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Who said: “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”
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Number Quotes (90 quotes)

'Conservation' (the conservation law) means this ... that there is a number, which you can calculate, at one moment—and as nature undergoes its multitude of changes, this number doesn't change. That is, if you calculate again, this quantity, it'll be the same as it was before. An example is the conservation of energy: there's a quantity that you can calculate according to a certain rule, and it comes out the same answer after, no matter what happens, happens.
'The Great Conservation Principles', The Messenger Series of Lectures, No. 3, Cornell University, 1964. From transcript of BBC programme (11 Dec 1964).
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230(231-1) ... is the greatest perfect number known at present, and probably the greatest that ever will be discovered; for; as they are merely curious without being useful, it is not likely that any person will attempt to find a number beyond it.
In An Elementary Investigation of the Theory of Numbers (1811), 43. The stated number, which evaluates as 2305843008139952128 was discovered by Euler in 1772 as the eighth known perfect number. It has 19 digits. By 2013, the 48th perfect number found had 34850340 digits.
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Die ganzen Zahlen hat der liebe Gatt gemacht, alles andere ist Menschenwerk.
The dear God has made the whole numbers, all the rest is man's work.
Speech at the Berlin meeting of the Society of German Scientists and Doctors in 1886, published in Jahreshericht der Deutschen Mathematiker-Vereinigung. Trans. obituary of Kronecker by H. E. Weber, Year Book of the Gennan Mathematics Association, 1893, 19.
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Replying to G. H. Hardy's suggestion that the number of a taxi (1729) was “dull”: No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways, the two ways being 13 + 123 and 93 + 103.
Proceedings of the London Mathematical Society (26 May 1921).
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Tolle numerum omnibus rebus et omnia pereunt.
Take from all things their number and all shall perish.
Etymologies [c.600], Book III, chapter 4, quoted in E. Grant (ed.), A Source Book in Medieval Science (1974), trans. E. Brehaut (1912), revised by E. Grant, 5.
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A googleplex is precisely as far from infinity as is the number 1 ... No matter what number you have in mind, infinity is larger.
In Cosmos (1980, 2011), 181.
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A poet is, after all, a sort of scientist, but engaged in a qualitative science in which nothing is measurable. He lives with data that cannot be numbered, and his experiments can be done only once. The information in a poem is, by definition, not reproducible. ... He becomes an equivalent of scientist, in the act of examining and sorting the things popping in [to his head], finding the marks of remote similarity, points of distant relationship, tiny irregularities that indicate that this one is really the same as that one over there only more important. Gauging the fit, he can meticulously place pieces of the universe together, in geometric configurations that are as beautiful and balanced as crystals.
In The Medusa and the Snail: More Notes of a Biology Watcher (1974, 1995), 107.
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Accurate and minute measurement seems to the non-scientific imagination, a less lofty and dignified work than looking for something new. But nearly all the grandest discoveries of science have been but the rewards of accurate measurement and patient long-continued labour in the minute sifting of numerical results.
Presidential inaugural address, to the General Meeting of the British Association, Edinburgh (2 Aug 1871). In Report of the Forty-First Meeting of the British Association for the Advancement of Science (1872) By xci.
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All is number
Quoted in Robert J. Scully, The Demon and the Quantum (2007), 7.

All the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers.
from Faraday's Lines of Force (1856)
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All the modern higher mathematics is based on a calculus of operations, on laws of thought. All mathematics, from the first, was so in reality; but the evolvers of the modern higher calculus have known that it is so. Therefore elementary teachers who, at the present day, persist in thinking about algebra and arithmetic as dealing with laws of number, and about geometry as dealing with laws of surface and solid content, are doing the best that in them lies to put their pupils on the wrong track for reaching in the future any true understanding of the higher algebras. Algebras deal not with laws of number, but with such laws of the human thinking machinery as have been discovered in the course of investigations on numbers. Plane geometry deals with such laws of thought as were discovered by men intent on finding out how to measure surface; and solid geometry with such additional laws of thought as were discovered when men began to extend geometry into three dimensions.
Lectures on the Logic of Arithmetic (1903), Preface, 18-19.
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Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians.
The Principles of Mathematics (1903), 451.
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Astronomy affords the most extensive example of the connection of physical sciences. In it are combined the sciences of number and quantity, or rest and motion. In it we perceive the operation of a force which is mixed up with everything that exists in the heavens or on earth; which pervades every atom, rules the motion of animate and inanimate beings, and is a sensible in the descent of the rain-drop as in the falls of Niagara; in the weight of the air, as in the periods of the moon.
On the Connexion of the Physical Sciences (1858), 1.
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Bistromathics itself is simply a revolutionary new way of understanding the behavior of numbers. Just as Einstein observed that space was not an absolute but depended on the observer's movement in space, and that time was not an absolute, but depended on the observer's movement in time, so it is now realized that numbers are not absolute, but depend on the observer's movement in restaurants.
Life, the Universe and Everything (1982, 1995), 47.
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Borel makes the amusing supposition of a million monkeys allowed to play upon the keys of a million typewriters. What is the chance that this wanton activity should reproduce exactly all of the volumes which are contained in the library of the British Museum? It certainly is not a large chance, but it may be roughly calculated, and proves in fact to be considerably larger than the chance that a mixture of oxygen and nitrogen will separate into the two pure constituents. After we have learned to estimate such minute chances, and after we have overcome our fear of numbers which are very much larger or very much smaller than those ordinarily employed, we might proceed to calculate the chance of still more extraordinary occurrences, and even have the boldness to regard the living cell as a result of random arrangement and rearrangement of its atoms. However, we cannot but feel that this would be carrying extrapolation too far. This feeling is due not merely to a recognition of the enormous complexity of living tissue but to the conviction that the whole trend of life, the whole process of building up more and more diverse and complex structures, which we call evolution, is the very opposite of that which we might expect from the laws of chance.
The Anatomy of Science (1926), 158-9.
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Chemistry works with an enormous number of substances, but cares only for some few of their properties; it is an extensive science. Physics on the other hand works with rather few substances, such as mercury, water, alcohol, glass, air, but analyses the experimental results very thoroughly; it is an intensive science. Physical chemistry is the child of these two sciences; it has inherited the extensive character from chemistry. Upon this depends its all-embracing feature, which has attracted so great admiration. But on the other hand it has its profound quantitative character from the science of physics.
In Theories of Solutions (1912), xix.
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De Morgan was explaining to an actuary what was the chance that a certain proportion of some group of people would at the end of a given time be alive; and quoted the actuarial formula, involving p [pi], which, in answer to a question, he explained stood for the ratio of the circumference of a circle to its diameter. His acquaintance, who had so far listened to the explanation with interest, interrupted him and exclaimed, 'My dear friend, that must be a delusion, what can a circle have to do with the number of people alive at a given time?'
Mathematical Recreations and Problems (1896), 180; See also De Morgan's Budget of Paradoxes (1872), 172.
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Defendit numerus: There is safety in numbers.
Anonymous
Latin proverb, first recorded in English about 1550. In James Roy Newman (ed.) The World of Mathematics (1956), Vol. 3, 1452.
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He telleth the number of stars; he calleth them all by their names.
Bible
Psalms 147:4 in The Holy Bible (1746), 549.
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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 never could do anything with figures, never had any talent for mathematics, never accomplished anything in my efforts at that rugged study, and to-day the only mathematics I know is multiplication, and the minute I get away up in that, as soon as I reach nine times seven— [He lapsed into deep thought, trying to figure nine times seven. Mr. McKelway whispered the answer to him.] I've got it now. It's eighty-four. Well, I can get that far all right with a little hesitation. After that I am uncertain, and I can't manage a statistic.
Speech at the New York Association for Promoting the Interests of the Blind (29 Mar 1906). In Mark Twain and William Dean Howells (ed.), Mark Twain's Speeches? (1910), 323.
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I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”
Quoted in G.H. Hardy, Ramanujan; Twelve Lectures on Subjects Suggested by his Life and Work (1940, reprint 1999), 12.
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Iamblichus in his treatise On the Arithmetic of Nicomachus observes p. 47- “that certain numbers were called amicable by those who assimilated the virtues and elegant habits to numbers.” He adds, “that 284 and 220 are numbers of this kind; for the parts of each are generative of each other according to the nature of friendship, as was shown by Pythagoras. For some one asking him what a friend was, he answered, another I (ετεϑος εγω) which is demonstrated to take place in these numbers.” [“Friendly” thus: Each number is equal to the sum of the factors of the other.]
In Theoretic Arithmetic (1816), 122. (Factors of 284 are 1, 2, 4 ,71 and 142, which give the sum 220. Reciprocally, factors of 220 are 1, 2, 4, 5, 10, 11 ,22, 44, 55 and 110, which give the sum 284.) Note: the expression “alter ego” is Latin for “the other I.”
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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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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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If you are surprised at the number of our maladies, count our cooks.
In Noble Words and Noble Deeds (1877), 239.
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In place of infinity we usually put some really big number, like 15.
Perhaps referring to the programmer's hexadecimal counting scheme which has 16 digits (0-0 followed by digits A-F), useful in binary context as a power of 2.
Anonymous
Attributed to a Computer Science Professor on various web pages. Webmaster has found no print source for this wording and comments, but its originality makes it worthy of inclusion here. Webmaster comments: perhaps one of those infinite number of monkeys typed it! Please make contact if you know a primary print source.
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It is a right, yes a duty, to search in cautious manner for the numbers, sizes, and weights, the norms for everything [God] has created. For He himself has let man take part in the knowledge of these things ... For these secrets are not of the kind whose research should be forbidden; rather they are set before our eyes like a mirror so that by examining them we observe to some extent the goodness and wisdom of the Creator.
Epitome of Copernican Astronomy. In Michael B. Foster, Mystery and Philosophy, 61. Cited by Max Casper and Doris Hellman, trans., ed. Kepler (1954), 381. Cited by Gerald J. Galgan, Interpreting the Present: Six Philosophical Essays (1993), 105. Gerald J. Galgan
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It is agreed that all sound which is the material of music is of three sorts. First is harmonica, which consists of vocal music; second is organica, which is formed from the breath; third is rhythmica, which receives its numbers from the beat of the fingers. For sound is produced either by the voice, coming through the throat; or by the breath, coming through the trumpet or tibia, for example; or by touch, as in the case of the cithara or anything else that gives a tuneful sound on being struck.
Etymologies [c.600], Book III, chapter 19, quoted in E. Grant (ed.), A Source Book in Medieval Science (1974), trans. E. Brehaut (1912), revised by E. Grant, 10.
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It is known that there is an infinite number of worlds, but that not every one is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is as near to nothing as makes no odds, so if every planet in the Universe has a populations of zero then the entire population of the Universe must also be zero, and any people you may actually meet from time to time are merely the products of a deranged imagination.
'Fit the Fifth', The Original Hitchhiker Radio Script, 102. In Alma E. Cavazos-Gaither, Mathematically Speaking (1998), 58.
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It is not of the essence of mathematics to be conversant with the ideas of number and quantity. Whether as a general habit of mind it would be desirable to apply symbolic processes to moral argument, is another question.
An Investigation of the Laws of Thought (1854), 12.
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It is strange that we know so little about the properties of numbers. They are our handiwork, yet they baffle us; we can fathom only a few of their intricacies. Having defined their attributes and prescribed their behavior, we are hard pressed to perceive the implications of our formulas.
'The Mysteries of Arithmetic: Commentary', The World of Mathematics (2000), Vol. 1, 497.
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It seems to me, that the only Objects of the abstract Sciences or of Demonstration is Quantity and Number, and that all Attempts to extend this more perfect Species of Knowledge beyond these Bounds are mere Sophistry and Illusion.
An Enquiry Concerning Human Understanding (1748), 252.
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Just as mathematics aims to study such entities as numbers, functions, spaces, etc., the subject matter of metamathematics is mathematics itself.
In 'Mathematics: A Non-Technical Exposition', American Scientist (3 Jul 1954), 42, No. 3, 490.
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Lord Kelvin was so satisfied with this triumph of science that he declared himself to be as certain of the existence of the ether as a man can be about anything.... “When you can measure what you are speaking about, and express it in numbers, you know something about it....” Thus did Lord Kelvin lay down the law. And though quite wrong, this time he has the support of official modern Science. It is NOT true that when you can measure what you are speaking about, you know something about it. The fact that you can measure something doesn't even prove that that something exists.... Take the ether, for example: didn't they measure the ratio of its elasticity to its density?
In Science is a Sacred Cow (1950), 69-70; 85.
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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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Measure, time and number are nothing but modes of thought or rather of imagination.
Letter to Ludvicus Meyer (20 Apr 1663), in Correspondence of Spinoza (2003), 118.
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Number is divided into even and odd. Even number is divided into the following: evenly even, evenly uneven, and unevenly uneven. Odd number is divided into the following: prime and incomposite, composite, and a third intermediate class (mediocris) which in a certain way is prime and incomposite but in another way secondary and composite.
Etymologies [c.600], Book III, chapter 5, quoted in E. Grant (ed.), A Source Book in Medieval Science (1974), trans. E. Brehaut (1912), revised by E. Grant, 5.
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Number is the within of all things.
Attributed as a concept rather than actual words (none of the original writings of Pythagoras have survived). In L. A. Michael, The Principles of Existence & Beyond (2007), 16, but webmaster is unable to validate or find in quote dictionaries.
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Number is therefore the most primitive instrument of bringing an unconscious awareness of order into consciousness.
In Creation Myths (1995), 326.
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Number, the most excellent of all inventions.
Aeschylus
Spoken by the character Prometheus in play, 'Prometheus Bound', as translated by G.M. Cookson, in Four Plays of Aeschylus (1922), 182.
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Numbers are a fearful thing.
Euripides
Spoken by the character Hecuba in the play, 'Hecuba', as translated by Edward P. Coleridge, in The Plays of Euripides (1907), Vol. 2, 157.
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Numbers written on restaurant checks [bills] within the confines of restaurants do not follow the same mathematical laws as numbers written on any other pieces of paper in any other parts of the Universe.
This single statement took the scientific world by storm. It completely revolutionized it. So many mathematical conferences got held in such good restaurants that many of the finest minds of a generation died of obesity and heart failure and the science of math was put back by years.
Life, the Universe and Everything (1982, 1995), 49.
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O comfortable allurement, O ravishing perswasion, to deal with a Science, whose subject is so Auncient, so pure, so excellent, so surmounting all creatures... By Numbers propertie ... we may... arise, clime, ascend, and mount up (with Speculative winges) in spirit, to behold in the Glas of creation, the Forme of Formes, the Exemplar Number of all things Numerable... Who can remaine, therefore, unpersuaded, to love, allow, and honor the excellent sciehce of Arithmatike?
John Dee
'Mathematicall Preface', in H. Billingsley, trans. The Elements of Geometry of the most Aunceint Philosopher Euclide of Megara (1570), in J. L. Hellbron, Weighing Imponderables and Other Quantitative Science around 1800 (1993), 2.
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Physical changes take place continuously, while chemical changes take place discontinuously. Physics deals chiefly with continuous varying quantities, while chemistry deals chiefly with whole numbers.
Treatise on Thermodynamics (1897), trans. Alexander Ogg (1903), 22, footnote.
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Physics is NOT a body of indisputable and immutable Truth; it is a body of well-supported probable opinion only .... Physics can never prove things the way things are proved in mathematics, by eliminating ALL of the alternative possibilities. It is not possible to say what the alternative possibilities are.... Write down a number of 20 figures; if you multiply this by a number of, say, 30 figures, you would arrive at some enormous number (of either 49 or 50 figures). If you were to multiply the 30-figure number by the 20-figure number you would arrive at the same enormous 49- or 50-figure number, and you know this to be true without having to do the multiplying. This is the step you can never take in physics.
In Science is a Sacred Cow (1950), 68, 88, 179.
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Population, when unchecked, increases in a geometrical ratio. Subsistence increases only in an arithmetical ratio. A slight acquaintance with numbers will show the immensity of the first power in comparison of the second.
An Essay on the Principle of Population (1798). In E. A. Wrigley and David Souden (eds.), The Works of Thomas Malthus (1986), Vol. 1, 9.
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Referring to the decimal system of numeration or its equivalent (with some base other than 10): To what heights would science now be raised if Archimedes had made that discovery!
Gauss regarded this oversight as the greatest calamity in the history of science.
Quoted in James Roy Newman, The World of Mathematics, 328.
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Standard mathematics has recently been rendered obsolete by the discovery that for years we have been writing the numeral five backward. This has led to reevaluation of counting as a method of getting from one to ten. Students are taught advanced concepts of Boolean algebra, and formerly unsolvable equations are dealt with by threats of reprisals.
Getting Even (1978), 44.
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Statistician: A man who believes figures don't lie but admits that, under analysis some of them won't stand up either.
Evan Esar
The Dictionary of Humorous Quotations (1949). In Robert Harris Shutler, Mathematics 436 - Finely Explained (2004), 3.
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Statistics: The only science that enables different experts using the same figures to draw different conclusions.
Evan Esar
The Dictionary of Humorous Quotations (1949). In Robert Harris Shutler, Mathematics 436 - Finely Explained (2004), 3.
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That this subject [of imaginary magnitudes] has hitherto been considered from the wrong point of view and surrounded by a mysterious obscurity, is to be attributed largely to an ill-adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question.
Theoria Residiorum Biquadraticorum, Commentario secunda', Werke (1863), Vol. 2. Quoted in Robert Edouard Moritz, Memorabilia Mathematica (1914), 282.

The Qualities then that are in Bodies rightly considered, are of Three sorts.
First, the Bulk, Figure, Number, Situation, and Motion, or Rest of their solid Parts; those are in them, whether we perceive them or no; and when they are of that size, that we can discover them, we have by these an Idea of the thing, as it is in it self, as is plain in artificial things. These I call primary Qualities.
Secondly, The Power that is in any Body, by Reason of its insensible primary Qualities, to operate after a peculiar manner on any of our Senses, and thereby produce in us the different Ideas of several Colours, Sounds, Smells, Tastes, etc. These are usually called sensible Qualities.
Thirdly, The Power that is in any Body, by Reason of the particular Constitution of its primary Qualities, to make such a change in the Bulk, Figure, Texture, and Motion of another Body, as to make it operate on our Senses, differently from what it did before. Thus the Sun has a Power to make Wax white, and Fire to make Lead fluid. These are usually called Powers.
An Essay Concerning Human Understanding (1690). Edited by Peter Nidditch (1975), Book 2, Chapter 8, Section 23, 140-1.
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The answer to the Great Question of … Life, the Universe and Everything … is Forty-two
The Hitch Hiker's Guide to the Galaxy (1979), Chapter 27.
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The faculty for remembering is not diminished in proportion to what one has learnt, just as little as the number of moulds in which you cast sand lessens its capacity for being cast in new moulds.
Religion: a Dialogue, and Other Essays (1890), 99.
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The first nonabsolute number is the number of people for whom the table is reserved. This will vary during the course of the first three telephone calls to the restaurant, and then bear no apparent relation to the number of people who actually turn up, or to the number of people who subsequently join them after the show/match/party/gig, or to the number of people who leave when they see who else has turned up.
The second nonabsolute number is the given time of arrival, which is now known to be one of the most bizarre of mathematical concepts, a recipriversexcluson, a number whose existence can only be defined as being anything other than itself. In other words, the given time of arrival is the one moment of time at which it is impossible that any member of the party will arrive. Recipriversexclusons now play a vital part in many branches of math, including statistics and accountancy and also form the basic equations used to engineer the Somebody Else's Problem field.
The third and most mysterious piece of nonabsoluteness of all lies in the relationship between the number of items on the check [bill], the cost of each item, the number of people at the table and what they are each prepared to pay for. (The number of people who have actually brought any money is only a subphenomenon of this field.)
Life, the Universe and Everything (1982, 1995), 47-48.
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The judicial mind is too commonly characterized by a regard for a fourth decimal as the equal of a whole number.
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The law of conservation rigidly excludes both creation and annihilation. Waves may change to ripples, and ripples to waves,—magnitude may be substituted for number, and number for magnitude,—asteroids may aggregate to suns, suns may resolve themselves into florae and faunae, and florae and faunae melt in air,—the flux of power is eternally the same. It rolls in music through the ages, and all terrestrial energy,—the manifestations of life, as well as the display of phenomena, are but the modulations of its rhythm.
Conclusion to lecture 12 (10 Apr 1862) at the Royal Institution, collected in Heat Considered as a Mode of Motion: Being a Course of Twelve Lectures (1863), 449.
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The method of producing these numbers is called a sieve by Eratosthenes, since we take the odd numbers mingled and indiscriminate and we separate out of them by this method of production, as if by some instrument or sieve, the prime and incomposite numbers by themselves, and the secondary and composite numbers by themselves, and we find separately those that are mixed.
Nicomachus, Introduction to Arithmetic, 1.13.2. Quoted in Morris R. Cohen and I. E. Drabkin, A Sourcebook in Greek Science (1948), 19-20.
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The methods of theoretical physics should be applicable to all those branches of thought in which the essential features are expressible with numbers.
Nobel Prize Banquet Speech (10 Dec1933). In Carl Gustaf Santesson (Ed.), Les Prix Nobel en 1933 (1935), 78
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The northern ocean is beautiful, ... and beautiful the delicate intricacy of the snowflake before it melts and perishes, but such beauties are as nothing to him who delights in numbers, spurning alike the wild irrationality of life and baffling complexity of nature’s laws.
In Kandelman's Krim: A Realistic Fantasy (1957), 101.
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The present rate of progress [in X-ray crystallography] is determined, not so much by the lack of problems to investigate or the limited power of X-ray analysis, as by the restricted number of investigators who have had a training in the technique of the new science, and by the time it naturally takes for its scientific and technical importance to become widely appreciated.
Concluding remark in Lecture (1936) on 'Forty Years of Crystal Physics', collected in Needham and Pagel (eds.) in Background to Modern Science: Ten Lectures at Cambridge Arranged by the History of Science Committee, (1938), 89.
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The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length... Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.
Disquisitiones Arithmeticae (1801), Article 329

The qualities of number appear to lead to the apprehension of truth.
Plato
The Republic 7 525b, trans. P. Shorey (1935), Vol. 2, Book 7, 161.
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The rudest numerical scales, such as that by which the mineralogists distinguish different degrees of hardness, are found useful. The mere counting of pistils and stamens sufficed to bring botany out of total chaos into some kind of form. It is not, however, so much from counting as from measuring, not so much from the conception of number as from that of continuous quantity, that the advantage of mathematical treatment comes. Number, after all, only serves to pin us down to a precision in our thoughts which, however beneficial, can seldom lead to lofty conceptions, and frequently descend to pettiness.
On the Doctrine of Chances, with Later Reflections (1878), 61-2.
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The starting point of Darwin’s theory of evolution is precisely the existence of those differences between individual members of a race or species which morphologists for the most part rightly neglect. The first condition necessary, in order that any process of Natural Selection may begin among a race, or species, is the existence of differences among its members; and the first step in an enquiry into the possible effect of a selective process upon any character of a race must be an estimate of the frequency with which individuals, exhibiting any given degree of abnormality with respect to that, character, occur. The unit, with which such an enquiry must deal, is not an individual but a race, or a statistically representative sample of a race; and the result must take the form of a numerical statement, showing the relative frequency with which the various kinds of individuals composing the race occur.
Biometrika: A Joumal for the Statistical Study of Biological Problems (1901), 1, 1-2.
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The total number of people who understand relativistic time, even after eighty years since the advent of special relativity, is still much smaller than the number of people who believe in horoscopes.
'Classical to Quantum: A Generalized Phase Transition', in A. van der Merwe et al. (eds.) Microphysical Reality and Quantum Formalism (1987), 145.
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The transfinite numbers are in a sense the new irrationalities [ ... they] stand or fall with the finite irrational numbers.
Gesammelte Abhandlungen (1932),395, trans. Ivor Grattan-Guinness.
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There is more danger of numerical sequences continued indefinitely than of trees growing up to heaven. Each will some time reach its greatest height.
Grundgesetz der Arithmetik(1893), Vol. 2, Section 60, In P. Greach and M. Black (eds., Translations from the Philosophical Writings of Gottlob Frege (1952), 204.
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This method is, to define as the number of a class the class of all classes similar to the given class. Membership of this class of classes (considered as a predicate) is a common property of all the similar classes and of no others; moreover every class of the set of similar classes has to the set of a relation which it has to nothing else, and which every class has to its own set. Thus the conditions are completely fulfilled by this class of classes, and it has the merit of being determinate when a class is given, and of being different for two classes which are not similar. This, then, is an irreproachable definition of the number of a class in purely logical terms.
The Principles of Mathematics (1903), 115.
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Those who are unacquainted with the details of scientific investigation have no idea of the amount of labour expended in the determination of those numbers on which important calculations or inferences depend. They have no idea of the patience shown by a Berzelius in determining atomic weights; by a Regnault in determining coefficients of expansion; or by a Joule in determining the mechanical equivalent of heat.
In Sound: A Course of Eight Lectures Delivered at the Royal Institution of Great Britain (1867), 26.
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Those who think 'Science is Measurement' should search Darwin's works for numbers and equations.
'David H. Hubel', in Larry R. Squire (ed.), The History of Neuroscience in Autobiography (1996), Vol. 1, 313.
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Through [the growing organism's] power of assimilation there is a constant encroachment of the organic upon the inorganic, a constant attempt to convert all available material into living substance, and to indefinitely multiply the total number of individual organisms.
In History of the Human Body (1919), 2.
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To a mathematician the eleventh means only a single unit: to the bushman who cannot count further than his ten fingers it is an incalculable myriad.
'Maxims for Revolutionists', in Man and Superman (1905), 236.
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To Nature nothing can be added; from Nature nothing can be taken away; the sum of her energies is constant, and the utmost man can do in the pursuit of physical truth, or in the applications of physical knowledge, is to shift the constituents of the never-varying total. The law of conservation rigidly excludes both creation and annihilation. Waves may change to ripples, and ripples to waves; magnitude may be substituted for number, and number for magnitude; asteroids may aggregate to suns, suns may resolve themselves into florae and faunae, and floras and faunas melt in air: the flux of power is eternally the same. It rolls in music through the ages, and all terrestrial energy—the manifestations of life as well as the display of phenomena—are but the modulations of its rhythm.
Conclusion of Heat Considered as a Mode of Motion: Being a Course of Twelve Lectures Delivered at the Royal Institution of Great Britain in the Season of 1862 (1863), 449.
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Truly I say to you, a single number has more genuine and permanent value than an expensive library full of hypotheses.
Letter to Griesinger (20 Jul 1844). In Jacob J. Weyrauch (ed.), Kleinere Schriften und Briefe von Robert Milyer, nebst Mittheilungen aus seinem Leben (1893), 226. Trans. Kenneth L. Caneva, Robert Mayer and the Conservation of Energy (1993), 37.
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Undeveloped though the science [of chemistry] is, it already has great power to bring benefits. Those accruing to physical welfare are readily recognized, as in providing cures, improving the materials needed for everyday living, moving to ameliorate the harm which mankind by its sheer numbers does to the environment, to say nothing of that which even today attends industrial development. And as we continue to improve our understanding of the basic science on which applications increasingly depend, material benefits of this and other kinds are secured for the future.
Speech at the Nobel Banquet (10 Dec 1983) for his Nobel Prize in Chemistry. In Wilhelm Odelberg (ed.), Les Prix Nobel: The Nobel Prizes (1984), 43.
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We can invent as many theories we like, and any one of them can be made to fit the facts. But that theory is always preferred which makes the fewest number of assumptions.
From interview by S.J. Woolf, 'Einstein’s Own Corner of Space’, New York Times (18 Aug 1929), Sunday Magazine, 2.
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We know that there is an infinite, and we know not its nature. As we know it to be false that numbers are finite, it is therefore true that there is a numerical infinity. But we know not of what kind; it is untrue that it is even, untrue that it is odd; for the addition of a unit does not change its nature; yet it is a number, and every number is odd or even (this certainly holds of every finite number). Thus we may quite well know that there is a God without knowing what He is.
Pensées (1670), Section 1, aphorism 223. In H. F. Stewart (ed.), Pascal's Pensées (1950), 117.
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We know the laws of trial and error, of large numbers and probabilities. We know that these laws are part of the mathematical and mechanical fabric of the universe, and that they are also at play in biological processes. But, in the name of the experimental method and out of our poor knowledge, are we really entitled to claim that everything happens by chance, to the exclusion of all other possibilities?
From Nobel Prize Lecture (Dec 1974), 'The Coming Age of the Cell'. Collected in Jan Lindsten (ed.) Nobel Lectures, Physiology or Medicine 1971-1980 (1992).
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We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori.
Letter to Friedrich Bessel (1830).

What about the magical number seven? What about the seven wonders of the world, the seven seas, the seven deadly sins, the seven daughters of Atlas in the Pleiades, the seven ages of man, the seven levels of hell, the seven primary colors, the seven notes of the musical scale, and the seven days of the week? What about the seven-point rating scale, the seven categories for absolute judgment, the seven objects in the span of attention, and the seven digits in the span of immediate memory? For the present I propose to withhold judgment. Perhaps there is something deep and profound behind all these sevens, something just calling out for us to discover it. But I suspect that it is only a pernicious, Pythagorean coincidence.
The Magical Number Seven, Plus or Minus Two (1956), 42-3.
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Whenever a man can get hold of numbers, they are invaluable: if correct, they assist in informing his own mind, but they are still more useful in deluding the minds of others. Numbers are the masters of the weak, but the slaves of the strong.
Passages From the Life of a Philosopher (1864), 410.

Wherever there is number, there is beauty.
Proclus
Quoted in Morris Kline, Mathematical Thought from Ancient to Modern Times (1990), Vol. 1, 131.
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Whether we like it or not, quantification in history is here to stay for reasons which the quantifiers themselves might not actively approve. We are becoming a numerate society: almost instinctively there seems now to be a greater degree of truth in evidence expressed numerically than in any literary evidence, no matter how shaky the statistical evidence, or acute the observing eye.
Is History Sick? (1973), 64.
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You know the formula m over naught equals infinity, m being any positive number? [m/0 = ∞]. Well, why not reduce the equation to a simpler form by multiplying both sides by naught? In which case you have m equals infinity times naught [m = ∞ × 0]. That is to say, a positive number is the product of zero and infinity. Doesn't that demonstrate the creation of the Universe by an infinite power out of nothing? Doesn't it?
In Point Counter Point (1928), 162.
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[Boswell]: Sir Alexander Dick tells me, that he remembers having a thousand people in a year to dine at his house: that is, reckoning each person as one, each time that he dined there. [Johnson]: That, Sir, is about three a day. [Boswell]: How your statement lessens the idea. [Johnson]: That, Sir, is the good of counting. It brings every thing to a certainty, which before floated in the mind indefinitely.
Entry for Fri 18 Apr 1783. In George Birkbeck-Hill (ed.), Boswell's Life of Johnson (1934-50), Vol. 4, 204.
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[Louis Rendu, Bishop of Annecy] collects observations, makes experiments, and tries to obtain numerical results; always taking care, however, so to state his premises and qualify his conclusions that nobody shall be led to ascribe to his numbers a greater accuracy than they merit. It is impossible to read his work, and not feel that he was a man of essentially truthful mind and that science missed an ornament when he was appropriated by the Church.
In The Glaciers of the Alps (1860), 299.
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[On why are numbers beautiful?] It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.
As quoted in Paul Hoffman, The Man who Loves Only Numbers (1998), 44.
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[This] may prove to be the beginning of some embracing generalization, which will throw light, not only on radioactive processes, but on elements in general and the Periodic Law.... Chemical homogeneity is no longer a guarantee that any supposed element is not a mixture of several of different atomic weights, or that any atomic weight is not merely a mean number.
From Chemical Society's Annual Reports (1910), Vol. 7, 285. As quoted in Francis Aston in Lecture (1936) on 'Forty Years of Atomic Theory', collected in Needham and Pagel (eds.) in Background to Modern Science: Ten Lectures at Cambridge Arranged by the History of Science Committee, (1938), 100. Cited in Alfred Walter Stewart, Recent Advances in Physical and Inorganic Chemistry (1920), 198.
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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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- 100 -
Sophie Germain
Gertrude Elion
Ernest Rutherford
James Chadwick
Marcel Proust
William Harvey
Johann Goethe
John Keynes
Carl Gauss
Paul Feyerabend
- 90 -
Antoine Lavoisier
Lise Meitner
Charles Babbage
Ibn Khaldun
Euclid
Ralph Emerson
Robert Bunsen
Frederick Banting
Andre Ampere
Winston Churchill
- 80 -
John Locke
Bronislaw Malinowski
Bible
Thomas Huxley
Alessandro Volta
Erwin Schrodinger
Wilhelm Roentgen
Louis Pasteur
Bertrand Russell
Jean Lamarck
- 70 -
Samuel Morse
John Wheeler
Nicolaus Copernicus
Robert Fulton
Pierre Laplace
Humphry Davy
Thomas Edison
Lord Kelvin
Theodore Roosevelt
Carolus Linnaeus
- 60 -
Francis Galton
Linus Pauling
Immanuel Kant
Martin Fischer
Robert Boyle
Karl Popper
Paul Dirac
Avicenna
James Watson
William Shakespeare
- 50 -
Stephen Hawking
Niels Bohr
Nikola Tesla
Rachel Carson
Max Planck
Henry Adams
Richard Dawkins
Werner Heisenberg
Alfred Wegener
John Dalton
- 40 -
Pierre Fermat
Edward Wilson
Johannes Kepler
Gustave Eiffel
Giordano Bruno
JJ Thomson
Thomas Kuhn
Leonardo DaVinci
Archimedes
David Hume
- 30 -
Andreas Vesalius
Rudolf Virchow
Richard Feynman
James Hutton
Alexander Fleming
Emile Durkheim
Benjamin Franklin
Robert Oppenheimer
Robert Hooke
Charles Kettering
- 20 -
Carl Sagan
James Maxwell
Marie Curie
Rene Descartes
Francis Crick
Hippocrates
Michael Faraday
Srinivasa Ramanujan
Francis Bacon
Galileo Galilei
- 10 -
Aristotle
John Watson
Rosalind Franklin
Michio Kaku
Isaac Asimov
Charles Darwin
Sigmund Freud
Albert Einstein
Florence Nightingale
Isaac Newton