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Who said: “God does not care about our mathematical difficulties. He integrates empirically.”
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Notion Quotes (59 quotes)

Archimedes was not free from the prevailing notion that geometry was degraded by being employed to produce anything useful. It was with difficulty that he was induced to stoop from speculation to practice. He was half ashamed of those inventions which were the wonder of hostile nations, and always spoke of them slightingly as mere amusements, as trifles in which a mathematician might be suffered to relax his mind after intense application to the higher parts of his science.
In Lord Bacon', Edinburgh Review (Jul 1887), in Critical and Miscellaneous Essays (1879), Vol. 1, 395.
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As physicists have arranged an extensive series of effects under the general term of Heat, so they have named another series Light, and a third they have called Electricity. We find ... that all these principles are capable of being produced through the medium of living bodies, for nearly all animals have the power of evolving heat; many insects, moreover, can voluntarily emit light; and the property of producing electricity is well evinced in the terrible shock of the electric eel, as well as in that of some other creatures. We are indeed in the habit of talking of the Electric fluid, or the Galvanic fluid, but this in reality is nothing but a licence of expression suitable to our finite and material notions.
In the Third Edition of Elements of Electro-Metallurgy: or The Art of Working in Metals by the Galvanic Fluid (1851), 1.
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As was the case for Nobel's own invention of dynamite, the uses that are made of increased knowledge can serve both beneficial and potentially harmful ends. Increased knowledge clearly implies increased responsibility. We reject the notion advocated in some quarters that man should stop eating from the tree of knowledge, as if that were humanly possible.
From Nobel Banquet Speech (10 Dec 1981), in Wilhelm Odelberg (ed.), Les Prix Nobel 1981 (1981), 44.
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Bernard Bolzano dispelled the clouds that throughout all the foregone centuries had enveloped the notion of Infinitude in darkness, completely sheared the great term of its vagueness without shearing it of its strength, and thus rendered it forever available for the purposes of logical discourse.
In Lectures on Science, Philosophy and Art (1908), 42.
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Does it not seem as if Algebra had attained to the dignity of a fine art, in which the workman has a free hand to develop his conceptions, as in a musical theme or a subject for a painting? It has reached a point where every properly developed algebraical composition, like a skillful landscape, is expected to suggest the notion of an infinite distance lying beyond the limits of the canvas.
In 'Lectures on the Theory of Reciprocants', Lecture XXI, American Journal of Mathematics (Jul 1886), 9, No. 3, 136.
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During a conversation with the writer in the last weeks of his life, Sylvester remarked as curious that notwithstanding he had always considered the bent of his mind to be rather analytical than geometrical, he found in nearly every case that the solution of an analytical problem turned upon some quite simple geometrical notion, and that he was never satisfied until he could present the argument in geometrical language.
In Proceedings London Royal Society, 63, 17.
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First get a clear notion of what you desire to accomplish and then in all probability you will succeed in doing it.
As quoted in Joseph Wickham Roe, English and American Tool Builders (1916), 48-49.
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Five centuries ago the printing press sparked a radical reshaping of the nature of education. By bringing a master’s words to those who could not hear a master’s voice, the technology of printing dissolved the notion that education must be reserved for those with the means to hire personal tutors. Today we are approaching a new technological revolution, one whose impact on education may be as far-reaching as that of the printing press: the emergence of powerful computers that are sufficiently inexpensive to be used by students for learning, play and exploration. It is our hope that these powerful but simple tools for creating and exploring richly interactive environments will dissolve the barriers to the production of knowledge as the printing press dissolved the barriers to its transmission.
As co-author with A.A. diSessa, from 'Preface', Turtle Geometry: The Computer as a Medium for Exploring Mathematics (1986), xiii.
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How can cosmic religious feeling be communicated from one person to another, if it can give rise to no definite notion of a God and no theology? In my view, it is the most important function of art and science to awaken this feeling and keep it alive in those who are receptive to it.
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I believe that the useful methods of mathematics are easily to be learned by quite young persons, just as languages are easily learned in youth. What a wondrous philosophy and history underlie the use of almost every word in every language—yet the child learns to use the word unconsciously. No doubt when such a word was first invented it was studied over and lectured upon, just as one might lecture now upon the idea of a rate, or the use of Cartesian co-ordinates, and we may depend upon it that children of the future will use the idea of the calculus, and use squared paper as readily as they now cipher. … When Egyptian and Chaldean philosophers spent years in difficult calculations, which would now be thought easy by young children, doubtless they had the same notions of the depth of their knowledge that Sir William Thomson might now have of his. How is it, then, that Thomson gained his immense knowledge in the time taken by a Chaldean philosopher to acquire a simple knowledge of arithmetic? The reason is plain. Thomson, when a child, was taught in a few years more than all that was known three thousand years ago of the properties of numbers. When it is found essential to a boy’s future that machinery should be given to his brain, it is given to him; he is taught to use it, and his bright memory makes the use of it a second nature to him; but it is not till after-life that he makes a close investigation of what there actually is in his brain which has enabled him to do so much. It is taken because the child has much faith. In after years he will accept nothing without careful consideration. The machinery given to the brain of children is getting more and more complicated as time goes on; but there is really no reason why it should not be taken in as early, and used as readily, as were the axioms of childish education in ancient Chaldea.
In Teaching of Mathematics (1902), 14.
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I cannot anyhow be contented to view this wonderful universe, and especially the nature of man, and to conclude that everything is the result of brute force. I am inclined to look at everything as resulting from designed laws, with the details, whether good or bad, left to the working out of what we call chance. Not that this notion at all satisfies me. I feel most deeply that the whole subject is too profound for the human intellect. A dog might as well speculate on the mind of Newton. Let each man hope and believe what he can.
Letter to Asa Gray (22 May 1860). In Charles Darwin and Francis Darwin (ed.), Charles Darwin: His Life Told in an Autobiographical Chapter, and in a Selected Series of His Published Letters (1892), 236.
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I cannot conceive of a God who rewards and punishes his creatures, or has a will of the type of which we are conscious in ourselves. An individual who should survive his physical death is also beyond my comprehension, nor do I wish it otherwise; such notions are for the fears or absurd egoism of feeble souls.
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I have been arranging certain experiments in reference to the notion that Gravity itself may be practically and directly related by experiment to the other powers of matter and this morning proceeded to make them. It was almost with a feeling of awe that I went to work, for if the hope should prove well founded, how great and mighty and sublime in its hitherto unchangeable character is the force I am trying to deal with, and how large may be the new domain of knowledge that may be opened up to the mind of man.
In ‎Thomas Martin (ed.) Faraday’s Diary: Sept. 6, 1847 - Oct. 17, 1851 (1934), 156.
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I never said a word against eminent men of science. What I complain of is a vague popular philosophy which supposes itself to be scientific when it is really nothing but a sort of new religion and an uncommonly nasty one. When people talked about the fall of man, they knew they were talking about a mystery, a thing they didn’t understand. Now they talk about the survival of the fittest: they think they do understand it, whereas they have not merely no notion, they have an elaborately false notion of what the words mean.
In The Club of Queer Trades (1903, 1905), 241.
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I once spoke to a human geneticist who declared that the notion of intelligence was quite meaningless, so I tried calling him unintelligent. He was annoyed, and it did not appease him when I went on to ask how he came to attach such a clear meaning to the notion of lack of intelligence. We never spoke again.
In Advice to a Young Scientist (1979), 25, footnote 2.
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If we knew all the laws of Nature, we should need only one fact or the description of one actual phenomenon to infer all the particular results at that point. Now we know only a few laws, and our result is vitiated, not, of course, by any confusion or irregularity in Nature, but by our ignorance of essential elements in the calculation. Our notions of law and harmony are commonly confined to those instances which we detect, but the harmony which results from a far greater number of seemingly conflicting, but really concurring, laws which we have not detected, is still more wonderful. The particular laws are as our points of view, as to the traveler, a mountain outline varies with every step, and it has an infinite number of profiles, though absolutely but one form. Even when cleft or bored through, it is not comprehended in its entireness.
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In science it is no crime to be wrong, unless you are (inappropriately) laying claim to truth. What matters is that science as a whole is a self-correcting mechanism in which both new and old notions are constantly under scrutiny. In other words, the edifice of scientific knowledge consists simply of a body of observations and ideas that have (so far) proven resistant to attack, and that are thus accepted as working hypotheses about nature.
In The Monkey in the Mirror: Essays on the Science of What Makes Us Human (2003), 9.
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In the year 1666 he retired again from Cambridge... to his mother in Lincolnshire & whilst he was musing in a garden it came into his thought that the power of gravity (wch brought an apple from the tree to the ground) was not limited to a certain distance from the earth but that this power must extend much farther than was usually thought. Why not as high as the moon said he to himself & if so that must influence her motion & perhaps retain her in her orbit, whereupon he fell a calculating what would be the effect of that supposition but being absent from books & taking the common estimate in use among Geographers & our seamen before Norwood had measured the earth, that 60 English miles were contained in one degree of latitude on the surface of the Earth his computation did not agree with his theory & inclined him then to entertain a notion that together with the force of gravity there might be a mixture of that force wch the moon would have if it was carried along in a vortex.
[The earliest account of Newton, gravity and an apple.]
Memorandum of a conversation with Newton in August 1726. Quoted in Richard Westfall, Never at Rest: A Biography of Isaac Newton (1980), 154.
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It is not always the truth that tells us where to look for new knowledge. We don’t search for the penny under the lamp post where the light is. We know we are more likely to find it out there in the darkness. My favorite way of expressing this notion to graduate students who are trying to do very hard experiments is to remind them that “God loves the noise as much as he does the signal.”
In 'Physics and the APS in 1979', Physics Today (Apr 1980), 33, No. 4, 50.
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It is still believed, apparently, that there is some thing mysteriously laudable about achieving viable offspring. I have searched the sacred and profane scriptures, for many years, but have yet to find any ground for this notion. To have a child is no more creditable than to have rheumatism–and no more discreditable. Ethically, it is absolutely meaningless. And practically, it is mainly a matter of chance.
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It is the function of notions in science to be useful, to be interesting, to be verifiable and to acquire value from anyone of these qualities. Scientific notions have little to gain as science from being forced into relation with that formidable abstraction, “general truth.”
In paper delivered before the Royal College of Surgeons of England (15 Feb 1932), in 'The Commemoration of Great Men', British Medical Journal (1932), 1, 32. Collected in The Collected Papers of Wilfred Trotter, FRS (1941), 29.
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It is therefore easy to see why the churches have always fought science and persecuted its devotees. On the other hand, I maintain that the cosmic religious feeling is the strongest and noblest motive for scientific research. Only those who realize the immense efforts and, above all, the devotion without which pioneer work in theoretical science cannot be achieved are able to grasp the strength of the emotion out of which alone such work, remote as it is from the immediate realities of life, can issue. What a deep conviction of the rationality of the universe and what a yearning to understand, were it but a feeble reflection of the mind revealed in this world, Kepler and Newton must have had to enable them to spend years of solitary labor in disentangling the principles of celestial mechanics! Those whose acquaintance with scientific research is derived chiefly from its practical results easily develop a completely false notion of the mentality of the men who, surrounded by a skeptical world, have shown the way to kindred spirits scattered wide through the world and through the centuries. Only one who has devoted his life to similar ends can have a vivid realization of what has inspired these men and given them the strength to remain true to their purpose in spite of countless failures. It is cosmic religious feeling that gives a man such strength. A contemporary has said, not unjustly, that in this materialistic age of ours the serious scientific workers are the only profoundly religious people.
…...
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It was his [Leibnitz’s] love of method and order, and the conviction that such order and harmony existed in the real world, and that our success in understanding it depended upon the degree and order which we could attain in our own thoughts, that originally was probably nothing more than a habit which by degrees grew into a formal rule. This habit was acquired by early occupation with legal and mathematical questions. We have seen how the theory of combinations and arrangements of elements had a special interest for him. We also saw how mathematical calculations served him as a type and model of clear and orderly reasoning, and how he tried to introduce method and system into logical discussions, by reducing to a small number of terms the multitude of compound notions he had to deal with. This tendency increased in strength, and even in those early years he elaborated the idea of a general arithmetic, with a universal language of symbols, or a characteristic which would be applicable to all reasoning processes, and reduce philosophical investigations to that simplicity and certainty which the use of algebraic symbols had introduced into mathematics.
A mental attitude such as this is always highly favorable for mathematical as well as for philosophical investigations. Wherever progress depends upon precision and clearness of thought, and wherever such can be gained by reducing a variety of investigations to a general method, by bringing a multitude of notions under a common term or symbol, it proves inestimable. It necessarily imports the special qualities of number—viz., their continuity, infinity and infinite divisibility—like mathematical quantities—and destroys the notion that irreconcilable contrasts exist in nature, or gaps which cannot be bridged over. Thus, in his letter to Arnaud, Leibnitz expresses it as his opinion that geometry, or the philosophy of space, forms a step to the philosophy of motion—i.e., of corporeal things—and the philosophy of motion a step to the philosophy of mind.
In Leibnitz (1884), 44-45. [The first sentence is reworded to better introduce the quotation. —Webmaster]
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Kepler’s suggestion of gravitation with the inverse distance, and Bouillaud’s proposed substitution of the inverse square of the distance, are things which Newton knew better than his modern readers. I have discovered two anagrams on his name, which are quite conclusive: the notion of gravitation was not new; but Newton went on.
In Budget of Paradoxes (1872), 82.
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Let him look at that dazzling light hung aloft as an eternal lamp to lighten the universe; let him behold the earth, a mere dot compared with the vast circuit which that orb describes, and stand amazed to find that the vast circuit itself is but a very fine point compared with the orbit traced by the stars as they roll their course on high. But if our vision halts there, let imagination pass beyond; it will fail to form a conception long before Nature fails to supply material. The whole visible world is but an imperceptible speck in the ample bosom of Nature. No notion comes near it. Though we may extend our thought beyond imaginable space, yet compared with reality we bring to birth mere atoms. Nature is an infinite sphere whereof the centre is everywhere, the circumference nowhere. In short, imagination is brought to silence at the thought, and that is the most perceptible sign of the all-power of God.
Let man reawake and consider what he is compared with the reality of things; regard himself lost in this remote corner of Nature; and from the tiny cell where he lodges, to wit the Universe, weigh at their true worth earth, kingdoms, towns, himself. What is a man face to face with infinity?
Pensées (1670), Section 1, aphorism 43. In H. F. Stewart (ed.), Pascal’s Pensées (1950), 19.
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Our notion of symmetry is derived from the human face. Hence, we demand symmetry horizontally and in breadth only, not vertically nor in depth.
In Pensées (1670), Section 1, No. 28. As paraphrased in W.H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms (1966). From the more complete translation, “Symmetry is what we see at a glance; based on the fact that there is no reason for any difference, and based also on the face of man; whence it happens that symmetry is only wanted in breadth, not in height or depth,” in Blaise Pascal and W.F. Trotter (trans.), 'Thoughts', collected in Charles W. Eliot (ed.), The Harvard Classics (1910), Vol. 48, 15. From the French, “Symétrie, en ce qu’on voit d’une vue, fondée sur ce qu’il n’y a pas de raison de faire autrement: et fondée aussi sur la figure de l’homme, d’où il arrive qu’on ne veut la symétrie qu’en largeur, non en hauteur ni profondeur,” in Blaise Pascal and Léon Brunschvicg (ed.), Pensées de Blaise Pascal (1904), Vol. 1, 37-38.
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Probability is the most important concept in modern science, especially as nobody has the slightest notion of what it means.
In Lecture (1929), as quoted in E.T. Bell (ed.), Development of Mathematics (1940), 540.
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Scholars should always receive with thanks new suppositions about things, provided they possess some tincture of sense; another head may often make an important discovery prompted by nothing more than such a stimulus: the generally accepted way of explaining a thing no longer had any effect on his brain and could communicate to it no new notion.
Aphorism 81 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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Science is not only compatible with spirituality; it is a profound source of spirituality. When we recognize our place in an immensity of light-years and in the passage of ages, when we grasp the intricacy, beauty, and subtlety of life, then that soaring feeling, that sense of elation and humility combined, is surely spiritual ... The notion that science and spirituality are somehow mutually exclusive does a disservice to both.
In Demon-Haunted World: Science as a Candle in the Dark (1995), 29.
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Significant inventions are not mere accidents. The erroneous view [that they are] is widely held, and it is one that the scientific and technical community, unfortunately, has done little to dispel. Happenstance usually plays a part, to be sure, but there is much more to invention than the popular notion of a bolt out of the blue. Knowledge in depth and in breadth are virtual prerequisites. Unless the mind is thoroughly charged beforehand, the proverbial spark of genius, if it should manifest itself, probably will find nothing to ignite.
Speech, at award of Perkin Medal. As quoted in 'Introduction', Royston M. Roberts, Serendipity (1989), x.
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Suppose it were perfectly certain that the life and fortune of every one of us would, one day or other, depend upon his winning or losing a game of chess. Don't you think that we should all consider it to be a primary duty to learn at least the names and the moves of the pieces; to have a notion of a gambit, and a keen eye for all the means of giving and getting out of check? Do you not think that we should look with a disapprobation amounting to scorn upon the father who allowed his son, or the state which allowed its members, to grow up without knowing a pawn from a knight?
Yet, it is a very plain and elementary truth that the life, the fortune, and the happiness of every one of us, and, more or less, of those who are connected with us, do depend upon our knowing something of the rules of a game infinitely more difficult and complicated than chess. It is a game which has been played for untold ages, every man and woman of us being one of the two players in a game of his or her own. The chess-board is the world, the pieces are the phenomena of the universe, the rules of the game are what we call the laws of nature. The player on the other side is hidden from us. We know that his play is always fair, just, and patient. But also we know, to our cost, that he never overlooks a mistake, or makes the smallest allowance for ignorance. To the man who plays well the highest stakes are paid with that sort of overflowing generosity with which the strong shows delight in strength. And one who plays ill is checkmated—without haste, but without remorse.
Address to the South London Working Men’s College. 'A Liberal Education; and Where to Find It', in David Masson, (ed.), Macmillan’s Magazine (Mar 1868), 17, 369. Also in 'A Liberal Education and Where to Find it' (1868). In Collected Essays (1893), Vol. 3, 82.
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Symbolism is useful because it makes things difficult. Now in the beginning everything is self-evident, and it is hard to see whether one self-evident proposition follows from another or not. Obviousness is always the enemy to correctness. Hence we must invent a new and difficult symbolism in which nothing is obvious. … Thus the whole of Arithmetic and Algebra has been shown to require three indefinable notions and five indemonstrable propositions.
In International Monthly (1901), 4, 85-86.
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That mathematics “do not cultivate the power of generalization,”; … will be admitted by no person of competent knowledge, except in a very qualified sense. The generalizations of mathematics, are, no doubt, a different thing from the generalizations of physical science; but in the difficulty of seizing them, and the mental tension they require, they are no contemptible preparation for the most arduous efforts of the scientific mind. Even the fundamental notions of the higher mathematics, from those of the differential calculus upwards are products of a very high abstraction. … To perceive the mathematical laws common to the results of many mathematical operations, even in so simple a case as that of the binomial theorem, involves a vigorous exercise of the same faculty which gave us Kepler’s laws, and rose through those laws to the theory of universal gravitation. Every process of what has been called Universal Geometry—the great creation of Descartes and his successors, in which a single train of reasoning solves whole classes of problems at once, and others common to large groups of them—is a practical lesson in the management of wide generalizations, and abstraction of the points of agreement from those of difference among objects of great and confusing diversity, to which the purely inductive sciences cannot furnish many superior. Even so elementary an operation as that of abstracting from the particular configuration of the triangles or other figures, and the relative situation of the particular lines or points, in the diagram which aids the apprehension of a common geometrical demonstration, is a very useful, and far from being always an easy, exercise of the faculty of generalization so strangely imagined to have no place or part in the processes of mathematics.
In An Examination of Sir William Hamilton’s Philosophy (1878), 612-13.
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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 conception of correspondence plays a great part in modern mathematics. It is the fundamental notion in the science of order as distinguished from the science of magnitude. If the older mathematics were mostly dominated by the needs of mensuration, modern mathematics are dominated by the conception of order and arrangement. It may be that this tendency of thought or direction of reasoning goes hand in hand with the modern discovery in physics, that the changes in nature depend not only or not so much on the quantity of mass and energy as on their distribution or arrangement.
In History of European Thought in the Nineteenth Century (1903), Vol. 2, 736.
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The constructions of the mathematical mind are at the same time free and necessary. The individual mathematician feels free to define his notions and set up his axioms as he pleases. But the question is will he get his fellow-mathematician interested in the constructs of his imagination. We cannot help the feeling that certain mathematical structures which have evolved through the combined efforts of the mathematical community bear the stamp of a necessity not affected by the accidents of their historical birth. Everybody who looks at the spectacle of modern algebra will be struck by this complementarity of freedom and necessity.
In 'A Half-Century of Mathematics',The American Mathematical Monthly (Oct 1951), 58, No. 8, 538-539.
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The great thing [about Kant’s philosophy] was to form the idea that this one thing—mind or world—may well be capable of other forms of appearance that we cannot grasp and that do not imply the notions of space and time. This means an imposing liberation from our inveterate prejudice.
…...
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The notion of evolution predicts the nested pattern of relationships we find in the living world; supernatural creation, on the other hand, predicts nothing. It is concepts of this latter kind that are truly untestable.
In The Monkey in the Mirror: Essays on the Science of What Makes Us Human (2003), 15.
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The notion that the “balance of nature” is delicately poised and easily upset is nonsense. Nature is extraordinarily tough and resilient, interlaced with checks and balances, with an astonishing capacity for recovering from disturbances in equilibrium. The formula for survival is not power; it is symbiosis.
In Encounter (Mar 1976), 16. As quoted and cited in Alan Lindsay Mackay , A Dictionary of Scientific Quotations (2nd Ed., 1991), 13.
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The notion, which is really the fundamental one (and I cannot too strongly emphasise the assertion), underlying and pervading the whole of modern analysis and geometry, is that of imaginary magnitude in analysis and of imaginary space in geometry.
In Presidential Address, in Collected Works, Vol. 11, 434.
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The opinion appears to be gaining ground that this very general conception of functionality, born on mathematical ground, is destined to supersede the narrower notion of causation, traditional in connection with the natural sciences. As an abstract formulation of the idea of determination in its most general sense, the notion of functionality includes and transcends the more special notion of causation as a one-sided determination of future phenomena by means of present conditions; it can be used to express the fact of the subsumption under a general law of past, present, and future alike, in a sequence of phenomena. From this point of view the remark of Huxley that Mathematics “knows nothing of causation” could only be taken to express the whole truth, if by the term “causation” is understood “efficient causation.” The latter notion has, however, in recent times been to an increasing extent regarded as just as irrelevant in the natural sciences as it is in Mathematics; the idea of thorough-going determinancy, in accordance with formal law, being thought to be alone significant in either domain.
In Presidential Address British Association for the Advancement of Science, Sheffield, Section A, Nature (1 Sep 1910), 84, 290.
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The original Marxist notion of ideology was conveniently forgotten because it inconveniently did not exempt common sense and empiricism from the charge of ideology.
In Social Amnesia (1975), 6-7.
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The powerful notion of entropy, which comes from a very special branch of physics … is certainly useful in the study of communication and quite helpful when applied in the theory of language.
From 'The Growth of Science and the Structure of Culture', Daedalus (Winter 1958), 87, No. 1, 67.
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The thing about electronic games is that they are basically repetitive. After a while, the children get bored. They need something different. [Meccano construction toy kits] offer creativity, a notion of mechanics, discovery of the world around you.
As quoted in by Hugh Schofield in web article 'Meccano Revives French Production' (23 Dec 2010).
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There are three ruling ideas, three so to say, spheres of thought, which pervade the whole body of mathematical science, to some one or other of which, or to two or all three of them combined, every mathematical truth admits of being referred; these are the three cardinal notions, of Number, Space and Order.
Arithmetic has for its object the properties of number in the abstract. In algebra, viewed as a science of operations, order is the predominating idea. The business of geometry is with the evolution of the properties of space, or of bodies viewed as existing in space.
In 'A Probationary Lecture on Geometry, York British Association Report (1844), Part 2; Collected Mathematical Papers, Vol. 2, 5.
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There are, and always have been, destructive pseudo-scientific notions linked to race and religion; these are the most widespread and damaging. Hopefully, educated people can succeed in shedding light into these areas of prejudice and ignorance, for as Voltaire once said: “Men will commit atrocities as long as they believe absurdities.”
From an article in a periodical of Miami-Dade Junior College by Bernard Sussman, 'Exclusive Interview with Martin Gardner', Southwind (Fall 1968), 3, No. 1, 7-11. As quoted and cited in Dana Richards, 'Martin Gardner: A “Documentary”', collected in Elwyn R. Berlekamp and ‎Tom Rodgers (ed.) The Mathemagician and Pied Puzzler: A Collection in Tribute to Martin Gardner (1999), 11.
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This notion that “science” is something that belongs in a separate compartment of its own, apart from everyday life, is one that I should like to challenge. We live in a scientific age; yet we assume that knowledge of science is the prerogative of only a small number of human beings, isolated and priest-like in their laboratories. This is not true. It cannot be true. The materials of science are the materials of life itself. Science is part of the reality of living; it is the what, the how, and the why of everything in our experience. It is impossible to understand man without understanding his environment and the forces that have molded him physically and mentally.
Address upon receiving National Book Award at reception, Hotel Commodore, New York (27 Jan 1952). As cited in Linda Lear, Rachel Carson: Witness for Nature (1997), 218-219.
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Too much openness and you accept every notion, idea, and hypothesis—which is tantamount to knowing nothing. Too much skepticism—especially rejection of new ideas before they are adequately tested—and you're not only unpleasantly grumpy, but also closed to the advance of science. A judicious mix is what we need.
In 'Wonder and Skepticism', Skeptical Enquirer (Jan-Feb 1995), 19, No. 1.
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Vision, in my view, is the cause of the greatest benefit to us, inasmuch as none of the accounts now given concerning the Universe would ever have been given if men had not seen the stars or the sun or the heavens. But as it is, the vision of day and night and of months and circling years has created the art of number and has given us not only the notion of Time but also means of research into the nature of the Universe. From these we have procured Philosophy in all its range, than which no greater boon ever has come or will come, by divine bestowal, unto the race of mortals.
Plato
…...
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We don't know what we are talking about. Many of us believed that string theory was a very dramatic break with our previous notions of quantum theory. But now we learn that string theory, well, is not that much of a break. The state of physics today is like it was when we were mystified by radioactivity. They were missing something absolutely fundamental. We are missing perhaps something as profound as they were back then.
Closing address to the 23rd Solvay Conference in Physics, Brussels, Belgium (Dec 2005). Quoted in Ashok Sengupta, Chaos, Nonlinearity, Complexity: The Dynamical Paradigm of Nature (2006), vii. Cite in Alfred B. Bortz, Physics: Decade by Decade (2007), 206.
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We have heard much about the poetry of mathematics, but very little of it has yet been sung. The ancients had a juster notion of their poetic value than we. The most distinct and beautiful statements of any truth must take at last the mathematical form.
In A Week on the Concord and Merrimack Rivers (1862), 381.
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We have overcome the notion that mathematical truths have an existence independent and apart from our own minds. It is even strange to us that such a notion could ever have existed.
With co-author James R. Newman, Mathematics and the Imagination (1940), 359.
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We have overcome the notion that mathematical truths have an existence independent and apart from our own minds. It is even strange to us that such a notion could ever have existed. [Coauthor with James R. Newman]
In Edward Kasner and James Newman, Mathematics and the Imagination (1940, 1949), 359.
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We speak erroneously of “artificial” materials, “synthetics”, and so forth. The basis for this erroneous terminology is the notion that Nature has made certain things which we call natural, and everything else is “man-made”, ergo artificial. But what one learns in chemistry is that Nature wrote all the rules of structuring; man does not invent chemical structuring rules; he only discovers the rules. All the chemist can do is find out what Nature permits, and any substances that are thus developed or discovered are inherently natural. It is very important to remember that.
From 'The Comprehensive Man', Ideas and Integrities: A Spontaneous Autobiographical Disclosure (1963), 75-76.
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We … came up with the notion that not all web pages are created equal. People are, but not web pages.
Guest Lecture, UC Berkeley, 'Search Engines, Technology, and Business' (3 Oct 2005). At 7:43 in the YouTube video.
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Whatever may have been imputed to some other studies under the notion of insignificancy and loss of time, yet these [mathematics], I believe, never caused repentance in any, except it was for their remissness in the prosecution of them.
In 'On the Usefulness of Mathematics', Works (1840), Vol. 2, 69.
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You sometimes speak of gravity as essential and inherent to matter. Pray do not ascribe that notion to me, for the cause of gravity is what I do not pretend to know, and therefore would take more time to consider of it.
Letter to Dr. Bentley (17 Jan 1692). In Four Letters from Sir Isaac Newton to Doctor Bentley (1756), 20.
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You’ve got to be fairly solemn [about the environment]. I mean the mere notion that there are three times as many people on Earth as there were when I started making television. How can the Earth accommodate them? When people, including politicians, set their faces against looking at the consequences—it’s just unbelievable that anyone could ignore it.
'Sir David Attenborough interview' (aged 84), by Andrew Pettie in The Telegraph (23 Dec 2010).
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“One and one make two” assumes that the changes in the shift of circumstance are unimportant. But it is impossible for us to analyze this notion of unimportant change.
In Science and Philosophy (1948), 103.
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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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