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Who said: “Nature does nothing in vain when less will serve; for Nature is pleased with simplicity and affects not the pomp of superfluous causes.”
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Home > Category Index for Science Quotations > Category Index C > Category: Concrete

Concrete Quotes (32 quotes)

...the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly generality is, in essence, the same as a small and concrete special case.
I Want to be a Mathematician: an Automathography in Three Parts (1985), 324.
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Q: What did the fish say when he hit a concrete wall? A: Dam!
Anonymous
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A fractal is a mathematical set or concrete object that is irregular or fragmented at all scales.
Cited as from Fractals: Form, Chance, and Dimension (1977), by J.W. Cannon, in review of The Fractal Geometry of Nature (1982) in The American Mathematical Monthly (Nov 1984), 91, No. 9, 594.
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All of our experience indicates that life can manifest itself only in a concrete form, and that it is bound to certain substantial loci. These loci are cells and cell formations. But we are far from seeking the last and highest level of understanding in the morphology of these loci of life. Anatomy does not exclude physiology, but physiology certainly presupposes anatomy. The phenomena that the physiologist investigates occur in special organs with quite characteristic anatomical arrangements; the various morphological parts disclosed by the anatomist are the bearers of properties or, if you will, of forces probed by the physiologist; when the physiologist has established a law, whether through physical or chemical investigation, the anatomist can still proudly state: This is the structure in which the law becomes manifest.
In 'Cellular-Pathologie', Archiv für pathologische Anatomie und Physiologie und fur klinische Medizin (1855), 8, 19, as translated in LellandJ. Rather, 'Cellular Pathology', Disease, Life, and Man: Selected Essays by Rudolf Virchow (1958), 84.
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Arithmetic, as we shall see by and by, is overdone, in a certain sense, in our schools; just so far as the teaching is based upon the concrete, so far is it profitable; but when the book-makers begin to make it too abstract, as they very often do, it becomes a torture to both teacher and learners, or, at best, a branch of imaginary knowledge unconnected with real life.
From 'Introduction', Mathematical Teaching and its Modern Methods (1886), 10.
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Beavers bred in captivity, inhabiting a concrete pool, will, if given the timber, fatuously go through all the motions of damming an ancestral stream.
In short story Work Suspended (1943) collected in Evelyn Waugh: The Complete Short Stories (1998), 320.
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Chemistry is like a majestic skyscraper. The concrete secure foundation of chemistry consists of countless experimentally observed facts. The theories, principles and laws developed from these observations are like an elevator which runs from the bottom to the top of the edifice.
Ernest R. Toon and George L. Ellis (eds.), Foundations of Chemistry (1968), 1.
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Every 10 years, an area the size of Britain disappears under a jungle of concrete.
As quoted from BBC TV series, Planet Earth II in Joe Shute, 'David Attenborough at 90: ‘I think about my mortality every day’', The Telegraph (29 Oct 2016).
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I have long been interested in landscape history, and when younger and more robust I used to do much tramping of the English landscape in search of ancient field systems, drove roads, indications of prehistoric settlement. Towns and cities, too, which always retain the ghost of their earlier incarnations beneath today's concrete and glass.
From 'An Interview With Penelope Lively', in a Reading Guide to the book The Photograph on the publisher's Penguin website.
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I recall my own emotions: I had just been initiated into the mysteries of the complex number. I remember my bewilderment: here were magnitudes patently impossible and yet susceptible of manipulations which lead to concrete results. It was a feeling of dissatisfaction, of restlessness, a desire to fill these illusory creatures, these empty symbols, with substance. Then I was taught to interpret these beings in a concrete geometrical way. There came then an immediate feeling of relief, as though I had solved an enigma, as though a ghost which had been causing me apprehension turned out to be no ghost at all, but a familiar part of my environment.
In Tobias Dantzig and Joseph Mazur (ed.), 'The Two Realities', Number: The Language of Science (1930, ed. by Joseph Mazur 2007), 254.
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I think that the difference between pure and applied mathematics is social rather than scientific. A pure mathematician is paid for making mathematical discoveries. An applied mathematician is paid for the solution of given problems.
When Columbus set sail, he was like an applied mathematician, paid for the search of the solution of a concrete problem: find a way to India. His discovery of the New World was similar to the work of a pure mathematician.
In S.H. Lui, 'An Interview with Vladimir Arnol’d', Notices of the AMS (Apr 1997) 44, No. 4, 438. Reprinted from the Hong Kong Mathematics Society (Feb 1996).
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I think there is something more important than believing: Action! The world is full of dreamers, there aren’t enough who will move ahead and begin to take concrete steps to actualize their vision.
…...
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If a teacher is full of his subject, and can induce enthusiasm in his pupils; if his facts are concrete and naturally connected, the amount of material that an average child can assimilate without injury is as astonishing as is the little that will fag him if it is a trifle above or below or remote from him, or taught dully or incoherently.
In The North American Review (Mar 1883), No. 316, 289.
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If we consider what science already has enabled men to know—the immensity of space, the fantastic philosophy of the stars, the infinite smallness of the composition of atoms, the macrocosm whereby we succeed only in creating outlines and translating a measure into numbers without our minds being able to form any concrete idea of it—we remain astounded by the enormous machinery of the universe.
Address (10 Sep 1934) to the International Congress of Electro-Radio Biology, Venice. In Associated Press, 'Life a Closed Book, Declares Marconi', New York Times (11 Sep 1934), 15.
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It has long been a complaint against mathematicians that they are hard to convince: but it is a far greater disqualification both for philosophy, and for the affairs of life, to be too easily convinced; to have too low a standard of proof. The only sound intellects are those which, in the first instance, set their standards of proof high. Practice in concrete affairs soon teaches them to make the necessary abatement: but they retain the consciousness, without which there is no sound practical reasoning, that in accepting inferior evidence because there is no better to be had, they do not by that acceptance raise it to completeness.
In An Examination of Sir William Hamilton’s Philosophy (1878), 611.
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Metaphorical language is a species of natural language which we construct out of arbitrary but concrete words. That is why it is so pleasing.
Aphorism 78 in Notebook D (1773-1775), as translated by R.J. Hollingdale in Aphorisms (1990). Reprinted as The Waste Books (2000), 56.
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No problem can be solved until it is reduced to some simple form. The changing of a vague difficulty into a specific, concrete form is a very essential element in thinking.
Seen, for example, in The Grain and Feed Review (1931), 21, 34.
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Our knowledge of the external world must always consist of numbers, and our picture of the universe—the synthesis of our knowledge—must necessarily be mathematical in form. All the concrete details of the picture, the apples, the pears and bananas, the ether and atoms and electrons, are mere clothing that we ourselves drape over our mathematical symbols— they do not belong to Nature, but to the parables by which we try to make Nature comprehensible. It was, I think, Kronecker who said that in arithmetic God made the integers and man made the rest; in the same spirit, we may add that in physics God made the mathematics and man made the rest.
From Address (1934) to the British Association for the Advancement of Science, Aberdeen, 'The New World—Picture of Modern Physics'. Printed in Nature (Sep 1934) 134, No. 3384, 356. As quoted and cited in Wilbur Marshall Urban, Language and Reality: The Philosophy of Language and the Principles of Symbolism (2004), Vol. 15, 542.
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Our national flower is the concrete cloverleaf.
In Hugh Rawson and Margaret Miner (eds.), The Oxford Dictionary of American Quotations (), 30, citing the magazine Quote (8 Oct 1961).
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Sometime between 1740 and 1780, electricians were for the first time enabled to take the foundations for their field for granted. From that point they pushed on to more concrete and recondite problems.
From The Structure of Scientific Revolutions (1970, 2012), 21-22.
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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 belief that mathematics, because it is abstract, because it is static and cold and gray, is detached from life, is a mistaken belief. Mathematics, even in its purest and most abstract estate, is not detached from life. It is just the ideal handling of the problems of life, as sculpture may idealize a human figure or as poetry or painting may idealize a figure or a scene. Mathematics is precisely the ideal handling of the problems of life, and the central ideas of the science, the great concepts about which its stately doctrines have been built up, are precisely the chief ideas with which life must always deal and which, as it tumbles and rolls about them through time and space, give it its interests and problems, and its order and rationality. That such is the case a few indications will suffice to show. The mathematical concepts of constant and variable are represented familiarly in life by the notions of fixedness and change. The concept of equation or that of an equational system, imposing restriction upon variability, is matched in life by the concept of natural and spiritual law, giving order to what were else chaotic change and providing partial freedom in lieu of none at all. What is known in mathematics under the name of limit is everywhere present in life in the guise of some ideal, some excellence high-dwelling among the rocks, an “ever flying perfect” as Emerson calls it, unto which we may approximate nearer and nearer, but which we can never quite attain, save in aspiration. The supreme concept of functionality finds its correlate in life in the all-pervasive sense of interdependence and mutual determination among the elements of the world. What is known in mathematics as transformation—that is, lawful transfer of attention, serving to match in orderly fashion the things of one system with those of another—is conceived in life as a process of transmutation by which, in the flux of the world, the content of the present has come out of the past and in its turn, in ceasing to be, gives birth to its successor, as the boy is father to the man and as things, in general, become what they are not. The mathematical concept of invariance and that of infinitude, especially the imposing doctrines that explain their meanings and bear their names—What are they but mathematicizations of that which has ever been the chief of life’s hopes and dreams, of that which has ever been the object of its deepest passion and of its dominant enterprise, I mean the finding of the worth that abides, the finding of permanence in the midst of change, and the discovery of a presence, in what has seemed to be a finite world, of being that is infinite? It is needless further to multiply examples of a correlation that is so abounding and complete as indeed to suggest a doubt whether it be juster to view mathematics as the abstract idealization of life than to regard life as the concrete realization of mathematics.
In 'The Humanization of Teaching of Mathematics', Science, New Series, 35, 645-46.
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The business of concrete mathematics is to discover the equations which express the mathematical laws of the phenomenon under consideration; and these equations are the starting-point of the calculus, which must obtain from them certain quantities by means of others.
In Positive Philosophy, Bk. 1, chap. 2.
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The computer takes up where psychoanalysis left off. It takes the ideas of a decentered self and makes it more concrete by modeling mind as a multiprocessing machine.
The Second Self, ch. 9 (1984). Turkle was on the faculty of the Massachusetts Institute of Technology's program in Science, Technology and Society.
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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 Highways of America are built chiefly of politics, whereas the proper material is crushed rock or concrete.
From Letter (24 Sep 1912) to his friend, the publisher Elbert Hubbard. In Jane Watts Fisher, Fabulous Hoosier: A Story of American Achievement (1947), 79.
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The most important distinction between the two qualities [talent and genius] is this: one, in conception, follows mechanical processes; the other, vital. Talent feebly conceives objects with the senses and understanding; genius, fusing all its powers together in the alembic of an impassioned imagination, clutches every thing in the concrete, conceives objects as living realities, gives body to spiritual abstractions, and spirit to bodily appearances, and like
“A gate of steel
Fronting the sun, receives and renders back
His figure and his heat!”
In 'Genius', Wellman’s Miscellany (Dec 1871), 4, No. 6, 203. The quotation at the end is from Wiliam Shakespeare, Tr. & Cress. iii, 3.
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The only truly secure system is one that is powered off, cast in a block of concrete and sealed in a lead-lined room with armed guards—and even then I have my doubts.
As quoted in epigraph to A.K. Dewdney, 'Computer Recreations: Of Worms, Viruses and Core War' by A. K. Dewdney in Scientific American (Mar 1989), 110. Also on the koth.org website.
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There is no area in our minds reserved for superstition, such as the Greeks had in their mythology; and superstition, under cover of an abstract vocabulary, has revenged itself by invading the entire realm of thought. Our science is like a store filled with the most subtle intellectual devices for solving the most complex problems, and yet we are almost incapable of applying the elementary principles of rational thought. In every sphere, we seem to have lost the very elements of intelligence: the ideas of limit, measure, degree, proportion, relation, comparison, contingency, interdependence, interrelation of means and ends. To keep to the social level, our political universe is peopled exclusively by myths and monsters; all it contains is absolutes and abstract entities. This is illustrated by all the words of our political and social vocabulary: nation, security, capitalism, communism, fascism, order, authority, property, democracy. We never use them in phrases such as: There is democracy to the extent that… or: There is capitalism in so far as… The use of expressions like “to the extent that” is beyond our intellectual capacity. Each of these words seems to represent for us an absolute reality, unaffected by conditions, or an absolute objective, independent of methods of action, or an absolute evil; and at the same time we make all these words mean, successively or simultaneously, anything whatsoever. Our lives are lived, in actual fact, among changing, varying realities, subject to the casual play of external necessities, and modifying themselves according to specific conditions within specific limits; and yet we act and strive and sacrifice ourselves and others by reference to fixed and isolated abstractions which cannot possibly be related either to one another or to any concrete facts. In this so-called age of technicians, the only battles we know how to fight are battles against windmills.
From 'The Power of Words', collected in Siân Miles (ed.), Simone Weil: An Anthology (2000), 222-223.
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Think of Adam and Eve like an imaginary number, like the square root of minus one: you can never see any concrete proof that it exists, but if you include it in your equations, you can calculate all manner of things that couldn't be imagined without it.
In The Golden Compass (1995, 2001), 372-373.
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When one talked with M. Hermite, he never evoked a sensuous image, and yet you soon perceived that the most abstract entities were for him like living beings.
From La Valeur de la Science (1904), 32, as translated by George Bruce Halsted (trans.), in The Value of Science (1907), 24. From the French, “Quand on causait avec M. Hermite; jamais il n’évoquait une image sensible, et pourtant vous vous aperceviez bientôt que les entités les plus abstraites étaient, pour lui comme des êtres vivants.” Also as epigraph, “Talk with M. Hermite. He never evokes a concrete image, yet you soon perceive that the more abstract entities are to him like living creatures”, in Eric Temple Bell, Men of Mathematics, (1937), 448.
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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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- 90 -
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- 70 -
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- 60 -
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- 50 -
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- 40 -
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- 30 -
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