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Number Quotes (699 quotes)

The supreme task of the physicist is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction. There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can reach them. In this methodological uncertainty, one might suppose that there were any number of possible systems of theoretical physics all equally well justified; and this opinion is no doubt correct, theoretically. But the development of physics has shown that at any given moment, out of all conceivable constructions, a single one has always proved itself decidedly superior to all the rest.
Address (1918) for Max Planck's 60th birthday, at Physical Society, Berlin, 'Principles of Research' in Essays in Science (1934), 4.
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... every chemical combination is wholly and solely dependent on two opposing forces, positive and negative electricity, and every chemical compound must be composed of two parts combined by the agency of their electrochemical reaction, since there is no third force. Hence it follows that every compound body, whatever the number of its constituents, can be divided into two parts, one of which is positively and the other negatively electrical.
Essai sur la théorie des proportions chemiques (1819), 98. Quoted by Henry M. Leicester in article on Bessel in Charles Coulston Gillespie (editor), Dictionary of Scientific Biography (1981), Vol. 2, 94.
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...That day in the account of creation, or those days that are numbers according to its recurrence, are beyond the experience and knowledge of us mortal earthbound men. And if we are able to make any effort towards an understanding of those days, we ought not to rush forward with an ill considered opinion, as if no other reasonable and plausible interpretation could be offered.
iv.44
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1095 … Then after Easter on the eve of St. Ambrose, which is on 4 April [recte 3 April], almost everywhere in this country and almost the whole night, stars in very large numbers were seen to fall from heaven, not by ones or twos, but in such quick succession that they could not be counted.
In The Anglo-Saxon Chronicle, as translated in The Anglo-Saxon Chronicle, Issue 1624 (1975), 230. The Chronicle is the work of many successive hands at several monasteries across England. For the date recorded, This meteor shower could have been a display of the Lyrids (according to The Starseeker, reprinted in O. Vrazell, 'Astronomical Observations in the Anglo Saxon Chronicle', Journal of the Royal Astronomical Society of Canada Newsletter (Aug 1984), 78, 57.
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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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Den förslags-mening: att olika element förenade med ett lika antal atomer af ett eller flere andra gemensamma element … och att likheten i krystallformen bestämmes helt och hållet af antalet af atomer, och icke af elementens.
[Mitscherlich Law of Isomerism] The same number of atoms combined in the same way produces the same crystalline form, and the same crystalline form is independent of the chemical nature of the atoms, and is determined only by their number and relative position.
Original Swedish from 'Om Förhållandet emellan chemiska sammansättningen och krystallformen hos Arseniksyrade och Phosphorsyrade Salter', Kungl. Svenska vetenskapsakademiens handlingar (1821), 4. In English as expressed later by James F.W. Johnston, 'Report on the Recent Progress and present State of Chemical Science', to Annual Meeting at Oxford (1832), collected in Report of the First and Second Meetings of the British Association for the Advancement of Science (1833), 422. A Google raw translation of the Swedish is: “The present proposal-sense: that various elements associated with an equal number of atoms of one or several other common elements … and that the similarity in: crystal shape is determined entirely by the number of atoms, and not by the elements.”
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La théorie est l’hypothèse vérifiée, après qu’elle a été soumise au contrôle du raisonnement et de la critique expérimentale. La meilleure théorie est celle qui a été vérifiée par le plus grand nombre de faits. Mais une théorie, pour rester bonne, doit toujours se modifier avec les progrès de la science et demeurer constamment soumise à la vérification et à la critique des faits nouveaux qui apparaissent.
A theory is a verified hypothesis, after it has been submitted to the control of reason and experimental criticism. The soundest theory is one that has been verified by the greatest number of facts. But to remain valid, a theory must be continually altered to keep pace with the progress of science and must be constantly resubmitted to verification and criticism as new facts appear.
Original work in French, Introduction à l'Étude de la Médecine Expérimentale (1865), 385. English translation by Henry Copley Green in An Introduction to the Study of Experimental Medicine (1927, 1957), 220.
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Numero pondere et mensura Deus omnia condidit.
God created everything by number, weight and measure.
On more than one occasion, Newton wrote these Latin words as his autograph, with his signature below. A photo of one example, dated “London 11 Sep 1722,” can be seen in I. Hargitta (ed.), Symmetry 2: Unifying Human Understanding (2014), 837. Reprinted from M. and B. Rozsondai, 'Symmetry Aspects of Bookbindings', Computers Math. Applic. (1989), 17, No. 4-6, 837. On this piece of paper, Newton dedicated these words to a Hungarian student, Ferenc Páriz Pápai Jr., whose album is now held by the Department of Manuscripts and Rare Books of the Library of the Hungarian Academy of Sciences (Shelf-number Tort. naplók, kis 8⁰ 6). The sentiment existed long before Newton used it. In the Bible, Wisdom of Solomon, 11:20, it appears as “Pondere, mensura, numero Deus omnia fecit,” (Vulgate) which the King James Version translates as “thou hast ordered all things in measure and number and weight.” Another Newton signed autograph with his Latin quote, dated 13 Jul 1716, sold (4 Apr 1991) for DM 9500 (about $31,000). See Dept of English, Temple University, The Scriblerian and the Kit-Cats (1991), 24-25, 230. A few years later, the sale (31 Mar 1998) of Newton's autograph cost the buyer $46,000. See Frank Ryan, Darwin's Blind Spot: Evolution Beyond Natural Selection (2002), 11-12. Other authors, including mathematicians, used the same Biblical passage. In Gilles Personne de Roberval's Aristarchi Samii de Mundi Systemate (1644), he frequently uses the abbreviation “P.N.E.M.” standing for “Pondere, mensura et mensura.” It was adopted as the motto of the Smeatonian Society of Engineers as “Omnia in Numero Pondere et Mensura.”
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Quinquies exscriptus, maneat tot millibus annis.
(I wrote it out five times, may it last the same number of millennia.)
final line of Ars magna
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Srinivasa Ramanujan quote: Replying to G. H. Hardy's suggestion that the number of a taxi (1729) was “dull”: No, it is a very in
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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Science for the Citizen is ... also written for the large and growing number of adolescents, who realize that they will be the first victims of the new destructive powers of science misapplied.
Science for the Citizen: A Self-Educator based on the Social Background of Scientific Discovery (1938), Author's Confessions, 9.
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Surtout l’astronomie et l’anatomie sont les deux sciences qui nous offrent le plus sensiblement deux grands caractères du Créateur; l’une, son immensité, par les distances, la grandeur, et le nombre des corps célestes; l’autre, son intelligence infinie, par la méchanique des animaux.
Above all, astronomy and anatomy are the two sciences which present to our minds most significantly the two grand characteristics of the Creator; the one, His immensity, by the distances, size, and number of the heavenly bodies; the other, His infinite intelligence, by the mechanism of animate beings.
Original French and translation in Craufurd Tait Ramage (ed.) Beautiful Thoughts from French and Italian Authors (1866), 119-120.
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Thomasina: Every week I plot your equations dot for dot, x’s against y’s in all manner of algebraical relation, and every week they draw themselves as commonplace geometry, as if the world of forms were nothing but arcs and angles. God’s truth, Septimus, if there is an equation for a curve like a bell, there must be an equation for one like a bluebell, and if a bluebell, why not a rose? Do we believe nature is written in numbers?
Septimus: We do.
Thomasina: Then why do your shapes describe only the shapes of manufacture?
Septimus: I do not know.
Thomasina: Armed thus, God could only make a cabinet.
In the play, Acadia (1993), Scene 3, 37.
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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 biologist, if he wishes to know how many toes a cat has, does not "frame the hypothesis that the number of feline digital extremities is 4, or 5, or 6," he simply looks at a cat and counts. A social scientist prefers the more long-winded expression every time, because it gives an entirely spurious impression of scientificness to what he is doing.
In Science is a Sacred Cow (1950), 151.
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A comparatively small variety of species is found in the older rocks, although of some particular ones the remains are very abundant; ... Ascending to the next group of rocks, we find the traces of life become more abundant, the number of species extended.
Vestiges of the Natural History of Creation (1844), 60-1.
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A considerable number of persons are able to protect themselves against the outbreak of serious neurotic phenomena only through intense work.
From Observations on Ferenczi's paper on 'Sunday Neuroses' (1918). Quoted in Peter Bryan Warr, Work, Happiness, and Unhappiness (2007), 161.
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A depressing number of people seem to process everything literally. They are to wit as a blind man is to a forest, able to find every tree, but each one coming as a surprise.
Quoted in Kim Lim (ed.), 1,001 Pearls of Spiritual Wisdom: Words to Enrich, Inspire, and Guide Your Life (2014), 32
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A distinguished writer [Siméon Denis Poisson] has thus stated the fundamental definitions of the science:
“The probability of an event is the reason we have to believe that it has taken place, or that it will take place.”
“The measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or contrary, and all equally possible” (equally like to happen).
From these definitions it follows that the word probability, in its mathematical acceptation, has reference to the state of our knowledge of the circumstances under which an event may happen or fail. With the degree of information which we possess concerning the circumstances of an event, the reason we have to think that it will occur, or, to use a single term, our expectation of it, will vary. Probability is expectation founded upon partial knowledge. A perfect acquaintance with all the circumstances affecting the occurrence of an event would change expectation into certainty, and leave neither room nor demand for a theory of probabilities.
An Investigation of the Laws of Thought (1854), 243-244. The Poisson quote is footnoted as from Recherches sur la Probabilité des Jugemens.
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A doctor’s reputation is made by the number of eminent men who die under his care.
Statement (14 Sep 1950) at age 94 to his doctor. As quoted in Michael Holroyd, Bernard Shaw: The Lure of Fantasy: Vol. 3: 1918-1951 (1991).
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A fair number of people who go on to major in astronomy have decided on it certainly by the time they leave junior high, if not during junior high. I think it’s somewhat unusual that way. I think most children pick their field quite a bit later, but astronomy seems to catch early, and if it does, it sticks.
From interview by Rebecca Wright, 'Oral History Transcript' (15 Sep 2000), on NASA website.
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A first step in the study of civilization is to dissect it into details, and to classify these in their proper groups. Thus, in examining weapons, they are to be classed under spear, club, sling, bow and arrow, and so forth; among textile arts are to be ranged matting, netting, and several grades of making and weaving threads; myths are divided under such headings as myths of sunrise and sunset, eclipse-myths, earthquake-myths, local myths which account for the names of places by some fanciful tale, eponymic myths which account for the parentage of a tribe by turning its name into the name of an imaginary ancestor; under rites and ceremonies occur such practices as the various kinds of sacrifice to the ghosts of the dead and to other spiritual beings, the turning to the east in worship, the purification of ceremonial or moral uncleanness by means of water or fire. Such are a few miscellaneous examples from a list of hundreds … To the ethnographer, the bow and arrow is the species, the habit of flattening children’s skulls is a species, the practice of reckoning numbers by tens is a species. The geographical distribution of these things, and their transmission from region to region, have to be studied as the naturalist studies the geography of his botanical and zoological species.
In Primitive Culture (1871), Vol. 1, 7.
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A fossil hunter needs sharp eyes and a keen search image, a mental template that subconsciously evaluates everything he sees in his search for telltale clues. A kind of mental radar works even if he isn’t concentrating hard. A fossil mollusk expert has a mollusk search image. A fossil antelope expert has an antelope search image. … Yet even when one has a good internal radar, the search is incredibly more difficult than it sounds. Not only are fossils often the same color as the rocks among which they are found, so they blend in with the background; they are also usually broken into odd-shaped fragments. … In our business, we don’t expect to find a whole skull lying on the surface staring up at us. The typical find is a small piece of petrified bone. The fossil hunter’s search therefore has to have an infinite number of dimensions, matching every conceivable angle of every shape of fragment of every bone on the human body.
Describing the skill of his co-worker, Kamoya Kimeu, who discovered the Turkana Boy, the most complete specimen of Homo erectus, on a slope covered with black lava pebbles.
Richard Leakey and Roger Lewin, Origins Reconsidered: In Search of What Makes Us Human (1992), 26.
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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 hundred years ago … an engineer, Herbert Spencer, was willing to expound every aspect of life, with an effect on his admiring readers which has not worn off today.
Things do not happen quite in this way nowadays. This, we are told, is an age of specialists. The pursuit of knowledge has become a profession. The time when a man could master several sciences is past. He must now, they say, put all his efforts into one subject. And presumably, he must get all his ideas from this one subject. The world, to be sure, needs men who will follow such a rule with enthusiasm. It needs the greatest numbers of the ablest technicians. But apart from them it also needs men who will converse and think and even work in more than one science and know how to combine or connect them. Such men, I believe, are still to be found today. They are still as glad to exchange ideas as they have been in the past. But we cannot say that our way of life is well-fitted to help them. Why is this?
In 'The Unification of Biology', New Scientist (11 Jan 1962), 13, No. 269, 72.
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A large number of areas of the brain are involved when viewing equations, but when one looks at a formula rated as beautiful it activates the emotional brain—the medial orbito-frontal cortex—like looking at a great painting or listening to a piece of music. … Neuroscience can’t tell you what beauty is, but if you find it beautiful the medial orbito-frontal cortex is likely to be involved; you can find beauty in anything.
As quoted in James Gallagher, 'Mathematics: Why The Brain Sees Maths As Beauty,' BBC News (13 Feb 2014), on bbc.co.uk web site.
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A mathematician’s reputation rests on the number of bad proofs he has given.
As quoted in John E. Littlewood, A Mathematician’s Miscellany (1953), 41. [His meaning was that an important result is built upon earlier clumsy work. —Webmaster]
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A number of years ago, when I was a freshly-appointed instructor, I met, for the first time, a certain eminent historian of science. At the time I could only regard him with tolerant condescension.
I was sorry of the man who, it seemed to me, was forced to hover about the edges of science. He was compelled to shiver endlessly in the outskirts, getting only feeble warmth from the distant sun of science- in-progress; while I, just beginning my research, was bathed in the heady liquid heat up at the very center of the glow.
In a lifetime of being wrong at many a point, I was never more wrong. It was I, not he, who was wandering in the periphery. It was he, not I, who lived in the blaze.
I had fallen victim to the fallacy of the “growing edge;” the belief that only the very frontier of scientific advance counted; that everything that had been left behind by that advance was faded and dead.
But is that true? Because a tree in spring buds and comes greenly into leaf, are those leaves therefore the tree? If the newborn twigs and their leaves were all that existed, they would form a vague halo of green suspended in mid-air, but surely that is not the tree. The leaves, by themselves, are no more than trivial fluttering decoration. It is the trunk and limbs that give the tree its grandeur and the leaves themselves their meaning.
There is not a discovery in science, however revolutionary, however sparkling with insight, that does not arise out of what went before. “If I have seen further than other men,” said Isaac Newton, “it is because I have stood on the shoulders of giants.”
Adding A Dimension: Seventeen Essays on the History of Science (1964), Introduction.
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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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A quarter-horse jockey learns to think of a twenty-second race as if it were occurring across twenty minutes—in distinct parts, spaced in his consciousness. Each nuance of the ride comes to him as he builds his race. If you can do the opposite with deep time, living in it and thinking in it until the large numbers settle into place, you can sense how swiftly the initial earth packed itself together, how swiftly continents have assembled and come apart, how far and rapidly continents travel, how quickly mountains rise and how quickly they disintegrate and disappear.
Annals of the Former World
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A scientific invention consists of six (or some number) ideas, five of which are absurd but which, with the addition of the sixth and enough rearrangement of the combinations, results in something no one has thought of before.
As quoted in Robert Coughlan, 'Dr. Edward Teller’s Magnificent Obsession', Life (6 Sep 1954), 66.
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A single kind of red cell is supposed to have an enormous number of different substances on it, and in the same way there are substances in the serum to react with many different animal cells. In addition, the substances which match each kind of cell are different in each kind of serum. The number of hypothetical different substances postulated makes this conception so uneconomical that the question must be asked whether it is the only one possible. ... We ourselves hold that another, simpler, explanation is possible.
Landsteiner and Adriano Sturli, 'Hamagglutinine normaler Sera', Wiener klinische Wochenschrift (1902), 15, 38-40. Trans. Pauline M. H. Mazumdar.
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A statistician is someone who is good with numbers but lacks the personality to be an accountant.
[Or economist]
Anonymous
Found, for example, in A Prairie Home Companion Pretty Good Joke Book (4th ed., 2005), 216. Also seen as defining an Economist in Eighteen Annual Institute on Securities Regulation (1987), 131, which quotes it as “Tim Wirth used to say.”
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A superficial knowledge of mathematics may lead to the belief that this subject can be taught incidentally, and that exercises akin to counting the petals of flowers or the legs of a grasshopper are mathematical. Such work ignores the fundamental idea out of which quantitative reasoning grows—the equality of magnitudes. It leaves the pupil unaware of that relativity which is the essence of mathematical science. Numerical statements are frequently required in the study of natural history, but to repeat these as a drill upon numbers will scarcely lend charm to these studies, and certainly will not result in mathematical knowledge.
In Primary Arithmetic: First Year, for the Use of Teachers (1897), 26-27.
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A superficial knowledge of mathematics may lead to the belief that this subject can be taught incidentally, and that exercises akin to counting the petals of flowers or the legs of a grasshopper are mathematical. Such work ignores the fundamental idea out of which quantitative reasoning grows—the equality of magnitudes. It leaves the pupil unaware of that relativity which is the essence of mathematical science. Numerical statements are frequently required in the study of natural history, but to repeat these as a drill upon numbers will scarcely lend charm to these studies, and certainly will not result in mathematical knowledge.
In Primary Arithmetic: First Year, for the Use of Teachers (1897), 26-27.
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A surprising number [of novels] have been read aloud to me, and I like all if moderately good, and if they do not end unhappily—against which a law ought to be passed.
Francis Darwin (ed.), The Life and Letters of Charles Darwin, Including an Autobiographical Chapter (1888), Vol. 1, 101.
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A theory of physics is not an explanation; it is a system of mathematical oppositions deduced from a small number of principles the aim of which is to represent as simply, as completely, and as exactly as possible, a group of experimental laws.
As quoted in Philipp Frank, Modern Science and its Philosophy (1949), 15, which cites Théorie Physique; Son Objet—Son Structure (1906), 24.
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A vast number, perhaps the numerical majority, of animal forms cannot be shown unequivocally to possess mind.
In 'The Brain Collaborates With Psyche', Man On His Nature: The Gifford Lectures, Edinburgh 1937-8 (1940), 284.
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A young person who reads a science book is confronted with a number of facts, x = ma … ma - me² … You never see in the scientific books what lies behind the discovery—the struggle and the passion of the person, who made that discovery.
From 'Asking Nature', collected in Lewis Wolpert and Alison Richards (eds.), Passionate Minds: The Inner World of Scientists (1997), 197.
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A … difference between most system-building in the social sciences and systems of thought and classification of the natural sciences is to be seen in their evolution. In the natural sciences both theories and descriptive systems grow by adaptation to the increasing knowledge and experience of the scientists. In the social sciences, systems often issue fully formed from the mind of one man. Then they may be much discussed if they attract attention, but progressive adaptive modification as a result of the concerted efforts of great numbers of men is rare.
The Study of Man (1941), 19-20.
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According to our ancient Buddhist texts, a thousand million solar systems make up a galaxy. … A thousand million of such galaxies form a supergalaxy. … A thousand million supergalaxies is collectively known as supergalaxy Number One. Again, a thousand million supergalaxy Number Ones form a Supergalaxy Number Two. A thousand million supergalaxy Number Twos make up a supergalaxy Number Three, and of these, it is stated in the texts that there are a countless number in the universe.
In 'Reactions to Man’s Landing on the Moon Show Broad Variations in Opinions', The New York Times (21 Jul 1969), 6.
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According to the older view, for every single effect of a serum, there was a separate substance, or at least a particular chemical group... A normal serum contained as many different haemagglutinins as it agglutinated different cells. The situation was undoubtedly made much simpler if, to use the Ehrlich terminology... the separate haptophore groups can combine with an extremely large number of receptors in stepwise differing quantities as a stain does with different animal tissues, though not always with the same intensity. A normal serum would therefore visibly affect such a large number of different blood cells... not because it contained countless special substances, but because of the colloids of the serum, and therefore of the agglutinins by reason of their chemical constitution and the electrochemical properties resulting from it. That this manner of representation is a considerable simplification is clear; it also opens the way to direct experimental testing by the methods of structural chemistry.
'Die Theorien der Antikorperbildung ... ', Wiener klinische Wöchenschrift (1909), 22, 1623-1631. Trans. Pauline M. H. Mazumdar.
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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), xci.
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After the birth of printing books became widespread. Hence everyone throughout Europe devoted himself to the study of literature... Every year, especially since 1563, the number of writings published in every field is greater than all those produced in the past thousand years. Through them there has today been created a new theology and a new jurisprudence; the Paracelsians have created medicine anew and the Copernicans have created astronomy anew. I really believe that at last the world is alive, indeed seething, and that the stimuli of these remarkable conjunctions did not act in vain.
De Stella Nova, On the New Star (1606), Johannes Kepler Gesammelte Werke (1937- ), Vol. 1, 330-2. Quoted in N. Jardine, The Birth of History and Philosophy of Science: Kepler's A Defence of Tycho Against Ursus With Essays on its Provenance and Significance (1984), 277-8.
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Again, it [the Analytical Engine] might act upon other things besides number, were objects found whose mutual fundamental relations could be expressed by those of the abstract science of operations, and which should be also susceptible of adaptations to the action of the operating notation and mechanism of the engine. Supposing for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent.
In Richard Taylor (ed.), 'Translator’s Notes to M. Menabrea’s Memoir', Scientific Memoirs, Selected from the Transactions of Foreign Academies and Learned Societies and from Foreign Journals (1843), 3, Note A, 694. Her notes were appended to L.F. Menabrea, of Turin, Officer of the Military Engineers, 'Article XXIX: Sketch of the Analytical Engine invented by Charles Babbage Esq.', Bibliothèque Universelle de Gnve (Oct 1842), No. 82.
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All change is relative. The universe is expanding relatively to our common material standards; our material standards are shrinking relatively to the size of the universe. The theory of the “expanding universe” might also be called the theory of the “shrinking atom”. …
:Let us then take the whole universe as our standard of constancy, and adopt the view of a cosmic being whose body is composed of intergalactic spaces and swells as they swell. Or rather we must now say it keeps the same size, for he will not admit that it is he who has changed. Watching us for a few thousand million years, he sees us shrinking; atoms, animals, planets, even the galaxies, all shrink alike; only the intergalactic spaces remain the same. The earth spirals round the sun in an ever-decreasing orbit. It would be absurd to treat its changing revolution as a constant unit of time. The cosmic being will naturally relate his units of length and time so that the velocity of light remains constant. Our years will then decrease in geometrical progression in the cosmic scale of time. On that scale man’s life is becoming briefer; his threescore years and ten are an ever-decreasing allowance. Owing to the property of geometrical progressions an infinite number of our years will add up to a finite cosmic time; so that what we should call the end of eternity is an ordinary finite date in the cosmic calendar. But on that date the universe has expanded to infinity in our reckoning, and we have shrunk to nothing in the reckoning of the cosmic being.
We walk the stage of life, performers of a drama for the benefit of the cosmic spectator. As the scenes proceed he notices that the actors are growing smaller and the action quicker. When the last act opens the curtain rises on midget actors rushing through their parts at frantic speed. Smaller and smaller. Faster and faster. One last microscopic blurr of intense agitation. And then nothing.
In The Expanding Universe (1933) , 90-92.
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All is number
Quoted in Robert J. Scully, The Demon and the Quantum (2007), 7.
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All sorts of computer errors are now turning up. You'd be surprised to know the number of doctors who claim they are treating pregnant men.
Anonymous
Official of the Quebec Health Insurance Board, on Use of Computers in Quebec Province's Comprehensive Medical-care system. F. 19, 4:5. In Barbara Bennett and Linda Amster, Who Said What (and When, and Where, and How) in 1971: December-June, 1971 (1972), Vol. 1, 38. (Later sources cite Isaac Asimov.)
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All that can be said upon the number and nature of elements is, in my opinion, confined to discussions entirely of a metaphysical nature. The subject only furnishes us with indefinite problems, which may be solved in a thousand different ways, not one of which, in all probability, is consistent with nature. I shall therefore only add upon this subject, that if, by the term elements, we mean to express those simple and indivisible atoms of which matter is composed, it is extremely probable we know nothing at all about them; but, if we apply the term elements, or principles of bodies, to express our idea of the last point which analysis is capable of reaching, we must admit, as elements, all the substances into which we are capable, by any means, to reduce bodies by decomposition.
Elements of Chemistry (1790), trans. R. Kerr, Preface, xxiv.
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All the effects of Nature are only the mathematical consequences of a small number of immutable laws.
From the original French, “Tous les effets de la nature ne sont que résultats mathématiques d'un petit noinbre de lois immuables.”, in Oeuvres de Laplace, Vol. VII: Théorie des probabilités (1847), Introduction, cliv.
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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.
In Lectures on the Logic of Arithmetic (1903), Preface, 18-19.
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All the species recognized by Botanists came forth from the Almighty Creator's hand, and the number of these is now and always will be exactly the same, while every day new and different florists' species arise from the true species so-called by Botanists, and when they have arisen they finally revert to the original forms. Accordingly to the former have been assigned by Nature fixed limits, beyond which they cannot go: while the latter display without end the infinite sport of Nature.
Philosophia Botanica (1751), aphorism 310. Trans. Frans A. Stafleu, Linnaeus and the Linnaeans: The Spreading of their Ideas in Systematic Botany, 1735-1789 (1971), 90.
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Although we are mere sojourners on the surface of the planet, chained to a mere point in space, enduring but for a moment of time, the human mind is not only enabled to number worlds beyond the unassisted ken of mortal eye, but to trace the events of indefinite ages before the creation of our race, and is not even withheld from penetrating into the dark secrets of the ocean, or the interior of the solid globe; free, like the spirit which the poet described as animating the universe.
In Principles of Geology (1830).
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Although [Charles Darwin] would patiently go on repeating experiments where there was any good to be gained, he could not endure having to repeat an experiment which ought, if complete care had been taken, to have told its story at first—and this gave him a continual anxiety that the experiment should not be wasted; he felt the experiment to be sacred, however slight a one it was. He wished to learn as much as possible from an experiment, so that he did not confine himself to observing the single point to which the experiment was directed, and his power of seeing a number of other things was wonderful. ... Any experiment done was to be of some use, and ... strongly he urged the necessity of keeping the notes of experiments which failed, and to this rule he always adhered.
In Charles Darwin: His Life Told in an Autobiographical Chapter, and in a Selected Series of his Published Letters (1908), 92.
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Always preoccupied with his profound researches, the great Newton showed in the ordinary-affairs of life an absence of mind which has become proverbial. It is related that one day, wishing to find the number of seconds necessary for the boiling of an egg, he perceived, after waiting a minute, that he held the egg in his hand, and had placed his seconds watch (an instrument of great value on account of its mathematical precision) to boil!
This absence of mind reminds one of the mathematician Ampere, who one day, as he was going to his course of lectures, noticed a little pebble on the road; he picked it up, and examined with admiration the mottled veins. All at once the lecture which he ought to be attending to returned to his mind; he drew out his watch; perceiving that the hour approached, he hastily doubled his pace, carefully placed the pebble in his pocket, and threw his watch over the parapet of the Pont des Arts.
Popular Astronomy: a General Description of the Heavens (1884), translated by J. Ellard Gore, (1907), 93.
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Among innumerable footsteps of divine providence to be found in the works of nature, there is a very remarkable one to be observed in the exact balance that is maintained, between the numbers of men and women; for by this means is provided, that the species never may fail, nor perish, since every male may have its female, and of proportionable age. This equality of males and females is not the effect of chance but divine providence, working for a good end.
'An Argument for Divine Providence, taken from the Constant Regularity observ’d in the Births of both Sexes', Philosophical Transactions of the Royal Society, 1710-12, 27,186.
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Among the social sciences, economists are the snobs. Economics, with its numbers and graphs and curves, at least has the coloration and paraphernalia of a hard science. It's not just putting on sandals and trekking out to take notes on some tribe.
'A Cuba Policy That's Stuck On Plan A', opinion column in Washington Post (17 April 2009).
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An astronomer must be the wisest of men; his mind must be duly disciplined in youth; especially is mathematical study necessary; both an acquaintance with the doctrine of number, and also with that other branch of mathematics, which, closely connected as it is with the science of the heavens, we very absurdly call geometry, the measurement of the earth.
Plato
From the 'Epilogue to the Laws' (Epinomis), 988-990. As quoted in William Whewell, History of the Inductive Sciences from the Earliest to the Present Time (1837), Vol. 1, 161. (Although referenced to Plato’s Laws, the Epinomis is regarded as a later addition, not by Plato himself.)
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An author has always great difficulty in avoiding unnecessary and tedious detail on the one hand; while, on the other, he must notice such a number of facts as may convince a student, that he is not wandering in a wilderness of crude hypotheses or unsupported assumptions.
In A Geological Manual (1832), Preface, iii.
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An event experienced is an event perceived, digested, and assimilated into the substance of our being, and the ratio between the number of cases seen and the number of cases assimilated is the measure of experience.
Address, opening of 1932-3 session of U.C.H. Medical School (4 Oct 1932), 'Art and Science in medicine', The Collected Papers of Wilfred Trotter, FRS (1941), 98.
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An important fact, an ingenious aperçu, occupies a very great number of men, at first only to make acquaintance with it; then to understand it; and afterwards to work it out and carry it further.
In The Maxims and Reflections of Goethe (1906), 189.
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An Individual, whatever species it might be, is nothing in the Universe. A hundred, a thousand individuals are still nothing. The species are the only creatures of Nature, perpetual creatures, as old and as permanent as it. In order to judge it better, we no longer consider the species as a collection or as a series of similar individuals, but as a whole independent of number, independent of time, a whole always living, always the same, a whole which has been counted as one in the works of creation, and which, as a consequence, makes only a unity in Nature.
'De la Nature: Seconde Vue', Histoire Naturelle, Générale et Particulière, Avec la Description du Cabinet du Roi (1765), Vol. 13, i. Trans. Phillip R. Sloan.
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An undefined problem has an infinite number of solutions.
…...
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And since the portions of the great and the small are equal in number, so too all things would be in everything. Nor is it possible that they should exist apart, but all things have a portion of everything.
Simplicius, Commentary on Aristotle’s Physics, 164, 26-8. In G. S. Kirk, J. E. Raven and M. Schofield (eds.), The Presocratic Philosophers: A Critical History with a Selection of Texts (1983) , p. 365-6.
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And therefore, sir, as you desire to live,
A day or two before your laxative,
Take just three worms, nor under nor above,
Because the gods unequal numbers love.
These digestives prepare you for your purge,
Of fumetery, centaury, and spurge;
And of ground-ivy add a leaf or two.
All which within our yard or garden grow.
Eat these, and be, my lord, of better cheer:
Your father’s son was never born to fear.
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And yet surely to alchemy this right is due, that it may be compared to the husbandman whereof Æsop makes the fable, that when he died he told his sons that he had left unto them gold buried under the ground in his vineyard: and they digged over the ground, gold they found none, but by reason of their stirring and digging the mould about the roots of their vines, they had a great vintage the year following: so assuredly the search and stir to make gold hath brought to light a great number of good and fruitful inventions and experiments, as well for the disclosing of nature as for the use of man's life.
The Advancement of Learning (1605, 1712), Vol. 1, 15.
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Any conception which is definitely and completely determined by means of a finite number of specifications, say by assigning a finite number of elements, is a mathematical conception. Mathematics has for its function to develop the consequences involved in the definition of a group of mathematical conceptions. Interdependence and mutual logical consistency among the members of the group are postulated, otherwise the group would either have to be treated as several distinct groups, or would lie beyond the sphere of mathematics.
In 'Mathematics', Encyclopedia Britannica (9th ed.).
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Any experiment may be regarded as forming an individual of a 'population' of experiments which might be performed under the same conditions. A series of experiments is a sample drawn from this population.
Now any series of experiments is only of value in so far as it enables us to form a judgment as to the statistical constants of the population to which the experiments belong. In a great number of cases the question finally turns on the value of a mean, either directly, or as the mean difference between the two qualities.
If the number of experiments be very large, we may have precise information as to the value of the mean, but if our sample be small, we have two sources of uncertainty:— (I) owing to the 'error of random sampling' the mean of our series of experiments deviates more or less widely from the mean of the population, and (2) the sample is not sufficiently large to determine what is the law of distribution of individuals.
'The Probable Error of a Mean', Biometrika, 1908, 6, 1.
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Any time you wish to demonstrate something, the number of faults is proportional to the number of viewers.
Anonymous
Bye's First Law of Model Railroading. In Paul Dickson, The Official Rules, (1978), 23.
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Anyone of common mental and physical health can practice scientific research. … Anyone can try by patient experiment what happens if this or that substance be mixed in this or that proportion with some other under this or that condition. Anyone can vary the experiment in any number of ways. He that hits in this fashion on something novel and of use will have fame. … The fame will be the product of luck and industry. It will not be the product of special talent.
In Essays of a Catholic Layman in England (1931).
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Anyone who considers arithmetical methods of producing random digits is, of course, in the state of sin. For, as has been pointed out several times, there is no such thing as a random number—there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method.
In paper delivered at a symposium on the Monte Carlo method. 'Various Techniques Used in Connection with Random Digits', Journal of Research of the National Bureau of Standards, Appl. Math. Series, Vol. 3 (1951), 3, 36. Reprinted in John von Neumann: Collected Works (1963), Vol. 5, 700. Also often seen misquoted (?) as “Anyone who attempts to generate random numbers by deterministic means is, of course, living in a state of sin.”
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Architecture is geometry made visible in the same sense that music is number made audible.
In The Beautiful Necessity: Seven Essays on Theosophy and Architecture (1910),
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Arithmetic is numbers you squeeze from your head to your hand to your pencil to your paper till you get the answer.
From 'Arithmetic', Harvest Poems, 1910-1960 (1960), 115.
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Arithmetic is where numbers fly like pigeons in and out of your head.
From 'Arithmetic', Harvest Poems, 1910-1960 (1960), 115.
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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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As a little boy, I showed an abnormal aptitude for mathematics this gift played a horrible part in tussles with quinsy or scarlet fever, when I felt enormous spheres and huge numbers swell relentlessly in my aching brain.
In Speak, Memory: An Autobiography Revisited (1999), 2
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As I strayed into the study of an eminent physicist, I observed hanging against the wall, framed like a choice engraving, several dingy, ribbon-like strips of, I knew not what... My curiosity was at once aroused. What were they? ... They might be shreds of mummy-wraps or bits of friable bark-cloth from the Pacific, ... [or] remnants from a grandmother’s wedding dress... They were none of these... He explained that they were carefully-prepared photographs of portions of the Solar Spectrum. I stood and mused, absorbed in the varying yet significant intensities of light and shade, bordered by mystic letters and symbolic numbers. As I mused, the pale legend began to glow with life. Every line became luminous with meaning. Every shadow was suffused with light shining from behind, suggesting some mighty achievement of knowledge; of knowledge growing more daring in proportion to the remoteness of the object known; of knowledge becoming more positive in its answers, as the questions which were asked seemed unanswerable. No Runic legend, no Babylonish arrowhead, no Egyptian hieroglyph, no Moabite stone, could present a history like this, or suggest thoughts of such weighty import or so stimulate and exalt the imagination.
The Sciences of Nature Versus the Science of Man: A Plea for the Science of Man (1871), 7-9.
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As immoral and unethical as this may be [to clone a human], there is a real chance that could have had some success. This is a pure numbers game. If they have devoted enough resources and they had access to enough eggs, there is a distinct possibility. But, again, without any scientific data, one has to be extremely skeptical.
Commenting on the announcement of the purported birth of the first cloned human.
Transcript of TV interview by Sanjay Gupta aired on CNN (27 Dec 2002).
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As new areas of the world came into view through exploration, the number of identified species of animals and plants grew astronomically. By 1800 it had reached 70,000. Today more than 1.25 million different species, two-thirds animal and one-third plant, are known, and no biologist supposes that the count is complete.
In The Intelligent Man's Guide to Science: The Biological Sciences (1960), 654. Also in Isaac Asimov’s Book of Science and Nature Quotations (1988), 320.
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As regards the co-ordination of all ordinary properties of matter, Rutherford’s model of the atom puts before us a task reminiscent of the old dream of philosophers: to reduce the interpretation of the laws of nature to the consideration of pure numbers.
In Faraday Lecture (1930), Journal of the Chemical Society (Feb 1932), 349. As quoted and cited in Chen Ning Yang, Elementary Particles (1961), 7.
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As there is no study which may be so advantageously entered upon with a less stock of preparatory knowledge than mathematics, so there is none in which a greater number of uneducated men have raised themselves, by their own exertions, to distinction and eminence. … Many of the intellectual defects which, in such cases, are commonly placed to the account of mathematical studies, ought to be ascribed to the want of a liberal education in early youth.
In Elements of the Philosophy of the Human Mind (1827), Vol. 3, Chap. 1, Sec. 3, 183.
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As to the Christian religion, Sir, … there is a balance in its favor from the number of great men who have been convinced of its truth after a serious consideration of the question. Grotius was an acute man, a lawyer, a man accustomed to examine evidence, and he was convinced. Grotius was not a recluse, but a man of the world, who surely had no bias on the side of religion. Sir Isaac Newton set out an infidel, and came to be a very firm believer.
(1763). In George Birkbeck Hill (ed.), Boswell’s Life of Johnson (1799), Vol. 1, 524.
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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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At present we begin to feel impatient, and to wish for a new state of chemical elements. For a time the desire was to add to the metals, now we wish to diminish their number. They increase upon us continually, and threaten to enclose within their ranks the bounds of our fair fields of chemical science. The rocks of the mountain and the soil of the plain, the sands of the sea and the salts that are in it, have given way to the powers we have been able to apply to them, but only to be replaced by metals.
In his 16th Lecture of 1818, in Bence Jones, The Life and Letters of Faraday (1870), Vol. 1, 256-257.
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Atoms have a nucleus, made of protons and neutrons bound together. Around this nucleus shells of electrons spin, and each shell is either full or trying to get full, to balance with the number of protons—to balance the number of positive and negative charges. An atom is like a human heart, you see.
The Lunatics (1988). In Gary Westfahl, Science Fiction Quotations: From the Inner Mind to the Outer Limits (2006), 323.
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Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world could have more astonished a Greek mathematician than to learn that, under the influence of compulsory education, the whole population of Western Europe, from the highest to the lowest, could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility. … Our modern power of easy reckoning with decimal fractions is the most miraculous result of a perfect notation.
In Introduction to Mathematics (1911), 59.
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Before the promulgation of the periodic law the chemical elements were mere fragmentary incidental facts in nature; there was no special reason to expect the discovery of new elements, and the new ones which were discovered from time to time appeared to be possessed of quite novel properties. The law of periodicity first enabled us to perceive undiscovered elements at a distance which formerly were inaccessible to chemical vision, and long ere they were discovered new elements appeared before our eyes possessed of a number of well-defined properties.
In Faraday Lecture, delivered before the Fellows of the Chemical Society in the Theatre of the Royal Institution (4 Jun 1889), printed in Professor Mendeléeff, 'The Periodic Law of the Chemical Elements', Transactions of the Chemical Society (1889), 55, 648.
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Being also in accord with Goethe that discoveries are made by the age and not by the individual, I should consider the instances to be exceedingly rare of men who can be said to be living before their age, and to be the repository of knowledge quite foreign to the thought of the time. The rule is that a number of persons are employed at a particular piece of work, but one being a few steps in advance of the others is able to crown the edifice with his name, or, having the ability to generalise already known facts, may become in time to be regarded as their originator. Therefore it is that one name is remembered whilst those of coequals have long been buried in obscurity.
In Historical Notes on Bright's Disease, Addison's Disease, and Hodgkin's Disease', Guy's Hospital Reports (1877), 22, 259-260.
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Between the lowest and the highest degree of spiritual and corporal perfection, there is an almost infinite number of intermediate degrees. The succession of degrees comprises the Universal Chain. It unites all beings, ties together all worlds, embraces all the spheres. One SINGLE BEING is outside this chain, and this is HE who made it.
Contemplation de la nature (1764), Vol. I, 27. Trans. Stephen Jay Gould, Ontogeny and Phylogeny (1977), 23.
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Beyond a critical point within a finite space, freedom diminishes as numbers increase. ...The human question is not how many can possibly survive within the system, but what kind of existence is possible for those who do survive.
Dune
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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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Bridges would not be safer if only people who knew the proper definition of a real number were allowed to design them.
In 'Topological Theory of Defects', Review of Modern Physics (Jul 1979), 51, No. 3. 591–648. This is seen widely on the web (? mis-)attributed to H.L. Mencken, for which Webmaster has so far been unable to find any validation whatsoever.
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Bring out number, weight, and measure in a year of dearth.
In 'Proverbs', The Poems: With Specimens of the Prose Writings of William Blake (1885), 279.
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But as no two (theoreticians) agree on this (skin friction) or any other subject, some not agreeing today with what they wrote a year ago, I think we might put down all their results, add them together, and then divide by the number of mathematicians, and thus find the average coefficient of error. (1908)
In Artificial and Natural Flight (1908), 3. Quoted in John David Anderson, Jr., Hypersonic and High Temperature Gas Dynamics (2000), 335.
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But if anyone, well seen in the knowledge, not onely of Sacred and exotick History, but of Astronomical Calculation, and the old Hebrew Kalendar, shall apply himself to these studies, I judge it indeed difficult, but not impossible for such a one to attain, not onely the number of years, but even, of dayes from the Creation of the World.
In 'Epistle to the Reader', The Annals of the World (1658). As excerpted in Wallen Yep, Man Before Adam: A Correction to Doctrinal Theology, "The Missing Link Found" (2002), 18.
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But many of our imaginations and investigations of nature are futile, especially when we see little living animals and see their legs and must judge the same to be ten thousand times thinner than a hair of my beard, and when I see animals living that are more than a hundred times smaller and am unable to observe any legs at all, I still conclude from their structure and the movements of their bodies that they do have legs... and therefore legs in proportion to their bodies, just as is the case with the larger animals upon which I can see legs... Taking this number to be about a hundred times smaller, we therefore find a million legs, all these together being as thick as a hair from my beard, and these legs, besides having the instruments for movement, must be provided with vessels to carry food.
Letter to N. Grew, 27 Sep 1678. In The Collected Letters of Antoni van Leeuwenhoek (1957), Vol. 2, 391.
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But the nature of our civilized minds is so detached from the senses, even in the vulgar, by abstractions corresponding to all the abstract terms our languages abound in, and so refined by the art of writing, and as it were spiritualized by the use of numbers, because even the vulgar know how to count and reckon, that it is naturally beyond our power to form the vast image of this mistress called ‘Sympathetic Nature.’
The New Science, bk. 2, para. 378 (1744, trans. 1984).
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But, as we consider the totality of similarly broad and fundamental aspects of life, we cannot defend division by two as a natural principle of objective order. Indeed, the ‘stuff’ of the universe often strikes our senses as complex and shaded continua, admittedly with faster and slower moments, and bigger and smaller steps, along the way. Nature does not dictate dualities, trinities, quarterings, or any ‘objective’ basis for human taxonomies; most of our chosen schemes, and our designated numbers of categories, record human choices from a cornucopia of possibilities offered by natural variation from place to place, and permitted by the flexibility of our mental capacities. How many seasons (if we wish to divide by seasons at all) does a year contain? How many stages shall we recognize in a human life?
…...
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By destroying the biological character of phenomena, the use of averages in physiology and medicine usually gives only apparent accuracy to the results. From our point of view, we may distinguish between several kinds of averages: physical averages, chemical averages and physiological and pathological averages. If, for instance, we observe the number of pulsations and the degree of blood pressure by means of the oscillations of a manometer throughout one day, and if we take the average of all our figures to get the true or average blood pressure and to learn the true or average number of pulsations, we shall simply have wrong numbers. In fact, the pulse decreases in number and intensity when we are fasting and increases during digestion or under different influences of movement and rest; all the biological characteristics of the phenomenon disappear in the average. Chemical averages are also often used. If we collect a man's urine during twenty-four hours and mix all this urine to analyze the average, we get an analysis of a urine which simply does not exist; for urine, when fasting, is different from urine during digestion. A startling instance of this kind was invented by a physiologist who took urine from a railroad station urinal where people of all nations passed, and who believed he could thus present an analysis of average European urine! Aside from physical and chemical, there are physiological averages, or what we might call average descriptions of phenomena, which are even more false. Let me assume that a physician collects a great many individual observations of a disease and that he makes an average description of symptoms observed in the individual cases; he will thus have a description that will never be matched in nature. So in physiology, we must never make average descriptions of experiments, because the true relations of phenomena disappear in the average; when dealing with complex and variable experiments, we must study their various circumstances, and then present our most perfect experiment as a type, which, however, still stands for true facts. In the cases just considered, averages must therefore be rejected, because they confuse, while aiming to unify, and distort while aiming to simplify. Averages are applicable only to reducing very slightly varying numerical data about clearly defined and absolutely simple cases.
From An Introduction to the Study of Experimental Medicine (1865), as translated by Henry Copley Greene (1957), 134-135.
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By science, then, I understand the consideration of all subjects, whether of a pure or mixed nature, capable of being reduced to measurement and calculation. All things comprehended under the categories of space, time and number properly belong to our investigations; and all phenomena capable of being brought under the semblance of a law are legitimate objects of our inquiries.
In Report of the British Association for the Advancement of Science (1833), xxviii.
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Cayley was singularly learned in the work of other men, and catholic in his range of knowledge. Yet he did not read a memoir completely through: his custom was to read only so much as would enable him to grasp the meaning of the symbols and understand its scope. The main result would then become to him a subject of investigation: he would establish it (or test it) by algebraic analysis and, not infrequently, develop it so to obtain other results. This faculty of grasping and testing rapidly the work of others, together with his great knowledge, made him an invaluable referee; his services in this capacity were used through a long series of years by a number of societies to which he was almost in the position of standing mathematical advisor.
In Proceedings of London Royal Society (1895), 58, 11-12.
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Certain elements have the property of producing the same crystal form when in combination with an equal number of atoms of one or more common elements, and the elements, from his point of view, can be arranged in certain groups. For convenience I have called the elements belonging to the same group … isomorphous.
Originally published in 'Om Förhållandet emellan chemiska sammansättningen och krystallformen hos Arseniksyrade och Phosphorsyrade Salter', (On the Relation between the Chemical Composition and Crystal Form of Salts of Arsenic and Phosphoric Acids), Kungliga Svenska vetenskapsakademiens handlingar (1821), 4. In F. Szabadváry article on 'Eilhard Mitscherlich' in Charles Coulston Gillispie (ed.), Dictionary of Scientific Biography (1974), Vol. 9, 424; perhaps from J.R. Partington, A History of Chemistry, Vol. 4 (1964), 210.
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Charles Babbage proposed to make an automaton chess-player which should register mechanically the number of games lost and gained in consequence of every sort of move. Thus, the longer the automaton went on playing game, the more experienced it would become by the accumulation of experimental results. Such a machine precisely represents the acquirement of experience by our nervous organization.
In ‘Experimental Legislation’, Popular Science (Apr 1880), 16, 754-5.
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Chemistry is an art that has furnished the world with a great number of useful facts, and has thereby contributed to the improvement of many arts; but these facts lie scattered in many different books, involved in obscure terms, mixed with many falsehoods, and joined to a great deal of false philosophy; so that it is not great wonder that chemistry has not been so much studied as might have been expected with regard to so useful a branch of knowledge, and that many professors are themselves but very superficially acquainted with it. But it was particularly to be expected, that, since it has been taught in universities, the difficulties in this study should have been in some measure removed, that the art should have been put into form, and a system of it attempted—the scattered facts collected and arranged in a proper order. But this has not yet been done; chemistry has not yet been taught but upon a very narrow plan. The teachers of it have still confined themselves to the purposes of pharmacy and medicine, and that comprehends a small branch of chemistry; and even that, by being a single branch, could not by itself be tolerably explained.
John Thomson, An Account of the Life, Lectures and Writings of William Cullen, M.D. (1832), Vol. 1, 40.
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Chemistry is one of those branches of human knowledge which has built itself upon methods and instruments by which truth can presumably be determined. It has survived and grown because all its precepts and principles can be re-tested at any time and anywhere. So long as it remained the mysterious alchemy by which a few devotees, by devious and dubious means, presumed to change baser metals into gold, it did not flourish, but when it dealt with the fact that 56 g. of fine iron, when heated with 32 g. of flowers of sulfur, generated extra heat and gave exactly 88 g. of an entirely new substance, then additional steps could be taken by anyone. Scientific research in chemistry, since the birth of the balance and the thermometer, has been a steady growth of test and observation. It has disclosed a finite number of elementary reagents composing an infinite universe, and it is devoted to their inter-reaction for the benefit of mankind.
Address upon receiving the Perkin Medal Award, 'The Big Things in Chemistry', The Journal of Industrial and Engineering Chemistry (Feb 1921), 13, No. 2, 163.
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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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Come, see the north-wind’s masonry, Out of an unseen quarry evermore Furnished with tile, the fierce artificer Curves his white bastions with projected roof Round every windward stake, or tree, or door. Speeding, the myriad-handed, his wild work So fanciful, so savage, naught cares he For number or proportion.
…...
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Compare the length of a moment with the period of ten thousand years; the first, however minuscule, does exist as a fraction of a second. But that number of years, or any multiple of it that you may name, cannot even be compared with a limitless extent of time, the reason being that comparisons can be drawn between finite things, but not between finite and infinite.
The Consolation of Philosophy [before 524], Book II, trans. P. G. Walsh (1999), 36.
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Concerning the alchemist, Mamugnano, no one harbors doubts any longer about his daily experiments in changing quicksilver into gold. It was realized that his craft did not go beyond one pound of quicksilver… . Thus the belief is now held that his allegations to produce a number of millions have been a great fraud.
Anonymous
'Further Successes by Bragadini. From Vienna on the 26th day of January 1590'. As quoted in George Tennyson Matthews (ed.) News and Rumor in Renaissance Europe: The Fugger Newsletters (1959), 179. A handwritten collection of news reports (1568-1604) by the powerful banking and merchant house of Fugger in Ausburg.
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Conservation is the foresighted utilization, preservation. And/or renewal of forest, waters, lands and minerals, for the greatest good of the greatest number for the longest time.
…...
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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?”
In Mathematical Recreations and Problems (1896), 180; See also De Morgan’s Budget of Paradoxes (1872), 172.
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Defenders of the short-sighted men who in their greed and selfishness will, if permitted, rob our country of half its charm by their reckless extermination of all useful and beautiful wild things sometimes seek to champion them by saying the “the game belongs to the people.” So it does; and not merely to the people now alive, but to the unborn people. The “greatest good for the greatest number” applies to the number within the womb of time, compared to which those now alive form but an insignificant fraction. Our duty to the whole, including the unborn generations, bids us restrain an unprincipled present-day minority from wasting the heritage of these unborn generations. The movement for the conservation of wild life and the larger movement for the conservation of all our natural resources are essentially democratic in spirit, purpose, and method.
'Bird Reserves at the Mouth of the Mississippi', A Book-Lover's Holidays in the Open (1920), 300-301.
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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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Definition of Mathematics.—It has now become apparent that the traditional field of mathematics in the province of discrete and continuous number can only be separated from the general abstract theory of classes and relations by a wavering and indeterminate line. Of course a discussion as to the mere application of a word easily degenerates into the most fruitless logomachy. It is open to any one to use any word in any sense. But on the assumption that “mathematics” is to denote a science well marked out by its subject matter and its methods from other topics of thought, and that at least it is to include all topics habitually assigned to it, there is now no option but to employ “mathematics” in the general sense of the “science concerned with the logical deduction of consequences from the general premisses of all reasoning.”
In article 'Mathematics', Encyclopedia Britannica (1911, 11th ed.), Vol. 17, 880. In the 2006 DVD edition of the encyclopedia, the definition of mathematics is given as “The science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects.” [Premiss is a variant form of “premise”. —Webmaster]
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Dissection … teaches us that the body of man is made up of certain kinds of material, so differing from each other in optical and other physical characters and so built up together as to give the body certain structural features. Chemical examination further teaches us that these kinds of material are composed of various chemical substances, a large number of which have this characteristic that they possess a considerable amount of potential energy capable of being set free, rendered actual, by oxidation or some other chemical change. Thus the body as a whole may, from a chemical point of view, be considered as a mass of various chemical substances, representing altogether a considerable capital of potential energy.
From Introduction to A Text Book of Physiology (1876, 1891), Book 1, 1.
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Each nerve cell receives connections from other nerve cells at six sites called synapses. But here is an astonishing fact—there are about one million billion connections in the cortical sheet. If you were to count them, one connection (or synapse) per second, you would finish counting some thirty-two million years after you began. Another way of getting a feeling for the numbers of connections in this extraordinary structure is to consider that a large match-head’s worth of your brain contains about a billion connections. Notice that I only mention counting connections. If we consider how connections might be variously combined, the number would be hyperastronomical—on the order of ten followed by millions of zeros. (There are about ten followed by eighty zero’s worth of positively charged particles in the whole known universe!)
Bright and Brilliant Fire, On the Matters of the Mind (1992), 17.
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Each new machine or technique, in a sense, changes all existing machines and techniques, by permitting us to put them together into new combinations. The number of possible combinations rises exponentially as the number of new machines or techniques rises
Future Shock (1970).
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Each thing in the world has names or unnamed relations to everything else. Relations are infinite in number and kind. To be is to be related. It is evident that the understanding of relations is a major concern of all men and women. Are relations a concern of mathematics? They are so much its concern that mathematics is sometimes defined to be the science of relations.
In Mole Philosophy and Other Essays (1927), 94-95.
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Ecology has not yet explicitly developed the kind of cohesive, simplifying generalizations exemplified by, say, the laws of physics. Nevertheless there are a number of generalizations that are already evident in what we now know about the ecosphere and that can be organized into a kind of informal set of laws of ecology.
In The Closing Circle: Nature, Man, and Technology (2014).
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Edmund Wilson attacked the pedantry of scholarly editions of literary classics, which (he claimed) took the pleasure out of reading. Extensive footnotes … spoiled the reader’s pleasure in the text. Wilson’s friend Lewis Mumford had compared footnote numbers … to “barbed wire” keeping the reader at arm’s length.
In Academic Instincts (2001), 39.
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Education is like a diamond with many facets: It includes the basic mastery of numbers and letters that give us access to the treasury of human knowledge, accumulated and refined through the ages; it includes technical and vocational training as well as instruction in science, higher mathematics, and humane letters.
In Proclamation 5463, for Education Day (19 Apr 1986). Collected in Public Papers of the Presidents of the United States: Ronald Reagan, 1986 (1988), 490.
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ENGINEER, in the military art, an able expert man, who, by a perfect knowledge in mathematics, delineates upon paper, or marks upon the ground, all sorts of forts, and other works proper for offence and defence. He should understand the art of fortification, so as to be able, not only to discover the defects of a place, but to find a remedy proper for them; as also how to make an attack upon, as well as to defend, the place. Engineers are extremely necessary for these purposes: wherefore it is requisite that, besides being ingenious, they should be brave in proportion. When at a siege the engineers have narrowly surveyed the place, they are to make their report to the general, by acquainting him which part they judge the weakest, and where approaches may be made with most success. Their business is also to delineate the lines of circumvallation and contravallation, taking all the advantages of the ground; to mark out the trenches, places of arms, batteries, and lodgments, taking care that none of their works be flanked or discovered from the place. After making a faithful report to the general of what is a-doing, the engineers are to demand a sufficient number of workmen and utensils, and whatever else is necessary.
In Encyclopaedia Britannica or a Dictionary of Arts and Sciences (1771), Vol. 2, 497.
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Engineering is quite different from science. Scientists try to understand nature. Engineers try to make things that do not exist in nature. Engineers stress invention. To embody an invention the engineer must put his idea in concrete terms, and design something that people can use. That something can be a device, a gadget, a material, a method, a computing program, an innovative experiment, a new solution to a problem, or an improvement on what is existing. Since a design has to be concrete, it must have its geometry, dimensions, and characteristic numbers. Almost all engineers working on new designs find that they do not have all the needed information. Most often, they are limited by insufficient scientific knowledge. Thus they study mathematics, physics, chemistry, biology and mechanics. Often they have to add to the sciences relevant to their profession. Thus engineering sciences are born.
Y.C. Fung and P. Tong, Classical and Computational Solid Mechanics (2001), 1.
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England was nothing, compared to continental nations until she had become commercial … until about the middle of the last century, when a number of ingenious and inventive men, without apparent relation to each other, arose in various parts of the kingdom, succeeded in giving an immense impulse to all the branches of the national industry; the result of which has been a harvest of wealth and prosperity, perhaps without a parallel in the history of the world.
In Lives of the Engineers (1862, 1874), xvii.
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Enormous numbers of people are taken in, or at least beguiled and fascinated, by what seems to me to be unbelievable hocum, and relatively few are concerned with or thrilled by the astounding—yet true—facts of science, as put forth in the pages of, say, Scientific American.
Metamagical Themas (1985), 93.
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Euclidean mathematics assumes the completeness and invariability of mathematical forms; these forms it describes with appropriate accuracy and enumerates their inherent and related properties with perfect clearness, order, and completeness, that is, Euclidean mathematics operates on forms after the manner that anatomy operates on the dead body and its members. On the other hand, the mathematics of variable magnitudes—function theory or analysis—considers mathematical forms in their genesis. By writing the equation of the parabola, we express its law of generation, the law according to which the variable point moves. The path, produced before the eyes of the student by a point moving in accordance to this law, is the parabola.
If, then, Euclidean mathematics treats space and number forms after the manner in which anatomy treats the dead body, modern mathematics deals, as it were, with the living body, with growing and changing forms, and thus furnishes an insight, not only into nature as she is and appears, but also into nature as she generates and creates,—reveals her transition steps and in so doing creates a mind for and understanding of the laws of becoming. Thus modern mathematics bears the same relation to Euclidean mathematics that physiology or biology … bears to anatomy.
In Die Mathematik die Fackelträgerin einer neuen Zeit (1889), 38. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 112-113.
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Everything material which is the subject of knowledge has number, order, or position; and these are her first outlines for a sketch of the universe. If our feeble hands cannot follow out the details, still her part has been drawn with an unerring pen, and her work cannot be gainsaid. So wide is the range of mathematical sciences, so indefinitely may it extend beyond our actual powers of manipulation that at some moments we are inclined to fall down with even more than reverence before her majestic presence. But so strictly limited are her promises and powers, about so much that we might wish to know does she offer no information whatever, that at other moments we are fain to call her results but a vain thing, and to reject them as a stone where we had asked for bread. If one aspect of the subject encourages our hopes, so does the other tend to chasten our desires, and he is perhaps the wisest, and in the long run the happiest, among his fellows, who has learned not only this science, but also the larger lesson which it directly teaches, namely, to temper our aspirations to that which is possible, to moderate our desires to that which is attainable, to restrict our hopes to that of which accomplishment, if not immediately practicable, is at least distinctly within the range of conception.
From Presidential Address (Aug 1878) to the British Association, Dublin, published in the Report of the 48th Meeting of the British Association for the Advancement of Science (1878), 31.
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Exercising the right of occasional suppression and slight modification, it is truly absurd to see how plastic a limited number of observations become, in the hands of men with preconceived ideas.
Meteorographica (1863), 5.
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Exper. I. I made a small hole in a window-shutter, and covered it with a piece of thick paper, which I perforated with a fine needle. For greater convenience of observation I placed a small looking-glass without the window-shutter, in such a position as to reflect the sun's light, in a direction nearly horizontal, upon the opposite wall, and to cause the cone of diverging light to pass over a table on which were several little screens of card-paper. I brought into the sunbeam a slip of card, about one-thirtieth of an inch in breadth, and observed its shadow, either on the wall or on other cards held at different distances. Besides the fringes of colour on each side of the shadow, the shadow itself was divided by similar parallel fringes, of smaller dimensions, differing in number, according to the distance at which the shadow was observed, but leaving the middle of the shadow always white. Now these fringes were the joint effects of the portions of light passing on each side of the slip of card and inflected, or rather diffracted, into the shadow. For, a little screen being placed a few inches from the card, so as to receive either edge of the shadow on its margin, all the fringes which had before been observed in the shadow on the wall, immediately disappeared, although the light inflected on the other side was allowed to retain its course, and although this light must have undergone any modification that the proximity of the other edge of the slip of card might have been capable of occasioning... Nor was it for want of a sufficient intensity of light that one of the two portions was incapable of producing the fringes alone; for when they were both uninterrupted, the lines appeared, even if the intensity was reduced to one-tenth or one-twentieth.
'Experiments and Calculations Relative to Physical Optics' (read in 1803), Philosophical Transactions (1804), 94, 2-3.
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Facts are of not much use, considered as facts. They bewilder by their number and their apparent incoherency. Let them be digested into theory, however, and brought into mutual harmony, and it is another matter.
From article 'Electro-magnetic Theory II', in The Electrician (16 Jan 1891), 26, No. 661, 331.
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Facts are to the mind the same thing as food to the body. On the due digestion of facts depends the strength and wisdom of the one, just as vigor and health depend on the other. The wisest in council, the ablest in debate, and the most agreeable in the commerce of life is that man who has assimilated to his understanding the greatest number of facts.
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Far from becoming discouraged, the philosopher should applaud nature, even when she appears miserly of herself or overly mysterious, and should feel pleased that as he lifts one part of her veil, she allows him to glimpse an immense number of other objects, all worthy of investigation. For what we already know should allow us to judge of what we will be able to know; the human mind has no frontiers, it extends proportionately as the universe displays itself; man, then, can and must attempt all, and he needs only time in order to know all. By multiplying his observations, he could even see and foresee all phenomena, all of nature's occurrences, with as much truth and certainty as if he were deducing them directly from causes. And what more excusable or even more noble enthusiasm could there be than that of believing man capable of recognizing all the powers, and discovering through his investigations all the secrets, of nature!
'Des Mulets', Oeuvres Philosophiques, ed. Jean Piveteau (1954), 414. Quoted in Jacques Roger, The Life Sciences in Eighteenth-Century French Thought, ed. Keith R. Benson and trans. Robert Ellrich (1997), 458.
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Finally, from what we now know about the cosmos, to think that all this was created for just one species among the tens of millions of species who live on one planet circling one of a couple of hundred billion stars that are located in one galaxy among hundreds of billions of galaxies, all of which are in one universe among perhaps an infinite number of universes all nestled within a grand cosmic multiverse, is provincially insular and anthropocentrically blinkered. Which is more likely? That the universe was designed just for us, or that we see the universe as having been designed just for us?
…...
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For chemistry is no science form’d à priori; ’tis no production of the human mind, framed by reasoning and deduction: it took its rise from a number of experiments casually made, without any expectation of what follow’d; and was only reduced into an art or system, by collecting and comparing the effects of such unpremeditated experiments, and observing the uniform tendency thereof. So far, then, as a number of experimenters agree to establish any undoubted truth; so far they may be consider'd as constituting the theory of chemistry.
From 'The Author's Preface', in A New Method of Chemistry (1727), vi.
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For example, there are numbers of chemists who occupy themselves exclusively with the study of dyestuffs. They discover facts that are useful to scientific chemistry; but they do not rank as genuine scientific men. The genuine scientific chemist cares just as much to learn about erbium—the extreme rarity of which renders it commercially unimportant—as he does about iron. He is more eager to learn about erbium if the knowledge of it would do more to complete his conception of the Periodic Law, which expresses the mutual relations of the elements.
From 'Lessons from the History of Science: The Scientific Attitude' (c.1896), in Collected Papers (1931), Vol. 1, 20.
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For it is not number of Experiments, but weight to be regarded; & where one will do, what need many?
In 'Mr. Newton's Answer to the Precedent Letter, Sent to the Publisher', Philosophical Transactions (1665-1678), Vol. 11 (25 Sep 1676), No. 128, 703.
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For the harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty.
On Growth and Form (1917), Epilogue, 778-9.
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For us, an atom shall be a small, spherical, homogeneous body or an essentially indivisible, material point, whereas a molecule shall be a separate group of atoms in any number and of any nature.
Annales de Chimie 1833, 52, 133. Trans. W. H. Brock.
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From a mathematical standpoint it is possible to have infinite space. In a mathematical sense space is manifoldness, or combinations of numbers. Physical space is known as the 3-dimension system. There is the 4-dimension system, the 10-dimension system.
As quoted in 'Electricity Will Keep The World From Freezing Up', New York Times (12 Nov 1911), SM4.
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Gauss once said, “Mathematics is the queen of the sciences and number theory the queen of mathematics.” If this is true we may add that the Disquisitions is the Magna Charter of number theory.
In Allgemeine Deutsche Biographie (1878, 8, 435. As cited and translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 158.
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Geologists have not been slow to admit that they were in error in assuming that they had an eternity of past time for the evolution of the earth's history. They have frankly acknowledged the validity of the physical arguments which go to place more or less definite limits to the antiquity of the earth. They were, on the whole, disposed to acquiesce in the allowance of 100 millions of years granted to them by Lord Kelvin, for the transaction of the whole of the long cycles of geological history. But the physicists have been insatiable and inexorable. As remorseless as Lear's daughters, they have cut down their grant of years by successive slices, until some of them have brought the number to something less than ten millions. In vain have the geologists protested that there must somewhere be a flaw in a line of argument which tends to results so entirely at variance with the strong evidence for a higher antiquity, furnished not only by the geological record, but by the existing races of plants and animals. They have insisted that this evidence is not mere theory or imagination, but is drawn from a multitude of facts which become hopelessly unintelligible unless sufficient time is admitted for the evolution of geological history. They have not been able to disapprove the arguments of the physicists, but they have contended that the physicists have simply ignored the geological arguments as of no account in the discussion.
'Twenty-five years of Geological Progress in Britain', Nature, 1895, 51, 369.
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Having discovered … by observation and comparison that certain objects agree in certain respects, we generalise the qualities in which they coincide,—that is, from a certain number of individual instances we infer a general law; we perform an act of Induction. This induction is erroneously viewed as analytic; it is purely a synthetic process.
In Lecture VI of his Biennial Course, by William Hamilton and Henry L. Mansel (ed.) and John Veitch (ed.), Metaphysics (1860), Vol. 1, 101.
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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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Hence, a generative grammar must be a system of rules that can iterate to generate an indefinitely large number of structures. This system of rules can be analyzed into the three major components of a generative grammar: the syntactic, phonological, and semantic components... the syntactic component of a grammar must specify, for each sentence, a deep structure that determines its semantic interpretation and a surface structure that determines its phonetic interpretation. The first of these is interpreted by the semantic component; the second, by the phonological component.
Aspects of the Theory of Syntax (1965), 15-6.
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Here I am at the limit which God and nature has assigned to my individuality. I am compelled to depend upon word, language and image in the most precise sense, and am wholly unable to operate in any manner whatever with symbols and numbers which are easily intelligible to the most highly gifted minds.
In Letter to Naumann (1826), in Vogel, Goethe's Selbstzeugnisse (1903), 56.
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Here I shall present, without using Analysis [mathematics], the principles and general results of the Théorie, applying them to the most important questions of life, which are indeed, for the most part, only problems in probability. One may even say, strictly speaking, that almost all our knowledge is only probable; and in the small number of things that we are able to know with certainty, in the mathematical sciences themselves, the principal means of arriving at the truth—induction and analogy—are based on probabilities, so that the whole system of human knowledge is tied up with the theory set out in this essay.
Philosophical Essay on Probabilities (1814), 5th edition (1825), trans. Andrew I. Dale (1995), 1.
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His [Henry Cavendish’s] Theory of the Universe seems to have been, that it consisted solely of a multitude of objects which could be weighed, numbered, and measured; and the vocation to which he considered himself called was, to weigh, number and measure as many of those objects as his allotted three-score years and ten would permit. This conviction biased all his doings, alike his great scientific enterprises, and the petty details of his daily life.
In George Wilson, The Life of the Honourable Henry Cavendish: Including the Abstracts of his Important Scientific Papers (1851), 186.
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Historical theories are, after all, intellectual apple carts. They are quite likely to be upset. Nor should it be forgotten that they tend to attract, when they gain ascendancy, a fair number of apple-polishers
'Books of the Times'. New York Times (9 Dec 1965), 45.
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Homo sapiens is a compulsive communicator. Look at the number of people you see walking around talking on mobile phones. We seem to have an infinite capacity for communicating and being communicated with. I’m not sure how admirable it is, but it certainly demonstrates that we are social organisms.
From interview with Michael Bond, 'It’s a Wonderful Life', New Scientist (14 Dec 2002), 176, No. 2373, 48.
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Houston, that may have seemed like a very long final phase. The autotargeting was taking us right into a... crater, with a large number of big boulders and rocks ... and it required... flying manually over the rock field to find a reasonably good area.
…...
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How does it arise that, while the statements of geologists that other organic bodies existed millions of years ago are tacitly accepted, their conclusions as to man having existed many thousands of years ago should be received with hesitation by some geologists, and be altogether repudiated by a not inconsiderable number among the other educated classes of society?
'Anniversary Address of the Geological Society of London', Proceedings of the Geological Society of London (1861), 17, lxvii.
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I accepted the Copernican position several years ago and discovered from thence the causes of many natural effects which are doubtless inexplicable by the current theories. I have written up many reasons and refutations on the subject, but I have not dared until now to bring them into the open, being warned by the fortunes of Copernicus himself, our master, who procured for himself immortal fame among a few but stepped down among the great crowd (for this is how foolish people are to be numbered), only to be derided and dishonoured. I would dare publish my thoughts if there were many like you; but since there are not, I shall forbear.
Letter to Johannes Kepler, 4 Aug 1597. Quoted in G. de Santillana, Crime of Galileo (1955), 11.
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I am ill at these numbers.
In Hamlet (1603), Act 2, Scene 2. [Note: Not for arithmetic. The numbers relate to the number of syllables in each line to write poetry; he is a bad poet. —Webmaster]
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I am more and more convinced that the ant colony is not so much composed of separate individuals as that the colony is a sort of individual, and each ant like a loose cell in it. Our own blood stream, for instance, contains hosts of white corpuscles which differ little from free-swimming amoebae. When bacteria invade the blood stream, the white corpuscles, like the ants defending the nest, are drawn mechanically to the infected spot, and will die defending the human cell colony. I admit that the comparison is imperfect, but the attempt to liken the individual human warrior to the individual ant in battle is even more inaccurate and misleading. The colony of ants with its component numbers stands half way, as a mechanical, intuitive, and psychical phenomenon, between our bodies as a collection of cells with separate functions and our armies made up of obedient privates. Until one learns both to deny real individual initiative to the single ant, and at the same time to divorce one's mind from the persuasion that the colony has a headquarters which directs activity … one can make nothing but pretty fallacies out of the polity of the ant heap.
In An Almanac for Moderns (1935), 121
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I am of opinion, then, ... that, if there is any circumstance thoroughly established in geology, it is, that the crust of our globe has been subjected to a great and sudden revolution, the epoch of which cannot be dated much farther back than five or six thousand years ago; that this revolution had buried all the countries which were before inhabited by men and by the other animals that are now best known; that the same revolution had laid dry the bed of the last ocean, which now forms all the countries at present inhabited; that the small number of individuals of men and other animals that escaped from the effects of that great revolution, have since propagated and spread over the lands then newly laid dry; and consequently, that the human race has only resumed a progressive state of improvement since that epoch, by forming established societies, raising monuments, collecting natural facts, and constructing systems of science and of learning.
'Preliminary discourse', to Recherches sur les Ossemens Fossiles (1812), trans. R. Kerr Essay on the Theory of the Earth (1813), 171-2.
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I am particularly concerned to determine the probability of causes and results, as exhibited in events that occur in large numbers, and to investigate the laws according to which that probability approaches a limit in proportion to the repetition of events. That investigation deserves the attention of mathematicians because of the analysis required. It is primarily there that the approximation of formulas that are functions of large numbers has its most important applications. The investigation will benefit observers in identifying the mean to be chosen among the results of their observations and the probability of the errors still to be apprehended. Lastly, the investigation is one that deserves the attention of philosophers in showing how in the final analysis there is a regularity underlying the very things that seem to us to pertain entirely to chance, and in unveiling the hidden but constant causes on which that regularity depends. It is on the regularity of the main outcomes of events taken in large numbers that various institutions depend, such as annuities, tontines, and insurance policies. Questions about those subjects, as well as about inoculation with vaccine and decisions of electoral assemblies, present no further difficulty in the light of my theory. I limit myself here to resolving the most general of them, but the importance of these concerns in civil life, the moral considerations that complicate them, and the voluminous data that they presuppose require a separate work.
Philosophical Essay on Probabilities (1825), trans. Andrew I. Dale (1995), Introduction.
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I asked Fermi whether he was not impressed by the agreement between our calculated numbers and his measured numbers. He replied, “How many arbitrary parameters did you use for your calculations?" I thought for a moment about our cut-off procedures and said, “Four." He said, “I remember my friend Johnny von Neumann used to say, with four parameters I can fit an elephant, and with five I can make him wiggle his trunk.” With that, the conversation was over.
As given in 'A Meeting with Enrico Fermi', Nature (22 Jan 2004), 427, 297. As quoted in Steven A. Frank, Dynamics of Cancer: Incidence, Inheritance, and Evolution (2007), 87. Von Neumann meant nobody need be impressed when a complex model fits a data set well, because if manipulated with enough flexible parameters, any data set can be fitted to an incorrect model, even one that plots a curve on a graph shaped like an elephant!
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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 believe there are
15,747,724,136,275,002,577,605,653,961,181,555,468,044,717,
914,527,116,709,366,231,025,076,185,631,031,296
protons in the universe, and the same number of electrons.
From Tarner Lecture (1938), 'The Physical Universe', in The Philosophy of Physical Science (1939, 2012), 170. Note: the number is 136 x 2256.
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I can see him [Sylvester] now, with his white beard and few locks of gray hair, his forehead wrinkled o’er with thoughts, writing rapidly his figures and formulae on the board, sometimes explaining as he wrote, while we, his listeners, caught the reflected sounds from the board. But stop, something is not right, he pauses, his hand goes to his forehead to help his thought, he goes over the work again, emphasizes the leading points, and finally discovers his difficulty. Perhaps it is some error in his figures, perhaps an oversight in the reasoning. Sometimes, however, the difficulty is not elucidated, and then there is not much to the rest of the lecture. But at the next lecture we would hear of some new discovery that was the outcome of that difficulty, and of some article for the Journal, which he had begun. If a text-book had been taken up at the beginning, with the intention of following it, that text-book was most likely doomed to oblivion for the rest of the term, or until the class had been made listeners to every new thought and principle that had sprung from the laboratory of his mind, in consequence of that first difficulty. Other difficulties would soon appear, so that no text-book could last more than half of the term. In this way his class listened to almost all of the work that subsequently appeared in the Journal. It seemed to be the quality of his mind that he must adhere to one subject. He would think about it, talk about it to his class, and finally write about it for the Journal. The merest accident might start him, but once started, every moment, every thought was given to it, and, as much as possible, he read what others had done in the same direction; but this last seemed to be his real point; he could not read without finding difficulties in the way of understanding the author. Thus, often his own work reproduced what had been done by others, and he did not find it out until too late.
A notable example of this is in his theory of cyclotomic functions, which he had reproduced in several foreign journals, only to find that he had been greatly anticipated by foreign authors. It was manifest, one of the critics said, that the learned professor had not read Rummer’s elementary results in the theory of ideal primes. Yet Professor Smith’s report on the theory of numbers, which contained a full synopsis of Kummer’s theory, was Professor Sylvester’s constant companion.
This weakness of Professor Sylvester, in not being able to read what others had done, is perhaps a concomitant of his peculiar genius. Other minds could pass over little difficulties and not be troubled by them, and so go on to a final understanding of the results of the author. But not so with him. A difficulty, however small, worried him, and he was sure to have difficulties until the subject had been worked over in his own way, to correspond with his own mode of thought. To read the work of others, meant therefore to him an almost independent development of it. Like the man whose pleasure in life is to pioneer the way for society into the forests, his rugged mind could derive satisfaction only in hewing out its own paths; and only when his efforts brought him into the uncleared fields of mathematics did he find his place in the Universe.
In Florian Cajori, Teaching and History of Mathematics in the United States (1890), 266-267.
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I conceived and developed a new geometry of nature and implemented its use in a number of diverse fields. It describes many of the irregular and fragmented patterns around us, and leads to full-fledged theories, by identifying a family of shapes I call fractals.
The Fractal Geometry of Nature (1977, 1983), Introduction, xiii.
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I confess that Fermat’s Theorem as an isolated proposition has very little interest for me, for a multitude of such theorems can easily be set up, which one could neither prove nor disprove. But I have been stimulated by it to bring our again several old ideas for a great extension of the theory of numbers. Of course, this theory belongs to the things where one cannot predict to what extent one will succeed in reaching obscurely hovering distant goals. A happy star must also rule, and my situation and so manifold distracting affairs of course do not permit me to pursue such meditations as in the happy years 1796-1798 when I created the principal topics of my Disquisitiones arithmeticae. But I am convinced that if good fortune should do more than I expect, and make me successful in some advances in that theory, even the Fermat theorem will appear in it only as one of the least interesting corollaries.
In reply to Olbers' attempt in 1816 to entice him to work on Fermat's Theorem. The hope Gauss expressed for his success was never realised.
Letter to Heinrich Olbers (21 Mar 1816). Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 413.
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I count Maxwell and Einstein, Eddington and Dirac, among “real” mathematicians. The great modern achievements of applied mathematics have been in relativity and quantum mechanics, and these subjects are at present at any rate, almost as “useless” as the theory of numbers.
In A Mathematician's Apology (1940, 2012), 131.
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I do hate sums. There is no greater mistake than to call arithmetic an exact science. There are permutations and aberrations discernible to minds entirely noble like mine; subtle variations which ordinary accountants fail to discover; hidden laws of number which it requires a mind like mine to perceive. For instance, if you add a sum from the bottom up, and then from the top down, the result is always different. Again if you multiply a number by another number before you have had your tea, and then again after, the product will be different. It is also remarkable that the Post-tea product is more likely to agree with other people’s calculations than the Pre-tea result.
Letter to Mrs Arthur Severn (Jul 1878), collected in The Letters of a Noble Woman (Mrs. La Touche of Harristown) (1908), 50. Also in 'Gleanings Far and Near', Mathematical Gazette (May 1924), 12, 95.
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I had made considerable advance ... in calculations on my favourite numerical lunar theory, when I discovered that, under the heavy pressure of unusual matters (two transits of Venus and some eclipses) I had committed a grievous error in the first stage of giving numerical value to my theory. My spirit in the work was broken, and I have never heartily proceeded with it since.
[Concerning his calculations on the orbital motion of the Moon.]
Private note (29 Sep 1890). In George Biddell Airy and Wilfrid Airy (ed.), Autobiography of Sir George Biddell Airy (1896), 350.
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I have divers times examined the same matter (human semen) from a healthy man... not from a sick man... nor spoiled by keeping... for a long time and not liquefied after the lapse of some time... but immediately after ejaculation before six beats of the pulse had intervened; and I have seen so great a number of living animalcules... in it, that sometimes more than a thousand were moving about in an amount of material the size of a grain of sand... I saw this vast number of animalcules not all through the semen, but only in the liquid matter adhering to the thicker part.
Letter to W. Brouncker, President of the Royal Society, undated, Nov 1677. In The Collected Letters of Antoni van Leeuwenhoek (1957), Vol. 2, 283-4.
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I have little patience with scientists who take a board of wood, look for its thinnest part and drill a great number of holes where drilling is easy.
P. Frank in 'Einstein's Philosophy of Science', Reviews of Modern Physics (1949).
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I have often admired the mystical way of Pythagoras, and the secret magick of numbers.
In Religio Medici (1642, 1754), pt. 1, sec. 12, 28.
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I have often thought that an interesting essay might be written on the influence of race on the selection of mathematical methods. methods. The Semitic races had a special genius for arithmetic and algebra, but as far as I know have never produced a single geometrician of any eminence. The Greeks on the other hand adopted a geometrical procedure wherever it was possible, and they even treated arithmetic as a branch of geometry by means of the device of representing numbers by lines.
In A History of the Study of Mathematics at Cambridge (1889), 123
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I have said that mathematics is the oldest of the sciences; a glance at its more recent history will show that it has the energy of perpetual youth. The output of contributions to the advance of the science during the last century and more has been so enormous that it is difficult to say whether pride in the greatness of achievement in this subject, or despair at his inability to cope with the multiplicity of its detailed developments, should be the dominant feeling of the mathematician. Few people outside of the small circle of mathematical specialists have any idea of the vast growth of mathematical literature. The Royal Society Catalogue contains a list of nearly thirty- nine thousand papers on subjects of Pure Mathematics alone, which have appeared in seven hundred serials during the nineteenth century. This represents only a portion of the total output, the very large number of treatises, dissertations, and monographs published during the century being omitted.
In Presidential Address British Association for the Advancement of Science, Sheffield, Section A, Nature (1 Sep 1910), 84, 285.
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I have tried to avoid long numerical computations, thereby following Riemann’s postulate that proofs should be given through ideas and not voluminous computations.
In Report on Number Theory (1897). As given in epigraph, without citation, in Eberhard Zeidler and Juergen Quandt (trans.), Nonlinear Functional Analysis and its Applications: IV: Applications to Mathematical Physics (2013), 448.
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I have written many direct and indirect arguments for the Copernican view, but until now I have not dared to publish them, alarmed by the fate of Copernicus himself, our master. He has won for himself undying fame in the eyes of a few, but he has been mocked and hooted at by an infinite multitude (for so large is the number of fools). I would dare to come forward publicly with my ideas if there were more people of your [Johannes Kepler’s] way of thinking. As this is not the case, I shall refrain.
Letter to Kepler (4 Aug 1597). In James Bruce Ross (ed.) and Mary Martin (ed., trans.), 'Comrades in the Pursuit of Truth', The Portable Renaissance Reader (1953, 1981), 597-599. As quoted and cited in Merry E. Wiesner, Early Modern Europe, 1450-1789 (2013), 377.
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I hope that in 50 years we will know the answer to this challenging question: are the laws of physics unique and was our big bang the only one? … According to some speculations the number of distinct varieties of space—each the arena for a universe with its own laws—could exceed the total number of atoms in all the galaxies we see. … So do we live in the aftermath of one big bang among many, just as our solar system is merely one of many planetary systems in our galaxy? (2006)
In 'Martin Rees Forecasts the Future', New Scientist (18 Nov 2006), No. 2578.
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I imagined in the beginning, that a few experiments would determine the problem; but experience soon convinced me, that a very great number indeed were necessary before such an art could be brought to any tolerable degree of perfection.
Upon pursuing the ''
Preface to An Essay on Combustion with a View to a New Art of Dyeing and Painting (1794), iii. In Marilyn Bailey Ogilvie and Joy Dorothy Harvey, The Biographical Dictionary of Women in Science (2000), 478.
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I must … explain how I was led to concern myself with the pathogenic protozoa. … I was sent to Algeria and put in charge of a department of the hospital at Bone. A large number of my patients had malarial fevers and I was naturally led to study these fevers of which I had only seen rare and benign forms in France.
From Nobel Lecture (11 Dec 1907), 'Protozoa as Causes of Diseases', collected in Nobel Lectures, Physiology or Medicine 1901-1921 (1967, 1999), 264.
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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 observed on most collected stones the imprints of innumerable plant fragments which were so different from those which are growing in the Lyonnais, in the nearby provinces, and even in the rest of France, that I felt like collecting plants in a new world… The number of these leaves, the way they separated easily, and the great variety of plants whose imprints I saw, appeared to me just as many volumes of botany representing in the same quarry the oldest library of the world.
In 'Examen des causes des Impressions des Plantes marquees sur certaines Pierres des environs de Saint-Chaumont dans le Lionnais', Memoires de l’ Academie Royale des Sciences (1718), 364, as trans. by Albert V. and Marguerite Carozzi.
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I picture the vast realm of the sciences as an immense landscape scattered with patches of dark and light. The goal towards which we must work is either to extend the boundaries of the patches of light, or to increase their number. One of these tasks falls to the creative genius; the other requires a sort of sagacity combined with perfectionism.
Thoughts on the Interpretation of Nature and Other Philosophical Works (1753/4), ed. D. Adams (1999), Section XIV, 42.
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I recall my own emotions: I had just been initiated into the mysteries of the complex number. I remember my bewilderment: here were magnitudes patently impossible and yet susceptible of manipulations which lead to concrete results. It was a feeling of dissatisfaction, of restlessness, a desire to fill these illusory creatures, these empty symbols, with substance. Then I was taught to interpret these beings in a concrete geometrical way. There came then an immediate feeling of relief, as though I had solved an enigma, as though a ghost which had been causing me apprehension turned out to be no ghost at all, but a familiar part of my environment.
In Tobias Dantzig and Joseph Mazur (ed.), 'The Two Realities', Number: The Language of Science (1930, ed. by Joseph Mazur 2007), 254.
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I regarded as quite useless the reading of large treatises of pure analysis: too large a number of methods pass at once before the eyes. It is in the works of application that one must study them; one judges their utility there and appraises the manner of making use of them.
As reported by J. F. Maurice in Moniteur Universel (1814), 228.
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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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I remember one occasion when I tried to add a little seasoning to a review, but I wasn’t allowed to. The paper was by Dorothy Maharam, and it was a perfectly sound contribution to abstract measure theory. The domains of the underlying measures were not sets but elements of more general Boolean algebras, and their range consisted not of positive numbers but of certain abstract equivalence classes. My proposed first sentence was: “The author discusses valueless measures in pointless spaces.”
In I Want to be a Mathematician: An Automathography (1985), 120.
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I should like to call the number of atom groups, with which an elementary atom coordinates … to form a complex radical, the coordination number of the atom in question … We must differentiate between valence number and coordination number. The valence number indicates the maximum number of monovalent atoms which can be bound directly to the atom in question without the participation of other elementary atoms … Perhaps this concept [of coordination number] is destined to serve as a basis for the theory of the constitution of inorganic compounds, just as valence theory formed the basis for the constitutional theory of carbon compounds.
In 'Beitrag zur Konstitution anorganischer Verbindungen', Zeitschrift fur anorganische Chemie, (1893), 3, 267-330. Translated in George G. Kauffman (ed.), Classics in Coordination Chemistry: Part I: The Selected Papers of Alfred Werner (1968), 84-87.
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I should like to compare this rearrangement which the proteins undergo in the animal or vegetable organism to the making up of a railroad train. In their passage through the body parts of the whole may be left behind, and here and there new parts added on. In order to understand fully the change we must remember that the proteins are composed of Bausteine united in very different ways. Some of them contain Bausteine of many kinds. The multiplicity of the proteins is determined by many causes, first through the differences in the nature of the constituent Bausteine; and secondly, through differences in the arrangement of them. The number of Bausteine which may take part in the formation of the proteins is about as large as the number of letters in the alphabet. When we consider that through the combination of letters an infinitely large number of thoughts may be expressed, we can understand how vast a number of the properties of the organism may be recorded in the small space which is occupied by the protein molecules. It enables us to understand how it is possible for the proteins of the sex-cells to contain, to a certain extent, a complete description of the species and even of the individual. We may also comprehend how great and important the task is to determine the structure of the proteins, and why the biochemist has devoted himself with so much industry to their analysis.
'The Chemical Composition of the Cell', The Harvey Lectures (1911), 7, 45.
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I think if we had not repaired the telescope, it would have been the end of the space station, because space station requires a huge number of space walks. I think it was fair to use the Hubble space telescope as a test case for space walks, to say, “Can NASA really do what they say they can do up there?”
Interview (22 May 1997). On Academy of Achievement website.
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I think it’s going to be great if people can buy a ticket to fly up and see black sky and the stars. I’d like to do it myself - but probably after it has flown a serious number of times first!
…...
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I think, and I am not the only one who does, that it is important never to introduce any conception which may not be completely defined by a finite number of words. Whatever may be the remedy adopted, we can promise ourselves the joy of the physician called in to follow a beautiful pathological case [beau cas pathologique].
From address read at the general session of the Fourth International Congress of Mathematicians in Rome (10 Apr 1908). As translated in 'The Future of Mathematics: by Henri Poincaré', General Appendix, Annual Report of the Boars of Regents of The Smithsonian Institution: For the Year Ending June 1909 (1910), 140.
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I trust ... I have succeeded in convincing you that modern chemistry is not, as it has so long appeared, an ever-growing accumulation of isolated facts, as impossible for a single intellect to co-ordinate as for a single memory to grasp.
The intricate formulae that hang upon these walls, and the boundless variety of phenomena they illustrate, are beginning to be for us as a labyrinth once impassable, but to which we have at length discovered the clue. A sense of mastery and power succeeds in our minds to the sort of weary despair with which we at first contemplated their formidable array. For now, by the aid of a few general principles, we find ourselves able to unravel the complexities of these formulae, to marshal the compounds which they represent in orderly series; nay, even to multiply their numbers at our will, and in a great measure to forecast their nature ere we have called them into existence. It is the great movement of modern chemistry that we have thus, for an hour, seen passing before us. It is a movement as of light spreading itself over a waste of obscurity, as of law diffusing order throughout a wilderness of confusion, and there is surely in its contemplation something of the pleasure which attends the spectacle of a beautiful daybreak, something of the grandeur belonging to the conception of a world created out of chaos.
Concluding remark for paper presented at the Friday Discourse of the the Royal Institution (7 Apr 1865). 'On the Combining Power of Atoms', Proceedings of the Royal Institution (1865), 4, No. 42, 416.
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I want to put in something about Bernoulli’s numbers, in one of my Notes, as an example of how the implicit function may be worked out by the engine, without having been worked out by human head & hands first. Give me the necessary data & formulae.
Lovelace Papers, Bodleian Library, Oxford University, 42, folio 12 (6 Feb 1841). As quoted and cited in Dorothy Stein (ed.), 'This First Child of Mine', Ada: A Life and a Legacy (1985), 106-107.
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I was suffering from a sharp attack of intermittent fever, and every day during the cold and succeeding hot fits had to lie down for several hours, during which time I had nothing to do but to think over any subjects then particularly interesting me. One day something brought to my recollection Malthus's 'Principles of Population', which I had read about twelve years before. I thought of his clear exposition of 'the positive checks to increase'—disease, accidents, war, and famine—which keep down the population of savage races to so much lower an average than that of more civilized peoples. It then occurred to me that these causes or their equivalents are continually acting in the case of animals also; and as animals usually breed much more rapidly than does mankind, the destruction every year from these causes must be enormous in order to keep down the numbers of each species, since they evidently do not increase regularly from year to year, as otherwise the world would long ago have been densely crowded with those that breed most quickly. Vaguely thinking over the enormous and constant destruction which this implied, it occurred to me to ask the question, Why do some die and some live? The answer was clearly, that on the whole the best fitted live. From the effects of disease the most healthy escaped; from enemies, the strongest, swiftest, or the most cunning; from famine, the best hunters or those with the best digestion; and so on. Then it suddenly flashed upon me that this self-acting process would necessarily improve the race, because in every generation the inferior would inevitably be killed off and the superior would remain—that is, the fittest would survive.
[The phrase 'survival of the fittest,' suggested by the writings of Thomas Robert Malthus, was expressed in those words by Herbert Spencer in 1865. Wallace saw the term in correspondence from Charles Darwin the following year, 1866. However, Wallace did not publish anything on his use of the expression until very much later, and his recollection is likely flawed.]
My Life: A Record of Events and Opinions (1905), Vol. 1, 361-362, or in reprint (2004), 190.
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I was unable to devote myself to the learning of this al-jabr [algebra] and the continued concentration upon it, because of obstacles in the vagaries of Time which hindered me; for we have been deprived of all the people of knowledge save for a group, small in number, with many troubles, whose concern in life is to snatch the opportunity, when Time is asleep, to devote themselves meanwhile to the investigation and perfection of a science; for the majority of people who imitate philosophers confuse the true with the false, and they do nothing but deceive and pretend knowledge, and they do not use what they know of the sciences except for base and material purposes; and if they see a certain person seeking for the right and preferring the truth, doing his best to refute the false and untrue and leaving aside hypocrisy and deceit, they make a fool of him and mock him.
A. P. Youschkevitch and B. A. Rosenfeld, 'Al-Khayyami', in C. C. Gillispie (ed.), Dictionary of Scientific Biography (1973), Vol. 7, 324.
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I wish that one would be persuaded that psychological experiments, especially those on the complex functions, are not improved [by large studies]; the statistical method gives only mediocre results; some recent examples demonstrate that. The American authors, who love to do things big, often publish experiments that have been conducted on hundreds and thousands of people; they instinctively obey the prejudice that the persuasiveness of a work is proportional to the number of observations. This is only an illusion.
L' Études expérimentale de l'intelligence (1903), 299.
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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 a mixture of different kinds of electrified atoms is moving along in one stream, then when electric and magnetic forces are applied to the stream simultaneously, the different kinds of atoms are sorted out, and the original stream is divided up into a number of smaller streams separated from each other. The particles in any one of the smaller streams are all of the same kind.
From the Romanes Lecture (10 Jun 1914) delivered in the Sheldonian Theatre, published as The Atomic Theory (1914), 9.
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If all sentient beings in the universe disappeared, there would remain a sense in which mathematical objects and theorems would continue to exist even though there would be no one around to write or talk about them. Huge prime numbers would continue to be prime, even if no one had proved them prime.
In When You Were a Tadpole and I Was a Fish: And Other Speculations About This and That (), 124.
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If an event can be produced by a number n of different causes, the probabilities of the existence of these causes, given the event (prises de l'événement), are to each other as the probabilities of the event, given the causes: and the probability of each cause is equal to the probability of the event, given that cause, divided by the sum of all the probabilities of the event, given each of the causes.
'Mémoire sur la Probabilité des Causes par les Événements' (1774). In Oeuvres complètes de Laplace, 14 Vols. (1843-1912), Vol. 8, 29, trans. Charles Coulston Gillispie, Pierre-Simon Laplace 1749-1827: A Life in Exact Science (1997), 16.
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If any layman were to ask a number of archaeologists to give, on the spur of the moment, a definition of archaeology, I suspect that such a person might find the answers rather confusing. He would, perhaps, sympathize with Socrates who, when he hoped to learn from the poets and artisans something about the arts they practised, was forced to go away with the conviction that, though they might themselves be able to accomplish something, they certainly could give no clear account to others of what they were trying to do.
Opening statement in lecture at Columbia University (8 Jan 1908), 'Archaeology'. Published by the Columbia University Press (1908).
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If E is considered to be a continuously divisible quantity, this distribution is possible in infinitely many ways. We consider, however—this is the most essential point of the whole calculation—E to be composed of a well-defined number of equal parts and use thereto the constant of nature h = 6.55 ×10-27 erg sec. This constant multiplied by the common frequency ν of the resonators gives us the energy element ε in erg, and dividing E by ε we get the number P of energy elements which must be divided over the N resonators.
[Planck’s constant, as introduced in 1900; subsequently written e = hν.]
In 'On the theory of the energy distribution law of the normal spectrum', in D. ter Haar and Stephen G. Brush, trans., Planck’s Original Papers in Quantum Physics (1972), 40.
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If even in science there is no a way of judging a theory but by assessing the number, faith and vocal energy of its supporters, then this must be even more so in the social sciences: truth lies in power.
In Radio Lecture (30 Jun 1973) broadcast by the Open University, collected in Imre Lakatos, John Worrall (ed.) and Gregory Currie (ed.), 'Introduction: Science and Pseudoscience', The Methodology of Scientific Research Programmes (1978, 1980), Vol. 1, 9.
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If it is true as Whewell says, that the essence of the triumphs of Science and its progress consists in that it enables us to consider evident and necessary, views which our ancestors held to be unintelligible and were unable to comprehend, then the extension of the number concept to include the irrational, and we will at once add, the imaginary, is the greatest forward step which pure mathematics has ever taken.
In Theorie der Complexen Zahlensysteme (1867), 60. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 281. From the original German, “Wenn es wahr ist, dass, wie Whewell meint, das Wesen der Triumphe der Wissenschaft und ihres Fortschrittes darin besteht, dass wir veranlasst werden, Ansichten, welche unsere Vorfahren für unbegreiflich hielten und unfähig waren zu begreifen, für evident und nothwendig zu halten, so war die Erweiterung des Zahlenbegriffes auf das Irrationale, und wollen wir sogleich hinzufügen, das Imaginäre, der grösste Fortschritt, den die reine Mathematik jemals gemacht hat.”
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If one proves the equality of two numbers a and b by showing first that “a is less than or equal to b” and then “a is greater than or equal to b”, it is unfair, one should instead show that they are really equal by disclosing the inner ground for their equality.
As quoted, without citation, in biography by Hermann Wehl, Emmy Noether (1935), 18.
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If the arrangement of society is bad (as ours is), and a small number of people have power over the majority and oppress it, every victory over Nature will inevitably serve only to increase that power and that oppression.
In Science, Liberty and Peace by Aldous Huxley (1947).
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If the views we have ventured to advance be correct, we may almost consider {greek words} of the ancients to be realised in hydrogen, an opinion, by the by, not altogether new. If we actually consider the specific gravities of bodies in their gaseous state to represent the number of volumes condensed into one; or in other words, the number of the absolute weight of a single volume of the first matter ({greek words}) which they contain, which is extremely probable, multiples in weight must always indicate multiples in volume, and vice versa; and the specific gravities, or absolute weights of all bodies in a gaseous state, must be multiples of the specific gravity or absolute weight of the first matter, ({Greek words}), because all bodies in the gaseous state which unite with one another unite with reference to their volume.
'Correction of a Mistake in the Essay on the Relation between the Specific Gravities of Bodies in their Gaseous State and the Weights of their Atoms', Annals of Philosophy (1816), 7, 113.
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If the world has begun with a single quantum, the notions of space and would altogether fail to have any meaning at the beginning; they would only begin to have a sensible meaning when the original quantum had been divided into a sufficient number of quanta. If this suggestion is correct, the beginning of the world happened a little before the beginning of space and time. I think that such a beginning of the world is far enough from the present order of Nature to be not at all repugnant. It may be difficult to follow up the idea in detail as we are not yet able to count the quantum packets in every case. For example, it may be that an atomic nucleus must be counted as a unique quantum, the atomic number acting as a kind of quantum number. If the future development of quantum theory happens to turn in that direction, we could conceive the beginning of the universe in the form of a unique atom, the atomic weight of which is the total mass of the universe. This highly unstable atom would divide in smaller and smaller atoms by a kind of super-radioactive process.
In 'The Beginning of the World from the Point of View of Quantum Theory', Nature (1931), 127, 706.
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If the world may be thought of as a certain definite quantity of force and as a certain definite number of centers of force—and every other representation remains indefinite and therefore useless—it follows that, in the great dice game of existence, it must pass through calculable number of combinations. In infinite time, every possible combination would at some time or another be realized; more: it would be realized an infinite number of times. And since between every combination and its next recurrence all other possible combinations would have to take place, and each of these combination conditions of the entire sequence of combinations in the same series, a circular movement of absolutely identical series is thus demonstrated: the world as a circular movement that has already repeated itself infinitely often and plays its game in infinitum. This conception is not simply a mechanistic conception; for if it were that, it would not condition an infinite recurrence of identical cases, but a final state. Because the world has not reached this, mechanistic theory must be considered an imperfect and merely provisional hypothesis.
The Will to Power (Notes written 1883-1888), book 4, no. 1066. Trans. W. Kaufmann and R. J. Hollingdale and ed. W. Kaufmann (1968), 549.
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If there is such a thing as luck, then I must be the most unlucky fellow in the world. I’ve never once made a lucky strike in all my life. When I get after something that I need, I start finding everything in the world that I don’t need—one damn thing after another. I find ninety-nine things that I don’t need, and then comes number one hundred, and that—at the very last—turns out to be just what I had been looking for.
In Martin André Rosanoff, 'Edison in His Laboratory', Harper’s Magazine (Sep 1932), 406.
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If to-day you ask a physicist what he has finally made out the æther or the electron to be, the answer will not be a description in terms of billiard balls or fly-wheels or anything concrete; he will point instead to a number of symbols and a set of mathematical equations which they satisfy. What do the symbols stand for? The mysterious reply is given that physics is indifferent to that; it has no means of probing beneath the symbolism. To understand the phenomena of the physical world it is necessary to know the equations which the symbols obey but not the nature of that which is being symbolised. …this newer outlook has modified the challenge from the material to the spiritual world.
Swarthmore Lecture (1929) at Friends’ House, London, printed in Science and the Unseen World (1929), 30.
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If we assume that there is only one enzyme present to act as an oxidizing agent, we must assume for it as many different degrees of activity as are required to explain the occurrence of the various colors known to mendelize (three in mice, yellow, brown, and black). If we assume that a different enzyme or group of enzymes is responsible for the production of each pigment we must suppose that in mice at least three such enzymes or groups of enzymes exist. To determine which of these conditions occurs in mice is not a problem for the biologist, but for the chemist. The biologist must confine his attention to determining the number of distinct agencies at work in pigment formation irrespective of their chemical nature. These agencies, because of their physiological behavior, the biologist chooses to call 'factors,' and attempts to learn what he can about their functions in the evolution of color varieties.
Experimental Studies of the Inheritance of Color in Mice (1913), 17-18.
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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 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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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 we were capable of following the progress of increase of the number of the parts of the most perfect animal, as they first formed in succession, from the very first to its state of full perfection, we should probably be able to compare it with some one of the incomplete animals themselves, of every order of animals in the Creation, being at no stage different from some of the inferior orders; or, in other words, if we were to take a series of animals, from the more imperfect to the perfect, we should probably find an imperfect animal, corresponding with some stage of the most perfect.
R. Owen (ed.), John Hunter's Observations on Animal Development (1841), 14.
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If we wish to give an account of the atomic constitution of the aromatic compounds, we are bound to explain the following facts:
1) All aromatic compounds, even the most simple, are relatively richer in carbon than the corresponding compounds in the class of fatty bodies.
2) Among the aromatic compounds, as well as among the fatty bodies, a large number of homologous substances exist.
3) The most simple aromatic compounds contain at least six atoms of carbon.
4) All the derivatives of aromatic substances exhibit a certain family likeness; they all belong to the group of 'Aromatic compounds'. In cases where more vigorous reactions take place, a portion of the carbon is often eliminated, but the chief product contains at least six atoms of carbon These facts justify the supposition that all aromatic compounds contain a common group, or, we may say, a common nucleus consisting of six atoms of carbon. Within this nucleus a more intimate combination of the carbon atoms takes place; they are more compactly placed together, and this is the cause of the aromatic bodies being relatively rich in carbon. Other carbon atoms can be joined to this nucleus in the same way, and according to the same law, as in the case of the group of fatty bodies, and in this way the existence of homologous compounds is explained.
Bulletin de la Societé Chimique de France (1865), 1, 98. Trans. W. H. Brock.
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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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If you take a number and double it and double it again and then double it a few more times, the number gets bigger and bigger and goes higher and higher and only arithmetic can tell you what the number is when you decide to quit doubling.
From 'Arithmetic', Harvest Poems, 1910-1960 (1960), 115-116.
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If you walk along the street you will encounter a number of scientific problems. Of these, about 80 per cent are insoluble, while 19½ per cent are trivial. There is then perhaps half a per cent where skill, persistence, courage, creativity and originality can make a difference. It is always the task of the academic to swim in that half a per cent, asking the questions through which some progress can be made.
'The Making of a Scientist', Journal of the Royal Society of Arts, June 1983, 406.
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If “Number rules the universe” as Pythagoras asserted, Number is merely our delegate to the throne, for we rule Number.
In Men of Mathematics (1937), 16.
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Imaginary numbers are a fine and wonderful refuge of the divine spirit almost an amphibian between being and non-being. (1702)
[Alternate translation:] The Divine Spirit found a sublime outlet in that wonder of analysis, that portent of the ideal world, that amphibian between being and not-being, which we call the imaginary root of negative unity.
Quoted in Félix Klein, Elementary Mathematics From an Advanced Standpoint: Arithmetic, Algebra, Analysis (1924), 56. Alternate translation as quoted in Tobias Dantzig, Number, the Language of Science: a Critical Survey Written for the Cultured Non-Mathematician (1930), 204
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In 1963, when I assigned the name “quark” to the fundamental constituents of the nucleon, I had the sound first, without the spelling, which could have been “kwork.” Then, in one of my occasional perusals of Finnegans Wake, by James Joyce, I came across the word “quark” in the phrase “Three quarks for Muster Mark.” Since “quark” (meaning, for one thing, the cry of a gull) was clearly intended to rhyme with “Mark,” as well as “bark” and other such words, I had to find an excuse to pronounce it as “kwork.” But the book represents the dreams of a publican named Humphrey Chimpden Earwicker. Words in the text are typically drawn from several sources at once, like the “portmanteau words” in Through the Looking Glass. From time to time, phrases occur in the book that are partially determined by calls for drinks at the bar. I argued, therefore, that perhaps one of the multiple sources of the cry “Three quarks for Muster Mark” might be pronunciation for “Three quarts for Mister Mark,” in which case the pronunciation “kwork” would not be totally unjustified. In any case, the number three fitted perfectly the way quarks occur in nature.
The Quark and the Jaguar (1994), 180.
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In a class I was taking there was one boy who was much older than the rest. He clearly had no motive to work. I told him that, if he could produce for me, accurately to scale, drawings of the pieces of wood required to make a desk like the one he was sitting at, I would try to persuade the Headmaster to let him do woodwork during the mathematics hours—in the course of which, no doubt, he would learn something about measurement and numbers. Next day, he turned up with this task completed to perfection. This I have often found with pupils; it is not so much that they cannot do the work, as that they see no purpose in it.
In Mathematician's Delight (1943), 52.
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In a dispassionate comparison of the relative values of human and robotic spaceflight, the only surviving motivation for continuing human spaceflight is the ideology of adventure. But only a tiny number of Earth’s six billion inhabitants are direct participants. For the rest of us, the adventure is vicarious and akin to that of watching a science fiction movie.
In 'Is Human Spaceflight Obsolete?', Issues in Science and Technology (Summer 2004). [Note: published one year after the loss of seven lives in the Space Shuttle Columbia disaster. —Webmaster]
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In a great number of programmes I’m not a scientist—I’m simply a commentator. So I should claim no virtue for the fact that [people] seem to trust me, if that is indeed the case. It’s simply that I very seldom talk about something they can’t see. If I say a lion is attacking a wildebeest, they can see it is; if I were to say something about a proton, it might be different.
As quoted in Bill Parry, 'Sir David Attenborough in Conversation', The Biologist (Jun 2010), 57, No. 2, 93.
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In a great number of the cosmogonic myths the world is said to have developed from a great water, which was the prime matter. In many cases, as for instance in an Indian myth, this prime matter is indicated as a solution, out of which the solid earth crystallized out.
In Theories of Solutions (1912), 1.
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In a randomly infinite Universe, any event occurring here and now with finite probability must be occurring simultaneously at an infinite number of other sites in the Universe. It is hard to evaluate this idea any further, but one thing is certain: if it is true then it is certainly not original!
With co-author Frank Tipler, The Anthropic Cosmological Principle (1986).
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