing but a part of my food, my food is nothing but a part of the earth, the earth is nothing but a part of the solar system.

Casting off the dark fog of verbal philosophy and vulgar medicine, which inculcate names alone ... I tried a series of experiments to explain more clearly many phenomena, particularly those of physiology. In order that I might subject as far as possible the reasonings of the Galenists and Peripatetics to sensory criteria, I began, after trying experiments, to write dialogues in which a Galenist adduced the better-known and stronger reasons and arguments; these a mechanist surgeon refuted by citing to the contrary the experiments I had tried, and a third, neutral interlocutor weighed the reasons advanced by both and provided an opportunity for further progress.

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.

Changes That Have Occurred in the Globe: When we have seen with our own eyes a mountain progressing into a plain; that is to say, an immense boulder separating from this mountain and covering the fields; an entire castle broken into pieces over the ground; a river swallowed up which then bursts out from its abyss; clear marks of a vast amount of water having once flooded regions now inhabited, and a hundred vestiges of other transformations, then we are much more willing to believe that great changes altered the face of the earth, than a Parisian lady who knows only that the place where her house was built was once a cultivated field. However, a lady from Naples who has seen the buried ruins of Herculaneum, is much less subject to the bias which leads us to believe that everything has always been as it is today.

Chemistry is yet, indeed, a mere embryon. Its principles are contested; experiments seem contradictory; their subjects are so minute as to escape our senses; and their result too fallacious to satisfy the mind. It is probably an age too soon to propose the establishment of a system.

Chlorine is a deadly poison gas employed on European battlefields in World War I. Sodium is a corrosive metal which burns upon contact with water. Together they make a placid and unpoisonous material, table salt. Why each of these substances has the properties it does is a subject called chemistry.

Civilized people can talk about anything. For them no subject is taboo…. In civilized societies there will be no intellectual bogeys at sight of which great grown-up babies are expected to hide their eyes

Coming to the question of life being found on other planets, Professor Haldane apologized for discoursing, as a mere biologist, on a subject on which we had been expecting a lecture by a physicist [J. D. Bernal]. He mentioned three hypotheses:
(a) That life had a supernatural origin,
(b) That it originated from inorganic materials, and (c) That life is a constituent of the Universe and can only arise from pre-existing life. The first hypothesis, he said, should be taken seriously, and he would proceed to do so. From the fact that there are 400,000 species of beetle on this planet, but only 8,000 species of mammals, he concluded that the Creator, if he exists, has a special preference for beetles, and so we might be more likely to meet them than any other type of animal on a planet which would support life.

Communication of science as subject-matter has so far outrun in education the construction of a scientific habit of mind that to some extent the natural common sense of mankind has been interfered with to its detriment.

Concerning the gods, I have no means of knowing either that they exist or that they do not exist, nor what sort of form they may have; there are many reasons why knowledge on this subject is not possible, owing to the lack of evidence and the shortness of human life.

Cosmology, for centuries consisting of speculation based on a minimum of observational evidence and a maximum of philosophical predilection, became in the twentieth century an observational science, its theories now subject to verification or refutation to a degree previously unimaginable.

Cuvier had even in his address & manner the character of a superior Man, much general power & eloquence in conversation & great variety of information on scientific as well as popular subjects. I should say of him that he is the most distinguished man of talents I have ever known on the continent: but I doubt if He be entitled to the appellation of a Man of Genius.

Dead is when the chemists take over the subject.

Deaths, births, and marriages, considering how much they are separately dependent on the freedom of the human will, should seem to be subject to no law according to which any calculation could be made beforehand of their amount; and yet the yearly registers of these events in great countries prove that they go on with as much conformity to the laws of nature as the oscillations of the weather.

Definition of Mathematics.—It has now become apparent that the traditional field of mathematics in the province of discrete and continuous number can only be separated from the general abstract theory of classes and relations by a wavering and indeterminate line. Of course a discussion as to the mere application of a word easily degenerates into the most fruitless logomachy. It is open to any one to use any word in any sense. But on the assumption that “mathematics” is to denote a science well marked out by its subject matter and its methods from other topics of thought, and that at least it is to include all topics habitually assigned to it, there is now no option but to employ “mathematics” in the general sense of the “science concerned with the logical deduction of consequences from the general premisses of all reasoning.”

Descartes' immortal conclusion cogito ergo sum was recently subjected to destruction testing by a group of graduate researchers at Princeton led by Professors Montjuic and Lauterbrunnen, and now reads, in the Shorter Harvard Orthodoxy:
(a) I think, therefore I am; or
(b) Perhaps I thought, therefore I was; but
(c) These days, I tend to leave that side of things to my wife.

Despite rapid progress in the right direction, the program of the average elementary school has been primarily devoted to teaching the fundamental subjects, the three R’s, and closely related disciplines… Artificial exercises, like drills on phonetics, multiplication tables, and formal writing movements, are used to a wasteful degree. Subjects such as arithmetic, language, and history include content that is intrinsically of little value. Nearly every subject is enlarged unwisely to satisfy the academic ideal of thoroughness… Elimination of the unessential by scientific study, then, is one step in improving the curriculum.

Does it not seem as if Algebra had attained to the dignity of a fine art, in which the workman has a free hand to develop his conceptions, as in a musical theme or a subject for a painting? It has reached a point where every properly developed algebraical composition, like a skillful landscape, is expected to suggest the notion of an infinite distance lying beyond the limits of the canvas.

Einstein, twenty-six years old, only three years away from crude privation, still a patent examiner, published in the Annalen der Physik in 1905 five papers on entirely different subjects. Three of them were among the greatest in the history of physics. One, very simple, gave the quantum explanation of the photoelectric effect—it was this work for which, sixteen years later, he was awarded the Nobel prize. Another dealt with the phenomenon of Brownian motion, the apparently erratic movement of tiny particles suspended in a liquid: Einstein showed that these movements satisfied a clear statistical law. This was like a conjuring trick, easy when explained: before it, decent scientists could still doubt the concrete existence of atoms and molecules: this paper was as near to a direct proof of their concreteness as a theoretician could give. The third paper was the special theory of relativity, which quietly amalgamated space, time, and matter into one fundamental unity. This last paper contains no references and quotes no authority. All of them are written in a style unlike any other theoretical physicist’s. They contain very little mathematics. There is a good deal of verbal commentary. The conclusions, the bizarre conclusions, emerge as though with the greatest of ease: the reasoning is unbreakable. It looks as though he had reached the conclusions by pure thought, unaided, without listening to the opinions of others. To a surprisingly large extent, that is precisely what he had done.

Ethnologists regard man as the primitive element of tribes, races, and peoples. The anthropologist looks at him as a member of the fauna of the globe, belonging to a zoölogical classification, and subject to the same laws as the rest of the animal kingdom. To study him from the last point of view only would be to lose sight of some of his most interesting and practical relations; but to be confined to the ethnologist’s views is to set aside the scientific rule which requires us to proceed from the simple to the compound, from the known to the unknown, from the material and organic fact to the functional phenomenon.

Every discipline must be honored for reason other than its utility, otherwise it yields no enthusiasm for industry.
For both reasons, I consider mathematics the chief subject for the common school. No more highly honored exercise for the mind can be found; the buoyancy [Spannkraft] which it produces is even greater than that produced by the ancient languages, while its utility is unquestioned.

Every subject in Davy’s mind has the principle of Vitality. Living thoughts spring up like Turf under his feet.

Everything material which is the subject of knowledge has number, order, or position; and these are her first outlines for a sketch of the universe. If our feeble hands cannot follow out the details, still her part has been drawn with an unerring pen, and her work cannot be gainsaid. So wide is the range of mathematical sciences, so indefinitely may it extend beyond our actual powers of manipulation that at some moments we are inclined to fall down with even more than reverence before her majestic presence. But so strictly limited are her promises and powers, about so much that we might wish to know does she offer no information whatever, that at other moments we are fain to call her results but a vain thing, and to reject them as a stone where we had asked for bread. If one aspect of the subject encourages our hopes, so does the other tend to chasten our desires, and he is perhaps the wisest, and in the long run the happiest, among his fellows, who has learned not only this science, but also the larger lesson which it directly teaches, namely, to temper our aspirations to that which is possible, to moderate our desires to that which is attainable, to restrict our hopes to that of which accomplishment, if not immediately practicable, is at least distinctly within the range of conception.

Evolutionary plasticity can be purchased only at the ruthlessly dear price of continuously sacrificing some individuals to death from unfavourable mutations. Bemoaning this imperfection of nature has, however, no place in a scientific treatment of this subject.

Evolutionists sometimes take as haughty an attitude toward the next level up the conventional ladder of disciplines: the human sciences. They decry the supposed atheoretical particularism of their anthropological colleagues and argue that all would be well if only the students of humanity regarded their subject as yet another animal and therefore yielded explanatory control to evolutionary biologists.

Exobiology—a curious development in view of the fact that this “science” has yet to demonstrate that its subject matter exists!

Experience is never at fault; it is only your judgment that is in error in promising itself such results from experience as are not caused by our experiments. For having given a beginning, what follows from it must necessarily be a natural development of such a beginning, unless it has been subject to a contrary influence, while, if it is affected by any contrary influence, the result which ought to follow from the aforesaid beginning will be found to partake of this contrary influence in a greater or less degree in proportion as the said influence is more or less powerful than the aforesaid beginning.

Extremely hazardous is the desire to explain everything, and to supply whatever appears a gap in history—for in this propensity lies the first cause and germ of all those violent and arbitrary hypotheses which perplex and pervert the science of history far more than the open avowal of our ignorance, or the uncertainty of our knowledge: hypotheses which give an oblique direction, or an exaggerated and false extension, to a view of the subject originally not incorrect.

Faraday, … by his untiring faithfulness in keeping his diary, contributes to our understanding the objects of his scientific research in magnetism, electricity and light, but he also makes us understand the scientist himself, as a living subject, the mind in action.

First, as concerns the success of teaching mathematics. No instruction in the high schools is as difficult as that of mathematics, since the large majority of students are at first decidedly disinclined to be harnessed into the rigid framework of logical conclusions. The interest of young people is won much more easily, if sense-objects are made the starting point and the transition to abstract formulation is brought about gradually. For this reason it is psychologically quite correct to follow this course.
Not less to be recommended is this course if we inquire into the essential purpose of mathematical instruction. Formerly it was too exclusively held that this purpose is to sharpen the understanding. Surely another important end is to implant in the student the conviction that correct thinking based on true premises secures mastery over the outer world. To accomplish this the outer world must receive its share of attention from the very beginning.
Doubtless this is true but there is a danger which needs pointing out. It is as in the case of language teaching where the modern tendency is to secure in addition to grammar also an understanding of the authors. The danger lies in grammar being completely set aside leaving the subject without its indispensable solid basis. Just so in Teaching of Mathematics it is possible to accumulate interesting applications to such an extent as to stunt the essential logical development. This should in no wise be permitted, for thus the kernel of the whole matter is lost. Therefore: We do want throughout a quickening of mathematical instruction by the introduction of applications, but we do not want that the pendulum, which in former decades may have inclined too much toward the abstract side, should now swing to the other extreme; we would rather pursue the proper middle course.

For between true Science, and erroneous Doctrines, Ignorance is in the middle. Naturall sense and imagination, are not subject to absurdity. Nature it selfe cannot erre: and as men abound in copiousnesses of language; so they become more wise, or more mad than ordinary. Nor is it possible without Letters for any man to become either excellently wise, or (unless his memory be hurt by disease, or ill constitution of organs) excellently foolish. For words are wise men's counters, they do but reckon by them; but they are the money of fools that value them by the authority of an Aristotle, a Cicero, or a Thomas, or any other Doctor whatsoever, if but a man.

For ourselves, we may take as a basic assumption, clear from a survey of particular cases, that natural things are some or all of them subject to change.

For, Mathematical Demonstrations being built upon the impregnable Foundations of Geometry and Arithmetick, are the only Truths, that can sink into the Mind of Man, void of all Uncertainty; and all other Discourses participate more or less of Truth, according as their Subjects are more or less capable of Mathematical Demonstration.

Fourier’s Theorem … is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics. To mention only sonorous vibrations, the propagation of electric signals along a telegraph wire, and the conduction of heat by the earth’s crust, as subjects in their generality intractable without it, is to give but a feeble idea of its importance.

Freudian psychoanalytical theory is a mythology that answers pretty well to Levi-Strauss's descriptions. It brings some kind of order into incoherence; it, too, hangs together, makes sense, leaves no loose ends, and is never (but never) at a loss for explanation. In a state of bewilderment it may therefore bring comfort and relief … give its subject a new and deeper understanding of his own condition and of the nature of his relationship to his fellow men. A mythical structure will be built up around him which makes sense and is believable-in, regardless of whether or not it is true.

From the age of 13, I was attracted to physics and mathematics. My interest in these subjects derived mostly from popular science books that I read avidly. Early on I was fascinated by theoretical physics and determined to become a theoretical physicist. I had no real idea what that meant, but it seemed incredibly exciting to spend one's life attempting to find the secrets of the universe by using one's mind.

From the infinitely great down to the infinitely small, all things are subject to [the laws of nature]. The sun and the planets follow the laws discovered by Newton and Laplace, just as the atoms in their combinations follow the laws of chemistry, as living creatures follow the laws of biology. It is only the imperfections of the human mind which multiply the divisions of the sciences, separating astronomy from physics or chemistry, the natural sciences from the social sciences. In essence, science is one. It is none other than the truth.

General preparatory instruction must continue to be the aim in the instruction at the higher institutions of learning. Exclusive selection and treatment of subject matter with reference to specific avocations is disadvantageous.

Genetics was, I would say, the first part of biology to become a pretty good theoretical subject, based on the theory of the gene and patterns of inheritance of characteristics.

Geology has its peculiar difficulties, from which all other sciences are exempt. Questions in chemistry may be settled in the laboratory by experiment. Mathematical and philosophical questions may be discussed, while the materials for discussion are ready furnished by our own intellectual reflections. Plants, animals and minerals, may be arranged in the museum, and all questions relating to their intrinsic principles may be discussed with facility. But the relative positions, the shades of difference, the peculiar complexions, whether continuous or in subordinate beds, are subjects of enquiry in settling the character of rocks, which can be judged of while they are in situ only.

Good scholars struggle to understand the world in an integral way (pedants bite off tiny bits and worry them to death). These visions of reality ... demand our respect, for they are an intellectual’s only birthright. They are often entirely wrong and always flawed in serious ways, but they must be understood honorably and not subjected to mayhem by the excision of patches.

Good work is no done by “humble” men. It is one of the first duties of a professor, for example, in any subject, to exaggerate a little both the importance of his subject and his own importance in it. A man who is always asking “Is what I do worth while?” and “Am I the right person to do it?” will always be ineffective himself and a discouragement to others. He must shut his eyes a little and think a little more of his subject and himself than they deserve. This is not too difficult: it is harder not to make his subject and himself ridiculous by shutting his eyes too tightly.

He is a learned man that understands one subject, a very learned man that understands two.

He [Samuel Johnson] bid me always remember this, that after a system is well settled upon positive evidence, a few objections ought not to shake it. “The human mind is so limited that it cannot take in all parts of a subject; so that there may be objections raised against anything. There are objections against a plenum, and objections against a vacuum. Yet one of them must certainly be true.”

Here the most sublime scene ever witnessed in the operating room was presented when the patient placed himself voluntarily upon the table, which was to become the altar of future fame. … The heroic bravery of the man who voluntarily placed himself upon the table, a subject for the surgeon’s knife, should be recorded and his name enrolled upon parchment, which should be hung upon the walls of the surgical amphitheatre in which the operation was performed. His name was Gilbert Abbott.
Description of the first public demonstration of ether at the Massachussetts General Hospital (16 Oct 1846).

His subject is the “Origin of Species,” & not the origin of Organization; & it seems a needless mischief to have opened the latter speculation at all.

Historically, Statistics is no more than State Arithmetic, a system of computation by which differences between individuals are eliminated by the taking of an average. It has been used—indeed, still is used—to enable rulers to know just how far they may safely go in picking the pockets of their subjects.

How can he [Thomas Edison] call it a wonderful success when everyone acquainted with the subject will recognize it as a conspicuous failure?

How can you shorten the subject? That stern struggle with the multiplication table, for many people not yet ended in victory, how can you make it less? Square root, as obdurate as a hardwood stump in a pasture nothing but years of effort can extract it. You can’t hurry the process. Or pass from arithmetic to algebra; you can’t shoulder your way past quadratic equations or ripple through the binomial theorem. Instead, the other way; your feet are impeded in the tangled growth, your pace slackens, you sink and fall somewhere near the binomial theorem with the calculus in sight on the horizon. So died, for each of us, still bravely fighting, our mathematical training; except for a set of people called “mathematicians”—born so, like crooks.

How could science be any enemy of religion when God commanded man to be a scientist the day He told him to rule the earth and subject it?

However, if we consider that all the characteristics which have been cited are only differences in degree of structure, may we not suppose that this special condition of organization of man has been gradually acquired at the close of a long period of time, with the aid of circumstances which have proved favorable? What a subject for reflection for those who have the courage to enter into it!

I accepted the Copernican position several years ago and discovered from thence the causes of many natural effects which are doubtless inexplicable by the current theories. I have written up many reasons and refutations on the subject, but I have not dared until now to bring them into the open, being warned by the fortunes of Copernicus himself, our master, who procured for himself immortal fame among a few but stepped down among the great crowd (for this is how foolish people are to be numbered), only to be derided and dishonoured. I would dare publish my thoughts if there were many like you; but since there are not, I shall forbear.

I admitted, that the world had existed millions of years. I am astonished at the ignorance of the masses on these subjects. Hugh Miller has it right when he says that 'the battle of evidences must now be fought on the field of the natural sciences.'

I am a strong advocate for free thought on all subjects, yet it appears to me (whether rightly or wrongly) that direct arguments against christianity & theism produce hardly any effect on the public; & freedom of thought is best promoted by the gradual illumination of men's minds, which follow[s] from the advance of science. It has, therefore, been always my object to avoid writing on religion, & I have confined myself to science. I may, however, have been unduly biassed by the pain which it would give some members of my family, if I aided in any way direct attacks on religion.

I am never content until I have constructed a mechanical model of the subject I am studying. If I succeed in making one, I understand. Otherwise, I do not. [Attributed; source unverified.]

I am not, personally, a believer or a religious man in any sense of institutional commitment or practice. But I have a great respect for religion, and the subject has always fascinated me, beyond almost all others (with a few exceptions, like evolution and paleontology).

I am of opinion, then, ... that, if there is any circumstance thoroughly established in geology, it is, that the crust of our globe has been subjected to a great and sudden revolution, the epoch of which cannot be dated much farther back than five or six thousand years ago; that this revolution had buried all the countries which were before inhabited by men and by the other animals that are now best known; that the same revolution had laid dry the bed of the last ocean, which now forms all the countries at present inhabited; that the small number of individuals of men and other animals that escaped from the effects of that great revolution, have since propagated and spread over the lands then newly laid dry; and consequently, that the human race has only resumed a progressive state of improvement since that epoch, by forming established societies, raising monuments, collecting natural facts, and constructing systems of science and of learning.

I am of the decided opinion, that mathematical instruction must have for its first aim a deep penetration and complete command of abstract mathematical theory together with a clear insight into the structure of the system, and doubt not that the instruction which accomplishes this is valuable and interesting even if it neglects practical applications. If the instruction sharpens the understanding, if it arouses the scientific interest, whether mathematical or philosophical, if finally it calls into life an esthetic feeling for the beauty of a scientific edifice, the instruction will take on an ethical value as well, provided that with the interest it awakens also the impulse toward scientific activity. I contend, therefore, that even without reference to its applications mathematics in the high schools has a value equal to that of the other subjects of instruction.

I am only a physicist with nothing material to show for my labours. I have never even seen the ionosphere, although I have worked on the subject for thirty years. That does show how lucky people can be. If there had been no ionosphere I would not have been standing here this morning.

I am particularly concerned to determine the probability of causes and results, as exhibited in events that occur in large numbers, and to investigate the laws according to which that probability approaches a limit in proportion to the repetition of events. That investigation deserves the attention of mathematicians because of the analysis required. It is primarily there that the approximation of formulas that are functions of large numbers has its most important applications. The investigation will benefit observers in identifying the mean to be chosen among the results of their observations and the probability of the errors still to be apprehended. Lastly, the investigation is one that deserves the attention of philosophers in showing how in the final analysis there is a regularity underlying the very things that seem to us to pertain entirely to chance, and in unveiling the hidden but constant causes on which that regularity depends. It is on the regularity of the main outcomes of events taken in large numbers that various institutions depend, such as annuities, tontines, and insurance policies. Questions about those subjects, as well as about inoculation with vaccine and decisions of electoral assemblies, present no further difficulty in the light of my theory. I limit myself here to resolving the most general of them, but the importance of these concerns in civil life, the moral considerations that complicate them, and the voluminous data that they presuppose require a separate work.