Agnostics

"The primary use of knowledge is for such guidance of conduct under all circumstances as shall make living complete. All other uses of knowledge are secondary."

"The Republican form of government is the highest form of government; but because of this it requires the highest type of human nature — a type nowhere at present existing."

"The ultimate result of shielding men from the effects of folly, is to fill the world with fools."

"Strong as it looks at the outset, State-agency perpetually disappoints every one. Puny as are its first stages, private efforts daily achieve results that astound the world."

"Unlike private enterprise which quickly modifies its actions to meet emergencies — unlike the shopkeeper who promptly finds the wherewith to satisfy a sudden demand — unlike the railway company which doubles its trains to carry a special influx of passengers; the law-made instrumentality lumbers on under all varieties of circumstances at its habitual rate. By its very nature it is fitted only for average requirements, and inevitably fails under unusual requirements."

"Under the natural course of things each citizen tends towards his fittest function. Those who are competent to the kind of work they undertake, succeed, and, in the average of cases, are advanced in proportion to their efficiency; while the incompetent, society soon finds out, ceases to employ, forces to try something easier, and eventually turns to use."

"The saying that beauty is but skin deep is but a skin-deep saying."

"Be it or be it not true that Man is shapen in iniquity and conceived in sin, it is unquestionably true that Government is begotten of aggression, and by aggression. In small undeveloped societies where for ages complete peace has continued, there exists nothing like what we call Government: no coercive agency, but mere honorary headship, if any headship at all. In these exceptional communities, unaggressive and from special causes unaggressed upon, there is so little deviation from the virtues of truthfulness, honesty, justice, and generosity, that nothing beyond an occasional expression of public opinion by informally-assembled elders is needful. Conversely, we find proofs that, at first recognized but temporarily during leadership in war, the authority of a chief is permanently established by continuity of war; and grows strong where successful aggression ends in subjection of neighboring tribes. And thence onwards, examples furnished by all races put beyond doubt the truth, that the coercive power of the chief, developing into king, and king of kings (a frequent title in the ancient East), becomes great in proportion as conquest becomes habitual and the union of subdued nations extensive. Comparisons disclose a further truth which should be ever present to us — the truth that the aggressiveness of the ruling power inside a society increases with its aggressiveness outside the society. As, to make an efficient army, the soldiers in their several grades must be subordinate to the commander; so, to make an efficient fighting community, must the citizens be subordinate to the ruling power. They must furnish recruits to the extent demanded, and yield up whatever property is required. An obvious implication is that the ethics of Government, originally identical with the ethics of war, must long remain akin to them; and can diverge from them only as warlike activities and preparations become less."

"What is essential to the idea of a slave? We primarily think of him as one who is owned by another. To be more than nominal, however, the ownership must be shown by control of the slave's actions — a control which is habitually for the benefit of the controller. That which fundamentally distinguishes the slave is that he labours under coercion to satisfy another's desires. The relation admits of sundry gradations. Remembering that originally the slave is a prisoner whose life is at the mercy of his captor, it suffices here to note that there is a harsh form of slavery in which, treated as an animal, he has to expend his entire effort for his owner's advantage. Under a system less harsh, though occupied chiefly in working for his owner, he is allowed a short time in which to work for himself, and some ground on which to grow extra food. A further amelioration gives him power to sell the produce of his plot and keep the proceeds. Then we come to the still more moderated form which commonly arises where, having been a free man working on his own land, conquest turns him into what we distinguish as a serf; and he has to give to his owner each year a fixed amount of labour or produce, or both: retaining the rest himself. Finally, in some cases, as in Russia before serfdom was abolished, he is allowed to leave his owner's estate and work or trade for himself elsewhere, under the condition that he shall pay an annual sum. What is it which, in these cases, leads us to qualify our conception of the slavery as more or less severe? Evidently the greater or smaller extent to which effort is compulsorily expended for the benefit of another instead of for self-benefit. If all the slave's labour is for his owner the slavery is heavy, and if but little it is light. Take now a further step. Suppose an owner dies, and his estate with its slaves comes into the hands of trustees; or suppose the estate and everything on it to be bought by a company; is the condition of the slave any the better if the amount of his compulsory labour remains the same? Suppose that for a company we substitute the community; does it make any difference to the slave if the time he has to work for others is as great, and the time left for himself is as small, as before? The essential question is—How much is he compelled to labour for other benefit than his own, and how much can he labour for his own benefit? The degree of his slavery varies according to the ratio between that which he is forced to yield up and that which he is allowed to retain; and it matters not whether his master is a single person or a society. If, without option, he has to labour for the society, and receives from the general stock such portion as the society awards him, he becomes a slave to the society."

"Influences of various kinds conspire to increase corporate action and decrease individual action. And the change is being on all sides aided by schemers, each of whom thinks only of his pet plan and not at all of the general reorganization which his plan, joined with others such, are working out. It is said that the French Revolution devoured its own children. Here, an analogous catastrophe seems not unlikely. The numerous socialistic changes made by Act of Parliament, joined with the numerous others presently to be made, will by-and-by be all merged in State-socialism—swallowed in the vast wave which they have little by little raised. "But why is this change described as 'the coming slavery'?," is a question which many will still ask. The reply is simple. All socialism involves slavery."

"We have unmistakable proof that throughout all past time, there has been a ceaseless devouring of the weak by the strong."

"The current opinion that science and poetry are opposed is a delusion. ... Think you that a drop of water, which to the vulgar eye is but a drop of water, loses any thing in the eye of the physicist who knows that its elements are held together by a force which, if suddenly liberated, would produce a flash of lightning? Think you that what is carelessly looked upon by the uninitiated as a mere snow-flake does not suggest higher associations to one who has seen through a microscope the wondrously varied and elegant forms of snow-crystals? Think you that the rounded rock marked with parallel scratches calls up as much poetry in an ignorant mind as in the mind of a geologist, who knows that over this rock a glacier slid a million years ago? The truth is, that those who have never entered upon scientific pursuits know not a tithe of the poetry by which they are surrounded."

"With a higher moral nature will come a restriction on the multiplication of the inferior."

"It cannot but happen that those individuals whose functions are most out of equilibrium with the modified aggregate of external forces, will be those to die; and that those will survive whose functions happen to be most nearly in equilibrium with the modified aggregate of external forces. But this survival of the fittest, implies multiplication of the fittest. Out of the fittest thus multiplied, there will, as before, be an overthrowing of the moving equilibrium wherever it presents the least opposing force to the new incident force."

"A self-respecting man is a man without a country. A fatherland is birdlime..."

"Hilbert was on the verge of retirement. He was the dignified chairman of the mathematical society's meetings, though he no longer came up with those caustic quips that people would repeat afterwards, imitating as best they could his Baltic accent. It is a pity these were not recorded before it was too late. The samples cited in English translation, in Constance Reid's biography of Hilbert, give only the palest idea of his biting wit."

"Hilbert's problems have the characteristics of any good founding document. Each one is a short essay on its subject, not overly specific, and yet Hilbert makes his intent remarkably clear. He leaves room for change and adjustment. Hilbert's goal was to foster the pursuit of mathematics."

"A more thorough study of Euclid's axioms and postulates proved them to be inadequate for the deduction of Euclid's geometry. ...Hilbert and others succeeded in filling the gap by stating explicitly a complete system of postulates for Euclidean and non-Euclidean geometries alike. Among the postulates missing in Euclid's list was the celebrated postulate of Archimedes, according to which, by placing an indefinite number of equal lengths end to end along a line, we should eventually pass any point arbitrarily selected on the line. Hilbert, by denying this postulate, just as Lobatchewski and Riemann had denied Euclid's parallel postulate, succeeded in constructing a new geometry known as non-Archimedean. It was perfectly consistent but much stranger than the classical non-Euclidean varieties. Likewise, it was proved possible to posit a system of postulates which would yield Euclidean or non-Euclidean geometries of any number of dimensions; hence, so far as rational requirements of the mind were concerned, there was no reason to limit geometry to three dimensions."

"David Hilbert—the undisputed, foremost living mathematician in the world and lifelong close friend and collaborator of the by then deceased Minkowski—had already presented to the Göttingen Academy his own version of the same equations a few days earlier [than Einstein]. Although Minkowski and Hilbert accomplished their most important achievements in pure mathematical fields, their respective contributions to relativity should in no sense be seen as merely occasional excursions into the field of theoretical physics. Minkowski and Hilbert were motivated by much more than a desire to apply their exceptional mathematical abilities opportunistically... On the contrary, Minkowski's and Hilbert's contributions to relativity are best understood as an organic part of their overall scientific careers."

"More decisive than any other influence for the young Hilbert at Königsberg was his friendship with Adolf Hurwitz and Minkowski. He got his thorough mathematical training less from lectures, teachers or books, than from conversation."

"Physics is too difficult for physicists!"

"The various branches of geometry are all interrelated closely and quite often unexpectedly. This shows up in many places in the book. Even so... it was necessary to make each chapter...self-contained... We hope that... we have rendered each chapter taken by itself... understandable and interesting. We want to take the reader on a leisurely walk... in the big garden that is geometry, so that each may pick for himself a bouquet to his liking."

"[M]athematics is not a popular subject... The reason for this is to be found in the common superstition that [it] is but a continuation... of the fine art of arithmetic, of juggling with numbers. [We] combat that superstition, by offering, instead of formulas, figures that may be looked at and that may easily be supplemented by models which the reader may construct. This book... bring[s] about a greater enjoyment of mathematics, by making it easier... to penetrate the essence of mathematics without... a laborious course of studies."

"[O]ur purpose is to give a presentation of geometry... in its visual, intuitive aspects. With the aid of visual imagination we can illuminate the manifold facts and problems... beyond this, it is possible... to depict the geometric outline of the methods of investigation and proof, without... entering into the details... In this manner, geometry being as many-faceted as it is and being related to the most diverse branches of mathematics, we may even obtain a summarizing survey of mathematics as a whole, and a valid idea of the variety of problems and the wealth of ideas it contains. Thus a presentation of geometry in large brushstrokes... and based on the approach through visual intuition, should contribute to a more just appreciation of mathematics by a wider range of people than just the specialists."

"In mathematics, as in any scientific research, we find two tendencies... [T]he tendency toward abstraction seeks to crystallize the logical relations inherent in the maze of material in a systematic and orderly manner. On the other hand, the tendency toward intuitive understanding fosters a more immediate grasp of the objects... a live rapport with them... which stresses the concrete meaning of their relations. ...[I]ntuitive understanding plays a major role in geometry. ...[S]uch concrete intuition is of great value not only for the research worker, but... for anyone who wishes to study and appreciate the results of research in geometry."

"An old French mathematician said: A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street. This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us."

"The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. That it may completely fulfil this high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples!"

"Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments. We also notice that, the farther a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separate branches of the science. So it happens that, with the extension of mathematics, its organic character is not lost but only manifests itself the more clearly."

"This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus."

"If we do not succeed in solving a mathematical problem, the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. After finding this standpoint, not only is this problem frequently more accessible to our investigation, but at the same time we come into possession of a method which is applicable also to related problems."

"To new concepts correspond, necessarily, new signs. These we choose in such a way that they remind us of the phenomena which were the occasion for the formation of the new concepts."

"It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem. I should say first of all, this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning."

"A mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution."

"History teaches the continuity of the development of science. We know that every age has its own problems, which the following age either solves or casts aside as profitless and replaces by new ones. If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems which the science of today sets and whose solution we expect from the future. To such a review of problems the present day, lying at the meeting of the centuries, seems to me well adapted. For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future."

"Good, he did not have enough imagination to become a mathematician."

"I do not see that the sex of the candidate is an argument against her admission as a Privatdozent. After all, the Senate is not a bath-house."

"Immer mit den einfachsten Beispielen anfangen."

"Keep computations to the lowest level of the multiplication table."

"Sometimes it happens that a man's circle of horizon becomes smaller and smaller, and as the radius approaches zero it concentrates on one point. And then that becomes his point of view."

"But he (Galileo) was not an idiot,... Only an idiot could believe that scientific truth needs martyrdom — that may be necessary in religion, but scientific results prove themselves in time."

"One of the supreme achievements of purely intellectual human activity."

"If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?"

"If one were to bring ten of the wisest men in the world together and ask them what was the most stupid thing in existence, they would not be able to discover anything so stupid as astrology."

""Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs."

"Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country."

"One can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it."

"The art of doing mathematics consists in finding that special case which contains all the germs of generality."

"Wir dürfen nicht denen glauben, die heute mit philosophischer Miene und überlegenem Tone den Kulturuntergang prophezeien und sich in dem Ignorabimus gefallen. Für uns gibt es kein Ignorabimus, und meiner Meinung nach auch für die Naturwissenschaft überhaupt nicht. Statt des törichten Ignorabimus heiße im Gegenteil unsere Losung:"

"Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics."

"As Cioran correctly points out, a principal danger of being overcivilized is that one all too easily relapses, out of sheer exhaustion and the unsatisfied need to be "stimulated," into a vulgar and passive barbarism. Thus, "the man who unmasks his fictions" through an indiscriminate pursuit of the lucidity that is promoted by modern liberal culture "renounces his own resources and, in a sense, himself. Consequently, he will accept other fictions which will deny him, since they will not have cropped up from his own depth." There, he concludes, "no man concerned with his own equilibrium may exceed a certain degree of lucidity and analysis.""