Single Quotes (118 quotes)
A great surgeon performs operations for stone by a single method; later he makes a statistical summary of deaths and recoveries, and he concludes from these statistics that the mortality law for this operation is two out of five. Well, I say that this ratio means literally nothing scientifically and gives us no certainty in performing the next operation; for we do not know whether the next case will be among the recoveries or the deaths. What really should be done, instead of gathering facts empirically, is to study them more accurately, each in its special determinism. We must study cases of death with great care and try to discover in them the cause of mortal accidents so as to master the cause and avoid the accidents.
A single and distinct luminous body causes stronger relief in the objects than a diffused light; as may be seen by comparing one side of a landscape illuminated by the sun, and one overshadowed by clouds, and illuminated only by the diffused light of the atmosphere.
A single ray of light from a distant star falling upon the eye of a tyrant in bygone times, may have altered the course of his life, may have changed the destiny of nations, may have transformed the surface of the globe, so intricate, so inconceivably com
All interesting issues in natural history are questions of relative frequency, not single examples. Everything happens once amidst the richness of nature. But when an unanticipated phenomenon occurs again and again–finally turning into an expectation–then theories are overturned.
All of the books in the world contain no more information than is broadcast as video in a single large American city in a single year. Not all bits have equal value.
All the events which occur upon the earth result from Law: even those actions which are entirely dependent on the caprices of the memory, or the impulse of the passions, are shown by statistics to be, when taken in the gross, entirely independent of the human will. As a single atom, man is an enigma; as a whole, he is a mathematical problem. As an individual, he is a free agent; as a species, the offspring of necessity.
All the unhappiness of men arises from one single fact, that they cannot stay quietly in their own chamber.
And now, as a germination of planetary dimensions, comes the thinking layer which over its full extent develops and intertwines its fibres, not to confuse and neutralise them but to reinforce them in the living unity of a single tissue.
Are God and Nature then at strife,
That Nature lends such evil dreams?
So careful of the type she seems,
So careless of the single life…
So careful of the type, but no.
From scarped cliff and quarried stone
She cries, “A thousand types are gone;
I care for nothing, all shall go.”
That Nature lends such evil dreams?
So careful of the type she seems,
So careless of the single life…
So careful of the type, but no.
From scarped cliff and quarried stone
She cries, “A thousand types are gone;
I care for nothing, all shall go.”
As a single footstep will not make a path on the earth, so a single thought will not make a pathway in the mind. To make a deep physical path, we walk again and again. To make a deep mental path, we must think over and over the kind of thoughts we wish to dominate our lives.
As the human fetus develops, its changing form seems to retrace the whole of human evolution from the time we were cosmic dust to the time we were single-celled organisms in the primordial sea to the time we were four-legged, land-dwelling reptiles and beyond, to our current status as largebrained, bipedal mammals. Thus, humans seem to be the sum total of experience since the beginning of the cosmos.
At the sight of a single bone, of a single piece of bone, I recognize and reconstruct the portion of the whole from which it would have been taken. The whole being to which this fragment belonged appears in my mind's eye.
Because a child of one doubles its age after the passage of a single year, it can be said to be aging rapidly.
Before the seas and lands had been created, before the sky that covers everything, Nature displayed a single aspect only throughout the cosmos; Chaos was its name, a shapeless, unwrought mass of inert bulk and nothing more, with the discordant seeds of disconnected elements all heaped together in anarchic disarray.
Do there exist many worlds, or is there but a single world? This is one of the most noble and exalted questions in the study of Nature.
Endowed with two qualities, which seemed incompatible with each other, a volcanic imagination and a pertinacity of intellect which the most tedious numerical calculations could not daunt, Kepler conjectured that the movements of the celestial bodies must be connected together by simple laws, or, to use his own expression, by harmonic laws. These laws he undertook to discover. A thousand fruitless attempts, errors of calculation inseparable from a colossal undertaking, did not prevent him a single instant from advancing resolutely toward the goal of which he imagined he had obtained a glimpse. Twenty-two years were employed by him in this investigation, and still he was not weary of it! What, in reality, are twenty-two years of labor to him who is about to become the legislator of worlds; who shall inscribe his name in ineffaceable characters upon the frontispiece of an immortal code; who shall be able to exclaim in dithyrambic language, and without incurring the reproach of anyone, “The die is cast; I have written my book; it will be read either in the present age or by posterity, it matters not which; it may well await a reader, since God has waited six thousand years for an interpreter of his words.”
Every cent we earn from Crocodile Hunter goes straight back into conservation. Every single cent.
For more than half a century, Martin Gardner has been the single brightest beacon defending rationality and good science against the mysticism and anti-intellectualism that surround us.
Generality of points of view and of methods, precision and elegance in presentation, have become, since Lagrange, the common property of all who would lay claim to the rank of scientific mathematicians. And, even if this generality leads at times to abstruseness at the expense of intuition and applicability, so that general theorems are formulated which fail to apply to a single special case, if furthermore precision at times degenerates into a studied brevity which makes it more difficult to read an article than it was to write it; if, finally, elegance of form has well-nigh become in our day the criterion of the worth or worthlessness of a proposition,—yet are these conditions of the highest importance to a wholesome development, in that they keep the scientific material within the limits which are necessary both intrinsically and extrinsically if mathematics is not to spend itself in trivialities or smother in profusion.
God is infinite, so His universe must be too. Thus is the excellence of God magnified and the greatness of His kingdom made manifest; He is glorified not in one, but in countless suns; not in a single earth, a single world, but in a thousand thousand, I say in an infinity of worlds.
High in the North in a land called Svithjod there is a mountain. It is a hundred miles long and a hundred miles high and once every thousand years a little bird comes to this mountain to sharpen its beak. When the mountain has thus been worn away a single day of eternity will have passed
I am a horse for single harness, not cut out for tandem or teamwork.
I am born into an environment–I know not whence I came nor whither I go nor who I am. This is my situation as yours, every single one of you. The fact that everyone always was in this same situation, and always will be, tells me nothing. Our burning question as to the whence and whither–all we can ourselves observe about it is the present environment. That is why we are eager to find out about it as much as we can. That is science, learning, knowledge; it is the true source of every spiritual endeavour of man. We try to find out as much as we can about the spatial and temporal surroundings of the place in which we find ourselves put by birth.
I am much occupied with the investigation of the physical causes [of motions in the Solar System]. My aim in this is to show that the celestial machine is to be likened not to a divine organism but rather to a clockwork … insofar as nearly all the manifold movements are carried out by means of a single, quite simple magnetic force. This physical conception is to be presented through calculation and geometry.
I am nothing more than a single narrow gasping lung, floating over the mists and summits.
I believe in “intelligence,” and I believe also that there are inherited differences in intellectual ability, but I do not believe that intelligence is a simple scalar endowment that can be quantified by attaching a single figure to it—an I.Q. or the like.
I cannot think of a single field in biology or medicine in which we can claim genuine understanding, and it seems to me the more we learn about living creatures, especially ourselves, the stranger life becomes.
I have mentioned mathematics as a way to settle in the mind a habit of reasoning closely and in train; not that I think it necessary that all men should be deep mathematicians, but that, having got the way of reasoning which that study necessarily brings the mind to, they might be able to transfer it to other parts of knowledge, as they shall have occasion. For in all sorts of reasoning, every single argument should be managed as a mathematical demonstration; the connection and dependence of ideas should be followed till the mind is brought to the source on which it bottoms, and observes the coherence all along; …
I trust I may be enabled in the treatment of patients always to act with a single eye to their good.
I wouldn’t miss this opportunity for anything. For the chance to work on these conservation issues, to serve my country, to work for this president, I’d do it all over again, every single minute.
If all the parts of the universe are interchained in a certain measure, any one phenomenon will not be the effect of a single cause, but the resultant of causes infinitely numerous.
If there is a lesson in our story it is that the manipulation, according to strictly self-consistent rules, of a set of symbols representing one single aspect of the phenomena may produce correct, verifiable predictions, and yet completely ignore all other aspects whose ensemble constitutes reality.
If we compare a mathematical problem with an immense rock, whose interior we wish to penetrate, then the work of the Greek mathematicians appears to us like that of a robust stonecutter, who, with indefatigable perseverance, attempts to demolish the rock gradually from the outside by means of hammer and chisel; but the modern mathematician resembles an expert miner, who first constructs a few passages through the rock and then explodes it with a single blast, bringing to light its inner treasures.
In order that an inventory of plants may be begun and a classification of them correctly established, we must try to discover criteria of some sort for distinguishing what are called “species”. After a long and considerable investigation, no surer criterion for determining species had occurred to me than distinguishing features that perpetuate themselves in propagation from seed. Thus, no matter what variations occur in the individuals or the species, if they spring from the seed of one and the same plant, they are accidental variations and not such as to distinguish a species. For these variations do not perpetuate themselves in subsequent seeding. Thus, for example, we do not regard caryophylli with full or multiple blossoms as a species distinct from caryophylli with single blossoms, because the former owe their origin to the seed of the latter and if the former are sown from their own seed, they once more produce single-blossom caryophylli. But variations that never have as their source seed from one and the same species may finally be regarded as distinct species. Or, if you make a comparison between any two plants, plants which never spring from each other's seed and never, when their seed is sown, are transmuted one into the other, these plants finally are distinct species. For it is just as in animals: a difference in sex is not enough to prove a difference of species, because each sex is derived from the same seed as far as species is concerned and not infrequently from the same parents; no matter how many and how striking may be the accidental differences between them; no other proof that bull and cow, man and woman belong to the same species is required than the fact that both very frequently spring from the same parents or the same mother. Likewise in the case of plants, there is no surer index of identity of species than that of origin from the seed of one and the same plant, whether it is a matter of individuals or species. For animals that differ in species preserve their distinct species permanently; one species never springs from the seed of another nor vice versa.
— John Ray
In scientific investigations it is grievously wrong to pander to the public’s impatience for results, or to let them think that for discovery it is necessary only to set up a great manufactory and a system of mass production. If in treatment team work is effective, in research it is the individual who counts first and above all. No great thought has ever sprung from anything but a single mind, suddenly conceiving. Throughout the whole world there has been too violent a forcing of the growth of ideas; too feverish a rush to perform experiments and publish conclusions. A year of vacation for calm detachment with all the individual workers thinking it all over in a desert should be proclaimed.
In the social equation, the value of a single life is nil; in the cosmic equation, it is infinite… Not only communism, but any political movement which implicitly relies on purely utilitarian ethics, must become a victim to the same fatal error. It is a fallacy as naïve as a mathematical teaser, and yet its consequences lead straight to Goya’s Disasters, to the reign of the guillotine, the torture chambers of the Inquisition, or the cellars of the Lubianka.
It is easy to make out three areas where scientists will be concentrating their efforts in the coming decades. One is in physics, where leading theorists are striving, with the help of experimentalists, to devise a single mathematical theory that embraces all the basic phenomena of matter and energy. The other two are in biology. Biologists—and the rest of us too—would like to know how the brain works and how a single cell, the fertilized egg cell, develops into an entire organism
It is important to realize that life on this planet has spent about three-quarters of its existence in single-celled form, and even today the majority of organisms still exist as single cells. The evolutionary pressure to become complex is evidently not very great.
It is impossible to put together a single prescription that will cure all ailing bodies.
It is not Cayley’s way to analyze concepts into their ultimate elements. … But he is master of the empirical utilization of the material: in the way he combines it to form a single abstract concept which he generalizes and then subjects to computative tests, in the way the newly acquired data are made to yield at a single stroke the general comprehensive idea to the subsequent numerical verification of which years of labor are devoted. Cayley is thus the natural philosopher among mathematicians.
It is not a simple life to be a single cell, although I have no right to say so, having been a single cell so long ago myself that I have no memory at all of that stage in my life.
It is not surprising, in view of the polydynamic constitution of the genuinely mathematical mind, that many of the major heros of the science, men like Desargues and Pascal, Descartes and Leibnitz, Newton, Gauss and Bolzano, Helmholtz and Clifford, Riemann and Salmon and Plücker and Poincaré, have attained to high distinction in other fields not only of science but of philosophy and letters too. And when we reflect that the very greatest mathematical achievements have been due, not alone to the peering, microscopic, histologic vision of men like Weierstrass, illuminating the hidden recesses, the minute and intimate structure of logical reality, but to the larger vision also of men like Klein who survey the kingdoms of geometry and analysis for the endless variety of things that flourish there, as the eye of Darwin ranged over the flora and fauna of the world, or as a commercial monarch contemplates its industry, or as a statesman beholds an empire; when we reflect not only that the Calculus of Probability is a creation of mathematics but that the master mathematician is constantly required to exercise judgment—judgment, that is, in matters not admitting of certainty—balancing probabilities not yet reduced nor even reducible perhaps to calculation; when we reflect that he is called upon to exercise a function analogous to that of the comparative anatomist like Cuvier, comparing theories and doctrines of every degree of similarity and dissimilarity of structure; when, finally, we reflect that he seldom deals with a single idea at a tune, but is for the most part engaged in wielding organized hosts of them, as a general wields at once the division of an army or as a great civil administrator directs from his central office diverse and scattered but related groups of interests and operations; then, I say, the current opinion that devotion to mathematics unfits the devotee for practical affairs should be known for false on a priori grounds. And one should be thus prepared to find that as a fact Gaspard Monge, creator of descriptive geometry, author of the classic Applications de l’analyse à la géométrie; Lazare Carnot, author of the celebrated works, Géométrie de position, and Réflections sur la Métaphysique du Calcul infinitesimal; Fourier, immortal creator of the Théorie analytique de la chaleur; Arago, rightful inheritor of Monge’s chair of geometry; Poncelet, creator of pure projective geometry; one should not be surprised, I say, to find that these and other mathematicians in a land sagacious enough to invoke their aid, rendered, alike in peace and in war, eminent public service.
It is now necessary to indicate more definitely the reason why mathematics not only carries conviction in itself, but also transmits conviction to the objects to which it is applied. The reason is found, first of all, in the perfect precision with which the elementary mathematical concepts are determined; in this respect each science must look to its own salvation .... But this is not all. As soon as human thought attempts long chains of conclusions, or difficult matters generally, there arises not only the danger of error but also the suspicion of error, because since all details cannot be surveyed with clearness at the same instant one must in the end be satisfied with a belief that nothing has been overlooked from the beginning. Every one knows how much this is the case even in arithmetic, the most elementary use of mathematics. No one would imagine that the higher parts of mathematics fare better in this respect; on the contrary, in more complicated conclusions the uncertainty and suspicion of hidden errors increases in rapid progression. How does mathematics manage to rid itself of this inconvenience which attaches to it in the highest degree? By making proofs more rigorous? By giving new rules according to which the old rules shall be applied? Not in the least. A very great uncertainty continues to attach to the result of each single computation. But there are checks. In the realm of mathematics each point may be reached by a hundred different ways; and if each of a hundred ways leads to the same point, one may be sure that the right point has been reached. A calculation without a check is as good as none. Just so it is with every isolated proof in any speculative science whatever; the proof may be ever so ingenious, and ever so perfectly true and correct, it will still fail to convince permanently. He will therefore be much deceived, who, in metaphysics, or in psychology which depends on metaphysics, hopes to see his greatest care in the precise determination of the concepts and in the logical conclusions rewarded by conviction, much less by success in transmitting conviction to others. Not only must the conclusions support each other, without coercion or suspicion of subreption, but in all matters originating in experience, or judging concerning experience, the results of speculation must be verified by experience, not only superficially, but in countless special cases.
It seems to me that every phenomenon, every fact, itself is the really interesting object. Whoever explains it, or connects it with other events, usually only amuses himself or makes sport of us, as, for instance, the naturalist or historian. But a single action or event is interesting, not because it is explainable, but because it is true.
It was cold. Space, the air we breathed, the yellow rocks, were deadly cold. There was something ultimate, passionless, and eternal in this cold. It came to us as a single constant note from the depths of space. We stood on the very boundary of life and death.
Just as Americans have discovered the hidden energy costs in a multitude of products—in refrigerating a steak, for example, on its way to the butcher—they are about to discover the hidden water costs. Beginning with the water that irrigated the corn that was fed to the steer, the steak may have accounted for 3,500 gallons. The water that goes into a 1,000-pound steer would float a destroyer. It takes 14,935 gallons of water to grow a bushel of wheat, 60,000 gallons to produce a ton of steel, 120 gallons to put a single egg on the breakfast table.
Leibnitz’s discoveries lay in the direction in which all modern progress in science lies, in establishing order, symmetry, and harmony, i.e., comprehensiveness and perspicuity,—rather than in dealing with single problems, in the solution of which followers soon attained greater dexterity than himself.
Man cannot change a single law of nature, but can put himself into such relations to natural laws that he can profit by them.
Man does not live by bread alone, there are other wants to be supplied, and even in a practical point of view, a single thought may be fraught with a thousand useful inventions.
Mathematics … above all other subjects, makes the student lust after knowledge, fills him, as it were, with a longing to fathom the cause of things and to employ his own powers independently; it collects his mental forces and concentrates them on a single point and thus awakens the spirit of individual inquiry, self-confidence and the joy of doing; it fascinates because of the view-points which it offers and creates certainty and assurance, owing to the universal validity of its methods. Thus, both what he receives and what he himself contributes toward the proper conception and solution of a problem, combine to mature the student and to make him skillful, to lead him away from the surface of things and to exercise him in the perception of their essence. A student thus prepared thirsts after knowledge and is ready for the university and its sciences. Thus it appears, that higher mathematics is the best guide to philosophy and to the philosophic conception of the world (considered as a self-contained whole) and of one’s own being.
Mathematics, from the earliest times to which the history of human reason can reach, has followed, among that wonderful people of the Greeks, the safe way of science. But it must not be supposed that it was as easy for mathematics as for logic, in which reason is concerned with itself alone, to find, or rather to make for itself that royal road. I believe, on the contrary, that there was a long period of tentative work (chiefly still among the Egyptians), and that the change is to be ascribed to a revolution, produced by the happy thought of a single man, whose experiments pointed unmistakably to the path that had to be followed, and opened and traced out for the most distant times the safe way of a science. The history of that intellectual revolution, which was far more important than the passage round the celebrated Cape of Good Hope, and the name of its fortunate author, have not been preserved to us. … A new light flashed on the first man who demonstrated the properties of the isosceles triangle (whether his name was Thales or any other name), for he found that he had not to investigate what he saw in the figure, or the mere concepts of that figure, and thus to learn its properties; but that he had to produce (by construction) what he had himself, according to concepts a priori, placed into that figure and represented in it, so that, in order to know anything with certainty a priori, he must not attribute to that figure anything beyond what necessarily follows from what he has himself placed into it, in accordance with the concept.
Maxwell's equations… originally consisted of eight equations. These equations are not “beautiful.” They do not possess much symmetry. In their original form, they are ugly. …However, when rewritten using time as the fourth dimension, this rather awkward set of eight equations collapses into a single tensor equation. This is what a physicist calls “beauty.”
My visceral perception of brotherhood harmonizes with our best modern biological knowledge ... Many people think (or fear) that equality of human races represents a hope of liberal sentimentality probably squashed by the hard realities of history. They are wrong. This essay can be summarized in a single phrase, a motto if you will: Human equality is a contingent fact of history. Equality is not true by definition; it is neither an ethical principle (though equal treatment may be) nor a statement about norms of social action. It just worked out that way. A hundred different and plausible scenarios for human history would have yielded other results (and moral dilemmas of enormous magnitude). They didn’t happen.
Nature has not arranged her productions on a single and direct line. They branch at every step, and in every direction, and he who attempts to reduce them into departments is left to do it by the lines of his own fancy.
New scientific ideas never spring from a communal body, however organized, but rather from the head of an individually inspired researcher who struggles with his problems in lonely thought and unites all his thought on one single point which is his whole world for the moment.
No man can be truly called an entomologist, sir; the subject is too vast for any single human intelligence to grasp.
Nothing is more symptomatic of the enervation, of the decompression of the Western imagination, than our incapacity to respond to the landings on the Moon. Not a single great poem, picture, metaphor has come of this breathtaking act, of Prometheus’ rescue of Icarus or of Phaeton in flight towards the stars.
Now, it may be stretching an analogy to compare epidemics of cholera—caused by a known agent—with that epidemic of violent crime which is destroying our cities. It is unlikely that our social problems can be traced to a single, clearly defined cause in the sense that a bacterial disease is ‘caused’ by a microbe. But, I daresay, social science is about as advanced in the late twentieth century as bacteriological science was in the mid nineteenth century. Our forerunners knew something about cholera; they sensed that its spread was associated with misdirected sewage, filth, and the influx of alien poor into crowded, urban tenements. And we know something about street crime; nowhere has it been reported that a member of the New York Stock Exchange has robbed ... at the point of a gun. Indeed, I am naively confident that an enlightened social scientist of the next century will be able to point out that we had available to us at least some of the clues to the cause of urban crime.
Owing to his lack of knowledge, the ordinary man cannot attempt to resolve conflicting theories of conflicting advice into a single organized structure. He is likely to assume the information available to him is on the order of what we might think of as a few pieces of an enormous jigsaw puzzle. If a given piece fails to fit, it is not because it is fraudulent; more likely the contradictions and inconsistencies within his information are due to his lack of understanding and to the fact that he possesses only a few pieces of the puzzle. Differing statements about the nature of things, differing medical philosophies, different diagnoses and treatments—all of these are to be collected eagerly and be made a part of the individual's collection of puzzle pieces. Ultimately, after many lifetimes, the pieces will fit together and the individual will attain clear and certain knowledge.
Philosophy [the universe] is written in that great book which ever lies before our eyes ... We cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in the mathematical language ... without whose help it is humanly impossible to comprehend a single word of it, and without which one wanders in vain through a dark labyrinth.
Science cannot avert a single thunderbolt.
Science is neither a single tradition, nor the best tradition there is, except for people who have become accustomed to its presence, its benefits and its disadvantages. In a democracy it should be separated from the state just as churches are now separated from the state.
Science means simplification. It substitutes a single rule for a million miscellaneous observations.
Sites need to be able to interact in one single, universal space.
Such is the substance of my faith; and if I were to sum up my credo in a single word, it would be that proud motto of Fustel de Coulanges, Quaero, I seek to learn.
Suppose physics soon succeeds, as Stephen Hawking and a few other physicists hope and believe, in reducing physics to a single equation or a small set of equations that will “explain” all of nature’s fundamental laws. We can then ask the unanswerable question, "Why this set of equations?”
Tactics used by many practitioners of pseudoscience: make a large number of vaguely scientific arguments in the hope of making the desired conclusion seem inevitable. It is essential to recognize that a disconnected assemblage of weak arguments does not create a single, strong scientific argument.
That mathematics “do not cultivate the power of generalization,”; … will be admitted by no person of competent knowledge, except in a very qualified sense. The generalizations of mathematics, are, no doubt, a different thing from the generalizations of physical science; but in the difficulty of seizing them, and the mental tension they require, they are no contemptible preparation for the most arduous efforts of the scientific mind. Even the fundamental notions of the higher mathematics, from those of the differential calculus upwards are products of a very high abstraction. … To perceive the mathematical laws common to the results of many mathematical operations, even in so simple a case as that of the binomial theorem, involves a vigorous exercise of the same faculty which gave us Kepler’s laws, and rose through those laws to the theory of universal gravitation. Every process of what has been called Universal Geometry—the great creation of Descartes and his successors, in which a single train of reasoning solves whole classes of problems at once, and others common to large groups of them—is a practical lesson in the management of wide generalizations, and abstraction of the points of agreement from those of difference among objects of great and confusing diversity, to which the purely inductive sciences cannot furnish many superior. Even so elementary an operation as that of abstracting from the particular configuration of the triangles or other figures, and the relative situation of the particular lines or points, in the diagram which aids the apprehension of a common geometrical demonstration, is a very useful, and far from being always an easy, exercise of the faculty of generalization so strangely imagined to have no place or part in the processes of mathematics.
The actual evolution of mathematical theories proceeds by a process of induction strictly analogous to the method of induction employed in building up the physical sciences; observation, comparison, classification, trial, and generalisation are essential in both cases. Not only are special results, obtained independently of one another, frequently seen to be really included in some generalisation, but branches of the subject which have been developed quite independently of one another are sometimes found to have connections which enable them to be synthesised in one single body of doctrine. The essential nature of mathematical thought manifests itself in the discernment of fundamental identity in the mathematical aspects of what are superficially very different domains. A striking example of this species of immanent identity of mathematical form was exhibited by the discovery of that distinguished mathematician … Major MacMahon, that all possible Latin squares are capable of enumeration by the consideration of certain differential operators. Here we have a case in which an enumeration, which appears to be not amenable to direct treatment, can actually be carried out in a simple manner when the underlying identity of the operation is recognised with that involved in certain operations due to differential operators, the calculus of which belongs superficially to a wholly different region of thought from that relating to Latin squares.
The advantage is that mathematics is a field in which one’s blunders tend to show very clearly and can be corrected or erased with a stroke of the pencil. It is a field which has often been compared with chess, but differs from the latter in that it is only one’s best moments that count and not one’s worst. A single inattention may lose a chess game, whereas a single successful approach to a problem, among many which have been relegated to the wastebasket, will make a mathematician’s reputation.
The advantage which science gained by Gauss’ long-lingering method of publication is this: What he put into print is as true and important today as when first published; his publications are statutes, superior to other human statutes in this, that nowhere and never has a single error been detected in them. This justifies and makes intelligible the pride with which Gauss said in the evening of his life of the first larger work of his youth: “The Disquisitiones arithmeticae belong to history.”
The ancients had a taste, let us say rather a passion, for the marvellous, which caused … grouping together the lofty deeds of a great number of heroes, whose names they have not even deigned to preserve, and investing the single personage of Hercules with them. … In our own time the public delight in blending fable with history. In every career of life, in the pursuit of science especially, they enjoy a pleasure in creating Herculeses.
The astronomers said, ‘Give us matter and a little motion and we will construct the universe. It is not enough that we should have matter, we must also have a single impulse, one shove to launch the mass and generate the harmony of the centrifugal and centripetal forces.’ ... There is no end to the consequences of the act. That famous aboriginal push propagates itself through all the balls of the system, and through every atom of every ball.
The average gambler will say “The player who stakes his whole fortune on a single play is a fool, and the science of mathematics can not prove him to be otherwise.” The reply is obvious: “The science of mathematics never attempts the impossible, it merely shows that other players are greater fools.”
The coastal zone may be the single most important portion of our planet. The loss of its biodiversity may have repercussions far be-yond our worst fears.
The colors are stunning. In a single view, I see - looking out at the edge of the earth: red at the horizon line, blending to orange and yellow, followed by a thin white line, then light blue, gradually turning to dark blue and various gradually darker shades of gray, then black and a million stars above. It’s breathtaking.
The eventual goal of science is to provide a single theory that describes the whole universe.
The first effect of the mind growing cultivated is that processes once multiple get to be performed in a single act. Lazarus has called this the progressive “condensation” of thought. ... Steps really sink from sight. An advanced thinker sees the relations of his topics is such masses and so instantaneously that when he comes to explain to younger minds it is often hard ... Bowditch, who translated and annotated Laplace's Méchanique Céleste, said that whenever his author prefaced a proposition by the words “it is evident,” he knew that many hours of hard study lay before him.
The first question which you will ask and which I must try to answer is this, “What is the use of climbing Mount Everest ?” and my answer must at once be, “It is no use.” There is not the slightest prospect of any gain whatsoever. Oh, we may learn a little about the behavior of the human body at high altitudes, and possibly medical men may turn our observation to some account for the purposes of aviation. But otherwise nothing will come of it. We shall not bring back a single bit of gold or silver, not a gem, nor any coal or iron. We shall not find a single foot of earth that can be planted with crops to raise food. It’s no use. So, if you cannot understand that there is something in man which responds to the challenge of this mountain and goes out to meet it, that the struggle is the struggle of life itself upward and forever upward, then you won’t see why we go. What we get from this adventure is just sheer joy. And joy is, after all, the end of life. We do not live to eat and make money. We eat and make money to be able to enjoy life. That is what life means and what life is for.
The full story of successful organ transplantation in man weaves together three separate pathways: the study of renal disease, skin grafting in twins, and surgical determination. A leitmotif permeates each of these pathways, i.e. a single event or report was critical for medical progress.
The golden age of mathematics—that was not the age of Euclid, it is ours. Ours is the age when no less than six international congresses have been held in the course of nine years. It is in our day that more than a dozen mathematical societies contain a growing membership of more than two thousand men representing the centers of scientific light throughout the great culture nations of the world. It is in our time that over five hundred scientific journals are each devoted in part, while more than two score others are devoted exclusively, to the publication of mathematics. It is in our time that the Jahrbuch über die Fortschritte der Mathematik, though admitting only condensed abstracts with titles, and not reporting on all the journals, has, nevertheless, grown to nearly forty huge volumes in as many years. It is in our time that as many as two thousand books and memoirs drop from the mathematical press of the world in a single year, the estimated number mounting up to fifty thousand in the last generation. Finally, to adduce yet another evidence of a similar kind, it requires not less than seven ponderous tomes of the forthcoming Encyclopaedie der Mathematischen Wissenschaften to contain, not expositions, not demonstrations, but merely compact reports and bibliographic notices sketching developments that have taken place since the beginning of the nineteenth century.
The greatest single achievement of nature to date was surely the invention of the molecule DNA.
The highest of the world’s mountains, it seems, has to make but a single gesture of magnificence to be the lord of all, vast in unchallenged and isolated supremacy.
The history of science teaches only too plainly the lesson that no single method is absolutely to be relied upon, that sources of error lurk where they are least expected, and that they may escape the notice of the most experienced and conscientious worker.
The individual feels the futility of human desires and aims and the sublimity and marvelous order which reveal themselves both in nature and in the world of thought. Individual existence impresses him as a sort of prison and he wants to experience the universe as a single significant whole. The beginnings of cosmic religious feeling already appear at an early stage of development, e.g., in many of the Psalms of David and in some of the Prophets. Buddhism, as we have learned especially from the wonderful writings of Schopenhauer, contains a much stronger element of this. The religious geniuses of all ages have been distinguished by this kind of religious feeling, which knows no dogma and no God conceived in man’s image; so that there can be no church whose central teachings are based on it. Hence it is precisely among the heretics of every age that we find men who were filled with this highest kind of religious feeling and were in many cases regarded by their contemporaries as atheists, sometimes also as saints. Looked at in this light, men like Democritus, Francis of Assisi, and Spinoza are closely akin to one another.
The iron labor of conscious logical reasoning demands great perseverance and great caution; it moves on but slowly, and is rarely illuminated by brilliant flashes of genius. It knows little of that facility with which the most varied instances come thronging into the memory of the philologist or historian. Rather is it an essential condition of the methodical progress of mathematical reasoning that the mind should remain concentrated on a single point, undisturbed alike by collateral ideas on the one hand, and by wishes and hopes on the other, and moving on steadily in the direction it has deliberately chosen.
The last few meters up to the summit no longer seem so hard. On reaching the top, I sit down and let my legs dangle into space. I don’t have to climb anymore. I pull my camera from my rucksack and, in my down mittens, fumble a long time with the batteries before I have it working properly. Then I film Peter. Now, after the hours of torment, which indeed I didn’t recognize as torment, now, when the monotonous motion of plodding upwards is at an end, and I have nothing more to do than breathe, a great peace floods my whole being. I breathe like someone who has run the race of his life and knows that he may now rest forever. I keep looking all around, because the first time I didn’t see anything of the panorama I had expected from Everest, neither indeed did I notice how the wind was continually chasing snow across the summit. In my state of spiritual abstraction, I no longer belong to myself and to my eyesight. I am nothing more than a single, narrow, gasping lung, floating over the mists and the summits.
The line separating investment and speculation, which is never bright and clear, becomes blurred still further when most market participants have recently enjoyed triumphs. Nothing sedates rationality like large doses of effortless money. After a heady experience of that kind, normally sensible people drift into behavior akin to that of Cinderella at the ball. They know that overstaying the festivities—that is, continuing to speculate in companies that have gigantic valuations relative to the cash they are likely to generate in the future—will eventually bring on pumpkins and mice. But they nevertheless hate to miss a single minute of what is one helluva party. Therefore, the giddy participants all plan to leave just seconds before midnight. There’s a problem, though: They are dancing in a room in which the clocks have no hands.
The meaning of life is contained in every single expression of life. It is present in the infinity of forms and phenomena that exist in all of creation.
The nature of the atoms, and the forces called into play in their chemical union; the interactions between these atoms and the non-differentiated ether as manifested in the phenomena of light and electricity; the structures of the molecules and molecular systems of which the atoms are the units; the explanation of cohesion, elasticity, and gravitation—all these will be marshaled into a single compact and consistent body of scientific knowledge.
The night spread out of the east in a great flood, quenching the red sunlight in a single minute. We wriggled by breathless degrees deep into our sleeping bags. Our sole thought was of comfort; we were not alive to the beauty or the grandeur of our position; we did not reflect on the splendor of our elevation. A regret I shall always have is that I did not muster up the energy to spend a minute or two stargazing. One peep I did make between the tent flaps into the night, and I remember dimly an appalling wealth of stars, not pale and remote as they appear when viewed through the moisture-laden air of lower levels, but brilliant points of electric blue fire standing out almost stereoscopically. It was a sight an astronomer would have given much to see, and here were we lying dully in our sleeping bags concerned only with the importance of keeping warm and comfortable.
The object of geometry in all its measuring and computing, is to ascertain with exactness the plan of the great Geometer, to penetrate the veil of material forms, and disclose the thoughts which lie beneath them? When our researches are successful, and when a generous and heaven-eyed inspiration has elevated us above humanity, and raised us triumphantly into the very presence, as it were, of the divine intellect, how instantly and entirely are human pride and vanity repressed, and, by a single glance at the glories of the infinite mind, are we humbled to the dust.
The problem for a writer of a text-book has come now, in fact, to be this—to write a book so neatly trimmed and compacted that no coach, on looking through it, can mark a single passage which the candidate for a minimum pass can safely omit. Some of these text-books I have seen, where the scientific matter has been, like the lady’s waist in the nursery song, compressed “so gent and sma’,” that the thickness barely, if at all, surpasses what is devoted to the publisher’s advertisements. We shall return, I verily believe, to the Compendium of Martianus Capella. The result of all this is that science, in the hands of specialists, soars higher and higher into the light of day, while educators and the educated are left more and more to wander in primeval darkness.
The solution of problems is one of the lowest forms of mathematical research, … yet its educational value cannot be overestimated. It is the ladder by which the mind ascends into higher fields of original research and investigation. Many dormant minds have been aroused into activity through the mastery of a single problem.
The statement that there is no single scientific method has become a truism only rather recently.
The sun rises. In that short phrase, in a single fact, is enough information to keep biology, physics, and philosophy busy for all the rest of time.
The supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.
The swelling and towering omnibuses, the huge trucks and wagons and carriages, the impetuous hansoms and the more sobered four-wheelers, the pony-carts, donkey-carts, hand-carts, and bicycles which fearlessly find their way amidst the turmoil, with foot-passengers winding in and out, and covering the sidewalks with their multitude, give the effect of a single monstrous organism, which writhes swiftly along the channel where it had run in the figure of a flood till you were tired of that metaphor. You are now a molecule of that vast organism.
The theory of punctuated equilibrium, proposed by Niles Eldredge and myself, is not, as so often misunderstood, a radical claim for truly sudden change, but a recognition that ordinary processes of speciation, properly conceived as glacially slow by the standard of our own life-span, do not resolve into geological time as long sequences of insensibly graded intermediates (the traditional, or gradualistic, view), but as geologically ‘sudden’ origins at single bedding planes.
The Wright Brothers created the single greatest cultural force since the invention of writing. The airplane became the first World Wide Web, bringing people, languages, ideas, and values together.
There is not, we believe, a single example of a medicine having been received permanently into the Materia Medica upon the sole ground of its physical, chemical, or physiological properties. Nearly every one has become a popular remedy before being adopted or even tried by physicians; by far the greater number were first employed in countries which were and are now in a state of scientific ignorance....
This fundamental discovery that all bodies owe their origin to arrangements of single initial corpuscular type is the beacon that lights the history of the universe to our eyes. In its own way, matter obeyed from the beginning that great law of biology to which we shall have to recur time and time again, the law of “complexification.”
This is one of the greatest advantages of modern geometry over the ancient, to be able, through the consideration of positive and negative quantities, to include in a single enunciation the several cases which the same theorem may present by a change in the relative position of the different parts of a figure. Thus in our day the nine principal problems and the numerous particular cases, which form the object of eighty-three theorems in the two books De sectione determinata of Appolonius constitute only one problem which is resolved by a single equation.
This single Stick, which you now behold ingloriously lying in that neglected Corner, I once knew in a flourishing State in a Forest: It was full of Sap, full of Leaves, and full of Boughs: But now, in vain does the busy Art of Man pretend to vie with Nature, by tying that withered Bundle of Twigs to its sapless Trunk: It is at best but the Reverse of what it was; a Tree turned upside down, the Branches on the Earth, and the Root in the Air.
Thus, we have three principles for increasing adequacy of data: if you must work with a single object, look for imperfections that record historical descent; if several objects are available, try to render them as stages of a single historical process; if processes can be directly observed, sum up their effects through time. One may discuss these principles directly or recognize the ‘little problems’ that Darwin used to exemplify them: orchids, coral reefs, and worms–the middle book, the first, and the last.
To pray is to ask that the laws of the universe be annulled on behalf of a single petitioner confessedly unworthy.
Today the greatest single source of wealth is between your ears.
Tonight, the moon came out, it was nearly full.
Way down here on earth, I could feel it’s pull.
The weight of gravity or just the lure of life,
Made me want to leave my only home tonight.
I’m just wondering how we know where we belong
Is it in the arc of the moon, leaving shadows on the lawn
In the path of fireflies and a single bird at dawn
Singing in between here and gone
Way down here on earth, I could feel it’s pull.
The weight of gravity or just the lure of life,
Made me want to leave my only home tonight.
I’m just wondering how we know where we belong
Is it in the arc of the moon, leaving shadows on the lawn
In the path of fireflies and a single bird at dawn
Singing in between here and gone
Truly I say to you, a single number has more genuine and permanent value than an expensive library full of hypotheses.
We build our personalities laboriously and through many years, and we cannot order fundamental changes just because we might value their utility; no button reading ‘positive attitude’ protrudes from our hearts, and no finger can coerce positivity into immediate action by a single and painless pressing.
We reverence ancient Greece as the cradle of western science. Here for the first time the world witnessed the miracle of a logical system which proceeded from step to step with such precision that every single one of its propositions was absolutely indubitable—I refer to Euclid’s geometry. This admirable triumph of reasoning gave the human intellect the necessary confidence in itself for its subsequent achievements. If Euclid failed to kindle your youthful enthusiasm, then you were not born to be a scientific thinker.
While the method of the natural sciences is... analytic, the method of the social sciences is better described as compositive or synthetic. It is the so-called wholes, the groups of elements which are structurally connected, which we learn to single out from the totality of observed phenomena... Insofar as we analyze individual thought in the social sciences the purpose is not to explain that thought, but merely to distinguish the possible types of elements with which we shall have to reckon in the construction of different patterns of social relationships. It is a mistake... to believe that their aim is to explain conscious action ... The problems which they try to answer arise only insofar as the conscious action of many men produce undesigned results... If social phenomena showed no order except insofar as they were consciously designed, there would indeed be no room for theoretical sciences of society and there would be, as is often argued, only problems of psychology. It is only insofar as some sort of order arises as a result of individual action but without being designed by any individual that a problem is raised which demands a theoretical explanation... people dominated by the scientistic prejudice are often inclined to deny the existence of any such order... it can be shown briefly and without any technical apparatus how the independent actions of individuals will produce an order which is no part of their intentions... The way in which footpaths are formed in a wild broken country is such an instance. At first everyone will seek for himself what seems to him the best path. But the fact that such a path has been used once is likely to make it easier to traverse and therefore more likely to be used again; and thus gradually more and more clearly defined tracks arise and come to be used to the exclusion of other possible ways. Human movements through the region come to conform to a definite pattern which, although the result of deliberate decision of many people, has yet not be consciously designed by anyone.
Why, then, are we surprised that comets, such a rare spectacle in the universe, are not known, when their return is at vast intervals?. … The time will come when diligent research over long periods will bring to light things which now lie hidden. A single lifetime, even though entirely devoted to the sky, would not be enough for the investigation of so vast a subject … And so this knowledge will be unfolded only through long successive ages. There will come a time when our descendants will be amazed that we did not know things that are so plain to them …. Many discoveries are reserved for ages still to come, when memory of us will have been effaced. Our universe is a sorry little affair unless it has in it something for every age to investigate … Nature does not reveal her mysteries once and for all. Someday there will be a man who will show in what regions comets have their orbit, why they travel so remote from other celestial bodies, how large they are and what sort they are.
With the extension of mathematical knowledge will it not finally become impossible for the single investigator to embrace all departments of this knowledge? In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which, at the same time, assist in understanding earlier theories and in casting aside some more complicated developments.
Without the slightest doubt there is something through which material and spiritual energy hold togehter and are complementary. In the last analysis, somehow or other, there must be a single energy operating in the world. And the first idea that occurs to us is that the 'soul' must be as it were the focal point of transformation at which, from all the points of nature, the forces of bodies converge, to become interiorised and sublimated in beauty and truth.
[It] is not the nature of things for any one man to make a sudden, violent discovery; science goes step by step and every man depends on the work of his predecessors. When you hear of a sudden unexpected discovery—a bolt from the blue—you can always be sure that it has grown up by the influence of one man or another, and it is the mutual influence which makes the enormous possibility of scientific advance. Scientists are not dependent on the ideas of a single man, but on the combined wisdom of thousands of men, all thinking of the same problem and each doing his little bit to add to the great structure of knowledge which is gradually being erected.
[On common water.] Its substance reaches everywhere; it touches the past and prepares the future; it moves under the poles and wanders thinly in the heights of air. It can assume forms of exquisite perfection in a snowflake, or strip the living to a single shining bone cast up by the sea.
[The] structural theory is of extreme simplicity. It assumes that the molecule is held together by links between one atom and the next: that every kind of atom can form a definite small number of such links: that these can be single, double or triple: that the groups may take up any position possible by rotation round the line of a single but not round that of a double link: finally that with all the elements of the first short period [of the periodic table], and with many others as well, the angles between the valencies are approximately those formed by joining the centre of a regular tetrahedron to its angular points. No assumption whatever is made as to the mechanism of the linkage. Through the whole development of organic chemistry this theory has always proved capable of providing a different structure for every different compound that can be isolated. Among the hundreds of thousands of known substances, there are never more isomeric forms than the theory permits.