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Who said: “I have no satisfaction in formulas unless I feel their arithmetical magnitude.”
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Advance Quotes (162 quotes)


A great department of thought must have its own inner life, however transcendent may be the importance of its relations to the outside. No department of science, least of all one requiring so high a degree of mental concentration as Mathematics, can be developed entirely, or even mainly, with a view to applications outside its own range. The increased complexity and specialisation of all branches of knowledge makes it true in the present, however it may have been in former times, that important advances in such a department as Mathematics can be expected only from men who are interested in the subject for its own sake, and who, whilst keeping an open mind for suggestions from outside, allow their thought to range freely in those lines of advance which are indicated by the present state of their subject, untrammelled by any preoccupation as to applications to other departments of science. Even with a view to applications, if Mathematics is to be adequately equipped for the purpose of coping with the intricate problems which will be presented to it in the future by Physics, Chemistry and other branches of physical science, many of these problems probably of a character which we cannot at present forecast, it is essential that Mathematics should be allowed to develop freely on its own lines.
In Presidential Address British Association for the Advancement of Science, Section A, (1910), Nature, 84, 286.
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A new era of ocean exploration can yield discoveries that will help inform everything from critical medical advances to sustainable forms of energy. Consider that AZT, an early treatment for HIV, is derived from a Caribbean reef sponge, or that a great deal of energy—from offshore wind, to OTEC (ocean thermal energy conservation), to wind and wave energy—is yet untapped in our oceans.
In 'Why Exploring the Ocean is Mankind’s Next Giant Leap', contributed to CNN 'Lightyears Blog' (13 Mar 2012).
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A science cannot be played with. If an hypothesis is advanced that obviously brings into direct sequence of cause and effect all the phenomena of human history, we must accept it, and if we accept it, we must teach it.
In The Degradation of the Democratic Dogma (1919), 131.
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A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data. God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe.
In Scientific American (May 1963). As quoted and cited in The Hutchinson Encyclopedia of Science (1998), 468.
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A troubling question for those of us committed to the widest application of intelligence in the study and solution of the problems of men is whether a general understanding of the social sciences will be possible much longer. Many significant areas of these disciplines have already been removed by the advances of the past two decades beyond the reach of anyone who does not know mathematics; and the man of letters is increasingly finding, to his dismay, that the study of mankind proper is passing from his hands to those of technicians and specialists. The aesthetic effect is admittedly bad: we have given up the belletristic “essay on man” for the barbarisms of a technical vocabulary, or at best the forbidding elegance of mathematical syntax.
Opening paragraph of 'The Study of Man: Sociology Learns the Language of Mathematics' in Commentary (1 Sep 1952). Reprinted in James Roy Newman, The World of Mathematics (1956), Vol. 2, 1294.
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All advances in science consist either in enlarging the range of experience or in expressing the regularities found or to be found in it.
Presidential Address, Royal Astronomical Society, London (13 Feb 1953), 'On Science and Modern Cosmology', Monthly Notices of the Royal Astronomical Society (1953), 113, No. 3, 393.
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All revolutionary advances in science may consist less of sudden and dramatic revelations than a series of transformations, of which the revolutionary significance may not be seen (except afterwards, by historians) until the last great step. In many cases the full potentiality and force of a most radical step in such a sequence of transformations may not even be manifest to its author.
The Newtonian Revolution (1980), 162.
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Almost all of the space program’s important advances in scientific knowledge have been accomplished by hundreds of robotic spacecraft in orbit about Earth and on missions to the distant planets Mercury, Venus, Mars, Jupiter, Saturn, Uranus, and Neptune. Robotic exploration of the planets and their satellites as well as of comets and asteroids has truly revolutionized our knowledge of the solar system.
In 'Is Human Spaceflight Obsolete?', Issues in Science and Technology (Summer 2004).
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As knowledge advances, science ceases to scoff at religion; and religion ceases to frown on science. The hour of mockery by the one, and of reproof by the other, is passing away. Henceforth, they will dwell together in unity and goodwill. They will mutually illustrate the wisdom, power, and grace of God. Science will adorn and enrich religion; and religion will ennoble and sanctify science.
In Tryon Edwards, A Dictionary of Thoughts (1908), 505.
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As long as algebra and geometry proceeded along separate paths, their advance was slow and their applications limited. But when these sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace toward perfection.
In Leçons Élémentaires sur la Mathematiques, Leçon cinquième. As quoted and cited in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 81.
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As man advances in civilisation, and small tribes are united into larger communities, the simplest reason would tell each individual that he ought to extend his social instincts and sympathies to all the members of the same nation, though personally unknown to him. This point being once reached, there is only an artificial barrier to prevent his sympathies extending to the men of all nations and races.
…...
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As science has supplanted its predecessors, so it may hereafter be superseded by some more perfect hypothesis, perhaps by some totally different way of looking at the phenomena—of registering the shadows on the screen—of which we in this generation can form no idea. The advance of knowledge is an infinite progression towards a goal that for ever recedes.
In The Golden Bough: A Study in Magic and Religion (1890, 1900), Vol. 3, 460.
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As to how far in advance of the first flight the man should know he’s going. I’m not in agreement with the argument that says word should be delayed until the last possible moment to save the pilot from developing a bad case of the jitters. If we don’t have the confidence to keep from getting clutched at that time, we have no business going at all. If I’m the guy going, I’ll be glad to get the dope as soon as possible. As for keeping this a big secret from us and having us all suited up and then saying to one man “you go” and stuffing him in and putting the lid on that thing and away he goes, well, we’re all big boys now.
As he wrote in an article for Life (14 Sep 1959), 38. In fact, he was the first to fly in Earth orbit on 20 Feb 1962, though Alan Shepard was picked for the earlier first suborbital flight.
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As we advance in life we learn the limits of our abilities.
Inaugural Address Delivered to the University of St. Andrews, March 19, 1869 (1869), 3.
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As we conquer peak after peak we see in front of us regions full of interest and beauty, but we do not see our goal, wo do not see the horizon; in the distance tower still higher peaks, which will yield to those who ascend them still wider prospects, and deepen the feeling, the truth of which is emphasised by every advance in science, that “Great are the Works of the Lord.”
In Presidential Address to the British Association, as quoted in Arthur L. Foley, 'Recent Developments in Physical Science, The Popular Science Monthly (1910), 456.
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Endowed with two qualities, which seemed incompatible with each other, a volcanic imagination and a pertinacity of intellect which the most tedious numerical calculations could not daunt, Kepler conjectured that the movements of the celestial bodies must be connected together by simple laws, or, to use his own expression, by harmonic laws. These laws he undertook to discover. A thousand fruitless attempts, errors of calculation inseparable from a colossal undertaking, did not prevent him a single instant from advancing resolutely toward the goal of which he imagined he had obtained a glimpse. Twenty-two years were employed by him in this investigation, and still he was not weary of it! What, in reality, are twenty-two years of labor to him who is about to become the legislator of worlds; who shall inscribe his name in ineffaceable characters upon the frontispiece of an immortal code; who shall be able to exclaim in dithyrambic language, and without incurring the reproach of anyone, “The die is cast; I have written my book; it will be read either in the present age or by posterity, it matters not which; it may well await a reader, since God has waited six thousand years for an interpreter of his words.”
In 'Eulogy on Laplace', in Smithsonian Report for the year 1874 (1875), 131-132.
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Engineers apply the theories and principles of science and mathematics to research and develop economical solutions to practical technical problems. Their work is the link between scientific discoveries and commercial applications. Engineers design products, the machinery to build those products, the factories in which those products are made, and the systems that ensure the quality of the product and efficiency of the workforce and manufacturing process. They design, plan, and supervise the construction of buildings, highways, and transit systems. They develop and implement improved ways to extract, process, and use raw materials, such as petroleum and natural gas. They develop new materials that both improve the performance of products, and make implementing advances in technology possible. They harness the power of the sun, the earth, atoms, and electricity for use in supplying the Nation’s power needs, and create millions of products using power. Their knowledge is applied to improving many things, including the quality of health care, the safety of food products, and the efficient operation of financial systems.
Bureau of Labor Statistics, Occupational Outlook Handbook (2000) as quoted in Charles R. Lord. Guide to Information Sources in Engineering (2000), 5. This definition has been revised and expanded over time in different issues of the Handbook.
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Euclid and Archimedes are allowed to be knowing, and to have demonstrated what they say: and yet whosoever shall read over their writings without perceiving the connection of their proofs, and seeing what they show, though he may understand all their words, yet he is not the more knowing. He may believe, indeed, but does not know what they say, and so is not advanced one jot in mathematical knowledge by all his reading of those approved mathematicians.
In Conduct of the Understanding, sect. 24.
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Euler was a believer in God, downright and straightforward. The following story is told by Thiebault, in his Souvenirs de vingt ans de séjour à Berlin, … Thiebault says that he has no personal knowledge of the truth of the story, but that it was believed throughout the whole of the north of Europe. Diderot paid a visit to the Russian Court at the invitation of the Empress. He conversed very freely, and gave the younger members of the Court circle a good deal of lively atheism. The Empress was much amused, but some of her counsellors suggested that it might be desirable to check these expositions of doctrine. The Empress did not like to put a direct muzzle on her guest’s tongue, so the following plot was contrived. Diderot was informed that a learned mathematician was in possession of an algebraical demonstration of the existence of God, and would give it him before all the Court, if he desired to hear it. Diderot gladly consented: though the name of the mathematician is not given, it was Euler. He advanced toward Diderot, and said gravely, and in a tone of perfect conviction:
Monsieur, (a + bn) / n = x, donc Dieu existe; repondez!

Diderot, to whom algebra was Hebrew, was embarrassed and disconcerted; while peals of laughter rose on all sides. He asked permission to return to France at once, which was granted.
In Budget of Paradoxes (1878), 251. [The declaration in French expresses, “therefore God exists; please answer!” This Euler-Diderot anecdote, as embellished by De Morgan, is generally regarded as entirely fictional. Diderot before he became an encyclopedist was an accomplished mathematician and fully capable of recognizing—and responding to—the absurdity of an algebraic expression in proving the existence of God. See B.H. Brown, 'The Euler-Diderot Anecdote', The American Mathematical Monthly (May 1942), 49, No. 5, 392-303. —Webmaster.]
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Even mistaken hypotheses and theories are of use in leading to discoveries. This remark is true in all the sciences. The alchemists founded chemistry by pursuing chimerical problems and theories which are false. In physical science, which is more advanced than biology, we might still cite men of science who make great discoveries by relying on false theories. It seems, indeed, a necessary weakness of our mind to be able to reach truth only across a multitude of errors and obstacles.
An Introduction to the Study of Experimental Medicine (1865, translation 1927, 1957), 170.
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Every great advance of science opens our eyes to facts which we had failed before to observe, and makes new demands on our powers of interpretation.
From The Grammar of Science (1892), 17.
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Evolution advances, not by a priori design, but by the selection of what works best out of whatever choices offer. We are the products of editing, rather than of authorship.
In 'The Origin of Optical Activity', Annals of the New York Academy of Science (1957), 69, 367.
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Exercise in the most rigorous thinking that is possible will of its own accord strengthen the sense of truth and right, for each advance in the ability to distinguish between correct and false thoughts, each habit making for rigour in thought development will increase in the sound pupil the ability and the wish to ascertain what is right in life and to defend it.
In Anleitung zum mathematischen Unterricht in den höheren Schulen (1906), 28.
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Experimental observations are only experience carefully planned in advance, and designed to form a secure basis of new knowledge.
In The Design of Experiments (1935, 1970), 7.
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Genius iz always in advance ov the times, and makes sum magnificent hits, but the world owes most ov its tributes to good hoss sense.
In The Complete Works of Josh Billings (1876), 79.
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George Stephenson, with a sagacity of mind in advance of the science of his day, answered, when asked what was the ultimate cause of motion of his locomotive engine, ‘that it went by the bottled-up rays of the sun.’
From 'Fuel', Lecture delivered to the British Association at Bradford, printed in Nature (25 Sep 1873), 8, 443.
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He should avail himself of their resources in such ways as to advance the expression of the spirit in the life of mankind. He should use them so as to afford to every human being the greatest possible opportunity for developing and expressing his distinctively human capacity as an instrument of the spirit, as a centre of sensitive and intelligent awareness of the objective universe, as a centre of love of all lovely things, and of creative action for the spirit.
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I am not ... asserting that humans are either genial or aggressive by inborn biological necessity. Obviously, both kindness and violence lie with in the bounds of our nature because we perpetrate both, in spades. I only advance a structural claim that social stability rules nearly all the time and must be based on an overwhelmingly predominant (but tragically ignored) frequency of genial acts, and that geniality is therefore our usual and preferred response nearly all the time ... The center of human nature is rooted in ten thousand ordinary acts of kindness that define our days.
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I am pleased, however, to see the efforts of hypothetical speculation, because by the collisions of different hypotheses, truth may be elicited and science advanced in the end.
Letter (5 Sep 1822) to Mr. George F. Hopkins. Collected in The Writings of Thomas Jefferson: Correspondence (1854), Vol. 7, 260.
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I do not think we can impose limits on research. Through hundreds of thousands of years, man’s intellectual curiosity has been essential to all the gains we have made. Although in recent times we have progressed from chance and hit-or-miss methods to consciously directed research, we still cannot know in advance what the results may be. It would be regressive and dangerous to trammel the free search for new forms of truth.
In Margaret Mead and Rhoda Bubendey Métraux (ed.), Margaret Mead, Some Personal Views (1979), 89.
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I don’t believe in evolution, like people believe in God … Science and technology are not advanced by people who believe, but by people who don’t know but are doing their best to find out.
In Terry Pratchett, Ian Stewart and Jack Cohen, The Science of Discworld (2014), 41.
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I had made considerable advance ... in calculations on my favourite numerical lunar theory, when I discovered that, under the heavy pressure of unusual matters (two transits of Venus and some eclipses) I had committed a grievous error in the first stage of giving numerical value to my theory. My spirit in the work was broken, and I have never heartily proceeded with it since.
[Concerning his calculations on the orbital motion of the Moon.]
Private note (29 Sep 1890). In George Biddell Airy and Wilfrid Airy (ed.), Autobiography of Sir George Biddell Airy (1896), 350.
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I have attempted to form a judgment as to the conditions for evolution based on the statistical consequences of Mendelian heredity. The most general conclusion is that evolution depends on a certain balance among its factors. There must be a gene mutation, but an excessive rate gives an array of freaks, not evolution; there must be selection, but too severe a process destroys the field of variability, and thus the basis for further advance; prevalence of local inbreeding within a species has extremely important evolutionary consequences, but too close inbreeding leads merely to extinction. A certain amount of crossbreeding is favorable but not too much. In this dependence on balance the species is like a living organism. At all levels of organization life depends on the maintenance of a certain balance among its factors.
In Proceedings of the Sixth International Congress of Genetics: Ithaca, New York, 1932 (1932) Vol. 1, 365.
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I have been asked whether I would agree that the tragedy of the scientist is that he is able to bring about great advances in our knowledge, which mankind may then proceed to use for purposes of destruction. My answer is that this is not the tragedy of the scientist; it is the tragedy of mankind.
S. R. Weart and G. W. Sallard (eds.), Leo Szilard: His Version of the Facts (1978), 229.
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I have said that mathematics is the oldest of the sciences; a glance at its more recent history will show that it has the energy of perpetual youth. The output of contributions to the advance of the science during the last century and more has been so enormous that it is difficult to say whether pride in the greatness of achievement in this subject, or despair at his inability to cope with the multiplicity of its detailed developments, should be the dominant feeling of the mathematician. Few people outside of the small circle of mathematical specialists have any idea of the vast growth of mathematical literature. The Royal Society Catalogue contains a list of nearly thirty- nine thousand papers on subjects of Pure Mathematics alone, which have appeared in seven hundred serials during the nineteenth century. This represents only a portion of the total output, the very large number of treatises, dissertations, and monographs published during the century being omitted.
In Presidential Address British Association for the Advancement of Science, Section A, (1910), Nature, 84, 285.
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I learned this, at least, by my experiment: that if one advances confidently in the direction of his dreams, and endeavors to live the life which he has imagined, he will meet with a success unexpected in common hours.
In Walden: or, Life in the Woods (1854, 1893), 496.
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I never pick up an item without thinking of how I might improve it. I never perfected an invention that I did not think about in terms of the service it might give others. I want to save and advance human life, not destroy it. I am proud of the fact that I never invented weapons to kill. The dove is my emblem.
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If Louis Pasteur were to come out of his grave because he heard that the cure for cancer still had not been found, NIH would tell him, “Of course we'll give you assistance. Now write up exactly what you will be doing during the three years of your grant.” Pasteur would say, “Thank you very much,” and would go back to his grave. Why? Because research means going into the unknown. If you know what you are going to do in science, then you are stupid! This is like telling Michelangelo or Renoir that he must tell you in advance how many reds and how many blues he will buy, and exactly how he will put those colors together.
Interview for Saturday Evening Post (Jan/Feb 1981), 30.
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If there is technological advance without social advance, there is, almost automatically, an increase in human misery, in impoverishment.
The Other America (1962).
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If you throw a stone in a pond... the waves which strike against the shores are thrown back towards the spot where the stone struck; and on meeting other waves they never intercept each other’s course... In a small pond one and the same stroke gives birth to many motions of advance and recoil.
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In every section of the entire area where the word science may properly be applied, the limiting factor is a human one. We shall have rapid or slow advance in this direction or in that depending on the number of really first-class men who are engaged in the work in question. ... So in the last analysis, the future of science in this country will be determined by our basic educational policy.
Quoted in Vannevar Bush, Science, the Endless Frontier: A Report to the President, July 1945. In Transactions of the Kansas Academy of Science: Volumes 48-49, 246.
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In physiology, as in all other sciences, no discovery is useless, no curiosity misplaced or too ambitious, and we may be certain that every advance achieved in the quest of pure knowledge will sooner or later play its part in the service of man.
The Linacre Lecture on the Law of the Heart (1918), 147.
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In science the important thing is to modify and change one's ideas as science advances.
[Misattributed? See instead Claude Bernard]
Webmaster believes this is a quote by Claude Bernard, for whom examples date back to at least 1935, whereas Webmaster has found attribution to Spencer only as early as 1997. If you know the primary source from either Spencer or Bernard, please contact Webmaster.
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In science the new is an advance; but in morals, as contradicting our inner ideals and historic idols, it is ever a retrogression.
Levana, or, The Doctrine of Education translated from the German (1880), 123.
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In science, each of us knows that what he has accomplished will be antiquated in ten, twenty, fifty years. That is the fate to which science is subjected; it is the very meaning of scientific work, to which it is devoted in a quite specific sense, as compared with other spheres of culture for which in general the same holds. Every scientific “fulfilment” raises new “questions”; it asks to be “surpassed” and outdated. Whoever wishes to serve science has to resign himself to this fact. Scientific works certainly can last as “gratifications” because of their artistic quality, or they may remain important as a means of training. Yet they will be surpassed scientifically—let that be repeated—for it is our common fate and, more our common goal. We cannot work without hoping that others will advance further than we have. In principle, this progress goes on ad infinitum.
Max Weber
From a Speech (1918) presented at Munich University, published in 1919, and collected in 'Wissenschaft als Beruf', Gessammelte Aufsätze zur Wissenschaftslehre (1922), 524-525. As given in H.H. Gerth and C. Wright-Mills (translators and eds.), 'Science as a Vocation', Max Weber: Essays in Sociology (1946), 138. A different translation of a shorter excerpt from this quote, beginning “[In] the realm of science, …” is also on the Max Weber Quotes web page on this site.
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In that pure enjoyment experienced on approaching to the ideal, in that eagerness to draw aside the veil from the hidden truth, and even in that discord which exists between the various workers, we ought to see the surest pledges of further scientific success. Science thus advances, discovering new truths, and at the same time obtaining practical results.
In The Principles of Chemistry (1891), Vol. 1, preface, footnote, ix, as translated from the Russian 5th edition by George Kamensky, edited by A. J. Greenaway.
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In the distance tower still higher peaks which will yield to those who ascend them still wider prospects, and deepen the feeling whose truth is emphasized by every advance in science: that “Great are the Works of the Lord.”
From Inaugural Address to the Annual Meeting of the British Association for the Advancement of Science at Winnipeg. Collected in 'The British Association at Winnipeg',Nature (26 Aug 1909), 81. No. 2078, 257.
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In the main, Bacon prophesied the direction of subsequent progress. But he “anticipated” the advance. He did not see that the new science was for a long time to be worked in the interest of old ends of human exploitation. He thought that it would rapidly give man new ends. Instead, it put at the disposal of a class the means to secure their old ends of aggrandizement at the expense of another class. The industrial revolution followed, as he foresaw, upon a revolution in scientific method. But it is taking the revolution many centuries to produce a new mind.
In Democracy and Education: An Introduction to the Philosophy of Education (1916), 330-331.
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In the world of physics we watch a shadowgraph performance of the drama of familiar life. The shadow of my elbow rests on the shadow table as the shadow ink flows over the shadow paper. It is all symbolic, and as a symbol the physicist leaves it. ... The frank realization that physical science is concerned with a world of shadows is one of the most significant of recent advances.
In The Nature of the Physical World (1928, 2005), xiv-xv.
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Indeed, we need not look back half a century to times which many now living remember well, and see the wonderful advances in the sciences and arts which have been made within that period. Some of these have rendered the elements themselves subservient to the purposes of man, have harnessed them to the yoke of his labors and effected the great blessings of moderating his own, of accomplishing what was beyond his feeble force, and extending the comforts of life to a much enlarged circle, to those who had before known its necessaries only.
From paper 'Report of the Commissioners Appointed to Fix the Site of the University of Virginia' (Dec 1818), reprinted in Annual Report of the Board of Visitors of the University of Virginia for the Fiscal Year Ending May 31, 1879 (1879), 10. Collected in Commonwealth of Virginia, Annual Reports of Officers, Boards, and Institutions of the Commonwealth of Virginia, for the Year Ending September 30, 1879 (1879).
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It is characteristic of science that the full explanations are often seized in their essence by the percipient scientist long in advance of any possible proof.
The Origin of Life, 1967
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It is easy to follow in the sacred writings of the Jewish people the development of the religion of fear into the moral religion, which is carried further in the New Testament. The religions of all civilized peoples, especially those of the Orient, are principally moral religions. An important advance in the life of a people is the transformation of the religion of fear into the moral religion.
…...
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It is no way derogatory to Newton, or Kepler, or Galileo, that Science in these days should have advanced far beyond them. Rather is this itself their crown of glory. Their works are still bearing fruit, and will continue to do so. The truths which they discovered are still living in our knowledge, pregnant with infinite consequences.
Co-author with his brother Augustus William Hare Guesses At Truth, By Two Brothers: Second Edition: With Large Additions (1848), Second Series, 251. (The volume is introduced as “more than three fourths new.” This quote is identified as by Julius; Augustus had died in 1833.)
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It is not surprising that science has made comparatively little advance among us, but that … it should have made so much.
In Letter (3 Feb 1873) to the Committee of Arrangements, in Proceedings of the Farewell Banquet to Professor Tyndall (4 Feb 1873), 19. Reprinted as 'On the Importance of the Cultivation of Science', The Popular Science Monthly (1873), Vol. 2, 646.
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It may be that ... when the advance of destructive weapons enables everyone to kill everybody else nobody will want to kill anyone at all. [Referring to the hydrogen bomb.]
Parliamentary debate concerning the hydrogen bomb (Nov 1953). In Robert Rhodes James, ed. Winston Churchill: His Complete Speeches, 1897-1963 (1974), Vol. 6, p.8505.
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It was my Uncle George who discovered that alcohol was a food well in advance of modern medical thought.
'The Delayed Exit of Claude and Eustace', The Inimitable Jeeves (2011), 193.
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I’ve always been inspired by Dr. Martin Luther King, who articulated his Dream of an America where people are judged not by skin color but “by the content of their character.” In the scientific world, people are judged by the content of their ideas. Advances are made with new insights, but the final arbitrator of any point of view are experiments that seek the unbiased truth, not information cherry picked to support a particular point of view.
In letter (1 Feb 2013) to Energy Department employees announcing his decision not to serve a second term.
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Mathematicians deal with possible worlds, with an infinite number of logically consistent systems. Observers explore the one particular world we inhabit. Between the two stands the theorist. He studies possible worlds but only those which are compatible with the information furnished by observers. In other words, theory attempts to segregate the minimum number of possible worlds which must include the actual world we inhabit. Then the observer, with new factual information, attempts to reduce the list further. And so it goes, observation and theory advancing together toward the common goal of science, knowledge of the structure and observation of the universe.
Lecture to Sigma Xi, 'The Problem of the Expanding Universe' (1941), printed in Sigma Xi Quarterly (1942), 30, 104-105. Reprinted in Smithsonian Institution Report of the Board of Regents (1943), 97, 123. As cited by Norriss S. Hetherington in 'Philosophical Values and Observation in Edwin Hubble's Choice of a Model of the Universe', Historical Studies in the Physical Sciences (1982), 13, No. 1, 63.
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Mathematics accomplishes really nothing outside of the realm of magnitude; marvellous, however, is the skill with which it masters magnitude wherever it finds it. We recall at once the network of lines which it has spun about heavens and earth; the system of lines to which azimuth and altitude, declination and right ascension, longitude and latitude are referred; those abscissas and ordinates, tangents and normals, circles of curvature and evolutes; those trigonometric and logarithmic functions which have been prepared in advance and await application. A look at this apparatus is sufficient to show that mathematicians are not magicians, but that everything is accomplished by natural means; one is rather impressed by the multitude of skilful machines, numerous witnesses of a manifold and intensely active industry, admirably fitted for the acquisition of true and lasting treasures.
In Werke [Kehrbach] (1890), Bd. 5, 101. As quoted, cited and translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 13.
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Mathematics has often been characterized as the most conservative of all sciences. This is true in the sense of the immediate dependence of new upon old results. All the marvellous new advancements presuppose the old as indispensable steps in the ladder. … Inaccessibility of special fields of mathematics, except by the regular way of logically antecedent acquirements, renders the study discouraging or hateful to weak or indolent minds.
In Number and its Algebra (1896), 136.
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Most classifications, whether of inanimate objects or of organisms, are hierarchical. There are “higher” and “lower” categories, there are higher and lower ranks. What is usually overlooked is that the use of the term “hierarchy” is ambiguous, and that two fundamentally different kinds of arrangements have been designated as hierarchical. A hierarchy can be either exclusive or inclusive. Military ranks from private, corporal, sergeant, lieutenant, captain, up to general are a typical example of an exclusive hierarchy. A lower rank is not a subdivision of a higher rank; thus, lieutenants are not a subdivision of captains. The scala naturae, which so strongly dominated thinking from the sixteenth to the eighteenth century, is another good illustration of an exclusive hierarchy. Each level of perfection was considered an advance (or degradation) from the next lower (or higher) level in the hierarchy, but did not include it.
The Growth of Biological Thought: Diversity, Evolution and Inheritance (1982), 205-6.
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Nature! … Incessant life, development, and movement are in her, but she advances not. She changes for ever and ever, and rests not a moment. Quietude is inconceivable to her, and she has laid her curse upon rest. She is firm. Her steps are measured, her exceptions rare, her laws unchangeable.
As quoted by T.H. Huxley, in Norman Lockyer (ed.), 'Nature: Aphorisms by Goethe', Nature (1870), 1, 9.
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No country in the world is so much indebted for its progress in power and intelligence to science than ours, and yet no country does so little to encourage or advance it.
Address (2 Jun 1874) at the Laying of the Cornerstone of the American Museum of Natural History, in Fifth and Sixth Annual Reports of the American Museum of Natural History (1 Dec 1874), 47.
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No great advance has ever been made in science, politics, or religion, without controversy.
From'The Faith Once Delivered to the Saints', Sermon VII collected in Sermons Delivered on Various Occasions (1828), 259.
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Now, it may be stretching an analogy to compare epidemics of cholera—caused by a known agent—with that epidemic of violent crime which is destroying our cities. It is unlikely that our social problems can be traced to a single, clearly defined cause in the sense that a bacterial disease is ‘caused’ by a microbe. But, I daresay, social science is about as advanced in the late twentieth century as bacteriological science was in the mid nineteenth century. Our forerunners knew something about cholera; they sensed that its spread was associated with misdirected sewage, filth, and the influx of alien poor into crowded, urban tenements. And we know something about street crime; nowhere has it been reported that a member of the New York Stock Exchange has robbed ... at the point of a gun. Indeed, I am naively confident that an enlightened social scientist of the next century will be able to point out that we had available to us at least some of the clues to the cause of urban crime.
'Cholera at the Harvey,' Woods Hole Cantata: Essays on Science and Society (1985).
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One of the principal results of civilization is to reduce more and more the limits within which the different elements of society fluctuate. The more intelligence increases the more these limits are reduced, and the nearer we approach the beautiful and the good. The perfectibility of the human species results as a necessary consequence of all our researches. Physical defects and monstrosities are gradually disappearing; the frequency and severity of diseases are resisted more successfully by the progress of modern science; the moral qualities of man are proving themselves not less capable of improvement; and the more we advance, the less we shall have need to fear those great political convulsions and wars and their attendant results, which are the scourges of mankind.
…...
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Our advanced and fashionable thinkers are, naturally, out on a wide swing of the pendulum, away from the previous swing of the pendulum.... They seem to have an un-argue-out-able position, as is the manner of sophists, but this is no guarantee that they are right.
In Science is a Sacred Cow (1950), 177-78.
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Pathology, probably more than any other branch of science, suffers from heroes and hero-worship. Rudolf Virchow has been its archangel and William Welch its John the Baptist, while Paracelsus and Cohnheim have been relegated to the roles of Lucifer and Beelzebub. … Actually, there are no heroes in Pathology—all of the great thoughts permitting advance have been borrowed from other fields, and the renaissance of pathology stems not from pathology itself but from the philosophers Kant and Goethe.
Quoted from an address to a second year class, in Levin L. Waters, obituary for Harry S. N. Greene, M.D., in Yale Journal of Biology and Medicine (Feb-Apr 1971), 43:4-5, 207.
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Quite distinct from the theoretical question of the manner in which mathematics will rescue itself from the perils to which it is exposed by its own prolific nature is the practical problem of finding means of rendering available for the student the results which have been already accumulated, and making it possible for the learner to obtain some idea of the present state of the various departments of mathematics. … The great mass of mathematical literature will be always contained in Journals and Transactions, but there is no reason why it should not be rendered far more useful and accessible than at present by means of treatises or higher text-books. The whole science suffers from want of avenues of approach, and many beautiful branches of mathematics are regarded as difficult and technical merely because they are not easily accessible. … I feel very strongly that any introduction to a new subject written by a competent person confers a real benefit on the whole science. The number of excellent text-books of an elementary kind that are published in this country makes it all the more to be regretted that we have so few that are intended for the advanced student. As an example of the higher kind of text-book, the want of which is so badly felt in many subjects, I may mention the second part of Prof. Chrystal’s Algebra published last year, which in a small compass gives a great mass of valuable and fundamental knowledge that has hitherto been beyond the reach of an ordinary student, though in reality lying so close at hand. I may add that in any treatise or higher text-book it is always desirable that references to the original memoirs should be given, and, if possible, short historic notices also. I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history.
In Presidential Address British Association for the Advancement of Science, Section A (1890), Nature, 42, 466.
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Realizing how often ingenious speculation in the complex biological world has led nowhere and how often the real advances in biology as well as in chemistry, physics and astronomy have kept within the bounds of mechanistic interpretation, we geneticists should rejoice, even with our noses on the grindstone (which means both eyes on the objectives), that we have at command an additional means of testing whatever original ideas pop into our heads.
'The Rise of Genetics', Science (1932), 1969, 264.
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Religion has run out of justifications. Thanks to the telescope and the microscope, it no longer offers an explanation of anything important. Where once it used to able, by its total command of a worldview, to prevent the emergence of rivals, it can now only impede and retard—or try to turn back—the measureable advances that we have made.
In God is not Great: How Religion Poisons Everything (2007, 2009), 282.
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Science advances through tentative answers to a series of more and more subtle questions which reach deeper and deeper into the essence of natural phenomena.
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Science and knowledge are subject, in their extension and increase, to laws quite opposite to those which regulate the material world. Unlike the forces of molecular attraction, which cease at sensible distances; or that of gravity, which decreases rapidly with the increasing distance from the point of its origin; the farther we advance from the origin of our knowledge, the larger it becomes, and the greater power it bestows upon its cultivators, to add new fields to its dominions.
In 'Future Prospects', On the Economy of Machinery and Manufactures (1st ed., 1832), chap. 32, 277-278.
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Science and technology have freed humanity from many burdens and given us this new perspective and great power. This power can be used for the good of all. If wisdom governs our actions; but if the world is mad or foolish, it can destroy itself just when great advances and triumphs are almost without its grasp.
As quoted in Suranjan Das 'The Nehru Years in Indian politics', Edinburgh Papers on South Asian Studies (16 Nov 2001), 16, 230. As cited in M.J. Vinod and Meena Deshpande, Contemporary Political Theory (2013), 507. Vinod and Deshpande introduce the quote by writing “Nehru was largely instrumental for building a scientific temper and culture in India” and “emphasized the need for building national laboratories and research institutes.”
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Science can be thought of as a large pool of knowledge, fed by a steady flow from the tap of basic research. Every now and then the water is dipped out and put to use, but one never knows which part of the water will be needed. This confuses the funding situation for basic science, because usually no specific piece of scientific work can be justified in advance; one cannot know which is going to be decisive. Yet history shows that keeping water flowing into the pool is a very worthwhile enterprise.
In 'Technology Development', Science (1983), 220, 576-580. As quoted and cited in H. Charles Romesburg, Best Research Practices (2009), 213.
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Science differs from politics or religion, in precisely this one discipline: we agree in advance to simply reject our own findings when they have been shown to be in error.
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Science is a human activity, and the best way to understand it is to understand the individual human beings who practise it. Science is an art form and not a philosophical method. The great advances in science usually result from new tools rather than from new doctrines. ... Every time we introduce a new tool, it always leads to new and unexpected discoveries, because Nature's imagination is richer than ours.
Concluding remark from 'The Scientist As Rebel' American Mathemtical Monthly (1996), 103, 805. Reprinted in The Scientist as Rebel (2006), 17-18, identified as originally written for a lecture (1992), then published as an essay in the New York Review.
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Science is advancing faster than ever, and on a broader front.
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Science is, and must be, culturally embedded; what else could the product of human passion be?... Culture is not the enemy of objectivity but a matrix that can either aid or retard advancing knowledge.
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Science never makes an advance until philosophy authorizes it to do so.
Essay on Freud (1937). Quoted in Charles W. Misner, Kip S. Thorne and John Archibald Wheeler, Gravitation (1973), 1208.
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Science now finds itself in paradoxical strife with society: admired but mistrusted; offering hope for the future but creating ambiguous choice; richly supported yet unable to fulfill all its promise; boasting remarkable advances but criticized for not serving more directly the goals of society.
How to Win the Nobel Prize: An Unexpected Life in Science (2004), xi.
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Science only advances by renouncing its past.
Webmaster is suspicious of the authenticity of this quote. An epigraph, without citation, in Chandler Burr, Separate Creation: The Search for the Biological Origins of Sexual Orientation (1997), 309. Seen in only a few recent places. It does not seem to appear in earlier decades, as might be expected for such a snappy quote. Perhaps it is a restatement of another quote. Please contact if you know the primary source.
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Science … is perpetually advancing. It is like a torch in the sombre forest of mystery. Man enlarges every day the circle of light which spreads round him, but at the same time, and in virtue of his very advance, he finds himself confronting, at an increasing number of points, the darkness of the Unknown.
In Einstein and the Universe; A Popular Exposition of the Famous Theory (1922), xvi.
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Science, in the immediate, produces knowledge and, indirectly, means of action. It leads to methodical action if definite goals are set up in advance. For the function of setting up goals and passing statements of value transcends its domain. While it is true that science, to the extent of its grasp of causative connections, may reach important conclusions as to the compatibility and incompatibility of goals and evaluations, the independent and fundamental definitions regarding goals and values remain beyond science’s reach.
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Scientific inquiry would thus he conceived of as analogous to terrestrial exploration, whose product—geography—yields results of continually smaller significance which fill in ever more minute gaps in our information. In such a view, later investigations yield findings of ever smaller importance, with each successive accretion making a relatively smaller contribution to what has already come to hand. The advance of science leads, step by diminished step, toward a fixed and final view of things.
In The Limits Of Science (1984, Rev. 1999), 67.
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Secondly, the study of mathematics would show them the necessity there is in reasoning, to separate all the distinct ideas, and to see the habitudes that all those concerned in the present inquiry have to one another, and to lay by those which relate not to the proposition in hand, and wholly to leave them out of the reckoning. This is that which, in other respects besides quantity is absolutely requisite to just reasoning, though in them it is not so easily observed and so carefully practised. In those parts of knowledge where it is thought demonstration has nothing to do, men reason as it were in a lump; and if upon a summary and confused view, or upon a partial consideration, they can raise the appearance of a probability, they usually rest content; especially if it be in a dispute where every little straw is laid hold on, and everything that can but be drawn in any way to give color to the argument is advanced with ostentation. But that mind is not in a posture to find truth that does not distinctly take all the parts asunder, and, omitting what is not at all to the point, draws a conclusion from the result of all the particulars which in any way influence it.
In Conduct of the Understanding, Sect. 7.
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Simplification of modes of proof is not merely an indication of advance in our knowledge of a subject, but is also the surest guarantee of readiness for farther progress.
In Lord Kelvin and Peter Guthrie Tait Elements of Natural Philosophy (1873), Preface.
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So the dividing line between the wave or particle nature of matter and radiation is the moment “Now”. As this moment steadily advances through time, it coagulates a wavy future into a particle past.
In The Development of X-ray analysis (1975) 13.
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Stem cells are probably going to be extremely useful. But it isn’t a given, and even if it were, I don’t think the end justifies the means. I am not against stem cells, I think it’s great. Blanket objection is not very reasonable to me—any effort to control scientific advances is doomed to fail. You cannot stop the human mind from working.
From Cornelia Dean, 'A Conversation with Joseph E. Murray', New York Times (25 Sep 2001), F5.
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Technocrats are turning us into daredevils. The haphazard gambles they are imposing on us too often jeopardize our safety for goals that do not advance the human cause but undermine it. By staking our lives on their schemes, decision makers are not meeting the mandate of a democratic society; they are betraying it. They are not ennobling us; they are victimizing us. And, in acquiescing to risks that have resulted in irreversible damage to the environment, we ourselves are not only forfeiting our own rights as citizens. We are, in turn, victimizing the ultimate nonvolunteers: the defenseless, voiceless—voteless—children of the future.
In Jacques Cousteau and Susan Schiefelbein, The Human, the Orchid, and the Octopus: Exploring and Conserving Our Natural World (2007), 85.
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Telescopes are in some ways like time machines. They reveal galaxies so far away that their light has taken billions of years to reach us. We in astronomy have an advantage in studying the universe, in that we can actually see the past.
We owe our existence to stars, because they make the atoms of which we are formed. So if you are romantic you can say we are literally starstuff. If you’re less romantic you can say we’re the nuclear waste from the fuel that makes stars shine.
We’ve made so many advances in our understanding. A few centuries ago, the pioneer navigators learnt the size and shape of our Earth, and the layout of the continents. We are now just learning the dimensions and ingredients of our entire cosmos, and can at last make some sense of our cosmic habitat.
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Television is too powerful a force for the public good to be stopped by misleading propaganda. No one can retard TV's advance any more than carriage makers could stop the automobile, the cable the wireless, or silent pictures the talkies.
Address to Stockholders, 30th Annual Meeting of RCA Corporation, printed in 'Television Outlook is Bright', Radio Age: Research, Manufacturing, Communications, Broadcasting, Television (Jul 1949), 8, No. 4, 21.
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The advance of science has enabled man to communicate at twice the speed of sound while he still acts at half the speed of sense.
Anonymous
In Evan Esar, 20,000 Quips & Quotes (1968, 1995), 159.
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The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new, but to the continuous evolution of zoologic types which develop ceaselessly and end by becoming unrecognisable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past. One must not think then that the old-fashioned theories have been sterile and vain.
The Value of Science (1905), in The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method(1946), trans. by George Bruce Halsted, 208.
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The advance of technology, like the growth of population and industry, has also been proceeding exponentially.
Foreign Affairs (Summer 1972).
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The advances of biology during the past 20 years have been breathtaking, particularly in cracking the mystery of heredity. Nevertheless, the greatest and most difficult problems still lie ahead. The discoveries of the 1970‘s about the chemical roots of memory in nerve cells or the basis of learning, about the complex behavior of man and animals, the nature of growth, development, disease and aging will be at least as fundamental and spectacular as those of the recent past.
As quoted in 'H. Bentley Glass', New York Times (12 Jan 1970), 96.
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The ancients devoted a lifetime to the study of arithmetic; it required days to extract a square root or to multiply two numbers together. Is there any harm in skipping all that, in letting the school boy learn multiplication sums, and in starting his more abstract reasoning at a more advanced point? Where would be the harm in letting the boy assume the truth of many propositions of the first four books of Euclid, letting him assume their truth partly by faith, partly by trial? Giving him the whole fifth book of Euclid by simple algebra? Letting him assume the sixth as axiomatic? Letting him, in fact, begin his severer studies where he is now in the habit of leaving off? We do much less orthodox things. Every here and there in one’s mathematical studies one makes exceedingly large assumptions, because the methodical study would be ridiculous even in the eyes of the most pedantic of teachers. I can imagine a whole year devoted to the philosophical study of many things that a student now takes in his stride without trouble. The present method of training the mind of a mathematical teacher causes it to strain at gnats and to swallow camels. Such gnats are most of the propositions of the sixth book of Euclid; propositions generally about incommensurables; the use of arithmetic in geometry; the parallelogram of forces, etc., decimals.
In Teaching of Mathematics (1904), 12.
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The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.
In 'The Mathematician', Works of the Mind (1947), 1, No. 1. Collected in James Roy Newman (ed.), The World of Mathematics (1956), Vol. 4, 2055.
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The companies that can afford to do basic research (and can’t afford not to) are ones that dominate their markets. … It’s cheap insurance, since failing to do basic research guarantees that the next major advance will be owned by someone else.
In Accidental Empires (1992), 79.
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The custom of giving patients appointments weeks in advance, during which time their illness may become seriously aggravated, seems to me to fall short of the ideal doctor-patient relationship.
…...
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The edge of the sea is a strange and beautiful place. All through the long history of Earth it has been an area of unrest where waves have broken heavily against the land, where the tides have pressed forward over the continents, receded, and then returned. For no two suc-cessive days is the shore line precisely the same. Not only do the tides advance and retreat in their eternal rhythms, but the level of the sea itself is never at rest. It rises or falls as the glaciers melt or grow, as the floor of the deep ocean basins shifts under its increasing load of sediments, or as the Earth’s crust along the continental margins warps up or down in adjustment to strain and tension. Today a little more land may belong to the sea, tomorrow a little less. Always the edge of the sea remains an elusive and indefinable boundary.
The Edge of the Sea
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The essence of religion is inertia; the essence of science is change. It is the function of the one to preserve, it is the function of the other to improve. If, as in Egypt, they are firmly chained together, either science will advance, in which case the religion will be altered, or the religion will preserve its purity, and science will congeal.
In The Martyrdom of Man (1876), 41.
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The first successes were such that one might suppose all the difficulties of science overcome in advance, and believe that the mathematician, without being longer occupied in the elaboration of pure mathematics, could turn his thoughts exclusively to the study of natural laws.
From Preface to Traité de calcul différentiel et de calcul intégral (1864-70), i. Quoted in address to the section of Algebra and Analysis, International Congress of Arts and Sciences, St. Louis (22 Sep 1904), 'On the Development of Mathematical Analysis and its Relation to Certain Other Sciences,' as translated by M.W. Haskell in Bulletin of the American Mathematical Society (May 1905), 11, 408.
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The function of Art is to imitate Nature in her manner of operation. Our understanding of “her manner of operation” changes according to advances in the sciences.
John Cage
A Year from Monday (1969), 31.
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The further the spiritual evolution of mankind advances, the more certain it seems to me that the path to genuine religiosity does not lie through the fear of life, and the fear of death, and blind faith, but through striving after rational knowledge.
…...
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The glory of science is not that it discovers “truth”; rather it advances inexorably by discovering and correcting error.
In Francis Bello, Lawrence Lessing and George A.W. Boehm, Great American Scientists (1960, 1961), 116.
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The inherent unpredictability of future scientific developments—the fact that no secure inference can be drawn from one state of science to another—has important implications for the issue of the limits of science. It means that present-day science cannot speak for future science: it is in principle impossible to make any secure inferences from the substance of science at one time about its substance at a significantly different time. The prospect of future scientific revolutions can never be precluded. We cannot say with unblinking confidence what sorts of resources and conceptions the science of the future will or will not use. Given that it is effectively impossible to predict the details of what future science will accomplish, it is no less impossible to predict in detail what future science will not accomplish. We can never confidently put this or that range of issues outside “the limits of science”, because we cannot discern the shape and substance of future science with sufficient clarity to be able to say with any assurance what it can and cannot do. Any attempt to set “limits” to science—any advance specification of what science can and cannot do by way of handling problems and solving questions—is destined to come to grief.
The Limits of Science (1984), 102-3.
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The intricate edifice of verifiable fact and tested theory that has been patiently created in just a brief few hundred years is man’s most solid achievement on earth.
In Francis Bello, Lawrence Lessing and George A.W. Boehm, Great American Scientists (1960, 1961), 117.
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The law of gravitation is indisputably and incomparably the greatest scientific discovery ever made, whether we look at the advance which it involved, the extent of truth disclosed, or the fundamental and satisfactory nature of this truth.
In History of the Inductive Sciences, Bk. 7, chap. 8, sect. 6.
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The life and soul of science is its practical application, and just as the great advances in mathematics have been made through the desire of discovering the solution of problems which were of a highly practical kind in mathematical science, so in physical science many of the greatest advances that have been made from the beginning of the world to the present time have been made in the earnest desire to turn the knowledge of the properties of matter to some purpose useful to mankind.
From 'Electrical Units of Measurement', a lecture delivered at the Institution of Civil Engineers, London (3 May 1883), Popular Lectures and Addresses Vol. 1 (1891), 86-87.
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The mind God is looking for in man is a doubting, questioning mind, not a dogmatic mind; dogmatic reasoning is wrong reasoning. Dogmatic reason ties a huge rock to a man’s foot and stops him forever from advancing.
From the play Galileo Galilei (2001) .
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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 most important thing we can do is inspire young minds and to advance the kind of science, math and technology education that will help youngsters take us to the next phase of space travel.
As summarized on a CNN web page - without quotation marks - from a statement by Glenn about the fourth National Space Day (4 May 2000). 'All systems go for National Space Day' on CNN website.
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The progress of Science is generally regarded as a kind of clean, rational advance along a straight ascending line; in fact it has followed a zig-zag course, at times almost more bewildering than the evolution of political thought. The history of cosmic theories, in particular, may without exaggeration be called a history of collective obsessions and controlled schizophrenias; and the manner in which some of the most important individual discoveries were arrived at reminds one more of a sleepwalker’s performance than an electronic brain’s.
From 'Preface', in The Sleepwalkers: A History of Man’s Changing Vision of the Universe (1959), 15.
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The regularity with which we conclude that further advances in a particular field are impossible seems equaled only by the regularity with which events prove that we are of too limited vision. And it always seems to be those who have the fullest opportunity to know who are the most limited in view. What, then, is the trouble? I think that one answer should be: we do not realize sufficiently that the unknown is absolutely infinite, and that new knowledge is always being produced.
Quoted in Guy Suits, 'Willis Rodney Whitney', National Academy of Sciences, Biographical Memoirs (1960), 357.
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The rockets that have made spaceflight possible are an advance that, more than any other technological victory of the twentieth century, was grounded in science fiction… . One thing that no science fiction writer visualized, however, as far as I know, was that the landings on the Moon would be watched by people on Earth by way of television.
In Asimov on Physics (1976), 35. Also in Isaac Asimov’s Book of Science and Nature Quotations (1988), 307.
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The school of Plato has advanced the interests of the race as much through geometry as through philosophy. The modern engineer, the navigator, the astronomer, built on the truths which those early Greeks discovered in their purely speculative investigations. And if the poetry, statesmanship, oratory, and philosophy of our day owe much to Plato’s divine Dialogues, our commerce, our manufactures, and our science are equally indebted to his Conic Sections. Later instances may be abundantly quoted, to show that the labors of the mathematician have outlasted those of the statesman, and wrought mightier changes in the condition of the world. Not that we would rank the geometer above the patriot, but we claim that he is worthy of equal honor.
In 'Imagination in Mathematics', North American Review, 85, 228.
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The science and technology which have advanced man safely into space have brought about startling medical advances for man on earth. Out of space research have come new knowledge, techniques and instruments which have enabled some bedridden invalids to walk, the totally deaf to hear, the voiceless to talk, and, in the foreseeable future, may even make it possible for the blind to “see.”
'From Outer Space—Advances For Medicine on Earth', contributed in Lillian Levy, Space, Its Impact on Man and Society (1965, reprinted 1973), 117.
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The scientific method cannot lead mankind because it is based upon experiment, and every experiment postpones the present moment until one knows the result. We always come to each other and even to ourselves too late so soon as we wish to know in advance what to do.
As quoted in H.W. Auden, The Faber Book of Aphorisms (1962), 260.
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The seemingly useless or trivial observation made by one worker leads on to a useful observation by another: and so science advances, “creeping on from point to point.”
Lecture 6, collected in Light Visible and Invisible: A Series of Lectures Delivered at the Royal Institution of Great Britain, at Christmas, 1896 (1897), 276.
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The situation with regard to insulin is particularly clear. In many parts of the world diabetic children still die from lack of this hormone. ... [T]hose of us who search for new biological facts and for new and better therapeutic weapons should appreciate that one of the central problems of the world is the more equitable distribution and use of the medical and nutritional advances which have already been established. The observations which I have recently made in parts of Africa and South America have brought this fact very forcible to my attention.
'Studies on Diabetes and Cirrhosis', Proceedings, American Philosophical Society (1952) 96, No. 1, 29.
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The solutions put forth by imperialism are the quintessence of simplicity...When they speak of the problems of population and birth, they are in no way moved by concepts related to the interests of the family or of society...Just when science and technology are making incredible advances in all fields, they resort to technology to suppress revolutions and ask the help of science to prevent population growth. In short, the peoples are not to make revolutions, and women are not to give birth. This sums up the philosophy of imperialism.
From Fidel Castro (1968).
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The suppression of uncomfortable ideas may be common in religion or in politics, but it is not the path to knowledge; it has no in the endeavor of science. We do not know in advance who will discover fundamental insights.
In Cosmos (1985), 74.
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The thirteen books of Euclid must have been a tremendous advance, probably even greater than that contained in the Principia of Newton.
In Article 'Eucleides', in Smith, Dictionary of Greek and Roman Biology and Mythology (1902).
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The United States is the most powerful technically advanced country in the world to-day. Its influence on the shaping of international relations is absolutely incalculable. But America is a large country and its people have so far not shown much interest in great international problems, among which the problem of disarmament occupies first place today. This must be changed, if only in the essential interests of the Americans. The last war has shown that there are no longer any barriers between the continents and that the destinies of all countries are closely interwoven. The people of this country must realize that they have a great responsibility in the sphere of international politics. The part of passive spectator is unworthy of this country and is bound in the end to lead to disaster all round.
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There are no royal roads to knowledge, and we can only advance to new and important truths along the rugged path of experience, guided by cautious induction.
In 'Report of the Secretary', Annual Report of the Board of Regents of the Smithsonian Institution for 1856 (1857), 36.
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There are two objects of medical education: To heal the sick and to advance the science.
Collected Papers of the Mayo Clinic and Mayo Foundation (1913).
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There are, as we have seen, a number of different modes of technological innovation. Before the seventeenth century inventions (empirical or scientific) were diffused by imitation and adaption while improvement was established by the survival of the fittest. Now, technology has become a complex but consciously directed group of social activities involving a wide range of skills, exemplified by scientific research, managerial expertise, and practical and inventive abilities. The powers of technology appear to be unlimited. If some of the dangers may be great, the potential rewards are greater still. This is not simply a matter of material benefits for, as we have seen, major changes in thought have, in the past, occurred as consequences of technological advances.
Concluding paragraph of "Technology," in Dictionary of the History of Ideas (1973), Vol. 4, 364.
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There is no way to guarantee in advance what pure mathematics will later find application. We can only let the process of curiosity and abstraction take place, let mathematicians obsessively take results to their logical extremes, leaving relevance far behind, and wait to see which topics turn out to be extremely useful. If not, when the challenges of the future arrive, we won’t have the right piece of seemingly pointless mathematics to hand.
In 'The Unplanned Impact of Mathematics', Nature (14 Jul 2011), 475, No. 7355, 167.
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There is, however, no universal recipe for scientific advance. It is a matter of groping forward into terra incognita of the outer world by means of methods which should be adapted to the circumstances.
In 'The Scientific Character of Geology', The Journal of Geology (Jul 1961), 69, No. 4, 455.
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There may be instances of mere accidental discovery; but, setting these aside, the great advances made in the inductive sciences are, for the most part, preceded by a more or less probable hypothesis. The imagination, having some small light to guide it, goes first. Further observation, experiment, and reason follow.
Presidential Address to Anniversary meeting of the Royal Society (30 Nov 1859), Proceedings of the Royal Society of London (1860), 10, 166.
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They hold that the function of universities is to make learning repellent and thus to prevent its becoming dangerously common. And they discharge this beneficent function all the more efficiently because they do it unconsciously and automatically. The professors think they are advancing healthy intellectual assimilation and digestion when they are in reality little better than cancer on the stomach.
Samuel Butler, Henry Festing Jones (ed.), The Note-Books of Samuel Butler (1917), 32.
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This political movement has patently demonstrated that it will not defend the integrity of science in any case in which science runs afoul of its core political constituencies. In so doing, it has ceded any right to govern a technologically advanced and sophisticated nation.
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This will end the mythology of the dumb little Dutch boy with his stupid finger in the dike to save his country.
On completion of new, technologically advanced sea barrier in the Netherlands
NY Times 5 Oct 86
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To emphasize this opinion that mathematicians would be unwise to accept practical issues as the sole guide or the chief guide in the current of their investigations, ... let me take one more instance, by choosing a subject in which the purely mathematical interest is deemed supreme, the theory of functions of a complex variable. That at least is a theory in pure mathematics, initiated in that region, and developed in that region; it is built up in scores of papers, and its plan certainly has not been, and is not now, dominated or guided by considerations of applicability to natural phenomena. Yet what has turned out to be its relation to practical issues? The investigations of Lagrange and others upon the construction of maps appear as a portion of the general property of conformal representation; which is merely the general geometrical method of regarding functional relations in that theory. Again, the interesting and important investigations upon discontinuous two-dimensional fluid motion in hydrodynamics, made in the last twenty years, can all be, and now are all, I believe, deduced from similar considerations by interpreting functional relations between complex variables. In the dynamics of a rotating heavy body, the only substantial extension of our knowledge since the time of Lagrange has accrued from associating the general properties of functions with the discussion of the equations of motion. Further, under the title of conjugate functions, the theory has been applied to various questions in electrostatics, particularly in connection with condensers and electrometers. And, lastly, in the domain of physical astronomy, some of the most conspicuous advances made in the last few years have been achieved by introducing into the discussion the ideas, the principles, the methods, and the results of the theory of functions. … the refined and extremely difficult work of Poincare and others in physical astronomy has been possible only by the use of the most elaborate developments of some purely mathematical subjects, developments which were made without a thought of such applications.
In Presidential Address British Association for the Advancement of Science, Section A, (1897), Nature, 56, 377.
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Undoubtedly, the capstone of every mathematical theory is a convincing proof of all of its assertions. Undoubtedly, mathematics inculpates itself when it foregoes convincing proofs. But the mystery of brilliant productivity will always be the posing of new questions, the anticipation of new theorems that make accessible valuable results and connections. Without the creation of new viewpoints, without the statement of new aims, mathematics would soon exhaust itself in the rigor of its logical proofs and begin to stagnate as its substance vanishes. Thus, in a sense, mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs.
As quoted in Hermann Weyl, Unterrichtsblätter für Mathematik und Naturwissenschaften (1932), 38, 177-188. As translated by Abe Shenitzer, in 'Part I. Topology and Abstract Algebra as Two Roads of Mathematical Comprehension', The American Mathematical Monthly (May 1995), 102, No. 7, 453.
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Was it not the great philosopher and mathematician Leibnitz who said that the more knowledge advances the more it becomes possible to condense it into little books?
Opening remark in 'Introductory Note', Outline of Science: A Plain Story Simply Told (1922), Vol. 1, iii. Webmaster has not yet identified the quote in Leibnitz's original words (translated). If you know the primary source, please contact Webmaster.
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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 are like the inhabitants of an isolated valley in New Guinea who communicate with societies in neighboring valleys (quite different societies, I might add) by runner and by drum. When asked how a very advanced society will communicate, they might guess by an extremely rapid runner or by an improbably large drum. They might not guess a technology beyond their ken. And yet, all the while, a vast international cable and radio traffic passes over them, around them, and through them... We will listen for the interstellar drums, but we will miss the interstellar cables. We are likely to receive our first messages from the drummers of the neighboring galactic valleys - from civilizations only somewhat in our future. The civilizations vastly more advanced than we, will be, for a long time, remote both in distance and in accessibility. At a future time of vigorous interstellar radio traffic, the very advanced civilizations may be, for us, still insubstantial legends.
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We cannot but think there is something like a fallacy in Mr. Buckle’s theory that the advance of mankind is necessarily in the direction of science, and not in that of morals.
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What friends do with us and for us is a real part of our life; for it strengthens and advances our personality. The assault of our enemies is not part of our life ; it is only part of our experience ; we throw it off and guard ourselves against it as against frost, storm, rain, hail, or any other of the external evils which may be expected to happen.
In The Maxims and Reflections of Goethe (1906), 201.
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What matters in science is the body of findings and generalizations available today: a time-defined cross-section of the process of scientific discovery. I see the advance of science as self-erasing in the sense that only those elements survive that have become part of the active body of knowledge.
In A Slot Machine, A Broken Test Tube (1985), 123.
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What we do see through geological time is the emergence of more complex worlds. ... [W]hen within the animal we see the emergence of larger and more complex brains, sophisticated vocalizations, echolocation, electrical perception, advanced social systems including eusociality, viviparity, warm-bloodedness, and agriculture—all of which are convergent—then to me that sounds like progress.
Life's Solution, 307. In Vinoth Ramachandra, Subverting Global Myths: Theology and the Public Issues Shaping our World (2008), 184.
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When Cayley had reached his most advanced generalizations he proceeded to establish them directly by some method or other, though he seldom gave the clue by which they had first been obtained: a proceeding which does not tend to make his papers easy reading. …
His literary style is direct, simple and clear. His legal training had an influence, not merely upon his mode of arrangement but also upon his expression; the result is that his papers are severe and present a curious contrast to the luxuriant enthusiasm which pervades so many of Sylvester’s papers. He used to prepare his work for publication as soon as he carried his investigations in any subject far enough for his immediate purpose. … A paper once written out was promptly sent for publication; this practice he maintained throughout life. … The consequence is that he has left few arrears of unfinished or unpublished papers; his work has been given by himself to the world.
In Proceedings of London Royal Society (1895), 58, 23-24.
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When science advances religion goes along with it; science builds the altar at which religion prays.
In The People's Bible: Discourses Upon Holy Scripture: Vol. 1. Genesis (1885), 105.
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When the mathematician says that such and such a proposition is true of one thing, it may be interesting, and it is surely safe. But when he tries to extend his proposition to everything, though it is much more interesting, it is also much more dangerous. In the transition from one to all, from the specific to the general, mathematics has made its greatest progress, and suffered its most serious setbacks, of which the logical paradoxes constitute the most important part. For, if mathematics is to advance securely and confidently, it must first set its affairs in order at home.
With co-author James R. Newman, in Mathematics and the Imagination (1940), 219.
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While it is never safe to affirm that the future of Physical Science has no marvels in store even more astonishing than those of the past, it seems probable that most of the grand underlying principles have been firmly established and that further advances are to be sought chiefly in the rigorous application of these principles to all the phenomena which come under our notice.
'Spectroscopy, Molecular Orbitals, and Chemical Bonding', Nobel Lecture (12 Dec 1966). In Nobel Lectures: Chemistry 1963-1970 (1972), 159.
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While it is never safe to affirm that the future of Physical Science has no marvels in store even more astonishing than those of the past, it seems probable that most of the grand underlying principles have been firmly established, and that further advances are to be sought chiefly in the rigorous applications of these principles to all the phenomena which come under our notice. It is here that the science of measurement shows its importance—where the quantitative results are more to be desired than qualitative work. An eminent physicist has remarked that the future truths of Physical Science are to be looked for in the sixth place of decimals.
University of Chicago, Annual Register 1894-1895 (1894), 150. Michelson also incorporated these lines in his address, 'Some of the Objects and Methods of Physical Science', at the opening of the Physics and Electrical Engineering Laboratory at the University of Kansas, reprinted in The Electrical Engineer (1 Jan 1896), 21, No. 400, 9.
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Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?
Opening of Lecture (1900), 'Mathematische Probleme' (Mathematical Problems), to the International Congress of Mathematicians, Paris. From the original German reprinted in David Hilbert: Gesammelte Abhandlungen (Collected Treatises, 1970), Vol. 3. For full citation, see the quote that begins, “This conviction of the solvability…”, on the David Hilbert Quotes page on this website.
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Whoever has undergone the intense experience of successful advances made in [science], is moved by profound reverence for the rationality made manifest in existence.
…...
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Why it is that animals, instead of developing in a simple and straightforward way, undergo in the course of their growth a series of complicated changes, during which they often acquire organs which have no function, and which, after remaining visible for a short time, disappear without leaving a trace ... To the Darwinian, the explanation of such facts is obvious. The stage when the tadpole breathes by gills is a repetition of the stage when the ancestors of the frog had not advanced in the scale of development beyond a fish.
In The Works of Francis Maitland Balfour (1885), Vol. 1, 702.
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With the extension of mathematical knowledge will it not finally become impossible for the single investigator to embrace all departments of this knowledge? In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which, at the same time, assist in understanding earlier theories and in casting aside some more complicated developments.
In 'Mathematical Problems', Lecture at the International Congress of Mathematics, Paris, (8 Aug 1900). Translated by Dr. Maby Winton Newson in Bulletin of the American Mathematical Society (1902), 8, 479. As quoted and cited in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath's Quotation-book (1914), 94-95. It is reprinted in Jeremy Gray, The Hilbert Challenge (2000), 282.
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With time, I attempt to develop hypotheses that are more risky. I agree with [Karl] Popper that scientists need to be interested in risky hypotheses because risky hypotheses advance science by producing interesting thoughts and potential falsifications of theories (of course, personally, we always strive for verification—we love our theories after all; but we should be ready to falsify them as well.
'Grand Theories and Mid-Range Theories&3039;, essay in Ken G. Smith (ed.) and Michael A. Hitt (ed), Great Minds in Management: the Theory of Process Development (2005), 89.
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Without the death of forests by Ice Age advance, there would be no northern lakes.
Without the death of mountains, there would be no sand or soil.
In 'The Nested Emergent Nature of Divine Creativity', Thank God for Evolution: How the Marriage of Science and Religion Will Transform Your Life and Our World (2007), 99.
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Young men, … Live in the serene peace of laboratories and libraries. Say to yourselves first: “What have I done for my instruction?” and, as you gradually advance, “What have I done for my country?”
Acceptance speech (27 Dec 1892) when awarded a 70th birthday commemorative medal by the Academy of Sciences in the great theatre of the Sorbonne, as translated in René Vallery-Radot and Mrs R.L. Devonshire (trans.), The Life of Pasteur (1902), Vol. 2, 297-298. Pasteur addressed an audience that included “deep masses of students” and “boys from the lycées.”
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[Aristotle formal logic thus far (1787)] has not been able to advance a single step, and hence is to all appearances closed and completed.
In Preface to second edition (1787) of Critique Of Pure Reason (1781) as translated by Werner Pluhar (1996), 15. An earlier translation by N. Kemp-Smith (1933) is similar, but ends with “appearance a closed and completed body of doctrine.”
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[I find it as difficult] to understand a scientist who does not acknowledge the presence of a superior rationality behind the existence of the universe as it is to comprehend a theologian who would deny the advances of science.
Speech, Huntsville Ministerial Association, in Wernher Von Braun and Irene E. Powell-Willhite (ed.), The Voice of Dr. Wernher Von Braun: An Anthology (2007), 89.
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[I have a great] distaste for controversy…. I have often seen it do great harm, and yet remember few cases in natural knowledge where it has helped much either to pull down error or advance truth. Criticism, on the other hand, is of much value.
In letter (6 May 1841) to Robert Hare, an American Chemist, collected in Experimental Researches in Electricity (1844), Vol. 2, 275, as a footnote added to a reprint of 'On Dr. Hare’s Second Letter, and on the Chemical and Contact Theories of the Voltaic Battery', London and Edinburgh Philosophical Magazine (1843), 23.
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[It] is not the nature of things for any one man to make a sudden, violent discovery; science goes step by step and every man depends on the work of his predecessors. When you hear of a sudden unexpected discovery—a bolt from the blue—you can always be sure that it has grown up by the influence of one man or another, and it is the mutual influence which makes the enormous possibility of scientific advance. Scientists are not dependent on the ideas of a single man, but on the combined wisdom of thousands of men, all thinking of the same problem and each doing his little bit to add to the great structure of knowledge which is gradually being erected.
Concluding remark in Lecture ii (1936) on 'Forty Years of Physics', revised and prepared for publication by J.A. Ratcliffe, collected in Needham and Pagel (eds.), Background to Modern Science: Ten Lectures at Cambridge Arranged by the History of Science Committee, (1938), 73-74. Note that the words as prepared for publication may not be verbatim as spoken in the original lecture by the then late Lord Rutherford.
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[Science] is sort of a game. Any fundamental advances in our field are made by looking at it with the smile of a child who plays a game.
As quoted in Lewis Wolpert and Alison Richards (eds.), Passionate Minds: The Inner World of Scientists (1997), 4.
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[T]he history of science has proved that fundamental research is the lifeblood of individual progress and that the ideas that lead to spectacular advances spring from it.
In J. Edwin Holmström, Records and Research in Engineering and Industrial Science (1956), 7.
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…so slow is moral progress. True, we have the bicycle, the motor-car, the dirigible airship and other marvellous means of breaking our bones; but our morality is not one rung the higher for it all. One would even say that, the farther we proceed in our conquest of matter, the more our morality recedes. The most advanced of our inventions consists in bringing men down with grapeshot and explosives with the swiftness of the reaper mowing the corn.
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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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