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Who said: “God does not care about our mathematical difficulties. He integrates empirically.”
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Complete Quotes (204 quotes)

“Le génie n'est qu'une longue patience”, a dit Buffon. Cela est bien incomplet. Le génie, c'est l'impatience dans les idées et la patience dans les faits : une imagination vive et un jugement calme; quelque chose comme un liquide en ébullition dans un vase qui reste toujours froid.
“Genius is just enduring patience,” said Buffon. This is far from complete. Genius is impatience in ideas and patience with the facts: a lively imagination and a calm judgment, rather like a liquid boiling in a cup that remains cold.
In Recueil d'Œuvres de Léo Errera: Botanique Générale (1908), 198. Google translation by Webmaster.
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Die Gewohnheit einer Meinung erzeugt oft völlige Ueberzeugung von ihrer Richtigkeit, sie verbirgt die schwächeren Theile davon, und macht uns unfähig, die Beweise dagegen anzunehmen.
The habit of an opinion often leads to the complete conviction of its truth, it hides the weaker parts of it, and makes us incapable of accepting the proofs against it.
(1827). German text in Ira Freund, The Study of Chemical Composition (1904), 31. Translated form in Carl Schorlemmer, The Rise and Development of Organic Chemistry (1894), 49.
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A cell has a history; its structure is inherited, it grows, divides, and, as in the embryo of higher animals, the products of division differentiate on complex lines. Living cells, moreover, transmit all that is involved in their complex heredity. I am far from maintaining that these fundamental properties may not depend upon organisation at levels above any chemical level; to understand them may even call for different methods of thought; I do not pretend to know. But if there be a hierarchy of levels we must recognise each one, and the physical and chemical level which, I would again say, may be the level of self-maintenance, must always have a place in any ultimate complete description.
'Some Aspects of Biochemistry', The Irish Journal of Medical Science (1932), 79, 346.
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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.
Quoted in E. J. Bowen's obituary of Hinshelwood, Chemistry in Britain (1967), Vol. 3, 536.
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A common mistake that people make when trying to design something completely foolproof is to underestimate the ingenuity of complete fools.
In Mostly Harmless (1992), 135.
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A complete and generous education fits a man to perform justly, skilfully and magnanimously all the offices of peace and war.
Louis Klopsch, Many Thoughts of Many Minds (1896), 78.
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A complete survey of life on Earth may appear to be a daunting task. But compared with what has been dared and achieved in high-energy physics, molecular genetics, and other branches of “big science,” it is in the second or third rank.
In 'Edward O. Wilson: The Biological Diversity Crisis: A Challenge to Science', Issues in Science and Technology (Fall 1985), 2, No. 1, 26.
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A complete theory of evolution must acknowledge a balance between ‘external’ forces of environment imposing selection for local adaptation and ‘internal’ forces representing constraints of inheritance and development. Vavilov placed too much emphasis on internal constraints and downgraded the power of selection. But Western Darwinians have erred equally in practically ignoring (while acknowledging in theory) the limits placed on selection by structure and development–what Vavilov and the older biologists would have called ‘laws of form.’
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A fossil hunter needs sharp eyes and a keen search image, a mental template that subconsciously evaluates everything he sees in his search for telltale clues. A kind of mental radar works even if he isn’t concentrating hard. A fossil mollusk expert has a mollusk search image. A fossil antelope expert has an antelope search image. … Yet even when one has a good internal radar, the search is incredibly more difficult than it sounds. Not only are fossils often the same color as the rocks among which they are found, so they blend in with the background; they are also usually broken into odd-shaped fragments. … In our business, we don’t expect to find a whole skull lying on the surface staring up at us. The typical find is a small piece of petrified bone. The fossil hunter’s search therefore has to have an infinite number of dimensions, matching every conceivable angle of every shape of fragment of every bone on the human body.
Describing the skill of his co-worker, Kamoya Kimeu, who discovered the Turkana Boy, the most complete specimen of Homo erectus, on a slope covered with black lava pebbles.
Richard Leakey and Roger Lewin, Origins Reconsidered: In Search of What Makes Us Human (1992), 26.
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A great advantage of X-ray analysis as a method of chemical structure analysis is its power to show some totally unexpected and surprising structure with, at the same time, complete certainty.
In 'X-ray Analysis of Complicated Molecules', Nobel Lecture (11 Dec 1964). In Nobel Lectures: Chemistry 1942-1962 (1964), 83.
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A map of the moon... should be in every geological lecture room; for no where can we have a more complete or more magnificent illustration of volcanic operations. Our sublimest volcanoes would rank among the smaller lunar eminences; and our Etnas are but spitting furnaces.
'On the Volcanoes of the Moon', American Journal of Science, 1846, 2 (2nd Series), 347.
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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.
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A strange feeling of complete, almost solemn contentment suddenly overcame me when the descent module landed, rocked, and stilled. The weather was foul, but I smelled Earth, unspeakably sweet and intoxicating. And wind. Now utterly delightful; wind after long days in space.
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A theory of physics is not an explanation; it is a system of mathematical oppositions deduced from a small number of principles the aim of which is to represent as simply, as completely, and as exactly as possible, a group of experimental laws.
As quoted in Philipp Frank, Modern Science and its Philosophy (1949), 15, which cites Théorie Physique; Son Objet—Son Structure (1906), 24.
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A well-chosen anthology is a complete dispensary of medicine for the more common mental disorders, and may be used as much for prevention as cure.
'The Use of Poetry', On English Poetry (1922), 85.
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Absorbed in the special investigation, I paid no heed to the edifice which was meanwhile unconsciously building itself up. Having however completed the comparison of the fossil species in Paris, I wanted, for the sake of an easy revision of the same, to make a list according to their succession in geological formations, with a view of determining the characteristics more exactly and bringing them by their enumeration into bolder relief. What was my joy and surprise to find that the simplest enumeration of the fossil fishes according to their geological succession was also a complete statement of the natural relations of the families among themselves; that one might therefore read the genetic development of the whole class in the history of creation, the representation of the genera and species in the several families being therein determined; in one word, that the genetic succession of the fishes corresponds perfectly with their zoological classification, and with just that classification proposed by me.
Quoted in Elizabeth Cary Agassiz (ed.), Louis Agassiz: His Life and Correspondence (1885), Vol. I, 203-4.
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Adrenalin does not excite sympathetic ganglia when applied to them directly, as does nicotine. Its effective action is localised at the periphery. The existence upon plain muscle of a peripheral nervous network, that degenerates only after section of both the constrictor and inhibitory nerves entering it, and not after section of either alone, has been described. I find that even after such complete denervation, whether of three days' or ten months' duration, the plain muscle of the dilatator pupillae will respond to adrenalin, and that with greater rapidity and longer persistence than does the iris whose nervous relations are uninjured. Therefore it cannot be that adrenalin excites any structure derived from, and dependent for its persistence on, the peripheral neurone. But since adrenalin does not evoke any reaction from muscle that has at no time of its life been innervated by the sympathetic, the point at which the stimulus of the chemical excitant is received, and transformed into what may cause the change of tension of the muscle fibre, is perhaps a mechanism developed out of the muscle cell in response to its union with the synapsing sympathetic fibre, the function of which is to receive and transform the nervous impulse. Adrenalin might then be the chemical stimulant liberated on each occasion when the impulse arrives at the periphery.
'On the Action of Adrenalin', Proceedings of the Physiological Society, 21 May 1904, in The Journal of Physiology 1904, 31, xxi.
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After I had addressed myself to this very difficult and almost insoluble problem, the suggestion at length came to me how it could be solved with fewer and much simpler constructions than were formerly used, if some assumptions (which are called axioms) were granted me. They follow in this order.
There is no one center of all the celestial circles or spheres.
The center of the earth is not the center of the universe, but only of gravity and of the lunar sphere.
All the spheres revolve about the sun as their mid-point, and therefore the sun is the center of the universe.
The ratio of the earth's distance from the sun to the height of the firmament is so much smaller than the ratio of the earth's radius to its distance from the sun that the distance from the earth to the sun is imperceptible in comparison with the height of the firmament.
Whatever motion appears in the firmament arises not from any motion of the firmament, but from the earth's motion. The earth together with its circumjacent elements performs a complete rotation on its fixed poles in a daily motion, while the firmament and highest heaven abide unchanged.
What appears to us as motions of the sun arise not from its motion but from the motion of the earth and our sphere, with which we revolve about the sun like any other planet. The earth has, then, more than one motion.
The apparent retrograde and direct motion of the planets arises not from their motion but from the earth's. The motion of the earth alone, therefore, suffices to explain so many apparent inequalities in the heavens.
'The Commentariolus', in Three Copernican Treatises (c.1510), trans. E. Rosen (1939), 58-9.
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All fossil anthropoids found hitherto have been known only from mandibular or maxillary fragments, so far as crania are concerned, and so the general appearance of the types they represented had been unknown; consequently, a condition of affairs where virtually the whole face and lower jaw, replete with teeth, together with the major portion of the brain pattern, have been preserved, constitutes a specimen of unusual value in fossil anthropoid discovery. Here, as in Homo rhodesiensis, Southern Africa has provided documents of higher primate evolution that are amongst the most complete extant. Apart from this evidential completeness, the specimen is of importance because it exhibits an extinct race of apes intermediate between living anthropoids and man ... Whether our present fossil is to be correlated with the discoveries made in India is not yet apparent; that question can only be solved by a careful comparison of the permanent molar teeth from both localities. It is obvious, meanwhile, that it represents a fossil group distinctly advanced beyond living anthropoids in those two dominantly human characters of facial and dental recession on one hand, and improved quality of the brain on the other. Unlike Pithecanthropus, it does not represent an ape-like man, a caricature of precocious hominid failure, but a creature well advanced beyond modern anthropoids in just those characters, facial and cerebral, which are to be anticipated in an extinct link between man and his simian ancestor. At the same time, it is equally evident that a creature with anthropoid brain capacity and lacking the distinctive, localised temporal expansions which appear to be concomitant with and necessary to articulate man, is no true man. It is therefore logically regarded as a man-like ape. I propose tentatively, then, that a new family of Homo-simidæ be created for the reception of the group of individuals which it represents, and that the first known species of the group be designated Australopithecus africanus, in commemoration, first, of the extreme southern and unexpected horizon of its discovery, and secondly, of the continent in which so many new and important discoveries connected with the early history of man have recently been made, thus vindicating the Darwinian claim that Africa would prove to be the cradle of mankind.
'Australopithicus africanus: The Man-Ape of South Africa', Nature, 1925, 115, 195.
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All women scientists should marry, rear children, cook, and clean in order to achieve fulfillment, to be a complete woman.
Quoted in The Chemical Educator, vol. 7, No. 2, in a book review of Eugene Straus, Rosalyn Yalow, Nobel Laureate: Her Life and Work in Medicine.
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Although the works of the Creator may be in themselves all equally perfect, the animal is, as I see it, the most complete work of nature, and man is her masterpiece.
'Histoire des Animaux', Histoire Naturelle, Générale et Particulière, Avec la Description du Cabinet du Roi (1749), Vol. 2, 2. Quoted in Jacques Roger, The Life Sciences in Eighteenth -Century French Thought, ed. Keith R. Benson and trans. Robert Ellrich (1997), 437.
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Although [Charles Darwin] would patiently go on repeating experiments where there was any good to be gained, he could not endure having to repeat an experiment which ought, if complete care had been taken, to have told its story at first—and this gave him a continual anxiety that the experiment should not be wasted; he felt the experiment to be sacred, however slight a one it was. He wished to learn as much as possible from an experiment, so that he did not confine himself to observing the single point to which the experiment was directed, and his power of seeing a number of other things was wonderful. ... Any experiment done was to be of some use, and ... strongly he urged the necessity of keeping the notes of experiments which failed, and to this rule he always adhered.
In Charles Darwin: His Life Told in an Autobiographical Chapter, and in a Selected Series of his Published Letters (1908), 92.
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An old French geometer used to say that a mathematical theory was never to be considered complete till you had made it so clear that you could explain it to the first man you met in the street.
In Nature (1873), 8, 458.
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André Weil suggested that there is a logarithmic law at work: first-rate people attract other first-rate people, but second-rate people tend to hire third-raters, and third-rate people hire fifth-raters. If a dean or a president is genuinely interested in building and maintaining a high-quality university (and some of them are), then he must not grant complete self-determination to a second-rate department; he must, instead, use his administrative powers to intervene and set things right. That’s one of the proper functions of deans and presidents, and pity the poor university in which a large proportion of both the faculty and the administration are second-raters; it is doomed to diverge to minus infinity.
In I Want to be a Mathematician: an Automathography (1985), 123.
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Any conception which is definitely and completely determined by means of a finite number of specifications, say by assigning a finite number of elements, is a mathematical conception. Mathematics has for its function to develop the consequences involved in the definition of a group of mathematical conceptions. Interdependence and mutual logical consistency among the members of the group are postulated, otherwise the group would either have to be treated as several distinct groups, or would lie beyond the sphere of mathematics.
In 'Mathematics', Encyclopedia Britannica (9th ed.).
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Art matures. It is the formal elaboration of activity, complete in its own pattern. It is a cosmos of its own.
In Art Is Action: A Discussion of Nine Arts in a Modern World (1939), 29.
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As every circumstance relating to so capital a discovery as this (the greatest, perhaps, that has been made in the whole compass of philosophy, since the time of Sir Isaac Newton) cannot but give pleasure to all my readers, I shall endeavour to gratify them with the communication of a few particulars which I have from the best authority. The Doctor [Benjamin Franklin], after having published his method of verifying his hypothesis concerning the sameness of electricity with the matter lightning, was waiting for the erection of a spire in Philadelphia to carry his views into execution; not imagining that a pointed rod, of a moderate height, could answer the purpose; when it occurred to him, that, by means of a common kite, he could have a readier and better access to the regions of thunder than by any spire whatever. Preparing, therefore, a large silk handkerchief, and two cross sticks, of a proper length, on which to extend it, he took the opportunity of the first approaching thunder storm to take a walk into a field, in which there was a shed convenient for his purpose. But dreading the ridicule which too commonly attends unsuccessful attempts in science, he communicated his intended experiment to no body but his son, who assisted him in raising the kite.
The kite being raised, a considerable time elapsed before there was any appearance of its being electrified. One very promising cloud passed over it without any effect; when, at length, just as he was beginning to despair of his contrivance, he observed some loose threads of the hempen string to stand erect, and to avoid one another, just as if they had been suspended on a common conductor. Struck with this promising appearance, he inmmediately presented his knuckle to the key, and (let the reader judge of the exquisite pleasure he must have felt at that moment) the discovery was complete. He perceived a very evident electric spark. Others succeeded, even before the string was wet, so as to put the matter past all dispute, and when the rain had wetted the string, he collected electric fire very copiously. This happened in June 1752, a month after the electricians in France had verified the same theory, but before he had heard of any thing that they had done.
The History and Present State of Electricity, with Original Experiments (1767, 3rd ed. 1775), Vol. 1, 216-7.
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As he [Clifford] spoke he appeared not to be working out a question, but simply telling what he saw. Without any diagram or symbolic aid he described the geometrical conditions on which the solution depended, and they seemed to stand out visibly in space. There were no longer consequences to be deduced, but real and evident facts which only required to be seen. … So whole and complete was his vision that for the time the only strange thing was that anybody should fail to see it in the same way. When one endeavored to call it up again, and not till then, it became clear that the magic of genius had been at work, and that the common sight had been raised to that higher perception by the power that makes and transforms ideas, the conquering and masterful quality of the human mind which Goethe called in one word das Dämonische.
In Leslie Stephen and Frederick Pollock (eds.), Lectures and Essays by William Kingdon Clifford(1879), Vol. 1, Introduction, 4-5.
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As new areas of the world came into view through exploration, the number of identified species of animals and plants grew astronomically. By 1800 it had reached 70,000. Today more than 1.25 million different species, two-thirds animal and one-third plant, are known, and no biologist supposes that the count is complete.
In The Intelligent Man's Guide to Science: The Biological Sciences (1960), 654. Also in Isaac Asimov’s Book of Science and Nature Quotations (1988), 320.
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As soon as we touch the complex processes that go on in a living thing, be it plant or animal, we are at once forced to use the methods of this science [chemistry]. No longer will the microscope, the kymograph, the scalpel avail for the complete solution of the problem. For the further analysis of these phenomena which are in flux and flow, the investigator must associate himself with those who have labored in fields where molecules and atoms, rather than multicellular tissues or even unicellular organisms, are the units of study.
'Experimental and Chemical Studies of the Blood with an Appeal for More Extended Chemical Training for the Biological and Medical Investigator', Science (6 Aug 1915), 42, 176.
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Because the region of the Celestial World is of so great and such incredible magnitude as aforesaid, and since in what has gone before it was at least generally demonstrated that this comet continued within the limits of the space of the Aether, it seems that the complete explanation of the whole matter is not given unless we are also informed within narrower limits in what part of the widest Aether, and next to which orbs of the Planets [the comet] traces its path, and by what course it accomplishes this.
De Mundi Aetherei Recentioribus Phaenomenis (On Recent Phenomena in the Aetherial World) (1588). Quoted in M. Boas Hall, The Scientific Renaissance 1450-1630 (1962), 115.
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Besides accustoming the student to demand, complete proof, and to know when he has not obtained it, mathematical studies are of immense benefit to his education by habituating him to precision. It is one of the peculiar excellencies of mathematical discipline, that the mathematician is never satisfied with à peu près. He requires the exact truth. Hardly any of the non-mathematical sciences, except chemistry, has this advantage. One of the commonest modes of loose thought, and sources of error both in opinion and in practice, is to overlook the importance of quantities. Mathematicians and chemists are taught by the whole course of their studies, that the most fundamental difference of quality depends on some very slight difference in proportional quantity; and that from the qualities of the influencing elements, without careful attention to their quantities, false expectation would constantly be formed as to the very nature and essential character of the result produced.
In An Examination of Sir William Hamilton’s Philosophy (1878), 611. [The French phrase, à peu près means “approximately”. —Webmaster]
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Built up of carbon, hydrogen, oxygen, nitrogen, together with traces of a few other elements, yet of a complexity of structure that has hitherto resisted all attempts at complete analysis, protoplasm is at once the most enduring and the most easily destroyed of substances; its molecules are constantly breaking down to furnish the power for the manifestations of vital phenomena, and yet, through its remarkable property of assimilation, a power possessed by nothing else upon earth, it constantly builds up its substance anew from the surrounding medium.
In History of the Human Body (1919), 1.
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But ... the working scientist ... is not consciously following any prescribed course of action, but feels complete freedom to utilize any method or device whatever which in the particular situation before him seems likely to yield the correct answer. ... No one standing on the outside can predict what the individual scientist will do or what method he will follow.
…...
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But Truth is that besides which there is nothing: nothing to modify it, nothing to question it, nothing to form an exception: the all-inclusive, the complete — By Truth, I mean the Universal.
The Book of the Damned (1932). In The Complete Books of Charles Fort (1975), 9.
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Büchsel in his reminiscences from the life of a country parson relates that he sought his recreation in Lacroix’s Differential Calculus and thus found intellectual refreshment for his calling. Instances like this make manifest the great advantage which occupation with mathematics affords to one who lives remote from the city and is compelled to forego the pleasures of art. The entrancing charm of mathematics, which captivates every one who devotes himself to it, and which is comparable to the fine frenzy under whose ban the poet completes his work, has ever been incomprehensible to the spectator and has often caused the enthusiastic mathematician to be held in derision. A classic illustration is the example of Archimedes….
From Die Entwickelung der Mathematik im Zusammenhange mit der Ausbreitung der Kultur (1893), 22. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 186. From the original German, “Wenn Büchsel in seinen Erinnerungen aus dem Leben eines Landgeistlichen erzählt, dass er in der Differentialrechnung von Lacroix Erholung gesucht und geistige Erfrischung ftir seinen Beruf gefunden habe, so erkennen wir darin den grossen Vorzug, den die Beschaftigung mit der Mathematik für jemanden hat, der fern von einer Stadt lebt und auf ihre Kunstgenüsse verzichten muss. Der berückende Zauber der Mathematik, dem jeder unterliegt, der sich ihr ergiebt, und der dem holden Wahnsinn vergleichbar ist, unter dessen Bann der Dichter sein Work vollendet, ist dem betrachtenden Mitmenschen immer unbegreiflich gewesen und hat den begeisterten Mathematiker oft zum Gespött werden lassen. Als klassisches Beispiel wird jedem Schüler Archimedes…”
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Chromosomes … [contain] some kind of code-script the entire pattern of the individual’s future development and of its functioning in the mature state. Every complete set of chromosomes contains the full code.
In What is Life? : The Physical Aspect of the Living Cell (1944), 20.
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Communism is at once a complete system of proletarian ideology and a new social system. It is different from any other ideological and social system, and is the most complete, progressive, revolutionary, and rational system in human history.
In Mao Tse-Tung: On New Democracy: Talks at the Yenan Forum on Literature and Art (1967), 32.
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Dear Mr. Bell: … Sir Wm. Thomson … speaks with much enthusiasm of your achievement. What yesterday he would have declared impossible he has today seen realized, and he declares it the most wonderful thing he has seen in America. You speak of it as an embryo invention, but to him it seems already complete, and he declares that, before long, friends will whisper their secrets over the electric wire. Your undulating current he declares a great and happy conception.
Letter to Alexander Graham Bell (25 Jun 1876). Quoted in Alexander Graham Bell, The Bell Telephone: The Deposition of Alexander Graham Bell, in the Suit Brought by the United States to Annul the Bell Patents (1908), 101. Note: William Thomson is better known as Lord Kelvin.
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Descartes is the completest type which history presents of the purely mathematical type of mind—that in which the tendencies produced by mathematical cultivation reign unbalanced and supreme.
In An Examination of Sir William Hamilton’s Philosophy (1878), 626.
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Each workman would receive two or three important parts and would affix them together and pass them on to the next who would add a part and pass the growing article to another who would do the same … until the complete arm is put together.
Describing the first factory assembly line process. The product was Colt’s revolving chamber gun, nicknamed the “equalizer.”
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Even when all is known, the care of a man is not yet complete, because eating alone will not keep a man well; he must also take exercise. For food and exercise, while possessing opposite qualities, yet work together to produce health.
Regimen, in Hippocrates, trans. W. H. S. Jones (1931), Vol. 4, 229.
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Every complete set of chromosomes contains the full code; so there are, as a rule, two copies of the latter in the fertilized egg cell, which forms the earliest stage of the future individual. In calling the structure of the chromosome fibres a code-script we mean that the all-penetrating mind, once conceived by Laplace, to which every causal connection lay immediately open, could tell from their structure whether the egg would develop, under suitable conditions, into a black cock or into a speckled hen, into a fly or a maize plant, a rhododendron, a beetle, a mouse or a woman. To which we may add, that the appearances of the egg cells are very often remarkably similar; and even when they are not, as in the case of the comparatively gigantic eggs of birds and reptiles, the difference is not so much in the relevant structures as in the nutritive material which in these cases is added for obvious reasons.
But the term code-script is, of course, too narrow. The chromosome structures are at the same time instrumental in bringing about the development they foreshadow. They are law-code and executive power?or, to use another simile, they are architect's plan and builder’s craft-in one.
In What is Life? : The Physical Aspect of the Living Cell (1944), 20-21.
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Every investigator must before all things look upon himself as one who is summoned to serve on a jury. He has only to consider how far the statement of the case is complete and clearly set forth by the evidence. Then he draws his conclusion and gives his vote, whether it be that his opinion coincides with that of the foreman or not.
In The Maxims and Reflections of Goethe (1906), 190.
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Every new theory as it arises believes in the flush of youth that it has the long sought goal; it sees no limits to its applicability, and believes that at last it is the fortunate theory to achieve the 'right' answer. This was true of electron theory—perhaps some readers will remember a book called The Electrical Theory of the Universe by de Tunzelman. It is true of general relativity theory with its belief that we can formulate a mathematical scheme that will extrapolate to all past and future time and the unfathomed depths of space. It has been true of wave mechanics, with its first enthusiastic claim a brief ten years ago that no problem had successfully resisted its attack provided the attack was properly made, and now the disillusionment of age when confronted by the problems of the proton and the neutron. When will we learn that logic, mathematics, physical theory, are all only inventions for formulating in compact and manageable form what we already know, like all inventions do not achieve complete success in accomplishing what they were designed to do, much less complete success in fields beyond the scope of the original design, and that our only justification for hoping to penetrate at all into the unknown with these inventions is our past experience that sometimes we have been fortunate enough to be able to push on a short distance by acquired momentum.
The Nature of Physical Theory (1936), 136.
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Every scientist, through personal study and research, completes himself and his own humanity. ... Scientific research constitutes for you, as it does for many, the way for the personal encounter with truth, and perhaps the privileged place for the encounter itself with God, the Creator of heaven and earth. Science shines forth in all its value as a good capable of motivating our existence, as a great experience of freedom for truth, as a fundamental work of service. Through research each scientist grows as a human being and helps others to do likewise.
Address to the members of the Pontifical Academy of Sciences (13 Nov 2000). In L’Osservatore Romano (29 Nov 2000), translated in English edition, 5.
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Everything in nature is a puzzle until it finds its solution in man, who solves it in some way with God, and so completes the circle of creation.
The Appeal to Life (1891), 315.
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Felling a tree was possibly the original deed of appropriation of the natural earth by early mankind in Europe. Thousands of years ago,… man lifted a heavy flint tool and struck at the base of a tree. He may have wanted the tree for shelter and fuel, or possibly to make a bridge over a river or a path through a bog…. [E]ventually the tree crashed to the floor, and the first act in the slow possession of the land by its people was complete.
In The Man Who Made Things Out of Trees: The Ash in Human Culture and History (2015), Chap. 1.
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Finally in a large population, divided and subdivided into partially isolated local races of small size, there is a continually shifting differentiation among the latter (intensified by local differences in selection but occurring under uniform and static conditions) which inevitably brings about an indefinitely continuing, irreversible, adaptive, and much more rapid evolution of the species. Complete isolation in this case, and more slowly in the preceding, originates new species differing for the most part in nonadaptive parallel orthogenetic lines, in accordance with the conditions. It is suggested, in conclusion, that the differing statistical situations to be expected among natural species are adequate to account for the different sorts of evolutionary processes which have been described, and that, in particular, conditions in nature are often such as to bring about the state of poise among opposing tendencies on which an indefinitely continuing evolutionary process depends.
In 'Evolution In Mendelian Populations', Genetics, (1931), 16, 158.
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For example, there are numbers of chemists who occupy themselves exclusively with the study of dyestuffs. They discover facts that are useful to scientific chemistry; but they do not rank as genuine scientific men. The genuine scientific chemist cares just as much to learn about erbium—the extreme rarity of which renders it commercially unimportant—as he does about iron. He is more eager to learn about erbium if the knowledge of it would do more to complete his conception of the Periodic Law, which expresses the mutual relations of the elements.
From 'Lessons from the History of Science: The Scientific Attitude' (c.1896), in Collected Papers (1931), Vol. 1, 20.
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For some months the astronomer Halley and other friends of Newton had been discussing the problem in the following precise form: what is the path of a body attracted by a force directed toward a fixed point, the force varying in intensity as the inverse of the distance? Newton answered instantly, “An ellipse.” “How do you know?” he was asked. “Why, I have calculated it.” Thus originated the imperishable Principia, which Newton later wrote out for Halley. It contained a complete treatise on motion.
In The Handmaiden of the Sciences (1937), 37.
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Foreshadowings of the principles and even of the language of [the infinitesimal] calculus can be found in the writings of Napier, Kepler, Cavalieri, Pascal, Fermat, Wallis, and Barrow. It was Newton's good luck to come at a time when everything was ripe for the discovery, and his ability enabled him to construct almost at once a complete calculus.
In History of Mathematics (3rd Ed., 1901), 366.
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Fundamentally, as is readily seen, there exists neither force nor matter. Both are abstractions of things, such as they are, looked at from different standpoints. They complete and presuppose each other. Isolated they are meaningless. … Matter is not a go-cart, to and from which force, like a horse, can be now harnessed, now loosed. A particle of iron is and remains exactly the same thing, whether it shoot through space as a meteoric stone, dash along on the tire of an engine-wheel, or roll in a blood-corpuscle through the veins of a poet. … Its properties are eternal, unchangeable, untransferable.
From the original German text in 'Über die Lebenskraft', Preface to Untersuchungen über tierische Elektrizität (1848), xliii. As translated in Ludwig Büchner, Force and Matter: Or, Principles of the Natural Order of the Universe (1891), 1.
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Furthermore, it’s equally evident that what goes on is actually one degree better than self-reproduction, for organisms appear to have gotten more elaborate in the course of time. Today's organisms are phylogenetically descended from others which were vastly simpler than they are, so much simpler, in fact, that it’s inconceivable, how any kind of description of the latter, complex organism could have existed in the earlier one. It’s not easy to imagine in what sense a gene, which is probably a low order affair, can contain a description of the human being which will come from it. But in this case you can say that since the gene has its effect only within another human organism, it probably need not contain a complete description of what is to happen, but only a few cues for a few alternatives. However, this is not so in phylogenetic evolution. That starts from simple entities, surrounded by an unliving amorphous milieu, and produce, something more complicated. Evidently, these organisms have the ability to produce something more complicated than themselves.
From lecture series on self-replicating machines at the University of Illinois, Lecture 5 (Dec 1949), 'Re-evaluation of the Problems of Complicated Automata—Problems of Hierarchy and Evolution', Theory of Self-Reproducing Automata (1966).
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Genuine religion has its root deep down in the heart of humanity and in the reality of things. It is not surprising that by our methods we fail to grasp it: the actions of the Deity make no appeal to any special sense, only a universal appeal; and our methods are, as we know, incompetent to detect complete uniformity. There is a principle of Relativity here, and unless we encounter flaw or jar or change, nothing in us responds; we are deaf and blind therefore to the Immanent Grandeur, unless we have insight enough to recognise in the woven fabric of existence, flowing steadily from the loom in an infinite progress towards perfection, the ever-growing garment of a transcendent God.
Continuity: The Presidential Address to the British Association (1913), 92-93.
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Geological facts being of an historical nature, all attempts to deduce a complete knowledge of them merely from their still, subsisting consequences, to the exclusion of unexceptionable testimony, must be deemed as absurd as that of deducing the history of ancient Rome solely from the medals or other monuments of antiquity it still exhibits, or the scattered ruins of its empire, to the exclusion of a Livy, a Sallust, or a Tacitus.
Geological Essays (1799), 5.
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Geometry is the most complete science.
Quotation noted by Ivor Grattan-Guinness from unpublished Hilbert lecture course on Geometry.
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Governments and parliaments must find that astronomy is one of the sciences which cost most dear: the least instrument costs hundreds of thousands of dollars, the least observatory costs millions; each eclipse carries with it supplementary appropriations. And all that for stars which are so far away, which are complete strangers to our electoral contests, and in all probability will never take any part in them. It must be that our politicians have retained a remnant of idealism, a vague instinct for what is grand; truly, I think they have been calumniated; they should be encouraged and shown that this instinct does not deceive them, that they are not dupes of that idealism.
In Henri Poincaré and George Bruce Halsted (trans.), The Value of Science: Essential Writings of Henri Poincare (1907), 84.
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Here is the element or power of conduct, of intellect and knowledge, of beauty, and of social life and manners, and all needful to build up a complete human life. … We have instincts responding to them all, and requiring them all, and we are perfectly civilized only when all these instincts of our nature—all these elements in our civilization have been adequately recognized and satisfied.
Collected in Tryon Edwards (ed.), A Dictionary of Thoughts: Being a Cyclopedia of Laconic Quotations from the Best Authors Both Ancient and Modern (1891), 75.
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Hippocrates is an excellent geometer but a complete fool in everyday affairs.
Aristotle
…...
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How, indeed, can there be a response within to the impression from without when there is nothing within that is in relation of congenial vibration with that which is without? Inattention in such case is insusceptibility; and if this be complete, then to demand attention is very much like demanding of the eye that it should attend to sound-waves, and of the ear that it should attend to light-waves.
As quoted in William W. Speer, Primary Arithmetic: First Year, for the Use of Teachers (1902), 3.
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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.
In 'Ueber das Lehrziel im mathemalischen Unterricht der höheren Realanstalten', Jahresbericht der Deutschen Mathematiker Vereinigung, 2, 192. (The Annual Report of the German Mathematical Association. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 73.
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I attained a triumph so complete that it is now rare to meet an American with marks of small pox on his face... Benefits are valuable according to their duration and extent, like the showers from heaven, but the benign remedy Vaccination saves millions of lives every century, like the blessing of the sun, universal and everlasting.
(Remark made near the end of his life.)
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I first met J. Robert Oppenheimer on October 8, 1942, at Berkeley, Calif. There we discussed the theoretical research studies he was engaged in with respect to the physics of the bomb. Our discussions confirmed my previous belief that we should bring all of the widely scattered theoretical work together. … He expressed complete agreement, and it was then that the idea of the prompt establishment of a Los Alamos was conceived.”
In 'Some Recollections of July 16, 1945', Bulletin of the Atomic Scientists (Jun 1970), 26, No. 6, 21.
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I had this experience at the age of eight. My parents gave me a microscope. I don’t recall why, but no matter. I then found my own little world, completely wild and unconstrained, no plastic, no teacher, no books, no anything predictable. At first I did not know the names of the water-drop denizens or what they were doing. But neither did the pioneer microscopists. Like them, I graduated to looking at butterfly scales and other miscellaneous objects. I never thought of what I was doing in such a way, but it was pure science. As true as could be of any child so engaged, I was kin to Leeuwenhoek, who said that his work “was not pursued in order to gain the praise I now enjoy, but chiefly from a craving after knowledge, which I notice resides in me more that most other men.”
In The Creation: An Appeal to Save Life on Earth (2010), 143-144.
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I have destroyed almost the whole race of frogs, which does not happen in that savage Batrachomyomachia of Homer. For in the anatomy of frogs, which, by favour of my very excellent colleague D. Carolo Fracassato, I had set on foot in order to become more certain about the membranous substance of the lungs, it happened to me to see such things that not undeservedly I can better make use of that [saying] of Homer for the present matter—
“I see with my eyes a work trusty and great.”
For in this (frog anatomy) owing to the simplicity of the structure, and the almost complete transparency of the vessels which admits the eye into the interior, things are more clearly shown so that they will bring the light to other more obscure matters.
De Pulmonibus (1661), trans. James Young, Proceedings of the Royal Society of Medicine (1929-30), 23, 7.
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I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world... Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating. And that I have created something is undeniable: the question is about its value.
A Mathematician's Apology (1940), 90-1.
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I have never done anything “useful.” No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world... Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating. And that I have created something is undeniable: the question is about its value. [The things I have added to knowledge do not differ from] the creations of the other artists, great or small, who have left some kind of memorial beind them.
Concluding remarks in A Mathmatician's Apology (1940, 2012), 150-151.
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I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the “Law of Frequency of Error.” The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshaled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.
In Natural Inheritance (1894), 66.
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I observed that plants not only have a faculty to correct bad air in six to ten days, by growing in it…but that they perform this important office in a complete manner in a few hours; that this wonderful operation is by no means owing to the vegetation of the plant, but to the influence of light of the sun upon the plant.
In Tobias George Smollett (ed.), 'Experiments Upon Vegetables', The Critical Review, Or, Annals of Literature (1779), 48, 334.
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I should like to compare this rearrangement which the proteins undergo in the animal or vegetable organism to the making up of a railroad train. In their passage through the body parts of the whole may be left behind, and here and there new parts added on. In order to understand fully the change we must remember that the proteins are composed of Bausteine united in very different ways. Some of them contain Bausteine of many kinds. The multiplicity of the proteins is determined by many causes, first through the differences in the nature of the constituent Bausteine; and secondly, through differences in the arrangement of them. The number of Bausteine which may take part in the formation of the proteins is about as large as the number of letters in the alphabet. When we consider that through the combination of letters an infinitely large number of thoughts may be expressed, we can understand how vast a number of the properties of the organism may be recorded in the small space which is occupied by the protein molecules. It enables us to understand how it is possible for the proteins of the sex-cells to contain, to a certain extent, a complete description of the species and even of the individual. We may also comprehend how great and important the task is to determine the structure of the proteins, and why the biochemist has devoted himself with so much industry to their analysis.
'The Chemical Composition of the Cell', The Harvey Lectures (1911), 7, 45.
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I think the next [21st] century will be the century of complexity. We have already discovered the basic laws that govern matter and understand all the normal situations. We don’t know how the laws fit together, and what happens under extreme conditions. But I expect we will find a complete unified theory sometime this century. The is no limit to the complexity that we can build using those basic laws.
[Answer to question: Some say that while the twentieth century was the century of physics, we are now entering the century of biology. What do you think of this?]
'"Unified Theory" Is Getting Closer, Hawking Predicts', interview in San Jose Mercury News (23 Jan 2000), 29A. Answer quoted in Ashok Sengupta, Chaos, Nonlinearity, Complexity: The Dynamical Paradigm of Nature (2006), vii. Question included in Hans-Joachim Schellnhuber, Nicholas Stern and Mario Molina , Global Sustainability: a Nobel Cause (2010), 13. Cite from Brent Davis and Dennis J. Sumara, Complexity and Education: Inquiries Into Learning, Teaching, and Research (2006), 171.
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If we do discover a complete unified theory, it should be in time understandable in broad principle by everyone, not just a few scientists. Then we shall all, philosophers, scientists and just ordinary people, be able to take part in the discussion of why it is that we and the universe exist. If we find the answer to that, it would be the ultimate triumph of human reason—for then we would know the mind of God.
A Brief History of Time (1988), 191.
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In a class I was taking there was one boy who was much older than the rest. He clearly had no motive to work. I told him that, if he could produce for me, accurately to scale, drawings of the pieces of wood required to make a desk like the one he was sitting at, I would try to persuade the Headmaster to let him do woodwork during the mathematics hours—in the course of which, no doubt, he would learn something about measurement and numbers. Next day, he turned up with this task completed to perfection. This I have often found with pupils; it is not so much that they cannot do the work, as that they see no purpose in it.
In Mathematician's Delight (1943), 52.
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In England, more than in any other country, science is felt rather than thought. … A defect of the English is their almost complete lack of systematic thinking. Science to them consists of a number of successful raids into the unknown.
The Social Function of Science (1939), 197.
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In short, the greatest contribution to real security that science can make is through the extension of the scientific method to the social sciences and a solution of the problem of complete avoidance of war.
In "Science and Security", Science (25 Jun 1948), 107, 665. Written while Director of the U.S. National Bureau of Standards.
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In this respect mathematics fails to reproduce with complete fidelity the obvious fact that experience is not composed of static bits, but is a string of activity, or the fact that the use of language is an activity, and the total meanings of terms are determined by the matrix in which they are embedded.
In The Nature of Physical Theory (1936), 58.
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Intelligence is important in psychology for two reasons. First, it is one of the most scientifically developed corners of the subject, giving the student as complete a view as is possible anywhere of the way scientific method can be applied to psychological problems. Secondly, it is of immense practical importance, educationally, socially, and in regard to physiology and genetics.
From Intelligence: Its Structure, Growth and Action: Its Structure, Growth and Action (1987), 1.
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It has been asserted … that the power of observation is not developed by mathematical studies; while the truth is, that; from the most elementary mathematical notion that arises in the mind of a child to the farthest verge to which mathematical investigation has been pushed and applied, this power is in constant exercise. By observation, as here used, can only be meant the fixing of the attention upon objects (physical or mental) so as to note distinctive peculiarities—to recognize resemblances, differences, and other relations. Now the first mental act of the child recognizing the distinction between one and more than one, between one and two, two and three, etc., is exactly this. So, again, the first geometrical notions are as pure an exercise of this power as can be given. To know a straight line, to distinguish it from a curve; to recognize a triangle and distinguish the several forms—what are these, and all perception of form, but a series of observations? Nor is it alone in securing these fundamental conceptions of number and form that observation plays so important a part. The very genius of the common geometry as a method of reasoning—a system of investigation—is, that it is but a series of observations. The figure being before the eye in actual representation, or before the mind in conception, is so closely scrutinized, that all its distinctive features are perceived; auxiliary lines are drawn (the imagination leading in this), and a new series of inspections is made; and thus, by means of direct, simple observations, the investigation proceeds. So characteristic of common geometry is this method of investigation, that Comte, perhaps the ablest of all writers upon the philosophy of mathematics, is disposed to class geometry, as to its method, with the natural sciences, being based upon observation. Moreover, when we consider applied mathematics, we need only to notice that the exercise of this faculty is so essential, that the basis of all such reasoning, the very material with which we build, have received the name observations. Thus we might proceed to consider the whole range of the human faculties, and find for the most of them ample scope for exercise in mathematical studies. Certainly, the memory will not be found to be neglected. The very first steps in number—counting, the multiplication table, etc., make heavy demands on this power; while the higher branches require the memorizing of formulas which are simply appalling to the uninitiated. So the imagination, the creative faculty of the mind, has constant exercise in all original mathematical investigations, from the solution of the simplest problems to the discovery of the most recondite principle; for it is not by sure, consecutive steps, as many suppose, that we advance from the known to the unknown. The imagination, not the logical faculty, leads in this advance. In fact, practical observation is often in advance of logical exposition. Thus, in the discovery of truth, the imagination habitually presents hypotheses, and observation supplies facts, which it may require ages for the tardy reason to connect logically with the known. Of this truth, mathematics, as well as all other sciences, affords abundant illustrations. So remarkably true is this, that today it is seriously questioned by the majority of thinkers, whether the sublimest branch of mathematics,—the infinitesimal calculus—has anything more than an empirical foundation, mathematicians themselves not being agreed as to its logical basis. That the imagination, and not the logical faculty, leads in all original investigation, no one who has ever succeeded in producing an original demonstration of one of the simpler propositions of geometry, can have any doubt. Nor are induction, analogy, the scrutinization of premises or the search for them, or the balancing of probabilities, spheres of mental operations foreign to mathematics. No one, indeed, can claim preeminence for mathematical studies in all these departments of intellectual culture, but it may, perhaps, be claimed that scarcely any department of science affords discipline to so great a number of faculties, and that none presents so complete a gradation in the exercise of these faculties, from the first principles of the science to the farthest extent of its applications, as mathematics.
In 'Mathematics', in Henry Kiddle and Alexander J. Schem, The Cyclopedia of Education, (1877.) As quoted and cited in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 27-29.
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It has been said that no science is established on a firm basis unless its generalisations can be expressed in terms of number, and it is the special province of mathematics to assist the investigator in finding numerical relations between phenomena. After experiment, then mathematics. While a science is in the experimental or observational stage, there is little scope for discerning numerical relations. It is only after the different workers have “collected data” that the mathematician is able to deduce the required generalisation. Thus a Maxwell followed Faraday and a Newton completed Kepler.
In Higher Mathematics for Students of Chemistry and Physics (1902), 3.
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It is characteristic of our age to endeavour to replace virtues by technology. That is to say, wherever possible we strive to use methods of physical or social engineering to achieve goals which our ancestors thought attainable only by the training of character. Thus, we try so far as possible to make contraception take the place of chastity, and anaesthetics to take the place of fortitude; we replace resignation by insurance policies and munificence by the Welfare State. It would be idle romanticism to deny that such techniques and institutions are often less painful and more efficient methods of achieving the goods and preventing the evils which unaided virtue once sought to achieve and avoid. But it would be an equal and opposite folly to hope that the take-over of virtue by technology may one day be complete, so that the necessity for the laborious acquisition of the capacity for rational choice by individuals can be replaced by the painless application of the fruits of scientific discovery over the whole field of human intercourse and enterprise.
'Mental Health in Plato's Republic', in The Anatomy of the Soul: Historical Essays in the Philosophy of Mind (1973), 26.
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It is clear that in maize, seemingly blending is really segregating inheritance, but with entire absence of dominance, and it seems probably that the same will be found to be true among rabbits and other mammals; failure to observe it hitherto is probably due to the fact that the factors concerned are numerous. For the greater the number of factors concerned, the more nearly will the result obtained approximate a complete and permanent blend. As the number of factors approaches infinity, the result will become identical with a permanent blend.
Heredity: In Relation to Evolution and Animal Breeding (1911), 138-9.
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It is hard to imagine while strenuously walking in the heart of an equatorial rain forest, gasping for every breath in a stifling humid sauna, how people could have ever adapted to life under these conditions. It is not just the oppressive climate - the tall forest itself is dark, little light reaching the floor from the canopy, and you do not see any animals. It is a complete contrast to the herbivore-rich dry savannahs of tropical Africa. Yet there are many animals here, evident by the loud, continual noise of large cryptic insects and the constant threat of stepping on a deadly king cobra. This was my first impression of the rain forest in Borneo.
The Humans Who Went Extinct
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It is imperative in the design process to have a full and complete understanding of how failure is being obviated in order to achieve success. Without fully appreciating how close to failing a new design is, its own designer may not fully understand how and why a design works. A new design may prove to be successful because it has a sufficiently large factor of safety (which, of course, has often rightly been called a “factor of ignorance”), but a design's true factor of safety can never be known if the ultimate failure mode is unknown. Thus the design that succeeds (ie, does not fail) can actually provide less reliable information about how or how not to extrapolate from that design than one that fails. It is this observation that has long motivated reflective designers to study failures even more assiduously than successes.
In Design Paradigms: Case Histories of Error and Judgment in Engineering (1994), 31. books.google.comHenry Petroski - 1994
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It is more important to have beauty in one's equations than to have them fit experiment... It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress. If there is not complete agreement between the results of one's work and experiment, one should not allow oneself to be too discouraged, because the discrepancy may well be due to minor features that are not properly taken into account and that will get cleared up with further developments of the theory.
In 'The Evolution of the Physicist’s Picture of Nature', Scientific American, May 1963, 208, 47.
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It is not only a decided preference for synthesis and a complete denial of general methods which characterizes the ancient mathematics as against our newer Science [modern mathematics]: besides this extemal formal difference there is another real, more deeply seated, contrast, which arises from the different attitudes which the two assumed relative to the use of the concept of variability. For while the ancients, on account of considerations which had been transmitted to them from the Philosophie school of the Eleatics, never employed the concept of motion, the spatial expression for variability, in their rigorous system, and made incidental use of it only in the treatment of phonoromically generated curves, modern geometry dates from the instant that Descartes left the purely algebraic treatment of equations and proceeded to investigate the variations which an algebraic expression undergoes when one of its variables assumes a continuous succession of values.
In 'Untersuchungen über die unendlich oft oszillierenden und unstetigen Functionen', Ostwald’s Klassiker der exacten Wissenschaften (1905), No. 153, 44-45. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 115. From the original German, “Nicht allein entschiedene Vorliebe für die Synthese und gänzliche Verleugnung allgemeiner Methoden charakterisiert die antike Mathematik gegenüber unserer neueren Wissenschaft; es gibt neben diesem mehr äußeren, formalen, noch einen tiefliegenden realen Gegensatz, welcher aus der verschiedenen Stellung entspringt, in welche sich beide zu der wissenschaftlichen Verwendung des Begriffes der Veränderlichkeit gesetzt haben. Denn während die Alten den Begriff der Bewegung, des räumlichen Ausdruckes der Veränderlichkeit, aus Bedenken, die aus der philosophischen Schule der Eleaten auf sie übergegangen waren, in ihrem strengen Systeme niemals und auch in der Behandlung phoronomisch erzeugter Kurven nur vorübergehend verwenden, so datiert die neuere Mathematik von dem Augenblicke, als Descartes von der rein algebraischen Behandlung der Gleichungen dazu fortschritt, die Größenveränderungen zu untersuchen, welche ein algebraischer Ausdruck erleidet, indem eine in ihm allgemein bezeichnete Größe eine stetige Folge von Werten durchläuft.”
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It is only in mathematics, and to some extent in poetry, that originality may be attained at an early age, but even then it is very rare (Newton and Keats are examples), and it is not notable until adolescence is completed.
In A Study of British Genius (1904), 142
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It is unreasonable to expect science to produce a system of ethics—ethics are a kind of highway code for traffic among mankind—and the fact that in physics atoms which were yesterday assumed to be square are now assumed to be round is exploited with unjustified tendentiousness by all who are hungry for faith; so long as physics extends our dominion over nature, these changes ought to be a matter of complete indifference to you.
Letter to Oskar Pfister, 24 Feb 1928. Quoted in H. Meng and E. Freud (eds.), Psycho-Analysis and Faith: The Letters of Sigmund Freud and Oscar Pfister (1963), 123.
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It isn’t easy to become a fossil. … Only about one bone in a billion, it is thought, becomes fossilized. If that is so, it means that the complete fossil legacy of all the Americans alive today—that’s 270 million people with 206 bones each—will only be about 50 bones, one-quarter of a complete skeleton. That’s not to say, of course, that any of these bones will ever actually be found.
In A Short History of Nearly Everything (2003), 321-322.
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It need scarcely be pointed out that with such a mechanism complete isolation of portion of a species should result relatively rapidly in specific differentiation, and one that is not necessarily adaptive. The effective inter­group competition leading to adaptive advance may be between species rather than races. Such isolation is doubtless usually geographic in character at the outset but may be clinched by the development of hybrid sterility. The usual difference of the chromosome complements of related species puts the importance of chromosome aberration as an evolutionary process beyond question, but, as I see it, this importance is not in the character differences which they bring (slight in balanced types), but rather in leading to the sterility of hybrids and thus making permanent the isolation of two groups.
How far do the observations of actual species and their subdivisions conform to this picture? This is naturally too large a subject for more than a few suggestions.
That evolution involves non-adaptive differentiation to a large extent at the subspecies and even the species level is indicated by the kinds of differences by which such groups are actually distinguished by systematics. It is only at the subfamily and family levels that clear-cut adaptive differences become the rule. The principal evolutionary mechanism in the origin of species must thus be an essentially nonadaptive one.
In Proceedings of the Sixth International Congress of Genetics: Ithaca, New York, 1932 (1932) Vol. 1, 363-364.
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It [analysis] lacks at this point such plan and unity that it is really amazing that it can be studied by so many people. The worst is that it has not at all been treated with rigor. There are only a few propositions in higher analysis that have been demonstrated with complete rigor. Everywhere one finds the unfortunate manner of reasoning from the particular to the general, and it is very unusual that with such a method one finds, in spite of everything, only a few of what many be called paradoxes. It is really very interesting to seek the reason.
In my opinion that arises from the fact that the functions with which analysis has until now been occupied can, for the most part, be expressed by means of powers. As soon as others appear, something that, it is true, does not often happen, this no longer works and from false conclusions there flow a mass of incorrect propositions.
From a letter to his professor Hansteen in Christiania, Oslo in Correspondence (1902), 23 . In Umberto Bottazzini and Warren Van Egmond, The Higher Calculus (1986), 87-88.
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Johnson said that he could repeat a complete chapter of The Natural History of Iceland from the Danish of Horrebow, the whole of which was exactly thus: “There are no snakes to meet with throughout the whole island.
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Just as the musician is able to form an acoustic image of a composition which he has never heard played by merely looking at its score, so the equation of a curve, which he has never seen, furnishes the mathematician with a complete picture of its course. Yea, even more: as the score frequently reveals to the musician niceties which would escape his ear because of the complication and rapid change of the auditory impressions, so the insight which the mathematician gains from the equation of a curve is much deeper than that which is brought about by a mere inspection of the curve.
In Jahresbericht der Deutschen Mathematiker Vereiningung, 13, 864. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 190
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Kepler’s laws, although not rigidly true, are sufficiently near to the truth to have led to the discovery of the law of attraction of the bodies of the solar system. The deviation from complete accuracy is due to the facts, that the planets are not of inappreciable mass, that, in consequence, they disturb each other's orbits about the Sun, and, by their action on the Sun itself, cause the periodic time of each to be shorter than if the Sun were a fixed body, in the subduplicate ratio of the mass of the Sun to the sum of the masses of the Sun and Planet; these errors are appreciable although very small, since the mass of the largest of the planets, Jupiter, is less than 1/1000th of the Sun's mass.
In Isaac Newton and Percival Frost (ed.) Newton’s Principia: Sections I, II, III (1863), 216.
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Leaving aside genetic surgery applied humans, I foresee that the coming century will place in our hands two other forms of biological technology which are less dangerous but still revolutionary enough to transform the conditions of our existence. I count these new technologies as powerful allies in the attack on Bernal's three enemies. I give them the names “biological engineering” and “self-reproducing machinery.” Biological engineering means the artificial synthesis of living organisms designed to fulfil human purposes. Self-reproducing machinery means the imitation of the function and reproduction of a living organism with non-living materials, a computer-program imitating the function of DNA and a miniature factory imitating the functions of protein molecules. After we have attained a complete understanding of the principles of organization and development of a simple multicellular organism, both of these avenues of technological exploitation should be open to us.
From 3rd J.D. Bernal Lecture, Birkbeck College London (16 May 1972), The World, the Flesh and the Devil (1972), 6. Collected in The Scientist as Rebel (2006), 292. (The World, the Flesh & the Devil: An Enquiry into the Future of the Three Enemies of the Rational Soul is the title of a book by J. D Bernal, a scientist who pioneered X-ray crystallography.)
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Let us keep the discoveries and indisputable measurements of physics. But ... A more complete study of the movements of the world will oblige us, little by little, to turn it upside down; in other words, to discover that if things hold and hold together, it is only by reason of complexity, from above.
In Teilhard de Chardin and Bernard Wall (trans.), The Phenomenon of Man (1959, 2008), 43. Originally published in French as Le Phénomene Humain (1955).
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Let us now discuss the extent of the mathematical quality in Nature. According to the mechanistic scheme of physics or to its relativistic modification, one needs for the complete description of the universe not merely a complete system of equations of motion, but also a complete set of initial conditions, and it is only to the former of these that mathematical theories apply. The latter are considered to be not amenable to theoretical treatment and to be determinable only from observation.
From Lecture delivered on presentation of the James Scott prize, (6 Feb 1939), 'The Relation Between Mathematics And Physics', printed in Proceedings of the Royal Society of Edinburgh (1938-1939), 59, Part 2, 125.
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Life is too complicated to permit a complete understanding through the study of whole organisms. Only by simplifying a biological problem—breaking it down into a multitude of individual problems—can you get the answers.
From interview with Neil A. Campbell, in 'Crossing the Boundaries of Science', BioScience (Dec 1986), 36, No. 11, 738.
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Many billions of years will elapse before the smallest, youngest stars complete their nuclear burning and into white dwarfs. But with slow, agonizing finality perpetual night will surely fall.
In The Last Three Minutes: Conjectures About The Ultimate Fate Of The Universe (1994, 2008), 50.
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Mathematical knowledge, therefore, appears to us of value not only in so far as it serves as means to other ends, but for its own sake as well, and we behold, both in its systematic external and internal development, the most complete and purest logical mind-activity, the embodiment of the highest intellect-esthetics.
In 'Ueber Wert und angeblichen Unwert der Mathematik', Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 13, 381.
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Mathematics as we practice it is much more formally complete and precise than other sciences, but it is much less formally complete and precise for its content than computer programs.
In 'On Proof and Progress in Mathematics', For the Learning of Mathematics (Feb 1995), 15, No. 1, 33. Reprinted from Bulletin of the American Mathematical Society (1994), 30, No. 2, 170.
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Mathematics is the language of languages, the best school for sharpening thought and expression, is applicable to all processes in nature; and Germany needs mathematical gymnasia. Mathematics is God’s form of speech, and simplifies all things organic and inorganic. As knowledge becomes real, complete and great it approximates mathematical forms. It mediates between the worlds of mind and of matter.
Summarizing the ideas presented by Christian Heinrich Dillmann in Die Mathematik die Fackelträgerin einer neuen Zeit (1889). From book review, 'Recent Literature on Arithmetic and Arithmetical Teaching', in Granville Stanley Hall (ed.), The Pedagogical Seminary (1892), 2, 168. Dillmann’s book title translates as “Mathematics the Torchbearer of a New Era”. (However, Conant concluded that it was a “loosely-written, vague and incoherent book, which belies every anticipation awakened by its attractive title.”)
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Mathematics, as much as music or any other art, is one of the means by which we rise to a complete self-consciousness. The significance of mathematics resides precisely in the fact that it is an art; by informing us of the nature of our own minds it informs us of much that depends on our minds.
In Aspects of Science: Second Series (1926), 94.
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Mechanical Notation ... I look upon it as one of the most important additions I have made to human knowledge. It has placed the construction of machinery in the rank of a demonstrative science. The day will arrive when no school of mechanical drawing will be thought complete without teaching it.
Passages From the Life of a Philosopher (1864), 452.
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Medicine deals with the states of health and disease in the human body. It is a truism of philosophy that a complete knowledge of a thing can only be obtained by elucidating its causes and antecedents, provided, of course, such causes exist. In medicine it is, therefore, necessary that causes of both health and disease should be determined.
Avicenna
'Concerning the Subject-Matter of Medicine', in The Canon of Medicine, adapted by L. Bakhtiar (1999), 11.
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Medicine rests upon four pillars—philosophy, astronomy, alchemy, and ethics. The first pillar is the philosophical knowledge of earth and water; the second, astronomy, supplies its full understanding of that which is of fiery and airy nature; the third is an adequate explanation of the properties of all the four elements—that is to say, of the whole cosmos—and an introduction into the art of their transformations; and finally, the fourth shows the physician those virtues which must stay with him up until his death, and it should support and complete the three other pillars.
Vas Buch Paragranum (c.1529-30), in J. Jacobi (ed.), Paracelsus: Selected Writings (1951), 133-4.
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Mere poets are sottish as mere drunkards are, who live in a continual mist, without seeing or judging anything clearly. A man should be learned in several sciences, and should have a reasonable, philosophical and in some measure a mathematical head, to be a complete and excellent poet.
In Notes and Observations on The Empress of Morocco (1674), 70.
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Money, mechanization, algebra. The three monsters of contemporary civilization. Complete analogy.
In Gravity and Grace (1952), 209.
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Most of his [Euler’s] memoirs are contained in the transactions of the Academy of Sciences at St. Petersburg, and in those of the Academy at Berlin. From 1728 to 1783 a large portion of the Petropolitan transactions were filled by his writings. He had engaged to furnish the Petersburg Academy with memoirs in sufficient number to enrich its acts for twenty years—a promise more than fulfilled, for down to 1818 [Euler died in 1793] the volumes usually contained one or more papers of his. It has been said that an edition of Euler’s complete works would fill 16,000 quarto pages.
In History of Mathematics (1897), 263-264.
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My goal is simple. It is a complete understanding of the universe, why it is as it is and why it exists at all.
In the Washington Post, April 15, 1988.
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My method consists in allowing the mind to play freely for a very brief period, until a couple or so of ideas have passed through it, and then, while the traces or echoes of those ideas are still lingering in the brain, to turn the attention upon them with a sudden and complete awakening; to arrest, to scrutinise them, and to record their exact appearance... The general impression they have left upon me is like that which many of us have experienced when the basement of our house happens to be under thorough sanitary repairs, and we realise for the first time the complex system of drains and gas and water pipes, flues, bell-wires, and so forth, upon which our comfort depends, but which are usually hidden out of sight, and with whose existence, so long as they acted well, we had never troubled ourselves.
Inquiries into Human Faculty and its Development (1883),185-6.
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Nevertheless, it is necessary to remember that a planned economy is not yet socialism. A planned economy as such may be accompanied by the complete enslavement of the individual. The achievement of socialism requires the solution of some extremely difficult socio-political problems: how is it possible, in view of the far-reaching centralisation of political and economic power, to prevent bureaucracy from becoming all-powerful and overweening? How can the rights of the individual be protected and therewith a democratic counterweight to the power of bureaucracy be assured?
…...
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No doubt science cannot admit of compromises, and can only bring out the complete truth. Hence there must be controversy, and the strife may be, and sometimes must be, sharp. But must it even then be personal? Does it help science to attack the man as well as the statement? On the contrary, has not science the noble privilege of carrying on its controversies without personal quarrels?
In his collected writings of 1861, preface. Quoted in Proceedings of the Royal Society of London, Vol. 75, 300.
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No experiment is ever a complete failure. It can always be used as a bad example.
In John Cook, Steve Deger and Leslie Ann Gibson, The Book of Positive Quotations (2007), 669.
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No history of civilization can be tolerably complete which does not give considerable space to the explanation of scientific progress. If we had any doubts about this, it would suffice to ask ourselves what constitutes the essential difference between our and earlier civilizations. Throughout the course of history, in every period, and in almost every country, we find a small number of saints, of great artists, of men of science. The saints of to-day are not necessarily more saintly than those of a thousand years ago; our artists are not necessarily greater than those of early Greece; they are more likely to be inferior; and of course, our men of science are not necessarily more intelligent than those of old; yet one thing is certain, their knowledge is at once more extensive and more accurate. The acquisition and systematization of positive knowledge is the only human activity which is truly cumulative and progressive. Our civilization is essentially different from earlier ones, because our knowledge of the world and of ourselves is deeper, more precise, and more certain, because we have gradually learned to disentangle the forces of nature, and because we have contrived, by strict obedience to their laws, to capture them and to divert them to the gratification of our own needs.
Introduction to the History of Science (1927), Vol. 1, 3-4.
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No science is immune to the infection of politics and the corruption of power. … The time has come to consider how we might bring about a separation, as complete as possible, between Science and Government in all countries. I call this the disestablishment of science, in the same sense in which the churches have been disestablished and have become independent of the state.
In 'The Disestablishment of Science', Encounter (Jul 1971), 15.
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Nothing is more detestable to the physical anthropologist than... [the] wretched habit of cremating the dead. It involves not only a prodigal waste of costly fuel and excellent fertilizer, but also the complete destruction of physical historical data. On the other hand, the custom of embalming and mummification is most praiseworthy and highly to be recommended.
Up From the Ape (1931), 531.
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Now it is a well-known principle of zoological evolution that an isolated region, if large and sufficiently varied in its topography, soil, climate and vegetation, will give rise to a diversified fauna according to the law of adaptive radiation from primitive and central types. Branches will spring off in all directions to take advantage of every possible opportunity of securing food. The modifications which animals undergo in this adaptive radiation are largely of mechanical nature, they are limited in number and kind by hereditary, stirp or germinal influences, and thus result in the independent evolution of similar types in widely-separated regions under the law of parallelism or homoplasy. This law causes the independent origin not only of similar genera but of similar families and even of our similar orders. Nature thus repeats herself upon a vast scale, but the similarity is never complete and exact.
'The Geological and Faunal Relations of Europe and America during the Tertiary Period and the Theory of the Successive Invasions of an African Fauna', Science (1900), 11, 563-64.
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One can learn imitation history—kings and dates, but not the slightest idea of the motives behind it all; imitation literature—stacks of notes on Shakespeare’s phrases, and a complete destruction of the power to enjoy Shakespeare.
In Mathematician's Delight (1943), 8.
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Our experience shows that not everything that is observable and measurable is predictable, no matter how complete our past observations may have been.
In Presidential Address (8 Feb 1963), Quarterly Journal of the Royal Astronomical Society (Mar 1963), 4, 197.
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Our nature consists in motion; complete rest is death.
In Pensées (1670), Section 23, No. 67. From Blaise Pascal and W.F. Trotter (trans.), 'Thoughts', collected in Charles W. Eliot (ed.), The Harvard Classics (1910), Vol. 48, 50. Also seen translated as “…in movement, absolute rest…”. From the French, “Notre nature est dans le mouvement; le repos entier est la mort,” in Pensées de Blaise Pascal (1847), 315.
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Our only complete assurance of surviving World War III is to halt it before it starts.
Speech (10 Nov 1948) preceding Armistice Day, at Boston, Massachusetts.
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Pavlov’s data on the two fundamental antagonistic nervous processes—stimulation and inhibition—and his profound generalizations regarding them, in particular, that these processes are parts of a united whole, that they are in a state of constant conflict and constant transition of the one to the other, and his views on the dominant role they play in the formation of the higher nervous activity—all those belong to the most established natural—scientific validation of the Marxist dialectal method. They are in complete accord with the Leninist concepts on the role of the struggle between opposites in the evolution, the motion of matter.
In E. A. Asratyan, I. P. Pavlov: His Life and Work (1953), 153.
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Perfect behavior is born of complete indifference.
Diary entry for 21 Feb 1940, The Burning Brand: Diaries 1935-1950 (1961), 169.
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Perfect concordance among reformers is not to be expected; and men who are honestly struggling towards the light cannot hope to attain at one bound to the complete truth. There is always a danger lest the fascination of a new discovery should lead us too far. Men of science, being human, are apt, like lovers, to exaggerate the perfections and be a little blind to the faults of the object of their choice.
'The Anniversary Address of the President', Quarterly Journal of the Geological Society of London, 1885, 41, 55.
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Perhaps I occasionally sought to give, or inadvertently gave, to the student a sense of battle on the intellectual battlefield. If all you do is to give them a faultless and complete and uninhabited architectural masterpiece, then you do not help them to become builders of their own.
National Academy of Sciences, Biographical Memoirs Vol. 66 (1995), 86-87.
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Physical science comes nearest to that complete system of exact knowledge which all sciences have before them as an ideal. Some fall far short of it. The physicist who inveighs against the lack of coherence and the indefiniteness of theological theories, will probably speak not much less harshly of the theories of biology and psychology. They also fail to come up to his standard of methodology. On the other side of him stands an even superior being—the pure mathematician—who has no high opinion of the methods of deduction used in physics, and does not hide his disapproval of the laxity of what is accepted as proof in physical science. And yet somehow knowledge grows in all these branches. Wherever a way opens we are impelled to seek by the only methods that can be devised for that particular opening, not over-rating the security of our finding, but conscious that in this activity of mind we are obeying the light that is in our nature.
Swarthmore Lecture (1929) at Friends’ House, London, printed in Science and the Unseen World (1929), 77-78.
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Physical science enjoys the distinction of being the most fundamental of the experimental sciences, and its laws are obeyed universally, so far as is known, not merely by inanimate things, but also by living organisms, in their minutest parts, as single individuals, and also as whole communities. It results from this that, however complicated a series of phenomena may be and however many other sciences may enter into its complete presentation, the purely physical aspect, or the application of the known laws of matter and energy, can always be legitimately separated from the other aspects.
In Matter and Energy (1912), 9-10.
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Physical science is thus approaching the stage when it will be complete, and therefore uninteresting. Given the laws governing the motions of electrons and protons, the rest is merely geography—a collection of particular facts.
In What I Believe (1925), 2.
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Pure mathematics proves itself a royal science both through its content and form, which contains within itself the cause of its being and its methods of proof. For in complete independence mathematics creates for itself the object of which it treats, its magnitudes and laws, its formulas and symbols.
In Die Mathematik die Fackelträgerin einer neuen Zeit (1889), 94. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 11.
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Sample recommendation letter:
Dear Search Committee Chair,
I am writing this letter for Mr. John Smith who has applied for a position in your department. I should start by saying that I cannot recommend him too highly.
In fact, there is no other student with whom I can adequately compare him, and I am sure that the amount of mathematics he knows will surprise you.
His dissertation is the sort of work you don’t expect to see these days.
It definitely demonstrates his complete capabilities.
In closing, let me say that you will be fortunate if you can get him to work for you.
Sincerely,
A. D. Visor (Prof.)
In A Mathematician Reads the Newspaper (1995), 43
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Science and religion are in full accord, but science and faith are in complete discord.
In The Kahlil Gibran Reader: Inspirational Writings (2006), 41.
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Science itself, therefore, may be regarded as a minimal problem, consisting of the completest possible presentment of facts with the least possible expenditure of thought.
Ernst Mach and Thomas Joseph McCormick (trans.), The Science of Mechanics: a Critical and Historical Account of its Development (1919), 490.
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Science offers us complete mastery over our environment and our own destiny, yet instead of rejoicing we feel deeply afraid. Why should this be?
From transcript of BBC radio Reith Lecture (12 Nov 1967), 'A Runaway World', on the bbc.co.uk website.
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Scientific method is often defined as if it were a set procedure, to be learned, like a recipe, as if anyone could like a recipe, as if anyone could become a scientist simply by learning the method. This is as absurd ... [so I shall not] discuss scientific method, but rather the methods of scientists. We proceed by common sense and ingenuity. There are no rules, only the principles of integrity and objectivity, with a complete rejection of all authority except that of fact.
In Science in the Making (1957), 8.
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Several times every day I observed the portions of the polyp with a magnifying glass. On the 4th December, that is to say on the ninth day after having cut the polyp, I seemed in the morning to be able to perceive, on the edges of the anterior end of the second part (the part that had neither head nor arms), three little points arising from those edges. They immediately made me think of the horns that serve as the legs and arms of the polyp. Nevertheless I did not want to decide at once that these were actually arms that were beginning to grow. Throughout the next day I continually observed these points: this excited me extremely, and awaited with impatience the moment when I should know with certainty what they were. At last, on the following day, they were so big that there was no longer any room for doubt that they were actually arms growing at the anterior extremity of this second part. The next day two more arms started to grow out, and a few days later three more. The second part thus had eight of them, and they were all in a short time as long as those of the first part, that is to say as long as those the polyp possessed before it was cut. I then no longer found any difference between the second part and a polyp that had never been cut. I had remarked the same thing about the first part since the day after the operation. When I observed them with the magnifying glass with all the attention of which I was capable, each of the two appeared perceptibly to be a complete polyp, and they performed all the functions that were known to me: they extended, contracted, and walked.
Mémoires, pour servir à l'histoire d'un genre de polyps d'eau douce à bras en forme de cornes (1744), 7-16. Trans. John R. Baker, in Abraham Trembley of Geneva: Scientist and Philosopher 1710-1784 (1952), 32.
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The aim of science is, on the one hand, as complete a comprehension as possible of the connection between perceptible experiences in their totality, and, on the other hand, the achievement of this aim by employing a minimum of primary concepts and relations.
H. Cuny, Albert Einstein: The Man and his Theories (1963), 128.
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The belief that mathematics, because it is abstract, because it is static and cold and gray, is detached from life, is a mistaken belief. Mathematics, even in its purest and most abstract estate, is not detached from life. It is just the ideal handling of the problems of life, as sculpture may idealize a human figure or as poetry or painting may idealize a figure or a scene. Mathematics is precisely the ideal handling of the problems of life, and the central ideas of the science, the great concepts about which its stately doctrines have been built up, are precisely the chief ideas with which life must always deal and which, as it tumbles and rolls about them through time and space, give it its interests and problems, and its order and rationality. That such is the case a few indications will suffice to show. The mathematical concepts of constant and variable are represented familiarly in life by the notions of fixedness and change. The concept of equation or that of an equational system, imposing restriction upon variability, is matched in life by the concept of natural and spiritual law, giving order to what were else chaotic change and providing partial freedom in lieu of none at all. What is known in mathematics under the name of limit is everywhere present in life in the guise of some ideal, some excellence high-dwelling among the rocks, an “ever flying perfect” as Emerson calls it, unto which we may approximate nearer and nearer, but which we can never quite attain, save in aspiration. The supreme concept of functionality finds its correlate in life in the all-pervasive sense of interdependence and mutual determination among the elements of the world. What is known in mathematics as transformation—that is, lawful transfer of attention, serving to match in orderly fashion the things of one system with those of another—is conceived in life as a process of transmutation by which, in the flux of the world, the content of the present has come out of the past and in its turn, in ceasing to be, gives birth to its successor, as the boy is father to the man and as things, in general, become what they are not. The mathematical concept of invariance and that of infinitude, especially the imposing doctrines that explain their meanings and bear their names—What are they but mathematicizations of that which has ever been the chief of life’s hopes and dreams, of that which has ever been the object of its deepest passion and of its dominant enterprise, I mean the finding of the worth that abides, the finding of permanence in the midst of change, and the discovery of a presence, in what has seemed to be a finite world, of being that is infinite? It is needless further to multiply examples of a correlation that is so abounding and complete as indeed to suggest a doubt whether it be juster to view mathematics as the abstract idealization of life than to regard life as the concrete realization of mathematics.
In 'The Humanization of Teaching of Mathematics', Science, New Series, 35, 645-46.
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The chemist works along his own brilliant line of discovery and exposition; the astronomer has his special field to explore; the geologist has a well-defined sphere to occupy. It is manifest, however, that not one of these men can tell the whole tale, and make a complete story of creation. Another man is wanted. A man who, though not necessarily going into formal science, sees the whole idea, and speaks of it in its unity. This man is the theologian. He is not a chemist, an astronomer, a geologist, a botanist——he is more: he speaks of circles, not of segments; of principles, not of facts; of causes and purposes rather than of effects and appearances. Not that the latter are excluded from his study, but that they are so wisely included in it as to be put in their proper places.
In The People's Bible: Discourses Upon Holy Scripture: Vol. 1. Genesis (1885), 120.
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The concept of an independent system is a pure creation of the imagination. For no material system is or can ever be perfectly isolated from the rest of the world. Nevertheless it completes the mathematician’s “blank form of a universe” without which his investigations are impossible. It enables him to introduce into his geometrical space, not only masses and configurations, but also physical structure and chemical composition. Just as Newton first conclusively showed that this is a world of masses, so Willard Gibbs first revealed it as a world of systems.
The Order of Nature: An Essay (1917), 126.
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The demonstration that no possible combination of known substances, known forms of machinery and known forms of force, can be united in a practical machine by which men shall fly along distances through the air, seems to the writer as complete as it is possible for the demonstration to be.
Side-lights on Astronomy and Kindred Fields of Popular Science (1906), 345.
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The digestive canal is in its task a complete chemical factory. The raw material passes through a long series of institutions in which it is subjected to certain mechanical and, mainly, chemical processing, and then, through innumerable side-streets, it is brought into the depot of the body. Aside from this basic series of institutions, along which the raw material moves, there is a series of lateral chemical manufactories, which prepare certain reagents for the appropriate processing of the raw material.
Speech to the Society of Russian Physicians (Dec 1874). as translated in Daniel P. Todes, Pavlov’s Physiology Factory: Experiment, Interpretation, Laboratory Enterprise (2002), 155.
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The effects of general change in literature are most tellingly recorded not in alteration of the best products, but in the transformation of the most ordinary workaday books; for when potboilers adopt the new style, then the revolution is complete.
…...
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The enthusiasm of Sylvester for his own work, which manifests itself here as always, indicates one of his characteristic qualities: a high degree of subjectivity in his productions and publications. Sylvester was so fully possessed by the matter which for the time being engaged his attention, that it appeared to him and was designated by him as the summit of all that is important, remarkable and full of future promise. It would excite his phantasy and power of imagination in even a greater measure than his power of reflection, so much so that he could never marshal the ability to master his subject-matter, much less to present it in an orderly manner.
Considering that he was also somewhat of a poet, it will be easier to overlook the poetic flights which pervade his writing, often bombastic, sometimes furnishing apt illustrations; more damaging is the complete lack of form and orderliness of his publications and their sketchlike character, … which must be accredited at least as much to lack of objectivity as to a superfluity of ideas. Again, the text is permeated with associated emotional expressions, bizarre utterances and paradoxes and is everywhere accompanied by notes, which constitute an essential part of Sylvester’s method of presentation, embodying relations, whether proximate or remote, which momentarily suggested themselves. These notes, full of inspiration and occasional flashes of genius, are the more stimulating owing to their incompleteness. But none of his works manifest a desire to penetrate the subject from all sides and to allow it to mature; each mere surmise, conceptions which arose during publication, immature thoughts and even errors were ushered into publicity at the moment of their inception, with utmost carelessness, and always with complete unfamiliarity of the literature of the subject. Nowhere is there the least trace of self-criticism. No one can be expected to read the treatises entire, for in the form in which they are available they fail to give a clear view of the matter under contemplation.
Sylvester’s was not a harmoniously gifted or well-balanced mind, but rather an instinctively active and creative mind, free from egotism. His reasoning moved in generalizations, was frequently influenced by analysis and at times was guided even by mystical numerical relations. His reasoning consists less frequently of pure intelligible conclusions than of inductions, or rather conjectures incited by individual observations and verifications. In this he was guided by an algebraic sense, developed through long occupation with processes of forms, and this led him luckily to general fundamental truths which in some instances remain veiled. His lack of system is here offset by the advantage of freedom from purely mechanical logical activity.
The exponents of his essential characteristics are an intuitive talent and a faculty of invention to which we owe a series of ideas of lasting value and bearing the germs of fruitful methods. To no one more fittingly than to Sylvester can be applied one of the mottos of the Philosophic Magazine:
“Admiratio generat quaestionem, quaestio investigationem investigatio inventionem.”
In Mathematische Annalen (1898), 50, 155-160. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 176-178.
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The epoch of intense cold which preceded the present creation has been only a temporary oscillation of the earth’s temperature, more important than the century-long phases of cooling undergone by the Alpine valleys. It was associated with the disappearance of the animals of the diluvial epoch of the geologists, as still demonstrated by the Siberian mammoths; it preceded the uplifting of the Alps and the appearance of the present-day living organisms, as demonstrated by the moraines and the existence of fishes in our lakes. Consequently, there is complete separation between the present creation and the preceding ones, and if living species are sometimes almost identical to those buried inside the earth, we nevertheless cannot assume that the former are direct descendants of the latter or, in other words, that they represent identical species.
From Discours de Neuchâtel (1837), as translated by Albert V. Carozzi in Studies on Glaciers: Preceded by the Discourse of Neuchâtel (1967), lviii.
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The frequency of disastrous consequences in compound fracture, contrasted with the complete immunity from danger to life or limb in simple fracture, is one of the most striking as well as melancholy facts in surgical practice.
'On a New Method of Treating Compound Fracture, Abscesses, etc: With Observations on the Conditions of Supperation', Part I, The Lancet (1867), 326.
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The goal of science is clear—it is nothing short of the complete interpretation of the universe. But the goal is an ideal one—it marks the direction in which we move and strive, but never the point we shall actually reach.
From The Grammar of Science (1892), 17.
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The Grand Duke [of Tuscany] …after observing the Medicaean plants several times with me … has now invited me to attach myself to him with the annual salary of one thousand florins, and with the title of Philosopher and Principal Mathematicial to His Highness; without the duties of office to perform, but with the most complete leisure; so that I can complete my Treatises...
From a letter to Kepler. Quoted in John Elliot Drinkwater Bethune, Life of Galileo Galilei (1832), 63.
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The highest court is in the end one’s own conscience and conviction—that goes for you and for Einstein and every other physicist—and before any science there is first of all belief. For me, it is belief in a complete lawfulness in everything that happens.
Letter from Planck to Niels Bohr (19 Oct 1930). As cited in J. L. Heilbron, The Dilemmas of an Upright Man: Max Planck As Spokesman for German Science (1987), 143.
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The history of thought should warn us against concluding that because the scientific theory of the world is the best that has yet been formulated, it is necessarily complete and final. We must remember that at bottom the generalizations of science or, in common parlance, the laws of nature are merely hypotheses devised to explain that ever-shifting phantasmagoria of thought which we dignify with the high-sounding names of the world and the universe. In the last analysis magic, religion, and science are nothing but theories of thought.
In The Golden Bough: A Study in Magic and Religion (1890, 1900), Vol. 3, 460.
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The inhibitory nerves are of as fundamental importance in the economy of the body as the motor nerves. No evidence exists that the same nerve fibre is sometimes capable of acting as a motor nerve, sometimes as a nerve of inhibition, but on the contrary the latter nerves form a separate and complete nervous system subject to as definite anatomical and histological laws as the former.
'On the Structure, Distribution and Function of the Nerves which Innervate the Visceral and Vascular Systems', The Journal of Physiology, 1886, 7, 40.
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The laws of science are the permanent contributions to knowledge—the individual pieces that are fitted together in an attempt to form a picture of the physical universe in action. As the pieces fall into place, we often catch glimpses of emerging patterns, called theories; they set us searching for the missing pieces that will fill in the gaps and complete the patterns. These theories, these provisional interpretations of the data in hand, are mere working hypotheses, and they are treated with scant respect until they can be tested by new pieces of the puzzle.
In Commencement Address, California Institute of Technology (10 Jun 1938), 'Experiment and Experience'. Collected in abridged form in The Huntington Library Quarterly (Apr 1939), 2, No. 3, 244.
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The moon makes her circuit of the heaven in twenty-eight days plus about an hour, and with her return to the sign from which she set forth, completes a lunar month.
Vitruvius
In De Architectura, Book 9, Chap 1, Sec. 5. As translated in Morris Hicky Morgan (trans.), Vitruvius: The Ten Books on Architecture (1914), 258.
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The more a science advances, the more will it be possible to understand immediately results which formerly could be demonstrated only by means of lengthy intermediate considerations: a mathematical subject cannot be considered as finally completed until this end has been attained.
In Formensystem binärer Formen (1875), 2.
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The new mathematics is a sort of supplement to language, affording a means of thought about form and quantity and a means of expression, more exact, compact, and ready than ordinary language. The great body of physical science, a great deal of the essential facts of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of the great complex world-wide States that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and write.
Mankind in the Making (1903), 204.
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The opening of a foreign trade, by making them acquainted with new objects, or tempting them by the easier acquisition of things which they had not previously thought attainable, sometimes works a sort of industrial revolution in a country whose resources were previously undeveloped for want of energy and ambition in the people; inducing those who were satisfied with scanty comforts and little work to work harder for the gratification of their new tastes, and even to save, and accumulate capital, for the still more complete satisfaction of those tastes at a future time.
In Principles of Political Economy, with Some of Their Applications to Social Philosophy Vol. 1 (1873), Vol. 1, 351.
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The other line of argument, which leads to the opposite conclusion, arises from looking at artificial automata. Everyone knows that a machine tool is more complicated than the elements which can be made with it, and that, generally speaking, an automaton A, which can make an automaton B, must contain a complete description of B, and also rules on how to behave while effecting the synthesis. So, one gets a very strong impression that complication, or productive potentiality in an organization, is degenerative, that an organization which synthesizes something is necessarily more complicated, of a higher order, than the organization it synthesizes. This conclusion, arrived at by considering artificial automaton, is clearly opposite to our early conclusion, arrived at by considering living organisms.
From lecture series on self-replicating machines at the University of Illinois, Lecture 5 (Dec 1949), 'Re-evaluation of the Problems of Complicated Automata—Problems of Hierarchy and Evolution', Theory of Self-Reproducing Automata (1966).
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The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example, no mathematical truths apply merely to fish, or merely to stones, or merely to colours. So long as you are dealing with pure mathematics, you are in the realm of complete and absolute abstraction. … Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about.
In Science and the Modern World: Lowell Lectures, 1925 (1925), 31.
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The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example, no mathematical truths apply merely to fish, or merely to stones, or merely to colours. … Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about.
In 'Mathematics', Science and the Modern World (1926, 2011), 27.
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The progressive development of man is vitally dependent on invention. It is the most important product of his creative brain. Its ultimate purpose is the complete mastery of mind over the material world, the harnessing of the forces of nature to human needs.
Opening lines of 'My Inventions', Electrical Experimenter (1919). Reprinted in My Inventions: The Autobiography of Nikola Tesla (2011), 4.
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The propositions of mathematics have, therefore, the same unquestionable certainty which is typical of such propositions as “All bachelors are unmarried,” but they also share the complete lack of empirical content which is associated with that certainty: The propositions of mathematics are devoid of all factual content; they convey no information whatever on any empirical subject matter.
From 'On the Nature of Mathematical Truth', collected in Carl Hempel and James H. Fetzer (ed.), The Philosophy of Carl G. Hempel: Studies in Science, Explanation, and Rationality (2001), Chap. 1, 13. Also Carl Hempel, 'On the Nature of Mathematical Truth', collected in J.R. Newman (ed.), The World of Mathematics (1956), Vol. 3, 1631.
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The ravages committed by man subvert the relations and destroy the balance which nature had established between her organized and her inorganic creations; and she avenges herself upon the intruder, by letting loose upon her defaced provinces destructive energies hitherto kept in check by organic forces destined to be his best auxiliaries, but which he has unwisely dispersed and driven from the field of action. When the forest is gone, the great reservoir of moisture stored up in its vegetable mould is evaporated, and returns only in deluges of rain to wash away the parched dust into which that mould has been converted. The well-wooded and humid hills are turned to ridges of dry rock, which encumbers the low grounds and chokes the watercourses with its debris, and–except in countries favored with an equable distribution of rain through the seasons, and a moderate and regular inclination of surface–the whole earth, unless rescued by human art from the physical degradation to which it tends, becomes an assemblage of bald mountains, of barren, turfless hills, and of swampy and malarious plains. There are parts of Asia Minor, of Northern Africa, of Greece, and even of Alpine Europe, where the operation of causes set in action by man has brought the face of the earth to a desolation almost as complete as that of the moon; and though, within that brief space of time which we call “the historical period,” they are known to have been covered with luxuriant woods, verdant pastures, and fertile meadows, they are now too far deteriorated to be reclaimable by man, nor can they become again fitted for human use, except through great geological changes, or other mysterious influences or agencies of which we have no present knowledge, and over which we have no prospective control. The earth is fast becoming an unfit home for its noblest inhabitant, and another era of equal human crime and human improvidence, and of like duration with that through which traces of that crime and that improvidence extend, would reduce it to such a condition of impoverished productiveness, of shattered surface, of climatic excess, as to threaten the depravation, barbarism, and perhaps even extinction of the species.
Man and Nature, (1864), 42-3.
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The real purpose of scientific method is to make sure Nature hasn’t misled you into thinking you know something you don’t actually know. There’s not a mechanic or scientist or technician alive who hasn’t suffered from that one so much that he’s not instinctively on guard. … If you get careless or go romanticizing scientific information, giving it a flourish here and there, Nature will soon make a complete fool out of you.
In Zen and the Art of Motorcycle Maintenance: An inquiry into Values (1974), 100-101.
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The republic has no need of scientists [savants].
Apocryphal remark; supposedly the judge’s reply to Lavoisier when he appealed at his trial for more time to complete his scientific work.
Cited in H. Guerlac, 'Lavoisier', in Charles Gillispie (ed.), Dictionary of Scientific Biography (1973), 8, 85. However, Jean-Pierre Poirier and Rebecca Balinski (trans.) in Lavoisier: Chemist, Biologist, Economist (1998), 379, says about the remark that “It is now known that the statement is apocryphal” [meaning it is of questionable authenticity], and that “Jean Noel Halle, in the name of the Advisory Board for Arts and Trades, had sent the judges a long report detailing the services Lavoisier had rendered to the Republic, but Coffinal [president of the Revolutionary Tribunal] had refused to acknowledge it.” Thus, Lavoisier was guillotined on 8 May 1794. Coffinhal was himself guillotined on 6 Aug 1794.
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The Scientific Revolution turns us away from the older sayings to discover the lost authorization in Nature. What we have been through in these last four millennia is the slow inexorable profaning of our species. And in the last part of the second millennium A.D., that process is apparently becoming complete. It is the Great Human Irony of our noblest and greatest endeavor on this planet that in the quest for authorization, in our reading of the language of God in Nature, we should read there so clearly that we have been so mistaken.
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The scientific world-picture vouchsafes a very complete understanding of all that happens–it makes it just a little too understandable. It allows you to imagine the total display as that of a mechanical clockwork which, for all that science knows, could go on just the same as it does, without there being consciousness, will, endeavor, pain and delight and responsibility connected with it–though they actually are. And the reason for this disconcerting situation is just this: that for the purpose of constructing the picture of the external world, we have used the greatly simplifying device of cutting our own personality out, removing it; hence it is gone, it has evaporated, it is ostensibly not needed.
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The signs of liver inflammation [hepatitis] are eight in number, as follows: high fever, thirst, complete anorexia, a tongue which is initially red and then turns black, biliary vomitus initially yellow egg yolk in color, which later turns dark green, pain on the right side which ascends up to the clavicle. … Occasionally a mild cough may occur and a sensation of heaviness which is first felt on the right side and then spreads widely.
As quoted in Fred Rosner, The Medical Legacy of Moses Maimonides (1998), 53-54.
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The study of the serum of immunized animals forms a new chapter in the history of the struggle between the animal and infective agents, under which heading practical results of the highest importance are already inscribed. Any explanation of the phenomena is, however, still far from complete.
In Studies in Immunity (1909), 8.
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The transition from a paradigm in crisis to a new one from which a new tradition of normal science can emerge is far from a cumulative process, one achieved by an articulation or extension of the old paradigm. Rather it is a reconstruction of the field from new fundamentals, a reconstruction that changes some of the field's most elementary theoretical generalizations as well as many of its paradigm methods and applications. During the transition period there will be a large but never complete overlap between the problems that can be solved by the old and by the new paradigm. But there will also be a decisive difference in the modes of solution. When the transition is complete, the profession will have changed its view of the field, its methods, and its goals.
The Structure of Scientific Revolutions (1962), 84-5.
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The view of the moon that we’ve been having recently is really spectacular. It fills about three-quarters of the hatch window, and of course we can see the entire circumference even though part of it is in complete shadow and part of it is in earthshine. It’s a view worth the price of the trip.
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The volume now in press will make the new gospel of geology and mineralogy, and if I live to complete my mineralogical text-book, I shall do for the mineral what Darwin did for the organic world.
Letter to a friend (1886), as quoted in Paper read before the American Philosophical Society (1 Apr 1898) by James Douglas, printed as A Memoir of Thomas Sterry Hunt (1898), 12.
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The work of the inventor consists of conceptualizing, combining, and ordering what is possible according to the laws of nature. This inner working out which precedes the external has a twofold characteristic: the participation of the subconscious in the inventing subject; and that encounter with an external power which demands and obtains complete subjugation, so that the way to the solution is experienced as the fitting of one's own imagination to this power.
Philosophie der Technik (1927). 'Technology in Its Proper Sphere' translated by William Carroll. In Carl Mitcham (ed.) and Robert Mackey (ed.), Philosophy and Technology: Readings in the Philosophical Problems of Technology, (1972), Vol. 14, 321. In David Lovekin, Technique, Discourse, and Consciousness (1991), 73.
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Then if the first argument remains secure (for nobody will produce a neater one, than the length of the periodic time is a measure of the size of the spheres), the order of the orbits follows this sequence, beginning from the highest: The first and highest of all is the sphere of the fixed stars, which contains itself and all things, and is therefore motionless. It is the location of the universe, to which the motion and position of all the remaining stars is referred. For though some consider that it also changes in some respect, we shall assign another cause for its appearing to do so in our deduction of the Earth's motion. There follows Saturn, the first of the wandering stars, which completes its circuit in thirty years. After it comes Jupiter which moves in a twelve-year long revolution. Next is Mars, which goes round biennially. An annual revolution holds the fourth place, in which as we have said is contained the Earth along with the lunar sphere which is like an epicycle. In fifth place Venus returns every nine months. Lastly, Mercury holds the sixth place, making a circuit in the space of eighty days. In the middle of all is the seat of the Sun. For who in this most beautiful of temples would put this lamp in any other or better place than the one from which it can illuminate everything at the same time? Aptly indeed is he named by some the lantern of the universe, by others the mind, by others the ruler. Trismegistus called him the visible God, Sophocles' Electra, the watcher over all things. Thus indeed the Sun as if seated on a royal throne governs his household of Stars as they circle around him. Earth also is by no means cheated of the Moon's attendance, but as Aristotle says in his book On Animals the Moon has the closest affinity with the Earth. Meanwhile the Earth conceives from the Sun, and is made pregnant with annual offspring. We find, then, in this arrangement the marvellous symmetry of the universe, and a sure linking together in harmony of the motion and size of the spheres, such as could be perceived in no other way. For here one may understand, by attentive observation, why Jupiter appears to have a larger progression and retrogression than Saturn, and smaller than Mars, and again why Venus has larger ones than Mercury; why such a doubling back appears more frequently in Saturn than in Jupiter, and still more rarely in Mars and Venus than in Mercury; and furthermore why Saturn, Jupiter and Mars are nearer to the Earth when in opposition than in the region of their occultation by the Sun and re-appearance. Indeed Mars in particular at the time when it is visible throughout the night seems to equal Jupiter in size, though marked out by its reddish colour; yet it is scarcely distinguishable among stars of the second magnitude, though recognized by those who track it with careful attention. All these phenomena proceed from the same course, which lies in the motion of the Earth. But the fact that none of these phenomena appears in the fixed stars shows their immense elevation, which makes even the circle of their annual motion, or apparent motion, vanish from our eyes.
'Book One. Chapter X. The Order of the Heavenly Spheres', in Copernicus: On the Revolutions of the Heavenly Spheres (1543), trans. A. M. Duncan (1976), 49-51.
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There are three distinctions in the kinds of bodies, or three states, which have more especially claimed the attention of philosophical chemists; namely, those which are marked by the terms elastic fluids, liquids, and solids. A very familiar instance is exhibited to us in water, of a body, which, in certain circumstances, is capable of assuming all the three states. In steam we recognise a perfectly elastic fluid, in water, a perfect liquid, and in ice of a complete solid. These observations have tacitly led to the conclusion which seems universally adopted, that all bodies of sensible magnitude, whether liquid or solid, are constituted of a vast number of extremely small particles, or atoms of matter bound together by a force of attraction.
A New System of Chemical Philosophy (1808), Vol. 1, 141.
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There is probably no other science which presents such different appearances to one who cultivates it and to one who does not, as mathematics. To this person it is ancient, venerable, and complete; a body of dry, irrefutable, unambiguous reasoning. To the mathematician, on the other hand, his science is yet in the purple bloom of vigorous youth, everywhere stretching out after the “attainable but unattained” and full of the excitement of nascent thoughts; its logic is beset with ambiguities, and its analytic processes, like Bunyan’s road, have a quagmire on one side and a deep ditch on the other and branch off into innumerable by-paths that end in a wilderness.
In 'The Theory of Transformation Groups', (A review of Erster Abschnitt, Theorie der Transformationsgruppen (1888)), Bulletin New York Mathematical Society (1893), 2 (First series), 61.
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These works [the creation of the world] are recorded to have been completed in six days … because six is a perfect number … [and] the perfection of the works was signified by the number six.
From De Civitate Dei (early 400s), as translated by Marcus Dods, collected in The Works of Aurelius Augustine (1871), Vol. 1: 'Of the Perfection of the Number Six, which is the First of the Numbers Which is Composed of its Aliquot Parts', The City of God, Book 11, sect. 30, 474. Also seen translated as “Six is a number perfect in itself, and not because God created the world in six days; rather the contrary is true. God created the world in six days because this number is perfect, and it would remain perfect, even if the work of the six days did not exist.” [Note: 6 can be exactly divided (excluding itself) by 1, 2 or 3. It is a perfect number because those possible divisors also add up to a sum of 6.]
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Though the parallel is not complete, it is safe to say that science will never touch them unaided by its practical applications. Its wonders may be catalogued for purposes of education, they may be illustrated by arresting experiments, by numbers and magnitudes which startle or fatigue the imagination but they will form no familiar portion of the intellectual furniture of ordinary men unless they be connected, however remotely, with the conduct of ordinary life.
Decadence (1908), 53.
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Thought-economy is most highly developed in mathematics, that science which has reached the highest formal development, and on which natural science so frequently calls for assistance. Strange as it may seem, the strength of mathematics lies in the avoidance of all unnecessary thoughts, in the utmost economy of thought-operations. The symbols of order, which we call numbers, form already a system of wonderful simplicity and economy. When in the multiplication of a number with several digits we employ the multiplication table and thus make use of previously accomplished results rather than to repeat them each time, when by the use of tables of logarithms we avoid new numerical calculations by replacing them by others long since performed, when we employ determinants instead of carrying through from the beginning the solution of a system of equations, when we decompose new integral expressions into others that are familiar,—we see in all this but a faint reflection of the intellectual activity of a Lagrange or Cauchy, who with the keen discernment of a military commander marshalls a whole troop of completed operations in the execution of a new one.
In Populär-wissenschafliche Vorlesungen (1903), 224-225.
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To be whole. To be complete. Wildness reminds us what it means to be human, what we are connected to rather than what we are separate from.
Testimony before the Senate Subcommittee on Forest & Public Lands Management regarding the Utah Public Lands Management Act of 1995, Washington, D.C. (13 Jul 1995).
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To complete a PhD[,] I took courses in the history of philosophy. … As a result of my studies, I concluded that the traditional philosophy of science had little if anything to do with biology. … I had no use for a philosophy based on such an occult force as the vis vitalis. … But I was equally disappointed by the traditional philosophy of science, which was all based on logic, mathematics, and the physical sciences, and had adopted Descartes’ conclusion that an organism was nothing but a machine. This Cartesianism left me completely dissatisfied.
In 'Introduction', What Makes Biology Unique?: Considerations on the Autonomy of a Scientific Discipline (2007), 2.
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To complete a PhD[,] I took courses in the history of philosophy. … As a result of my studies, I concluded that the traditional philosophy of science had little if anything to do with biology. … I had no use for a philosophy based on such an occult force as the vis vitalis. … But I was equally disappointed by the traditional philosophy of science, which was all based on logic, mathematics, and the physical sciences, and had adopted Descartes’ conclusion that an organism was nothing but a machine. This Cartesianism left me completely dissatisfied.
In 'Introduction', What Makes Biology Unique?: Considerations on the Autonomy of a Scientific Discipline (2007), 2.
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To give a causal explanation of an event means to deduce a statement which describes it, using as premises of the deduction one or more universal laws, together with certain singular statements, the initial conditions ... We have thus two different kinds of statement, both of which are necessary ingredients of a complete causal explanation. (1959)
The Logic of Scientific Discovery: Logik Der Forschung (1959, 2002), 38.
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Under the... new hypothesis [of Continental Drift] certain geological concepts come to acquire a new significance amounting in a few cases to a complete inversion of principles, and the inquirer will find it necessary to re-orient his ideas. For the first time he will get glimpses... of a pulsating restless earth, all parts of which are in greater or less degree of movement in respect to the axis of rotation, having been so, moreover, throughout geological time. He will have to leave behind him—perhaps reluctantly—the dumbfounding spectacle of the present continental masses, firmly anchored to a plastic foundation yet remaining fixed in space; set thousands of kilometres apart, it may be, yet behaving in almost identical fashion from epoch to epoch and stage to stage like soldiers, at drill; widely stretched in some quarters at various times and astoundingly compressed in others, yet retaining their general shapes, positions and orientations; remote from one another through history, yet showing in their fossil remains common or allied forms of terrestrial life; possessed during certain epochs of climates that may have ranged from glacial to torrid or pluvial to arid, though contrary to meteorological principles when their existing geographical positions are considered -to mention but a few such paradoxes!
Our Wandering Continents: An Hypothesis of Continental Drifting (1937), 3.
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Unless we can “remember” ourselves, we are completely mechanical. Self-observation is possible only through self-remembering. These are the first steps in self-consciousness.
In On Love & Psychological Exercises: With Some Aphorisms & Other Essays (1998), 54.
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Very few, even among those who have taken the keenest interest in the progress of the revolution in natural knowledge set afoot by the publication of the “Origin of Species”; and who have watched, not without astonishment, the rapid and complete change which has been effected both inside and outside the boundaries of the scientific world in the attitude of men’s minds towards the doctrines which are expounded in that great work, can have been prepared for the extraordinary manifestation of affectionate regard for the man, and of profound reverence for the philosopher, which followed the announcement, on Thursday last, of the death of Mr Darwin.
'Obituary [of Charles Darwin]' (1882). In Collected Essays (1893), Vol. 2, 244.
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We are as yet got little farther than to the surface of things: yet ought we not to be discouraged; though we can never hope to attain to the complete knowledge of the texture, or constituent frame and nature of bodies, yet may we reasonably expect by this method of experiments, to make farther and farther advances abundantly sufficient to reward our pains.
In 'Preface', Statical Essays: Containing Hæmastatics (1769), Vol. 2, ii.
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We believe in the possibility of a theory which is able to give a complete description of reality, the laws of which establish relations between the things themselves and not merely between their probabilities ... God does not play dice.
…...
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We may now give the following more precise expression to our chief law of biogeny:— The evolution of the foetus (or ontogenesis) is a condensed and abbreviated recapitulation of the evolution of the stem (or phylogenesis); and this recapitulation is the more complete in proportion as the original development (or palingenesis) is preserved by a constant heredity; on the other hand, it becomes less complete in proportion as a varying adaptation to new conditions increases the disturbing factors in the development (or cenogenesis).
The Evolution of Man. Translated from the 5th edition of Anthropogenie by Joseph McCabe (1910), 8.
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We need another and a wiser and perhaps a more mystical concept of animals. Remote from universal nature, and living by complicated artifice, man in civilization surveys the creature through the glass of his knowledge and sees thereby a feather magnified and the whole image in distortion. We patronize them for their incompleteness, for their tragic fate of having taken form so far below ourselves. And therein we err, and greatly err. For the animal shall not be measured by man. In a world older and more complete than ours they move finished and complete, gifted with extensions of the senses we have lost or never attained, living by voices we shall never hear. They are not brethren, they are not underlings; they are other nations, caught with ourselves in the net of life and time, fellow prisoners of the splendour and travail of the earth.
The Outermost House (1928), 25.
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We often think that when we have completed our study of one we know all about two, because “two” is “one and one.” We forget that we still have to make a study of “and.”
…...
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We shall not have a complete theory until we can do more than merely say that “Things are as they are because they were as they were.”
From Inaugural Lecture (29 Apr 1980) as Lucasian Professor at Cambridge University, 'Is the End in Sight for Theoretical Physics?', collected in Black Holes and Baby Universes and Other Essays (1993), 51.
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What remains to be learned may indeed dwarf imagination. Nevertheless, the universe itself is closed and finite. … The uniformity of nature and the general applicability of natural laws set limits to knowledge. If there are just 100, or 105, or 110 ways in which atoms may form, then when one has identified the full range of properties of these, singly and in combination, chemical knowledge will be complete.
Presidential Address (28 Dec 1970) to the American Association for the Advancement of Science. 'Science: Endless Horizons or Golden Age?', Science (8 Jan 1971), 171, No. 3866, 24.
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When an investigator has developed a formula which gives a complete representation of the phenomena within a certain range, he may be prone to satisfaction. Would it not be wiser if he should say “Foiled again! I can find out no more about Nature along this line.”
Quoted in Astro-Physical Journal, 1945, 101, 133.
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When Ramanujan was sixteen, he happened upon a copy of Carr’s Synopsis of Mathematics. This chance encounter secured immortality for the book, for it was this book that suddenly woke Ramanujan into full mathematical activity and supplied him essentially with his complete mathematical equipment in analysis and number theory. The book also gave Ramanujan his general direction as a dealer in formulas, and it furnished Ramanujan the germs of many of his deepest developments.
In Mathematical Circles Squared (1972), 158. George Shoobridge Carr (1837-1914) wrote his Synopsis of Elementary Results in Mathematics in 1886.
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Whoever would not remain in complete ignorance of the resources which cause him to act; whoever would seize, at a single philosophical glance, the nature of man and animals, and their relations to external objects; whoever would establish, on the intellectual and moral functions, a solid doctrine of mental diseases, of the general and governing influence of the brain in the states of health and disease, should know, that it is indispensable, that the study of the organization of the brain should march side by side with that of its functions.
On the Organ of the Moral Qualities and Intellectual Faculties, and the Plurality of the Cerebral Organs (1835), 45-6.
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Why does a man want to be a scientist? There are many goals: fame, position, a thirst for understanding. The first two can be attained without intellectual integrity; the third cannot. … The thirst for knowledge, what Thomas Huxley called the ‘Divine dipsomania’, can only be satisfied by complete intellectual integrity. It seems to me the only one of the three goals that continues to reward the pursuer. He presses on, “knowing that Nature never did betray the heart that loved her”. Here is another kind of love, that has so many faces. Love is neither passion, nor pride, nor pity, nor blind adoration, but it can be any or all of these if they are transfigured by deep and unbiased understanding.
In Cecilia Payne-Gaposchkin: An Autobiography and Other Recollections (1996), 123.
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Why then does science work? The answer is that nobody knows. It is a complete mystery—perhaps the complete mystery&mdashwhy the human mind should be able to understand anything at all about the wider universe. ... Perhaps it is because our brains evolved through the working of natural law that they somehow resonate with natural law. ... But the mystery, really, is not that we are at one with the universe, but that we are so to some degree at odds with it, different from it, and yet can understand something about it. Why is this so?
Coming of Age in the Milky Way (1988), 385. In Vinoth Ramachandra, Subverting Global Myths: Theology and the Public Issues Shaping our World (2008), 185.
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You are not expected to complete the work in your lifetime. Nor must you refuse to do your unique part.
Talmud
…...
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You believe in the God who plays dice, and I in complete law and order in a world that objectively exists, and which I, in a wildly speculative way, am trying to capture. … Even the great initial success of the quantum theory does not make me believe in the fundamental dice-game, although I am well aware that our younger colleagues interpret this as a consequence of senility. No doubt the day will come when we will see whose instinctive attitude was the correct one.
Letter to Max Born (7 Sep 1944). In Born-Einstein Letters, 146. Einstein Archives 8-207. In Albert Einstein, Alice Calaprice, Freeman Dyson, The Ultimate Quotable Einstein (2011), 393-394. Often seen paraphrased as “I cannot believe that God plays dice with the cosmos.” Also see a related quote about God playing dice on the Stephen W. Hawking Quotes page of this website.
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[A]s you know, scientific education is fabulously neglected … This is an evil that is inherited, passed on from generation to generation. The majority of educated persons are not interested in science, and are not aware that scientific knowledge forms part of the idealistic background of human life. Many believe—in their complete ignorance of what science really is—that it has mainly the ancillary task of inventing new machinery, or helping to invent it, for improving our conditions of life. They are prepared to leave this task to the specialists, as they leave the repairing of their pipes to the plumber. If persons with this outlook decide upon the curriculum of our children, the result is necessarily such as I have just described it.
Opening remarks of the second of four public lectures for the Dublin Institute for Advanced Studies at University College, Dublin (Feb 1950), The Practical Achievements of Science Tending to Obliterate its True Import', collected in Science and Humanism: Physics in Our Time (1951). Reprinted in 'Nature and the Greeks' and 'Science and Humanism' (1996), 113.
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[Hermetic philosophy (Alchemy) was the only Art which might be able] to complete and bring to light not only medicine but also a universal Philosophy.
…quaeque sola non Medicinam tantum, sed et universam Philosophiam valde perficere et illustrare possit.
Original Latin in The Theological Works of Isaac Barrow (1859), 46. Translation, with citation, in B.J.T. Dobbs, The Foundations of Newton's Alchemy (1983), 96.
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[I shall not] discuss scientific method, but rather the methods of scientists. We proceed by common sense and ingenuity. There are no rules, only the principles of integrity and objectivity, with a complete rejection of all authority except that of fact.
In Science in the Making (1957), 9.
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[Nature] is complete, but never finished.
As quoted by T.H. Huxley, in Norman Lockyer (ed.), 'Nature: Aphorisms by Goethe', Nature (1870), 1, 10.
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… just as the astronomer, the physicist, the geologist, or other student of objective science looks about in the world of sense, so, not metaphorically speaking but literally, the mind of the mathematician goes forth in the universe of logic in quest of the things that are there; exploring the heights and depths for facts—ideas, classes, relationships, implications, and the rest; observing the minute and elusive with the powerful microscope of his Infinitesimal Analysis; observing the elusive and vast with the limitless telescope of his Calculus of the Infinite; making guesses regarding the order and internal harmony of the data observed and collocated; testing the hypotheses, not merely by the complete induction peculiar to mathematics, but, like his colleagues of the outer world, resorting also to experimental tests and incomplete induction; frequently finding it necessary, in view of unforeseen disclosures, to abandon one hopeful hypothesis or to transform it by retrenchment or by enlargement:—thus, in his own domain, matching, point for point, the processes, methods and experience familiar to the devotee of natural science.
In Lectures on Science, Philosophy and Art (1908), 26
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Carl Sagan Thumbnail In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. (1987) -- Carl Sagan
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