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Who said: “God does not care about our mathematical difficulties. He integrates empirically.”
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Abstract Quotes (27 quotes)

Combien de gens se font abstraits pour paraître profonds! La plupart des termes abstraits sont des ombres qui cachent des vides.
How many people become abstract in order to appear profound! Most abstract terms are shadows that conceal a void.
Quoted in M. Paul De Raynal, Pensées de J. Joubert (1862), 456.
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Une même expression, dont les géomètres avaient considéré les propriétés abstraites, … représente'aussi le mouvement de la lumière dans l’atmosphère, quelle détermine les lois de la diffusion de la chaleur dans la matière solide, et quelle entre dans toutes les questions principales de la théorie des probabilités.
The same expression whose abstract properties geometers had considered … represents as well the motion of light in the atmosphere, as it determines the laws of diffusion of heat in solid matter, and enters into all the chief problems of the theory of probability.
From Théorie Analytique de la Chaleur (1822), translated by Alexander Freeman in The Analytical Theory of Heat (1878), 7.
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Abstract as it is, science is but an outgrowth of life. That is what the teacher must continually keep in mind. … Let him explain … science is not a dead system—the excretion of a monstrous pedantism—but really one of the most vigorous and exuberant phases of human life.
In 'The Teaching of the History of Science', The Scientific Monthly (Sep 1918), 195-196.
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Again, it [the Analytical Engine] might act upon other things besides number, were objects found whose mutual fundamental relations could be expressed by those of the abstract science of operations, and which should be also susceptible of adaptations to the action of the operating notation and mechanism of the engine. Supposing for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent.
In Richard Taylor (ed.), 'Translator’s Notes to M. Menabrea’s Memoir', Scientific Memoirs, Selected from the Transactions of Foreign Academies and Learned Societies and from Foreign Journals (1843), 3, Note A, 694. Her notes were appended to L.F. Menabrea, of Turin, Officer of the Military Engineers, 'Article XXIX: Sketch of the Analytical Engine invented by Charles Babbage Esq.', Bibliothèque Universelle de Gnve (Oct 1842), No. 82.
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Another diversity of Methods is according to the subject or matter which is handled; for there is a great difference in delivery of the Mathematics, which are the most abstracted of knowledges, and Policy, which is the most immersed ... , yet we see how that opinion, besides the weakness of it, hath been of ill desert towards learning, as that which taketh the way to reduce learning to certain empty and barren generalities; being but the very husks and shells of sciences, all the kernel being forced out and expulsed with the torture and press of the method.
Advancement of Learning, Book 2. In James Spedding, The Works of Francis Bacon (1863), Vol. 6, 292-293 . Peter Pešić, explains that 'By Mathematics, he had in mind a sterile and rigid scheme of logical classifications, called dichotomies in his time,' inLabyrinth: A Search for the Hidden Meaning of Science (2001), 73.
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Before an experiment can be performed, it must be planned—the question to nature must be formulated before being posed. Before the result of a measurement can be used, it must be interpreted—nature's answer must be understood properly. These two tasks are those of the theorist, who finds himself always more and more dependent on the tools of abstract mathematics. Of course, this does not mean that the experimenter does not also engage in theoretical deliberations. The foremost classical example of a major achievement produced by such a division of labor is the creation of spectrum analysis by the joint efforts of Robert Bunsen, the experimenter, and Gustav Kirchoff, the theorist. Since then, spectrum analysis has been continually developing and bearing ever richer fruit.
'The Meaning and Limits of Exact Science', Science (30 Sep 1949), 110, No. 2857, 325. Advance reprinting of chapter from book Max Planck, Scientific Autobiography (1949), 110.
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Electric and magnetic forces. May they live for ever, and never be forgot, if only to remind us that the science of electromagnetics, in spite of the abstract nature of its theory, involving quantities whose nature is entirely unknown at the present, is really and truly founded on the observations of real Newtonian forces, electric and magnetic respectively.
From 'Electromagnetic Theory, CXII', The Electrician (23 Feb 1900), Vol. 44, 615.
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He that desires to learn Truth should teach himself by Facts and Experiments; by which means he will learn more in a Year than by abstract reasoning in an Age.
In Academical Lectures on the Theory of Physic (1751), Vol. 1. As quoted in Thomas Steele Hall, A Source Book in Animal Biology (1951), 485.
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If we take in our hand any Volume; of Divinity or School Metaphysics, for Instance; let us ask, Does it contain any abstract Reasoning concerning Quantity or Number? No. Does it contain any experimental Reasoning concerning Matter of Fact and Existence? No. Commit it then to the Flames: For it can contain nothing but Sophistry and Illusion.
An Enquiry Concerning Human Understanding (1748), 256.
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In our search after the Knowledge of Substances, our want of Ideas, that are suitable to such a way of proceeding, obliges us to a quite different method. We advance not here, as in the other (where our abstract Ideas are real as well as nominal Essences) by contemplating our Ideas, and considering their Relations and Correspondencies; that helps us very little, for the Reasons, and in another place we have at large set down. By which, I think it is evident, that Substances afford Matter of very little general Knowledge; and the bare Contemplation of their abstract Ideas, will carry us but a very little way in the search of Truth and Certainty. What then are we to do for the improvement of our Knowledge in Substantial beings? Here we are to take a quite contrary Course, the want of Ideas of their real essences sends us from our own Thoughts, to the Things themselves, as they exist.
An Essay Concerning Human Understanding (1690). Edited by Peter Nidditch (1975), Book 4, Chapter 12, Section 9, 644.
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In the mathematics I can report no deficience, except it be that men do not sufficiently understand this excellent use of the pure mathematics, in that they do remedy and cure many defects in the wit and faculties intellectual. For if the wit be too dull, they sharpen it; if too wandering, they fix it; if too inherent in the sense, they abstract it. So that as tennis is a game of no use in itself, but of great use in respect it maketh a quick eye and a body ready to put itself into all postures, so in the mathematics that use which is collateral and intervenient is no less worthy than that which is principal and intended.
The Advancement of Learning (1605), Book 2. Reprinted in The Two Books of Francis Bacon: Of the Proficience and Advancement of Learning, Divine and Human (2009), 97.
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It is impossible not to feel stirred at the thought of the emotions of man at certain historic moments of adventure and discovery—Columbus when he first saw the Western shore, Pizarro when he stared at the Pacific Ocean, Franklin when the electric spark came from the string of his kite, Galileo when he first turned his telescope to the heavens. Such moments are also granted to students in the abstract regions of thought, and high among them must be placed the morning when Descartes lay in bed and invented the method of co-ordinate geometry.
Quoted in James Roy Newman, The World of Mathematics (2000), Vol. 1, 239.
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Science only means knowledge; and for [Greek] ancients it did only mean knowledge. Thus the favorite science of the Greeks was Astronomy, because it was as abstract as Algebra. ... We may say that the great Greek ideal was to have no use for useful things. The Slave was he who learned useful things; the Freeman was he who learned useless things. This still remains the ideal of many noble men of science, in the sense they do desire truth as the great Greeks desired it; and their attitude is an external protest against vulgarity of utilitarianism.
'About Beliefs', in As I was Saying: A Book of Essays (1936), 65-66. Collected in G. K. Chesterton and Dale Ahlquist (ed.), In Defense of Sanity: The Best Essays of G.K. Chesterton (2011), 318.
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Such propositions are therefore called Eternal Truths, not because they are Eternal Truths, not because they are External Propositions actually formed, and antecedent to the Understanding, that at any time makes them; nor because they are imprinted on the Mind from any patterns, that are any where out of the mind, and existed before: But because, being once made, about abstract Ideas, so as to be true, they will, whenever they can be supposed to be made again at any time, past or to come, by a Mind having those Ideas, always actually be true. For names being supposed to stand perpetually for the same ideas, and the same ideas having immutably the same habitudes one to another, Propositions concerning any abstract Ideas that are once true, must needs be eternal Verities.
An Essay Concerning Human Understanding (1690). Edited by Peter Nidditch (1975), Book 4, Chapter 11, Section 14, 638-9.
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The advance from the simple to the complex, through a process of successive differentiations, is seen alike in the earliest changes of the Universe to which we can reason our way back, and in the earliest changes which we can inductively establish; it is seen in the geologic and climatic evolution of the Earth; it is seen in the unfolding of every single organism on its surface, and in the multiplication of kinds of organisms; it is seen in the evolution of Humanity, whether contemplated in the civilized individual, or in the aggregate of races; it is seen in the evolution of Society in respect alike of its political, its religious, and its economical organization; and it is seen in the evolution of all those endless concrete and abstract products of human activity which constitute the environment of our daily life. From the remotest past which Science can fathom, up to the novelties of yesterday, that in which Progress essentially consists, is the transformation of the homogeneous into the heterogeneous.
Progress: Its Law and Cause (1857), 35.
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The engineer is concerned to travel from the abstract to the concrete. He begins with an idea and ends with an object. He journeys from theory to practice. The scientist’s job is the precise opposite. He explores nature with his telescopes or microscopes, or much more sophisticated techniques, and feeds into a computer what he finds or sees in an attempt to define mathematically its significance and relationships. He travels from the real to the symbolic, from the concrete to the abstract. The scientist and the engineer are the mirror image of each other.
In The Development of Design (1981), 19-20.
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The great testimony of history shows how often in fact the development of science has emerged in response to technological and even economic needs, and how in the economy of social effort, science, even of the most abstract and recondite kind, pays for itself again and again in providing the basis for radically new technological developments. In fact, most people—when they think of science as a good thing, when they think of it as worthy of encouragement, when they are willing to see their governments spend substance upon it, when they greatly do honor to men who in science have attained some eminence-have in mind that the conditions of their life have been altered just by such technology, of which they may be reluctant to be deprived.
The Open Mind (1955), 89-90.
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The results of mathematics are seldom directly applied; it is the definitions that are really useful. Once you learn the concept of a differential equation, you see differential equations all over, no matter what you do. This you cannot see unless you take a course in abstract differential equations. What applies is the cultural background you get from a course in differential equations, not the specific theorems. If you want to learn French, you have to live the life of France, not just memorize thousands of words. If you want to apply mathematics, you have to live the life of differential equations. When you live this life, you can then go back to molecular biology with a new set of eyes that will see things you could not otherwise see.
In 'A Mathematician's Gossip', Indiscrete Thoughts (2008), 213.
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The universe is governed by science. But science tells us that we can’t solve the equations, directly in the abstract. We need to use the effective theory of Darwinian natural selection of those societies most likely to survive. We assign them higher value.
[Answer to question: What is the value in knowing “Why are we here?”]
'Stephen Hawking: "There is no heaven; it's a fairy story"', interview in newspaper The Guardian (15 May 2011).
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The world is very complicated and it is clearly impossible for the human mind to understand it completely. Man has therefore devised an artifice which permits the complicated nature of the world to be blamed on something which is called accidental and thus permits him to abstract a domain in which simple laws can be found.
In Floyd Merrell, Unthinking Thinking: Jorge Luis Borges, Mathematics, and the New Physics (1991), 156.
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There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.
From a page of quotations, without citations, in G.E. Martin The Foundations of Geometry and the Non-Euclidean Plane (1975), 225. If you know the primary source, please contact Webmaster.
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Time is that which is measured by a clock. This is a sound way of looking at things. A quantity like time, or any other physical measurement, does not exist in a completely abstract way. We find no sense in talking about something unless we specify how we measure it. It is the definition by the method of measuring a quantity that is the one sure way of avoiding talking nonsense about this kind of thing.
From Relativity and Common Sense: A New Approach to Einstein (1980), 65.
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We think of Euclid as of fine ice; we admire Newton as we admire the peak of Teneriffe. Even the intensest labors, the most remote triumphs of the abstract intellect, seem to carry us into a region different from our own—to be in a terra incognita of pure reasoning, to cast a chill on human glory.
In Estimates of Some Englishmen and Scotchmen (1856), 411-412
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We [may] answer the question: “Why is snow white?” by saying, “For the same reason that soap-suds or whipped eggs are white”—in other words, instead of giving the reason for a fact, we give another example of the same fact. This offering a similar instance, instead of a reason, has often been criticised as one of the forms of logical depravity in men. But manifestly it is not a perverse act of thought, but only an incomplete one. Furnishing parallel cases is the necessary first step towards abstracting the reason imbedded in them all.
In The Principles of Psychology (1918), Vol. 2, 363-364.
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When the most abstract and “useless” disciplines have been cultivated for a time, they are often seized upon as practical tools by other departments of science. I conceive that this is no accident, as if one bought a top hat for a wedding, and discovered later when a fire broke out, that it could be used as a water bucket.
In The World of Mathematics (1956), Vol.4, 2051.
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[S]ome physicists describe gravity in terms of ten dimensions all curled up. But those aren't real words—just placeholders, used to refer to parts of abstract equations.
In God's Debris: A Thought Experiment (2004), 20-21.
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[T]he 47th proposition in Euclid might now be voted down with as much ease as any proposition in politics; and therefore if Lord Hawkesbury hates the abstract truths of science as much as he hates concrete truth in human affairs, now is his time for getting rid of the multiplication table, and passing a vote of censure upon the pretensions of the hypotenuse.
In 'Peter Plymley's Letters', Essays Social and Political (1877), 530.
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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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