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Who said: “The path towards sustainable energy sources will be long and sometimes difficult. But America cannot resist this transition, we must lead it... That is how we will preserve our planet, commanded to our care by God. That�s what will lend meaning to the creed our fathers once declared.”
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Home > Category Index for Science Quotations > Category Index P > Category: Pure

Pure Quotes (299 quotes)

... we ought to have saints' days to commemorate the great discoveries which have been made for all mankind, and perhaps for all time—or for whatever time may be left to us. Nature ... is a prodigal of pain. I should like to find a day when we can take a holiday, a day of jubilation when we can fête good Saint Anaesthesia and chaste and pure Saint Antiseptic. ... I should be bound to celebrate, among others, Saint Penicillin...
Speech at Guildhall, London (10 Sep 1947). Collected in Winston Churchill and Randolph Spencer Churchill (ed.), Europe Unite: Speeches, 1947 and 1948 (1950), 138.
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…nature seems very conversant with the rules of pure mathematics, as our own mathematicians have formulated them in their studies, out of their own inner consciousness and without drawing to any appreciable extent on their experience of the outer world.
In The Mysterious Universe (1930), 113.
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‘I was reading an article about “Mathematics”. Perfectly pure mathematics. My own knowledge of mathematics stops at “twelve times twelve,” but I enjoyed that article immensely. I didn’t understand a word of it; but facts, or what a man believes to be facts, are always delightful. That mathematical fellow believed in his facts. So do I. Get your facts first, and’—the voice dies away to an almost inaudible drone—’then you can distort ‘em as much as you please.’
In 'An Interview with Mark Twain', in Rudyard Kipling, From Sea to Sea (1899), Vol. 2, 180.
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“But in the binary system,” Dale points out, handing back the squeezable glass, “the alternative to one isn’t minus one, it’s zero. That’s the beauty of it, mechanically.” “O.K. Gotcha. You’re asking me, What’s this minus one? I’ll tell you. It’s a plus one moving backward in time. This is all in the space-time foam, inside the Planck duration, don’t forget. The dust of points gives birth to time, and time gives birth to the dust of points. Elegant, huh? It has to be. It’s blind chance, plus pure math. They’re proving it, every day. Astronomy, particle physics, it’s all coming together. Relax into it, young fella. It feels great. Space-time foam.”
In Roger's Version: A Novel (1986), 304.
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[A plant] does not change itself gradually, but remains unaffected during all succeeding generations. It only throws off new forms, which are sharply contrasted with the parent, and which are from the very beginning as perfect and as constant, as narrowly defined, and as pure of type as might be expected of any species.
In Species and Varieties: Their Origin and Mutation (1905), 28-9.
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[Charles Kettering] is unique in that he combines in one individual the interest in pure science with the practical ability to apply knowledge in useful devices.
As quoted in book review, T.A. Boyd, 'Charles F. Kettering: Prophet of Progress', Science (30 Jan 1959), 256.
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[Experimental Physicist] Phys. I know that it is often a help to represent pressure and volume as height and width on paper; and so geometry may have applications to the theory of gases. But is it not going rather far to say that geometry can deal directly with these things and is not necessarily concerned with lengths in space?
[Mathematician] Math. No. Geometry is nowadays largely analytical, so that in form as well as in effect, it deals with variables of an unknown nature. …It is literally true that I do not want to know the significance of the variables x, y, z, t that I am discussing. …
Phys. Yours is a strange subject. You told us at the beginning that you are not concerned as to whether your propositions are true, and now you tell us you do not even care to know what you are talking about.
Math. That is an excellent description of Pure Mathematics, which has already been given by an eminent mathematician [Bertrand Russell].
In Space, Time and Gravitation: An Outline of the General Relativity Theory (1920, 1921), 14.
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[In mathematics] we behold the conscious logical activity of the human mind in its purest and most perfect form. Here we learn to realize the laborious nature of the process, the great care with which it must proceed, the accuracy which is necessary to determine the exact extent of the general propositions arrived at, the difficulty of forming and comprehending abstract concepts; but here we learn also to place confidence in the certainty, scope and fruitfulness of such intellectual activity.
In Ueber das Verhältnis der Naturwissenschaften zur Gesammtheit der Wissenschaft, Vorträge und Reden (1896), Bd. 1, 176. Also seen translated as “In mathematics we see the conscious logical activity of our mind in its purest and most perfect form; here is made manifest to us all the labor and the great care with which it progresses, the precision which is necessary to determine exactly the source of the established general theorems, and the difficulty with which we form and comprehend abstract conceptions; but we also learn here to have confidence in the certainty, breadth, and fruitfulness of such intellectual labor”, in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 20. From the original German, “Hier sehen wir die bewusste logische Thätigkeit unseres Geistes in ihrer reinsten und vollendetsten Form; wir können hier die ganze Mühe derselben kennen lernen, die grosse Vorsicht, mit der sie vorschreiten muss, die Genauigkeit, welche nöthig ist, um den Umfang der gewonnenen allgemeinen Sätze genau zu bestimmen, die Schwierigkeit, abstracte Begriffe zu bilden und zu verstehen; aber ebenso auch Vertrauen fassen lernen in die Sicherheit, Tragweite und Fruchtbarkeit solcher Gedankenarbeit.”
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[Isaac Newton] regarded the Universe as a cryptogram set by the Almighty—just as he himself wrapt the discovery of the calculus in a cryptogram when he communicated with Leibniz. By pure thought, by concentration of mind, the riddle, he believed, would be revealed to the initiate.
In 'Newton, the Man' (1946). In Geoffrey Keynes (ed.), Essays in Biography, 2nd edition (1951), 314.
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[King Hiero II] requested Archimedes to consider [whether a crown was pure gold or alloyed with silver]. The latter, while the case was still on his mind, happened to go to the bath, and on getting into a tub observed that the more his body sank into it the more water ran out over the tub. As this pointed out the way to explain the case in question, without a moment’s delay, and transported with joy, he jumped out of the tub and rushed home naked, crying with a loud voice that he had found what he was seeking; for as he ran he shouted repeatedly in Greek, “Eὕρηκα, εὕρηκα.”
Vitruvius
This famous anecdote, being written about two centuries after Archimedes, is of questionable authenticity, but Vitruvius provided the origin of the story as we know it. In De Architectura, Book 9, Introduction, Sec. 10. As translated in Morris Hicky Morgan (trans.), Vitruvius: The Ten Books on Architecture (1914), 254. Also seen translated as “While Archimedes was turning the problem over, he chanced to come to the place of bathing, and there, as he was sitting down in the tub, he noticed that the amount of water which flowed over the tub was equal to the amount by which his body was immersed. This showed him a means of solving the problem. … In his joy, he leapt out of the tub and, rushing naked towards his home, he cried out with a loud voice that he had found what he sought.” In Ivor Bulmer-Thomas, Selections Illustrating the History of Greek Mathematics (1939), 37.
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[Mathematics] is an independent world
Created out of pure intelligence.
In The Prelude, Book 6, lines 186-187. [Preceding lines refer to “laws of Nature” and “a treatise of geometry.” Wordsworth did not use the word “mathematics”, which is added parenthetically to give context to the quote.]
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[P]ure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.
In A Mathematician’s Apology (1940, reprint with Foreward by C.P. Snow 1992), 134.
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[Pure mathematics is] good to give chills in the spine to a certain number of people, me included. I don’t know what else it is good for, and I don’t care. But … like von Neumann said, one never knows whether someone is going to find another use for it.
In The Beauty of Doing Mathematics: Three Public Dialogues (1985), 49.
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[Pure research] is worth every penny it costs.
Quoted in 'Moon-Struck Scientist,' New York Times (27 Apr 1961), 42.
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[Relativist] Rel. There is a well-known proposition of Euclid which states that “Any two sides of a triangle are together greater than the third side.” Can either of you tell me whether nowadays there is good reason to believe that this proposition is true?
[Pure Mathematician] Math. For my part, I am quite unable to say whether the proposition is true or not. I can deduce it by trustworthy reasoning from certain other propositions or axioms, which are supposed to be still more elementary. If these axioms are true, the proposition is true; if the axioms are not true, the proposition is not true universally. Whether the axioms are true or not I cannot say, and it is outside my province to consider.
In Space, Time and Gravitation: An Outline of the General Relativity Theory (1920, 1921), 1.
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[The famous attack of Sir William Hamilton on the tendency of mathematical studies] affords the most express evidence of those fatal lacunae in the circle of his knowledge, which unfitted him for taking a comprehensive or even an accurate view of the processes of the human mind in the establishment of truth. If there is any pre-requisite which all must see to be indispensable in one who attempts to give laws to the human intellect, it is a thorough acquaintance with the modes by which human intellect has proceeded, in the case where, by universal acknowledgment, grounded on subsequent direct verification, it has succeeded in ascertaining the greatest number of important and recondite truths. This requisite Sir W. Hamilton had not, in any tolerable degree, fulfilled. Even of pure mathematics he apparently knew little but the rudiments. Of mathematics as applied to investigating the laws of physical nature; of the mode in which the properties of number, extension, and figure, are made instrumental to the ascertainment of truths other than arithmetical or geometrical—it is too much to say that he had even a superficial knowledge: there is not a line in his works which shows him to have had any knowledge at all.
In Examination of Sir William Hamilton's Philosophy (1878), 607.
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[Tom Bombadil is] an exemplar, a particular embodying of pure (real) natural science: the spirit that desires knowledge of other things, their history and nature, because they are ‘other’ and wholly independent of the enquiring mind, a spirit coeval with the rational mind, and entirely unconcerned with ‘doing’ anything with the knowledge: Zoology and Botany not Cattle-breeding or Agriculture. Even the Elves hardly show this: they are primarily artists.
From Letter draft to Peter Hastings (manager of a Catholic bookshop in Oxford, who wrote about his enthusiasm for Lord of the Rings) (Sep 1954). In Humphrey Carpenter (ed.) assisted by Christopher Tolkien, The Letters of J.R.R. Tolkien (1995, 2014), 192, Letter No. 153.
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[Walter] Baade, like all scientists of substance, had a set view of how things were put together, to be sure a view to be always challenged by the scientist himself, but defended as well against all less informed mortals who objected without simon-pure reasons.
From Owen Gingerich in Physics Today (1994), 47, No.12, 34-40. As quoted and cited in Donald Lynden-Bell and François Schweizer, 'Allan Rex Sandage: 18 June 1926—13 November 2010', Biographical Memoirs of Fellows of the Royal Society (2012), Vol. 58, 251.
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Strictly Germ-proof

The Antiseptic Baby and the Prophylactic Pup
Were playing in the garden when the Bunny gamboled up;
They looked upon the Creature with a loathing undisguised;—
It wasn't Disinfected and it wasn't Sterilized.

They said it was a Microbe and a Hotbed of Disease;
They steamed it in a vapor of a thousand-odd degrees;
They froze it in a freezer that was cold as Banished Hope
And washed it in permanganate with carbolated soap.

In sulphurated hydrogen they steeped its wiggly ears;
They trimmed its frisky whiskers with a pair of hard-boiled shears;
They donned their rubber mittens and they took it by the hand
And elected it a member of the Fumigated Band.

There's not a Micrococcus in the garden where they play;
They bathe in pure iodoform a dozen times a day;
And each imbibes his rations from a Hygienic Cup—
The Bunny and the Baby and the Prophylactic Pup.
Printed in various magazines and medical journals, for example, The Christian Register (11 Oct 1906), 1148, citing Women's Home Companion. (Making fun of the contemporary national passion for sanitation.)
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[At a musical concert:]
…the music’s pure algebra of enchantment.
In Louis Untermeyer, Modern American Poetry (1962), 430.
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But how shall we this union well expresse?
Naught tyes the soule: her subtiltie is such
She moves the bodie, which she doth possesse.
Yet no part toucheth, but by Vertue's touch.
Then dwels she not therein as in a tent;
Nor as a pilot in his Ship doth sit;
Nor as the spider in his web is pent;
Nor as the Waxe retaines the print in it;
Nor as a Vessell water doth containe;
Nor as one Liquor in another shed;
Nor as the heate dath in the fire remaine;
Nor as a voice throughout the ayre is spred;
But as the faire and cheerfull morning light,
Doth here, and there, her silver beames impart,
And in an instant doth her selfe unite
To the transparent Aire, in all, and part:
Still resting whole, when blowes the Aire devide;
Abiding pure, when th' Aire is most corrupted;
Throughout the Aire her beames dispersing wide,
And when the Aire is tost, not interrupted:
So doth the piercing Soule the body fill;
Being all in all, and all in part diffus'd;
Indivisible, incorruptible still,
Not forc't, encountred, troubled or confus'd.
And as the Sunne above the light doth bring,
Tough we behold it in the Aire below;
So from th'eternall light the Soule doth spring,
Though in the Bodie she her powers do show.
From 'Nosce Teipsum' (1599), in Claire Howard (ed.), The Poems of Sir John Davies (1941), 151-2.
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Engineering, too, owes its most useful materials to the achievements of chemists in identifying, separating, and transforming materials: structural steel for the framework of bridges and buildings, portland cement for roadways and aqueducts, pure copper for the electrical industries, aluminum alloys for automobiles and airplanes, porcelain for spark plugs and electrical insulators. The triumphs of engineering skill rest on a chemical foundation.
In Fundamental Chemistry, and Elementary Textbook for College Classes (1936), 8.
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Error has made man so deep, sensitive, and inventive that he has put forth such blossoms as religions and arts. Pure knowledge could not have been capable of it.
Human, All-To-Human, Vol. 1, 44-45. (1878), 140. In Willard Huntington Wright, What Nietzsche Taught? (1917), 78.
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Le savant n’étudie pas la nature parce que cela est utile; il l’étudie parce qu’il y prend plaisir et il y prend plaisir parce qu’elle est belle. Si la nature n’était pas belle, elle ne vaudrait pas la peine d’être connue, la vie ne vaudrait pas la peine d’être vécue.
The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and life would not be worth living. I am not speaking, of course, of the beauty which strikes the senses, of the beauty of qualities and appearances. I am far from despising this, but it has nothing to do with science. What I mean is that more intimate beauty which comes from the harmonious order of its parts, and which a pure intelligence can grasp.
In Science et Méthode (1920), 48, as translated by Francis Maitland, in Science and Method (1908, 1952), 15.
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Arthur Stanley Eddington quote: Proof is an idol before whom the pure mathematician tortures himself. In physics we are generall
Proof is an idol before whom the pure mathematician tortures himself. In physics we are generally content to sacrifice before the lesser shrine of Plausibility.
In Gifford Lecture (1927), Edinburgh, as collected in 'Science and Mysticism', The Nature of the Physical World (1928), 337.
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To the Memory of Fourier
Fourier! with solemn and profound delight,
Joy born of awe, but kindling momently
To an intense and thrilling ecstacy,
I gaze upon thy glory and grow bright:
As if irradiate with beholden light;
As if the immortal that remains of thee
Attuned me to thy spirit’s harmony,
Breathing serene resolve and tranquil might.
Revealed appear thy silent thoughts of youth,
As if to consciousness, and all that view
Prophetic, of the heritage of truth
To thy majestic years of manhood due:
Darkness and error fleeing far away,
And the pure mind enthroned in perfect day.
In R. Graves, Life of W. R. Hamilton (1882), Vol. l, 696.
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A casual glance at crystals may lead to the idea that they were pure sports of nature, but this is simply an elegant way of declaring one’s ignorance. With a thoughtful examination of them, we discover laws of arrangement. With the help of these, calculation portrays and links up the observed results. How variable and at the same time how precise and regular are these laws! How simple they are ordinarily, without losing anything of their significance! The theory which has served to develop these laws is based entirely on a fact, whose existence has hitherto been vaguely discerned rather than demonstrated. This fact is that in all minerals which belong to the same species, these little solids, which are the crystal elements and which I call their integrant molecules, have an invariable form, in which the faces lie in the direction of the natural fracture surfaces corresponding to the mechanical division of the crystals. Their angles and dimensions are derived from calculations combined with observation.
Traité de mineralogie … Publié par le conseil des mines (1801), Vol. 1, xiii-iv, trans. Albert V. and Marguerite Carozzi.
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A mind which has once imbibed a taste for scientific enquiry, and has learnt the habit of applying its principles readily to the cases which occur, has within itself an inexhaustable source of pure and exciting contemplations:— One would think that Shakespeare had such a mind in view when he describes a contemplative man as finding
    “Tongues in trees—books in running brooks—
    Sermons in stones—and good in everything.”
Accustomed to trace the operations of general causes and the exemplification of general laws, in circumstances where the uninformed and uninquiring eye, perceives neither novelty nor beauty, he walks in the midst of wonders; every object which falls in his way elucidates some principle, affords some instruction and impresses him with a sense of harmony and order. Nor is it a mere passive pleasure which is thus communicated. A thousand questions are continually arising in his mind, a thousand objects of enquiry presenting themselves, which keep his faculties in constant exercise, and his thoughts perpetually on the wing, so that lassitude is excluded from his life, and that craving after artificial excitement and dissipation of the mind, which leads so many into frivolous, unworthy, and destructive pursuits, is altogether eradicated from his bosom.
In Dionysius Lardner (ed.), Cabinet Cyclopaedia, Vol 1, Preliminary Discourse on the Study of Natural Philosophy (1831), 14-15.
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A mind which has once imbibed a taste for scientific enquiry, and has learnt the habit of applying its principles readily to the cases which occur, has within itself an inexhaustible source of pure and exciting contemplations.
In Dionysius Lardner (ed.), Cabinet Cyclopaedia, Vol 1, Preliminary Discourse on the Study of Natural Philosophy (1831), 14-15.
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A popular cliche in philosophy says that science is pure analysis or reductionism, like taking the rainbow to pieces; and art is pure synthesis, putting the rainbow together. This is not so. All imagination begins by analyzing nature.
In The Ascent of Man (1973).
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A short, broad man of tremendous vitality, the physical type of Hereward, the last of the English, and his brother-in-arms, Winter, Sylvester’s capacious head was ever lost in the highest cloud-lands of pure mathematics. Often in the dead of night he would get his favorite pupil, that he might communicate the very last product of his creative thought. Everything he saw suggested to him something new in the higher algebra. This transmutation of everything into new mathematics was a revelation to those who knew him intimately. They began to do it themselves. His ease and fertility of invention proved a constant encouragement, while his contempt for provincial stupidities, such as the American hieroglyphics for π and e, which have even found their way into Webster’s Dictionary, made each young worker apply to himself the strictest tests.
In Florian Cajori, Teaching and History of Mathematics in the United States (1890), 265.
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Although my Aachen colleagues and students at first regarded the “pure mathematician” with suspicion, I soon had the satisfaction of being accepted a useful member not merely in teaching but also engineering practice; thus I was requested to render expert opinions and to participate in the Ingenieurverein [engineering association].
As quoted in Paul Forman and Armin Hermann, 'Sommerfeld, Arnold (Johannes Wilhelm)', Biography in Dictionary of Scientific Biography (1975), Vol. 12, 527. Cited from 'Autobiographische Skizze', Gesammelte Schriften, Vol 4, 673–682.
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Among all highly civilized peoples the golden age of art has always been closely coincident with the golden age of the pure sciences, particularly with mathematics, the most ancient among them.
This coincidence must not be looked upon as accidental, but as natural, due to an inner necessity. Just as art can thrive only when the artist, relieved of the anxieties of existence, can listen to the inspirations of his spirit and follow in their lead, so mathematics, the most ideal of the sciences, will yield its choicest blossoms only when life’s dismal phantom dissolves and fades away, when the striving after naked truth alone predominates, conditions which prevail only in nations while in the prime of their development.
From Die Entwickelung der Mathematik im Zusammenhange mit der Ausbreitung der Kultur (1893), 4. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 191-192. From the original German, “Bei allen Kulturvölkern ist die Blüthezeit der Kunst auch immer zeitlich eng verbunden mit einer Blüthezeit der reinen Wissenschaften, insbesondere der ältesten unter ihnen, der Mathematik.
Dieses Zusammentreffen dürfte auch nicht ein zufälliges, sondern ein natürliches, ein Ergebniss innerer Notwendigkeit sein. Wie die Kunst nur gedeihen kann, wenn der Künstler, unbekümmert um die Bedrängnisse des Daseins, den Eingebungen seines Geistes lauschen und ihnen folgen kann, so kann die idealste Wissenschaft, die Mathematik, erst dann ihre schönsten Blüthen treiben, wenn des Erdenlebens schweres Traumbild sinkt und sinkt und sinkt, wenn das Streben nach der nackten Wahrheit allein bestimmend ist, was nur bei Nationen in der Vollkraft ihrer Entwickelung vorkommt.”
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An applied mathematician loves the theorem. A pure mathematician loves the proof.
Anonymous
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An evolution is a series of events that in itself as series is purely physical, — a set of necessary occurrences in the world of space and time. An egg develops into a chick; … a planet condenses from the fluid state, and develops the life that for millions of years makes it so wondrous a place. Look upon all these things descriptively, and you shall see nothing but matter moving instant after instant, each instant containing in its full description the necessity of passing over into the next. … But look at the whole appreciatively, historically, synthetically, as a musician listens to a symphony, as a spectator watches a drama. Now you shall seem to have seen, in phenomenal form, a story.
In The Spirit of Modern Philosophy: An Essay in the Form of Lectures (1892), 425.
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An eye critically nice will discern in every colour a tendency to some other colour, according as it is influenced by light, shade, depth or diluteness; nor is this the case only in the inherent colours of pigments, &c. but it is so also in the transient colours of the prism, &c. Hence blue in its depth inclines to purple; deep-yellow to orange, &c.; nor is it practicable to realize these colours to the satisfaction of the critical eye,-since perfect colours, like perfect geometrical figures, are pure ideals. My examples of colours are therefore quite as adequate to their office of illustrating and distinguishing, as the figure of an angle inclining to the acute or obtuse, instead of a perfect right angle, or middle form, would be in illustrating the conception of an angle in general.
In 'On Colors—In Answer to Mr. T. Hargreaves’s Strictures on the Work Entitled “Chromatics; or, An Essay on the Analogy and Harmony of Colours”', The Philosophical Magazine and Journal: Comprehending The Various Branches of Science, The Liberal and Fine Arts, Geology, Agriculture, Manufactures and Commerce (Oct 1817), 50, No. 234, 241.
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And as for other men, who worked in tank-rooms full of steam, and in some of which there were open vats near the level of the floor, their peculiar trouble was that they fell into the vats; and when they were fished out, there was never enough of them left to be worth exhibiting,—sometimes they would be overlooked for days, till all but the bones of them had gone out into the world as Durham's Pure Leaf Lard! This contributed to the passing of the Pure Food Act of 1906.
The Jungle (1906), 117.
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And indeed I am not humming,
Thus to sing of Cl-ke and C-ming,
Who all the universe surpasses
in cutting up and making gases;
With anatomy and chemics,
Metaphysics and polemics,
Analyzing and chirugery,
And scientific surgery …
H-slow's lectures on the cabbage
Useful are as roots of Babbage;
Fluxions and beet-root botany,
Some would call pure monotony.
Punch in Cambridge (28 Jan 1834). In Mark Weatherall, Gentlemen, Scientists, and Medicine at Cambridge 1800-1940 (2000), Vol. 3,77. The professors named were William Clark (anatomy), James Cumming (chemistry) and Johns Stephens Henslow (botany).
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And so the great truth, now a paradox, may become a commonplace, that man is greater than his surroundings, and that the production of a breed of men and women, even in our great cities, less prone to disease, and pain, more noble in aspect, more rational in habits, more exultant in the pure joy of living, is not only scientifically possible, but that even the partial fulfillment of this dream, if dream it be, is the most worthy object towards which the lover of his kind can devote the best energies of his life.
In 'The Breed of Man', The Nineteenth Century, (Oct 1900), 669, as collected in Martin Polley (ed.), The History of Sport in Britain, 1880-1914: Sport, Education, and Improvement (2004), Vol. 2, 181.
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Applied research generates improvements, not breakthroughs. Great scientific advances spring from pure research. Even scientists renowned for their “useful” applied discoveries often achieved success only when they abandoned their ostensible applied-science goal and allowed their minds to soar—as when Alexander Fleming, “just playing about,” refrained from throwing away green molds that had ruined his experiment, studied them, and discovered penicillin. Or when C. A. Clarke, a physician affiliated with the University of Liverpool, became intrigued in the 1950s by genetically created color patterns that emerged when he cross-bred butterflies as a hobby. His fascination led him—“by the pleasant route of pursuing idle curiosity”—to the successful idea for preventing the sometimes fatal anemia that threatened babies born of a positive-Rhesus-factor father and a negative-Rhesus-factor mother.
In Jacques Cousteau and Susan Schiefelbein, The Human, the Orchid, and the Octopus: Exploring and Conserving Our Natural World (2007), 214-215.
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Applied science, purposeful and determined, and pure science, playful and freely curious, continuously support and stimulate each other. The great nation of the future will be the one which protects the freedom of pure science as much as it encourages applied science.
From a radio talk, collected in Warren Weaver (ed.), The Scientists Speak (1946)
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Archimedes possessed so high a spirit, so profound a soul, and such treasures of highly scientific knowledge, that though these inventions [used to defend Syracuse against the Romans] had now obtained him the renown of more than human sagacity, he yet would not deign to leave behind him any commentary or writing on such subjects; but, repudiating as sordid and ignoble the whole trade of engineering, and every sort of art that lends itself to mere use and profit, he placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life; studies, the superiority of which to all others is unquestioned, and in which the only doubt can be whether the beauty and grandeur of the subjects examined, or the precision and cogency of the methods and means of proof, most deserve our admiration.
Plutarch
In John Dryden (trans.), Life of Marcellus.
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Aristotle ... imputed this symphony of the heavens ... this music of the spheres to Pythagorus. ... But Pythagoras alone of mortals is said to have heard this harmony ... If our hearts were as pure, as chaste, as snowy as Pythagoras' was, our ears would resound and be filled with that supremely lovely music of the wheeling stars.
'On the Music of the Spheres'. Second Prolusion. In John Milton and Merritt Yerkes Hughes (ed.), Complete Poems and Major Prose (1957, 2003), 603-604.
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As an exercise of the reasoning faculty, pure mathematics is an admirable exercise, because it consists of reasoning alone, and does not encumber the student with an exercise of judgment: and it is well to begin with learning one thing at a time, and to defer a combination of mental exercises to a later period.
In Annotations to Bacon’s Essays (1873), Essay 1, 493.
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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 pure truth is the polar star of our science [mathematics], so it is the great advantage of our science over others that it awakens more easily the love of truth in our pupils. … If Hegel justly said, “Whoever does not know the works of the ancients, has lived without knowing beauty,” Schellbach responds with equal right, “Who does not know mathematics, and the results of recent scientific investigation, dies without knowing truth.”
Max Simon
From Didaktik und Methodik des Rechnens und der Mathematik (1908), 37. As quoted and translated in J.W.A. Young, Teaching of Mathematics in the Elementary and the Secondary School (1907), 44. From the original German, “Wenn Hegel mit Recht sagt: ‘Wer die Werke der Alten nicht kennt, der hat gelebt, ohne die Schönheit gekannt zu haben’, so erwidert Schellbach mit nicht minderem Recht: ‘Wer die Math. und die Resultate der neueren Naturforschung nicht gekannt hat, der stirbt, ohne die Wahrheit zu kennen.’”
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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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Astronomy and Pure Mathematics are the magnetic poles toward which the compass of my mind ever turns.
In Letter to Bolyai (30 Jun 1803), in Franz Schmidt and Paul Stäckel, Briefwechsel zwischen Carl Friedrich Gauss und Wolfgang Bolyai, (1899), Letter XXIII , 55.
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At the origin, the [space travel] pioneers of the greatest adventure of all times were motivated by the drive to explore, by the pure spirit of conquest, by the lofty desire to open up new fields to human genius. … From their exceptional journeys, they all came back with the revelation of beauty. Beauty of the black sky, beauty and variety of our planet, beauty of the Earth seen from the Moon, girdled by a scintillating belt of equatorial thunderstorms. They all emphasize that our planet is one, that borderlines are artificial, that humankind is one single community on board spaceship Earth. They all insist that this fragile gem is at our mercy and that we must all endeavor to protect it.
Written for 'Foreword' to Kevin W. Kelley (ed.), The Home Planet (1988), paragraphs 6-7 (unpaginated).
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Babbage … gave the name to the [Cambridge] Analytical Society, which he stated was formed to advocate “the principles of pure d-ism as opposed to the dot-age of the university.”
In History of Mathematics (3rd Ed., 1901), 451.
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Bertrand, Darboux, and Glaisher have compared Cayley to Euler, alike for his range, his analytical power, and, not least, for his prolific production of new views and fertile theories. There is hardly a subject in the whole of pure mathematics at which he has not worked.
In Proceedings of London Royal Society (1895), 58, 21.
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Blood mixture and the result drop in the racial level is the sole cause of the dying out of old cultures; for men do not perish as a result of lost wars, but by the loss of that force of resistance which is continued only in pure blood. All who are not of good race in this world are chaff.
Mein Kampf (1925-26), American Edition (1943), 296. In William Lawrence Shirer, The Rise and Fall of the Third Reich (1990), 88.
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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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But it is precisely mathematics, and the pure science generally, from which the general educated public and independent students have been debarred, and into which they have only rarely attained more than a very meagre insight. The reason of this is twofold. In the first place, the ascendant and consecutive character of mathematical knowledge renders its results absolutely insusceptible of presentation to persons who are unacquainted with what has gone before, and so necessitates on the part of its devotees a thorough and patient exploration of the field from the very beginning, as distinguished from those sciences which may, so to speak, be begun at the end, and which are consequently cultivated with the greatest zeal. The second reason is that, partly through the exigencies of academic instruction, but mainly through the martinet traditions of antiquity and the influence of mediaeval logic-mongers, the great bulk of the elementary text-books of mathematics have unconsciously assumed a very repellant form,—something similar to what is termed in the theory of protective mimicry in biology “the terrifying form.” And it is mainly to this formidableness and touch-me-not character of exterior, concealing withal a harmless body, that the undue neglect of typical mathematical studies is to be attributed.
In Editor’s Preface to Augustus De Morgan and Thomas J. McCormack (ed.), Elementary Illustrations of the Differential and Integral Calculus (1899), v.
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But the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed.
From Herbert Spencer Lecture, at University of Oxford (10 Jun 1933), 'On the Methods of Theoretical Physics'. Printed in Philosophy of Science, (Apr 1934), 1, No. 2. Quoted and cited in epigraph, A. H. Louie, More Than Life Itself: A Synthetic Continuation in Relational Biology (2013), 81.
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But, contrary to the lady’s prejudices about the engineering profession, the fact is that quite some time ago the tables were turned between theory and applications in the physical sciences. Since World War II the discoveries that have changed the world are not made so much in lofty halls of theoretical physics as in the less-noticed labs of engineering and experimental physics. The roles of pure and applied science have been reversed; they are no longer what they were in the golden age of physics, in the age of Einstein, Schrödinger, Fermi and Dirac.
'The Age of Computing: a Personal Memoir', Daedalus (1992), 121, 120.
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But, further, no animal can live upon a mixture of pure protein, fat and carbohydrate, and even when the necessary inorganic material is carefully supplied, the animal still cannot flourish. The animal body is adjusted to live either upon plant tissues or the tissues of other animals, and these contain countless substances other than the proteins, carbohydrates and fats... In diseases such as rickets, and particularly in scurvy, we have had for long years knowledge of a dietetic factor; but though we know how to benefit these conditions empirically, the real errors in the diet are to this day quite obscure. They are, however, certainly of the kind which comprises these minimal qualitative factors that I am considering.
'The Analyst and the Medical Man', The Analyst (1906), 31, 395-6.
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By no amount of reasoning can we altogether eliminate all contingency from our world. Moreover, pure speculation alone will not enable us to get a determinate picture of the existing world. We must eliminate some of the conflicting possibilities, and this can be brought about only by experiment and observation.
Reason and Nature: an Essay on the Meaning of Scientific Method? (2nd Ed., 1964), 82.
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By research in pure science I mean research made without any idea of application to industrial matters but solely with the view of extending our knowledge of the Laws of Nature. I will give just one example of the ‘utility’ of this kind of research, one that has been brought into great prominence by the War—I mean the use of X-rays in surgery. Now, not to speak of what is beyond money value, the saving of pain, or, it may be, the life of the wounded, and of bitter grief to those who loved them, the benefit which the state has derived from the restoration of so many to life and limb, able to render services which would otherwise have been lost, is almost incalculable. Now, how was this method discovered? It was not the result of a research in applied science starting to find an improved method of locating bullet wounds. This might have led to improved probes, but we cannot imagine it leading to the discovery of X-rays. No, this method is due to an investigation in pure science, made with the object of discovering what is the nature of Electricity. The experiments which led to this discovery seemed to be as remote from ‘humanistic interest’ —to use a much misappropriated word—as anything that could well be imagined. The apparatus consisted of glass vessels from which the last drops of air had been sucked, and which emitted a weird greenish light when stimulated by formidable looking instruments called induction coils. Near by, perhaps, were great coils of wire and iron built up into electro-magnets. I know well the impression it made on the average spectator, for I have been occupied in experiments of this kind nearly all my life, notwithstanding the advice, given in perfect good faith, by non-scientific visitors to the laboratory, to put that aside and spend my time on something useful.
In Speech made on behalf of a delegation from the Conjoint Board of Scientific Studies in 1916 to Lord Crewe, then Lord President of the Council. In George Paget Thomson, J. J. Thomson and the Cavendish Laboratory in His Day (1965), 167-8.
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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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Chance alone is at the source of every innovaton, of all creation in the biosphere. Pure chance, only chance, absolute but blind liberty is at the root of the prodigious edifice that is evolution... It today is the sole conceivable hypothesis, the only one that squares with observed and tested fact.
Stating life began by the chance collision of particles of nucleic acid in the “prebiotic soup.”
In Jacques Monod and Austryn Wainhouse (trans.), Chance and Necessity: An Essay on the Natural Philosophy of Modern Biology (1971), 112-113.
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Chemistry has the same quickening and suggestive influence upon the algebraist as a visit to the Royal Academy, or the old masters may be supposed to have on a Browning or a Tennyson. Indeed it seems to me that an exact homology exists between painting and poetry on the one hand and modern chemistry and modern algebra on the other. In poetry and algebra we have the pure idea elaborated and expressed through the vehicle of language, in painting and chemistry the idea enveloped in matter, depending in part on manual processes and the resources of art for its due manifestation.
Attributed.
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Children are told that an apple fell on Isaac Newton’s head and he was led to state the law of gravity. This, of course, is pure foolishness. What Newton discovered was that any two particles in the universe attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. This is not learned from a falling apple, but by observing quantities of data and developing a mathematical theory that can be verified by additional data. Data gathered by Galileo on falling bodies and by Johannes Kepler on motions of the planets were invaluable aids to Newton. Unfortunately, such false impressions about science are not universally outgrown like the Santa Claus myth, and some people who don’t study much science go to their graves thinking that the human race took until the mid-seventeenth century to notice that objects fall.
In How to Tell the Liars from the Statisticians (1983), 127.
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Common to all these types is the anthropomorphic character of their conception of God. In general, only individuals of exceptional endowments, and exceptionally high-minded communities, rise to any considerable extent above this level. But there is a third stage of religious experience which belongs to all of them, even though it is rarely found in a pure form: I shall call it cosmic religious feeling. It is very difficult to elucidate this feeling to anyone who is entirely without it, especially as there is no anthropomorphic conception of God corresponding to it.
…...
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Doubtless the reasoning faculty, the mind, is the leading and characteristic attribute of the human race. By the exercise of this, man arrives at the properties of the natural bodies. This is science, properly and emphatically so called. It is the science of pure mathematics; and in the high branches of this science lies the truly sublime of human acquisition. If any attainment deserves that epithet, it is the knowledge, which, from the mensuration of the minutest dust of the balance, proceeds on the rising scale of material bodies, everywhere weighing, everywhere measuring, everywhere detecting and explaining the laws of force and motion, penetrating into the secret principles which hold the universe of God together, and balancing worlds against worlds, and system against system. When we seek to accompany those who pursue studies at once so high, so vast, and so exact; when we arrive at the discoveries of Newton, which pour in day on the works of God, as if a second fiat had gone forth from his own mouth; when, further, we attempt to follow those who set out where Newton paused, making his goal their starting-place, and, proceeding with demonstration upon demonstration, and discovery upon discovery, bring new worlds and new systems of worlds within the limits of the known universe, failing to learn all only because all is infinite; however we may say of man, in admiration of his physical structure, that “in form and moving he is express and admirable,” it is here, and here without irreverence, we may exclaim, “In apprehension how like a god!” The study of the pure mathematics will of course not be extensively pursued in an institution, which, like this [Boston Mechanics’ Institute], has a direct practical tendency and aim. But it is still to be remembered, that pure mathematics lie at the foundation of mechanical philosophy, and that it is ignorance only which can speak or think of that sublime science as useless research or barren speculation.
In Works (1872), Vol. 1, 180.
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Dr. Wallace, in his Darwinism, declares that he can find no ground for the existence of pure scientists, especially mathematicians, on the hypothesis of natural selection. If we put aside the fact that great power in theoretical science is correlated with other developments of increasing brain-activity, we may, I think, still account for the existence of pure scientists as Dr. Wallace would himself account for that of worker-bees. Their function may not fit them individually to survive in the struggle for existence, but they are a source of strength and efficiency to the society which produces them.
In Grammar of Science (1911), Part, 1, 221.
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During the last two centuries and a half, physical knowledge has been gradually made to rest upon a basis which it had not before. It has become mathematical. The question now is, not whether this or that hypothesis is better or worse to the pure thought, but whether it accords with observed phenomena in those consequences which can be shown necessarily to follow from it, if it be true
In Augustus De Morgan and Sophia Elizabeth De Morgan (ed.), A Budget of Paradoxes (1872), 2.
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Einstein, twenty-six years old, only three years away from crude privation, still a patent examiner, published in the Annalen der Physik in 1905 five papers on entirely different subjects. Three of them were among the greatest in the history of physics. One, very simple, gave the quantum explanation of the photoelectric effect—it was this work for which, sixteen years later, he was awarded the Nobel prize. Another dealt with the phenomenon of Brownian motion, the apparently erratic movement of tiny particles suspended in a liquid: Einstein showed that these movements satisfied a clear statistical law. This was like a conjuring trick, easy when explained: before it, decent scientists could still doubt the concrete existence of atoms and molecules: this paper was as near to a direct proof of their concreteness as a theoretician could give. The third paper was the special theory of relativity, which quietly amalgamated space, time, and matter into one fundamental unity.
This last paper contains no references and quotes no authority. All of them are written in a style unlike any other theoretical physicist’s. They contain very little mathematics. There is a good deal of verbal commentary. The conclusions, the bizarre conclusions, emerge as though with the greatest of ease: the reasoning is unbreakable. It looks as though he had reached the conclusions by pure thought, unaided, without listening to the opinions of others. To a surprisingly large extent, that is precisely what he had done.
In Variety of Men (1966), 100-101. First published in Commentary magazine.
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Every natural scientist who thinks with any degree of consistency at all will, I think, come to the view that all those capacities that we understand by the phrase psychic activities (Seelenthiitigkeiten) are but functions of the brain substance; or, to express myself a bit crudely here, that thoughts stand in the same relation to the brain as gall does to the liver or urine to the kidneys. To assume a soul that makes use of the brain as an instrument with which it can work as it pleases is pure nonsense; we would then be forced to assume a special soul for every function of the body as well.
Carl Vogt
In Physiologische Briefe für Gelbildete aIle Stünde (1845-1847), 3 parts, 206. as translated in Frederick Gregory, Scientific Materialism in Nineteenth Century Germany (1977), 64.
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Facts are not pure unsullied bits of information; culture also influences what we see and how we see it. Theories, moreover, are not inexorable inductions from facts. The most creative theories are often imaginative visions imposed upon facts; the source of imagination is also strongly cultural.
In The Mismeasure of Man (1981, 1996), 54.
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First of all, we ought to observe, that mathematical propositions, properly so called, are always judgments a priori, and not empirical, because they carry along with them necessity, which can never be deduced from experience. If people should object to this, I am quite willing to confine my statements to pure mathematics, the very concept of which implies that it does not contain empirical, but only pure knowledge a priori.
In Critique of Pure Reason (1900), 720.
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For me, the study of these laws is inseparable from a love of Nature in all its manifestations. The beauty of the basic laws of natural science, as revealed in the study of particles and of the cosmos, is allied to the litheness of a merganser diving in a pure Swedish lake, or the grace of a dolphin leaving shining trails at night in the Gulf of California.
Nobel Banquet Speech (10 Dec 1969), in Wilhelm Odelberg (ed.),Les Prix Nobel en 1969 (1970).
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For most scientists, I think the justification of their work is to be found in the pure joy of its creativeness; the spirit which moves them is closely akin to the imaginative vision which inspires an artist.
In Modern Science and Modern Man (1951), 58.
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Fractal is a word invented by Mandelbrot to bring together under one heading a large class of objects that have [played] … an historical role … in the development of pure mathematics. A great revolution of ideas separates the classical mathematics of the 19th century from the modern mathematics of the 20th. Classical mathematics had its roots in the regular geometric structures of Euclid and the continuously evolving dynamics of Newton. Modern mathematics began with Cantor’s set theory and Peano’s space-filling curve. Historically, the revolution was forced by the discovery of mathematical structures that did not fit the patterns of Euclid and Newton. These new structures were regarded … as “pathological,” .… as a “gallery of monsters,” akin to the cubist paintings and atonal music that were upsetting established standards of taste in the arts at about the same time. The mathematicians who created the monsters regarded them as important in showing that the world of pure mathematics contains a richness of possibilities going far beyond the simple structures that they saw in Nature. Twentieth-century mathematics flowered in the belief that it had transcended completely the limitations imposed by its natural origins.
Now, as Mandelbrot points out, … Nature has played a joke on the mathematicians. The 19th-century mathematicians may not have been lacking in imagination, but Nature was not. The same pathological structures that the mathematicians invented to break loose from 19th-century naturalism turn out to be inherent in familiar objects all around us.
From 'Characterizing Irregularity', Science (12 May 1978), 200, No. 4342, 677-678. Quoted in Benoit Mandelbrot, The Fractal Geometry of Nature (1977, 1983), 3-4.
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Frequently, I have been asked if an experiment I have planned is pure or applied science; to me it is more important to know if the experiment will yield new and probably enduring knowledge about nature. If it is likely to yield such knowledge, it is, in my opinion, good fundamental research; and this is more important than whether the motivation is purely aesthetic satisfaction on the part of the experimenter on the one hand or the improvement of the stability of a high-power transistor on the other.
Quoted in Richard R. Nelson, 'The Link Between Science and Invention: The Case of the Transistor,' The Rate and Direction of the Inventive Activity (1962). In Daniel S. Greenberg, The Politics of Pure Science (1999), 32, footnote.
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From Pythagoras (ca. 550 BC) to Boethius (ca AD 480-524), when pure mathematics consisted of arithmetic and geometry while applied mathematics consisted of music and astronomy, mathematics could be characterized as the deductive study of “such abstractions as quantities and their consequences, namely figures and so forth” (Aquinas ca. 1260). But since the emergence of abstract algebra it has become increasingly difficult to formulate a definition to cover the whole of the rich, complex and expanding domain of mathematics.
In 100 Years of Mathematics: a Personal Viewpoint (1981), 2.
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From the point of view of the pure morphologist the recapitulation theory is an instrument of research enabling him to reconstruct probable lines of descent; from the standpoint of the student of development and heredity the fact of recapitulation is a difficult problem whose solution would perhaps give the key to a true understanding of the real nature of heredity.
Form and Function: A Contribution to the History of Animal Morphology (1916), 312-3.
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Given a situation, a system with a Leerstelle [a gap], whether a given completion (Lueckenfuellung) does justice to the structure, is the “right” one, is often determined by the structure of the system, the situation. There are requirements, structurally determined; there are possible in pure cases unambiguous decisions as to which completion does justice to the situation, which does not, which violates the requirements and the situation.
From 'Some Problems in the Theory of Ethics', collected in Mary Henle (ed.), Documents of Gestalt Psychology (1961), 36.
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Given any domain of thought in which the fundamental objective is a knowledge that transcends mere induction or mere empiricism, it seems quite inevitable that its processes should be made to conform closely to the pattern of a system free of ambiguous terms, symbols, operations, deductions; a system whose implications and assumptions are unique and consistent; a system whose logic confounds not the necessary with the sufficient where these are distinct; a system whose materials are abstract elements interpretable as reality or unreality in any forms whatsoever provided only that these forms mirror a thought that is pure. To such a system is universally given the name MATHEMATICS.
In 'Mathematics', National Mathematics Magazine (Nov 1937), 12, No. 2, 62.
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Gödel proved that the world of pure mathematics is inexhaustible; no finite set of axioms and rules of inference can ever encompass the whole of mathematics; given any finite set of axioms, we can find meaningful mathematical questions which the axioms leave unanswered. I hope that an analogous Situation exists in the physical world. If my view of the future is correct, it means that the world of physics and astronomy is also inexhaustible; no matter how far we go into the future, there will always be new things happening, new information coming in, new worlds to explore, a constantly expanding domain of life, consciousness, and memory.
From Lecture 1, 'Philosophy', in a series of four James Arthur Lectures, 'Lectures on Time and its Mysteries' at New York University (Autumn 1978). Printed in 'Time Without End: Physics and Biology in an Open Universe', Reviews of Modern Physics (Jul 1979), 51, 449.
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Hardly a pure science, history is closer to animal husbandry than it is to mathematics, in that it involves selective breeding. The principal difference between the husbandryman and the historian is that the former breeds sheep or cows or such, and the latter breeds (assumed) facts. The husbandryman uses his skills to enrich the future; the historian uses his to enrich the past. Both are usually up to their ankles in bullshit.
Another Roadside Attraction (1990), 127.
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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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Here's to pure mathematics—may it never be of any use to anybody.
Anonymous
A toast, variously attributed as used of old at Cambridge University, or as used by G.N. Hardy (according to Arthur C. Clarke in 'The Joy of Maths', Greetings, Carbon-Based Bipeds!: Collected Essays, 1934-1998 (2001), 460).
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Hieron asked Archimedes to discover, without damaging it, whether a certain crown or wreath was made of pure gold, or if the goldsmith had fraudulently alloyed it with some baser metal. While Archimedes was turning the problem over in his mind, he chanced to be in the bath house. There, as he was sitting in the bath, he noticed that the amount of water that was flowing over the top of it was equal in volume to that part of his body that was immersed. He saw at once a way of solving the problem. He did not delay, but in his joy leaped out of the bath. Rushing naked through the streets towards his home, he cried out in a loud voice that he had found what he sought. For, as he ran, he repeatedly shouted in Greek; “Eureka! Eurekal I’ve found it! I’ve found it!”
Vitrivius Pollio, De Architectura, ix, prologue, section 10.
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I am absolutely convinced that no wealth in the world can help humanity forward, even in the hands of the most devoted worker. The example of great and pure individuals is the only thing that can lead us to noble thoughts and deeds. Money only appeals to selfishness and irresistibly invites abuse. Can anyone imagine Moses, Jesus or Gandhi armed with the moneybags of Carnegie?
…...
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I believe with Schopenhauer that one of the strongest motives that lead men to art and science is escape from everyday life with its painful crudity and hopeless dreariness, from the fetters of one’s own ever shifting desires. A finely tempered nature longs to escape from personal life into the world of objective perception and thought; this desire may be compared with the townsman’s irresistible longing to escape from his noisy, cramped surroundings into the silence of high mountains, where the eye ranges freely through the still, pure air and fondly traces out the restful contours apparently built for eternity.
Address at The Physical Society, Berlin (1918) for Max Planck’s 60th birthday, 'Principles of Research', collected in Essays in Science (1934) 2.
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I feel more comfortable with gorillas than people. I can anticipate what a gorilla's going to do, and they're purely motivated.
Preferring the “silence of the forest” to the noise of a cocktail party while participating in a symposium, 'What We Can Learn About Humankind From the Apes' at Sweet Briar College campus. As quoted by Nan Robertson in 'Three Who Have Chosen a Life in the Wild', New York Times (1 May 1981), B36.
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I had this experience at the age of eight. My parents gave me a microscope. I don’t recall why, but no matter. I then found my own little world, completely wild and unconstrained, no plastic, no teacher, no books, no anything predictable. At first I did not know the names of the water-drop denizens or what they were doing. But neither did the pioneer microscopists. Like them, I graduated to looking at butterfly scales and other miscellaneous objects. I never thought of what I was doing in such a way, but it was pure science. As true as could be of any child so engaged, I was kin to Leeuwenhoek, who said that his work “was not pursued in order to gain the praise I now enjoy, but chiefly from a craving after knowledge, which I notice resides in me more that most other men.”
In The Creation: An Appeal to Save Life on Earth (2010), 143-144.
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I have no patience with the doctrine of “pure science,”—that science is science only as it is uncontaminated by application in the arts of life: and I have no patience with the spirit that considers a piece of work to be legitimate only as it has direct bearing on the arts and affairs of men. We must discover all things that are discoverable and make a record of it: the application will take care of itself.
In 'The Survey Idea in Country Life', collected in John Phelan, Readings in Rural Sociology (1920), 480.
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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 know of no department of natural science more likely to reward a man who goes into it thoroughly than anthropology. There is an immense deal to be done in the science pure and simple, and it is one of those branches of inquiry which brings one into contact with the great problems of humanity in every direction.
…...
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I maintain that in every special natural doctrine only so much science proper is to be met with as mathematics; for… science proper, especially [science] of nature, requires a pure portion, lying at the foundation of the empirical, and based upon a priori knowledge of natural things. … To the possibility of a determinate natural thing, and therefore to cognise it à priori, is further requisite that the intuition corresponding à priori to the conception should be given; in other words, that the conception should be constructed. But the cognition of the reason through construction of conceptions is mathematical. A pure philosophy of nature in general, namely, one that only investigates what constitutes a nature in general, may thus be possible without mathematics; but a pure doctrine of nature respecting determinate natural things (corporeal doctrine and mental doctrine), is only possible by means of mathematics; and as in every natural doctrine only so much science proper is to be met with therein as there is cognition à priori, a doctrine of nature can only contain so much science proper as there is in it of applied mathematics.
From Preface to The Metaphysical Foundations of Natural Science (1786), as translated by Ernest Belford Boax, in Kant’s Prolegomena: And The Metaphysical Foundations of Natural Science (1883), 140.
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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 saw Eternity the other night,
Like a great Ring of pure and endless light,
All calm, as it was bright;
And round beneath it,
Time, in hours, days, years,
Driv’n by the spheres
Like a vast shadow mov’d; in which the world
And all her train were hurl’d.
In 'The World', in Silex Scintillans (1650), 91.
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I say it is impossible that so sensible a people [citizens of Paris], under such circumstances, should have lived so long by the smoky, unwholesome, and enormously expensive light of candles, if they had really known that they might have had as much pure light of the sun for nothing.
[Describing the energy-saving benefit of adopting daylight saving time. (1784)]
'An Economical Project', The Life and Miscellaneous Writings of Benjamin Franklin (1839), 58. A translation of this letter appeared in one of the Paris daily papers about 1784. He estimated, during six months, a saving of over 64 million pound weight of candles, worth over 96 million livres tournois.
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I then [in 1902] possessed one decigramme of very pure radium chloride. It had taken me almost four years to produce the kind of evidence which chemical science demands, that radium is truly a new element. … The demonstration that cost so much effort was the basis of the new science of radioactivity.
As translated by Charlotte and Vernon Kellogg in Marie Curie, 'Autobiographical Notes', Pierre Curie (1923), 188.
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I think that the difference between pure and applied mathematics is social rather than scientific. A pure mathematician is paid for making mathematical discoveries. An applied mathematician is paid for the solution of given problems.
When Columbus set sail, he was like an applied mathematician, paid for the search of the solution of a concrete problem: find a way to India. His discovery of the New World was similar to the work of a pure mathematician.
In S.H. Lui, 'An Interview with Vladimir Arnol’d', Notices of the AMS (Apr 1997) 44, No. 4, 438. Reprinted from the Hong Kong Mathematics Society (Feb 1996).
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I wish there was a verb to otter, ottering around in pure play, to honour Otter ludens, which plays in my mind long after I’ve seen one.
In 'Fifty Years On, the Silence of Rachel Carson’s Spring Consumes Us', The Guardian (25 Sep 2012),
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I wish they don’t forget to keep those treasures pure which they have in excellence over the west: their artistic building of life, the simplicity a nd modesty in personal need, and the pureness and calmness of Japanese soul.
…...
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I would clarify that by ‘animal’ I understand a being that has feeling and that is capable of exercising life functions through a principle called soul; that the soul uses the body's organs, which are true machines, by virtue of its being the principal cause of the action of each of the machine's parts; and that although the placement that these parts have with respect to one another does scarcely anything else through the soul's mediation than what it does in pure machines, the entire machine nonetheless needs to be activated and guided by the soul in the same way as an organ, which, although capable of rendering different sounds through the placement of the parts of which it is composed, nonetheless never does so except through the guidance of the organist.
'La Mechanique des Animaux', in Oeuvres Diverses de Physique et de Mechanique (1721), Vol. 1, 329. Quoted in Jacques Roger, Keith R. Benson (ed.), Robert Ellrich (trans.), The Life Sciences in Eighteenth-Century French Thought, (1997), 273-4.
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If diphtheria is a disease caused by a microorganism, it is essential that three postulates be fulfilled. The fulfilment of these postulates is necessary in order to demonstrate strictly the parasitic nature of a disease:
1) The organism must be shown to be constantly present in characteristic form and arrangement in the diseased tissue.
2) The organism which, from its behaviour appears to be responsible for the disease, must be isolated and grown in pure culture.
3) The pure culture must be shown to induce the disease experimentally.
An early statement of Koch's postulates.
Mittheilungen aus den Kaiserliche Gesundheitsamt (1884) Vol. 2. Trans. T. D. Brock, Robert Koch: A Life in Medicine and Bacteriology (1988), 180.
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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 the term education may be understood in so large a sense as to include all that belongs to the improvement of the mind, either by the acquisition of the knowledge of others or by increase of it through its own exertions, we learn by them what is the kind of education science offers to man. It teaches us to be neglectful of nothing — not to despise the small beginnings, for they precede of necessity all great things in the knowledge of science, either pure or applied.
'Science as a Branch of Education', lecture to the Royal Institution, 11 Jun 1858. Reprinted in The Mechanics Magazine (1858), 49, 11.
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If we had nothing but pecuniary rewards and worldly honours to look to, our profession would not be one to be desired. But in its practice you will find it to be attended with peculiar privileges, second to none in intense interest and pure pleasures. It is our proud office to tend the fleshly tabernacle of the immortal spirit, and our path, rightly followed, will be guided by unfettered truth and love unfeigned. In the pursuit of this noble and holy calling I wish you all God-speed.
Conclusion of Graduation Address, University of Edinburgh (1876). In John Vaughan, 'Lord Lister', The Living Age (1918), 297, 361.
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If we seek for the simplest arrangement, which would enable it [the eye] to receive and discriminate the impressions of the different parts of the spectrum, we may suppose three distinct sensations only to be excited by the rays of the three principal pure colours, falling on any given point of the retina, the red, the green, and the violet; while the rays occupying the intermediate spaces are capable of producing mixed sensations, the yellow those which belong to the red and green, and the blue those which belong to the green and violet.
'Chromatics', in Supplement to the Fourth, Fifth, and Sixth Editions of the Encyclopedia Britannica (1824), Vol. 3, 142.
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In every combustion there is disengagement of the matter of fire or of light. A body can burn only in pure air [oxygen]. There is no destruction or decomposition of pure air and the increase in weight of the body burnt is exactly equal to the weight of air destroyed or decomposed. The body burnt changes into an acid by addition of the substance that increases its weight. Pure air is a compound of the matter of fire or of light with a base. In combustion the burning body removes the base, which it attracts more strongly than does the matter of heat, which appears as flame, heat and light.
'Memoire sur la combustion en général', Mémoires de l'Académie des Sciences, 1777, 592. Reprinted in Oeuvres de Lavoisier (1864), Vol. 2, 225-33, trans. M. P. Crosland.
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In human beings pure masculinity or femininity is not to be found either in a psychological or biological sense.
Three Essays on the Theory of Sexuality (1905), In James Strachey (ed.), The Standard Edition of the Complete Psychological Works of Sigmund Freud (1953), Vol. 7, 220, fn 1.
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In mathematics two ends are constantly kept in view: First, stimulation of the inventive faculty, exercise of judgment, development of logical reasoning, and the habit of concise statement; second, the association of the branches of pure mathematics with each other and with applied science, that the pupil may see clearly the true relations of principles and things.
In 'Aim of the Mathematical Instruction', International Commission on Teaching of Mathematics, American Report: United States Bureau of Education: Bulletin 1912, No. 4, 7.
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In my intercourse with mankind, I have always found those who would thrust theory into practical matters to be, at bottom, men of no judgement and pure quacks.
Quoted in James Kip Finch, Engineering Classics of James Kip Finch.
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In nature hybrid species are usually sterile, but in science the reverse is often true. Hybrid subjects are often astonishingly fertile, whereas if a scientific discipline remains too pure it usually wilts.
In What Mad Pursuit: A Personal View of Scientific Discovery (1988), 150.
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In our days everything seems pregnant with its contrary. Machinery, gifted with the wonderful power of shortening and fructifying human labor, we behold starving and overworking it… . At the same pace that mankind masters nature, man seems to become enslaved to other men or his own infamy. Even the pure light of science seems unable to shine but on the dark background of ignorance.
Karl Marx
In Speech (14 Apr 1856) on the 4th Anniversary of the People’s Paper, collected in David McLellan (ed.), Karl Marx: Selected Writings (2000), 368.
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In physiology, as in all other sciences, no discovery is useless, no curiosity misplaced or too ambitious, and we may be certain that every advance achieved in the quest of pure knowledge will sooner or later play its part in the service of man.
The Linacre Lecture on the Law of the Heart (1918), 147.
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In pure mathematics we have a great structure of logically perfect deductions which constitutes an integral part of that great and enduring human heritage which is and should be largely independent of the perhaps temporary existence of any particular geographical location at any particular time. … The enduring value of mathematics, like that of the other sciences and arts, far transcends the daily flux of a changing world. In fact, the apparent stability of mathematics may well be one of the reasons for its attractiveness and for the respect accorded it.
In Fundamentals of Mathematics (1941), 463.
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In Pure Mathematics, where all the various truths are necessarily connected with each other, (being all necessarily connected with those hypotheses which are the principles of the science), an arrangement is beautiful in proportion as the principles are few; and what we admire perhaps chiefly in the science, is the astonishing variety of consequences which may be demonstrably deduced from so small a number of premises.
In Elements of the Philosophy of the Human Mind (1827), Vol. 3, Chap. 1, Sec. 8, 186.
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In so far as such developments utilise the natural energy running to waste, as in water power, they may be accounted as pure gain. But in so far as they consume the fuel resources of the globe they are very different. The one is like spending the interest on a legacy, and the other is like spending the legacy itself. ... [There is] a still hardly recognised coming energy problem.
Matter and Energy (1911), 139.
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In studying the fate of our forest king, we have thus far considered the action of purely natural causes only; but, unfortunately, man is in the woods, and waste and pure destruction are making rapid headway. If the importance of the forests were even vaguely understood, even from an economic standpoint, their preservation would call forth the most watchful attention of government
John Muir
In The Mountains of California (1894), 198.
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In that pure enjoyment experienced on approaching to the ideal, in that eagerness to draw aside the veil from the hidden truth, and even in that discord which exists between the various workers, we ought to see the surest pledges of further scientific success. Science thus advances, discovering new truths, and at the same time obtaining practical results.
In The Principles of Chemistry (1891), Vol. 1, preface, footnote, ix, as translated from the Russian 5th edition by George Kamensky, edited by A. J. Greenaway.
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In the light of fuller day,
Of purer science, holier laws.
Epicedium On the Death of a Certain Journal. In Poems (1856), 249.
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In the mathematical investigations I have usually employed such methods as present themselves naturally to a physicist. The pure mathematician will complain, and (it must be confessed) sometimes with justice, of deficient rigour. But to this question there are two sides. For, however important it may be to maintain a uniformly high standard in pure mathematics, the physicist may occasionally do well to rest content with arguments which are fairly satisfactory and conclusive from his point of view. To his mind, exercised in a different order of ideas, the more severe procedure of the pure mathematician may appear not more but less demonstrative. And further, in many cases of difficulty to insist upon the highest standard would mean the exclusion of the subject altogether in view of the space that would be required.
In Preface to second edition, The Theory of Sound (1894), Vol. 1, vii.
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In the mathematics I can report no deficience, except that it be that men do not sufficiently understand the excellent use of the pure mathematics, in that they do remedy and cure many defects in the wit and faculties intellectual. For if the wit be too dull, they sharpen it; if too wandering, they fix it; if too inherent in the sense, they abstract it. So that as tennis is a game of no use in itself, but of great use in respect it maketh a quick eye and a body ready to put itself into all postures; so in the mathematics, that use which is collateral and intervenient is no less worthy than that which is principal and intended.
As translated in John Fauvel and Jeremy Gray (eds.) A History of Mathematics: A Reader (1987), 290-291. From De Augmentis, Book 3, The Advancement of Learning (1605), Book 2. Reprinted in The Two Books of Francis Bacon: Of the Proficience and Advancement of Learning, Divine and Human (2009), 97.
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In the physical world, one cannot increase the size or quantity of anything without changing its quality. Similar figures exist only in pure geometry.
In W.H. Auden and ‎Louis Kronenberger, The Viking Book of Aphorisms: A Personal Selection, (1966), 98.
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In the pure mathematics we contemplate absolute truths which existed in the divine mind before the morning stars sang together, and which will continue to exist there when the last of their radiant host shall have fallen from heaven.
From Address (22 Apr 1857) for Inauguration of Washington University, St. Louis, Missouri, collected in 'Academical Education', Orations and Speeches on Various Occasions (1870), Vol. 3, 514. This is seen misattributed to Eric Temple Bell, who only quoted it, attributing it to Everett, in for example, Mathematics: Queen and Servant of Sciences (1938), 20.
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In the summer of 1937, … I told Banach about an expression Johnny [von Neumann] had once used in conversation with me in Princeton before stating some non-Jewish mathematician’s result, “Die Goim haben den folgendenSatzbewiesen” (The goys have proved the following theorem). Banach, who was pure goy, thought it was one of the funniest sayings he had ever heard. He was enchanted by its implication that if the goys could do it, Johnny and I ought to be able to do it better. Johnny did not invent this joke, but he liked it and we started using it.
In Adventures of a Mathematician (1976, 1991), 107. Von Neumann, who was raised in Budapest by a Jewish family, knew the Yiddish word “goy” was equivalent to “gentile” or a non-Jew. Stefan Banach, a Polish mathematician, was raised in a Catholic family, hence “pure goy”. Ulam thus gives us the saying so often elsewhere seen attributed to von Neumann without the context: “The goys have proved the following theorem.” It is seen anecdotally as stated by von Neumann to begin a classroom lecture.
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It [mathematics] is in the inner world of pure thought, where all entia dwell, where is every type of order and manner of correlation and variety of relationship, it is in this infinite ensemble of eternal verities whence, if there be one cosmos or many of them, each derives its character and mode of being,—it is there that the spirit of mathesis has its home and its life.
Is it a restricted home, a narrow life, static and cold and grey with logic, without artistic interest, devoid of emotion and mood and sentiment? That world, it is true, is not a world of solar light, not clad in the colours that liven and glorify the things of sense, but it is an illuminated world, and over it all and everywhere throughout are hues and tints transcending sense, painted there by radiant pencils of psychic light, the light in which it lies. It is a silent world, and, nevertheless, in respect to the highest principle of art—the interpenetration of content and form, the perfect fusion of mode and meaning—it even surpasses music. In a sense, it is a static world, but so, too, are the worlds of the sculptor and the architect. The figures, however, which reason constructs and the mathematic vision beholds, transcend the temple and the statue, alike in simplicity and in intricacy, in delicacy and in grace, in symmetry and in poise. Not only are this home and this life thus rich in aesthetic interests, really controlled and sustained by motives of a sublimed and supersensuous art, but the religious aspiration, too, finds there, especially in the beautiful doctrine of invariants, the most perfect symbols of what it seeks—the changeless in the midst of change, abiding things hi a world of flux, configurations that remain the same despite the swirl and stress of countless hosts of curious transformations.
In 'The Universe and Beyond', Hibbert Journal (1904-1906), 3, 314.
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It has been asserted … that the power of observation is not developed by mathematical studies; while the truth is, that; from the most elementary mathematical notion that arises in the mind of a child to the farthest verge to which mathematical investigation has been pushed and applied, this power is in constant exercise. By observation, as here used, can only be meant the fixing of the attention upon objects (physical or mental) so as to note distinctive peculiarities—to recognize resemblances, differences, and other relations. Now the first mental act of the child recognizing the distinction between one and more than one, between one and two, two and three, etc., is exactly this. So, again, the first geometrical notions are as pure an exercise of this power as can be given. To know a straight line, to distinguish it from a curve; to recognize a triangle and distinguish the several forms—what are these, and all perception of form, but a series of observations? Nor is it alone in securing these fundamental conceptions of number and form that observation plays so important a part. The very genius of the common geometry as a method of reasoning—a system of investigation—is, that it is but a series of observations. The figure being before the eye in actual representation, or before the mind in conception, is so closely scrutinized, that all its distinctive features are perceived; auxiliary lines are drawn (the imagination leading in this), and a new series of inspections is made; and thus, by means of direct, simple observations, the investigation proceeds. So characteristic of common geometry is this method of investigation, that Comte, perhaps the ablest of all writers upon the philosophy of mathematics, is disposed to class geometry, as to its method, with the natural sciences, being based upon observation. Moreover, when we consider applied mathematics, we need only to notice that the exercise of this faculty is so essential, that the basis of all such reasoning, the very material with which we build, have received the name observations. Thus we might proceed to consider the whole range of the human faculties, and find for the most of them ample scope for exercise in mathematical studies. Certainly, the memory will not be found to be neglected. The very first steps in number—counting, the multiplication table, etc., make heavy demands on this power; while the higher branches require the memorizing of formulas which are simply appalling to the uninitiated. So the imagination, the creative faculty of the mind, has constant exercise in all original mathematical investigations, from the solution of the simplest problems to the discovery of the most recondite principle; for it is not by sure, consecutive steps, as many suppose, that we advance from the known to the unknown. The imagination, not the logical faculty, leads in this advance. In fact, practical observation is often in advance of logical exposition. Thus, in the discovery of truth, the imagination habitually presents hypotheses, and observation supplies facts, which it may require ages for the tardy reason to connect logically with the known. Of this truth, mathematics, as well as all other sciences, affords abundant illustrations. So remarkably true is this, that today it is seriously questioned by the majority of thinkers, whether the sublimest branch of mathematics,—the infinitesimal calculus—has anything more than an empirical foundation, mathematicians themselves not being agreed as to its logical basis. That the imagination, and not the logical faculty, leads in all original investigation, no one who has ever succeeded in producing an original demonstration of one of the simpler propositions of geometry, can have any doubt. Nor are induction, analogy, the scrutinization of premises or the search for them, or the balancing of probabilities, spheres of mental operations foreign to mathematics. No one, indeed, can claim preeminence for mathematical studies in all these departments of intellectual culture, but it may, perhaps, be claimed that scarcely any department of science affords discipline to so great a number of faculties, and that none presents so complete a gradation in the exercise of these faculties, from the first principles of the science to the farthest extent of its applications, as mathematics.
In 'Mathematics', in Henry Kiddle and Alexander J. Schem, The Cyclopedia of Education, (1877.) As quoted and cited in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 27-29.
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It is curious to observe how differently these great men [Plato and Bacon] estimated the value of every kind of knowledge. Take Arithmetic for example. Plato, after speaking slightly of the convenience of being able to reckon and compute in the ordinary transactions of life, passes to what he considers as a far more important advantage. The study of the properties of numbers, he tells us, habituates the mind to the contemplation of pure truth, and raises us above the material universe. He would have his disciples apply themselves to this study, not that they may be able to buy or sell, not that they may qualify themselves to be shop-keepers or travelling merchants, but that they may learn to withdraw their minds from the ever-shifting spectacle of this visible and tangible world, and to fix them on the immutable essences of things.
Bacon, on the other hand, valued this branch of knowledge only on account of its uses with reference to that visible and tangible world which Plato so much despised. He speaks with scorn of the mystical arithmetic of the later Platonists, and laments the propensity of mankind to employ, on mere matters of curiosity, powers the whole exertion of which is required for purposes of solid advantage. He advises arithmeticians to leave these trifles, and employ themselves in framing convenient expressions which may be of use in physical researches.
In 'Lord Bacon', Edinburgh Review (Jul 1837). Collected in Critical and Miscellaneous Essays: Contributed to the Edinburgh Review (1857), Vol. 1, 394.
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It is not so long since, during one of the meetings of the Association, one of the leading English newspapers briefly described a sitting of this Section in the words, “Saturday morning was devoted to pure mathematics, and so there was nothing of any general interest:” still, such toleration is better than undisguised and ill-informed hostility.
In Report of the 67th meeting of the British Association for the Advancement of Science.
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It is not surprising, in view of the polydynamic constitution of the genuinely mathematical mind, that many of the major heros of the science, men like Desargues and Pascal, Descartes and Leibnitz, Newton, Gauss and Bolzano, Helmholtz and Clifford, Riemann and Salmon and Plücker and Poincaré, have attained to high distinction in other fields not only of science but of philosophy and letters too. And when we reflect that the very greatest mathematical achievements have been due, not alone to the peering, microscopic, histologic vision of men like Weierstrass, illuminating the hidden recesses, the minute and intimate structure of logical reality, but to the larger vision also of men like Klein who survey the kingdoms of geometry and analysis for the endless variety of things that flourish there, as the eye of Darwin ranged over the flora and fauna of the world, or as a commercial monarch contemplates its industry, or as a statesman beholds an empire; when we reflect not only that the Calculus of Probability is a creation of mathematics but that the master mathematician is constantly required to exercise judgment—judgment, that is, in matters not admitting of certainty—balancing probabilities not yet reduced nor even reducible perhaps to calculation; when we reflect that he is called upon to exercise a function analogous to that of the comparative anatomist like Cuvier, comparing theories and doctrines of every degree of similarity and dissimilarity of structure; when, finally, we reflect that he seldom deals with a single idea at a tune, but is for the most part engaged in wielding organized hosts of them, as a general wields at once the division of an army or as a great civil administrator directs from his central office diverse and scattered but related groups of interests and operations; then, I say, the current opinion that devotion to mathematics unfits the devotee for practical affairs should be known for false on a priori grounds. And one should be thus prepared to find that as a fact Gaspard Monge, creator of descriptive geometry, author of the classic Applications de l’analyse à la géométrie; Lazare Carnot, author of the celebrated works, Géométrie de position, and Réflections sur la Métaphysique du Calcul infinitesimal; Fourier, immortal creator of the Théorie analytique de la chaleur; Arago, rightful inheritor of Monge’s chair of geometry; Poncelet, creator of pure projective geometry; one should not be surprised, I say, to find that these and other mathematicians in a land sagacious enough to invoke their aid, rendered, alike in peace and in war, eminent public service.
In Lectures on Science, Philosophy and Art (1908), 32-33.
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It is not, indeed, strange that the Greeks and Romans should not have carried ... any ... experimental science, so far as it has been carried in our time; for the experimental sciences are generally in a state of progression. They were better understood in the seventeenth century than in the sixteenth, and in the eighteenth century than in the seventeenth. But this constant improvement, this natural growth of knowledge, will not altogether account for the immense superiority of the modern writers. The difference is a difference not in degree, but of kind. It is not merely that new principles have been discovered, but that new faculties seem to be exerted. It is not that at one time the human intellect should have made but small progress, and at another time have advanced far; but that at one time it should have been stationary, and at another time constantly proceeding. In taste and imagination, in the graces of style, in the arts of persuasion, in the magnificence of public works, the ancients were at least our equals. They reasoned as justly as ourselves on subjects which required pure demonstration.
History (May 1828). In Samuel Austin Allibone, Prose Quotations from Socrates to Macaulay (1880), 36.
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It is sometimes helpful to differentiate between the God of Miracles and the God of Order. When scientists use the word God, they usually mean the God of Order. …The God of Miracles intervenes in our affairs, performs miracles, destroys wicked cities, smites enemy armies, drowns the Pharaoh's troops, and avenges the pure and noble. …This is not to say that miracles cannot happen, only that they are outside what is commonly called science.
In 'Conclusion', Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension (1995), 330-331.
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It is the duty of every man of good will to strive steadfastly in his own little world to make this teaching of pure humanity a living force, so far as he can. If he makes an honest attempt in this direction without being crushed and trampled under foot by his contemporaries, he may consider himself and the community to which he belongs lucky.
…...
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It is through it [intuition] that the mathematical world remains in touch with the real world, and even if pure mathematics could do without it, we should still have to have recourse to it to fill up the gulf that separates the symbol from reality.
…...
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It may well be doubted whether, in all the range of science, there is any field so fascinating to the explorer—so rich in hidden treasures—so fruitful in delightful surprises—as that of Pure Mathematics. The charm lies chiefly, I think, in the absolute certainty of its results; for that is what, beyond all mental treasures, the human intellect craves for. Let us only be sure of something! More light, more light!
Written without pseudonym as Charles L. Dodgson. Opening remarks in Introduction to A New Theory of Parallels (1888, 1890), xv.
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It may well be doubted whether, in all the range of Science, there is any field so fascinating to the explorer—so rich in hidden treasures—so fruitful in delightful surprises—as that of Pure Mathematics. The charm lies chiefly, I think, in the absolute certainty of its results: for that is what, beyond all mental treasures, the human intellect craves for. Let us only be sure of something! More light, more light … “And if our fate be death, give light and let us die” This is the cry that, through all the ages, is going up from perplexed Humanity, and Science has little else to offer, that will really meet the demands of its votaries, than the conclusions of Pure Mathematics.
Opening of 'Introduction', A New Theory of Parallels (1890), xv. As a non-fiction work, the author’s name on the title page of this book was Charles Lutwidge Dodgson. Being better known for his works of fiction as Lewis Carroll, all quotes relating to this one person, published under either name, are gathered on this single web page under his pen name.
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Kriegman says … “Think binary. When matter meets antimatter, both vanish, into pure energy. But both existed; I mean, there was a condition we’ll call ‘existence.’ Think of one and minus one. Together they add up to zero, nothing, nada, niente, right? Picture them together, then picture them separating—peeling apart. … Now you have something, you have two somethings, where once you had nothing.”
In Roger's Version (1986), 304.
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Let NEWTON, pure Intelligence, whom GOD
To Mortals lent, to trace his boundless Works
From Laws sublimely simple, speak thy Fame
In all Philosophy.
The Seasons (1746), 130.
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Let there be light! said God; and forthwith light
Ethereal, first of things, quintessence, pure.
From Paradise Lost (1821), Book 7, 209.
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Mathematic is either Pure or Mixed: To Pure Mathematic belong those sciences which handle Quantity entirely severed from matter and from axioms of natural philosophy. These are two, Geometry and Arithmetic; the one handling quantity continued, the other dissevered. … Mixed Mathematic has for its subject some axioms and parts of natural philosophy, and considers quantity in so far as it assists to explain, demonstrate and actuate these.
In De Augmentis, Bk. 3; Advancement of Learning, Bk. 2.
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Mathematical knowledge, therefore, appears to us of value not only in so far as it serves as means to other ends, but for its own sake as well, and we behold, both in its systematic external and internal development, the most complete and purest logical mind-activity, the embodiment of the highest intellect-esthetics.
In 'Ueber Wert und angeblichen Unwert der Mathematik', Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 13, 381.
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Mathematicians attach great importance to the elegance of their methods and their results. This is not pure dilettantism. What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details. But this is exactly what yields great results, in fact the more we see this aggregate clearly and at a single glance, the better we perceive its analogies with other neighboring objects, consequently the more chances we have of divining the possible generalizations. Elegance may produce the feeling of the unforeseen by the unexpected meeting of objects we are not accustomed to bring together; there again it is fruitful, since it thus unveils for us kinships before unrecognized. It is fruitful even when it results only from the contrast between the simplicity of the means and the complexity of the problem set; it makes us then think of the reason for this contrast and very often makes us see that chance is not the reason; that it is to be found in some unexpected law. In a word, the feeling of mathematical elegance is only the satisfaction due to any adaptation of the solution to the needs of our mind, and it is because of this very adaptation that this solution can be for us an instrument. Consequently this esthetic satisfaction is bound up with the economy of thought.
In 'The Future of Mathematics', Monist, 20, 80. Translated from the French by George Bruce Halsted.
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Mathematics … belongs to every inquiry, moral as well as physical. Even the rules of logic, by which it is rigidly bound, could not be deduced without its aid. The laws of argument admit of simple statement, but they must be curiously transposed before they can be applied to the living speech and verified by observation. In its pure and simple form the syllogism cannot be directly compared with all experience, or it would not have required an Aristotle to discover it. It must be transmuted into all the possible shapes in which reasoning loves to clothe itself. The transmutation is the mathematical process in the establishment of the law.
From Memoir (1870) read before the National Academy of Sciences, Washington, printed in 'Linear Associative Algebra', American Journal of Mathematics (1881), 4, 97-98.
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Mathematics had never had more than a secondary interest for him [her husband, George Boole]; and even logic he cared for chiefly as a means of clearing the ground of doctrines imagined to be proved, by showing that the evidence on which they were supposed to give rest had no tendency to prove them. But he had been endeavoring to give a more active and positive help than this to the cause of what he deemed pure religion.
In Eleanor Meredith Cobham, Mary Everest Boole: Collected Works (1931), 40.
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Mathematics in its pure form, as arithmetic, algebra, geometry, and the applications of the analytic method, as well as mathematics applied to matter and force, or statics and dynamics, furnishes the peculiar study that gives to us, whether as children or as men, the command of nature in this its quantitative aspect; mathematics furnishes the instrument, the tool of thought, which we wield in this realm.
In Psychologic Foundations of Education (1898), 325.
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Mathematics is an infinity of flexibles forcing pure thought into a cosmos. It is an arc of austerity cutting realms of reason with geodesic grandeur.
In The American Mathematical Monthly (1949), 56, 19. Excerpted in John Ewing (ed,), A Century of Mathematics: Through the Eyes of the Monthly (1996), 186.
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Mathematics is distinguished from all other sciences except only ethics, in standing in no need of ethics. Every other science, even logic—logic, especially—is in its early stages in danger of evaporating into airy nothingness, degenerating, as the Germans say, into an anachrioid [?] film, spun from the stuff that dreams are made of. There is no such danger for pure mathematics; for that is precisely what mathematics ought to be.
In Charles S. Peirce, ‎Charles Hartshorne (ed.), ‎Paul Weiss (ed.), Collected Papers of Charles Sanders Peirce (1931), Vol. 4, 200.
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Mathematics is the life supreme. The life of the gods is mathematics. All divine messengers are mathematicians. Pure mathematics is religion. Its attainment requires a theophany.
In Schriften (1901), Bd.. 2, 223.
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Mathematics make the mind attentive to the objects which it considers. This they do by entertaining it with a great variety of truths, which are delightful and evident, but not obvious. Truth is the same thing to the understanding as music to the ear and beauty to the eye. The pursuit of it does really as much gratify a natural faculty implanted in us by our wise Creator as the pleasing of our senses: only in the former case, as the object and faculty are more spiritual, the delight is more pure, free from regret, turpitude, lassitude, and intemperance that commonly attend sensual pleasures.
In An Essay on the Usefulness of Mathematical Learning (1701), 3-4.
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Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.
Essay, 'The Study of Mathematics' (1902), collected in Philosophical Essays (1910), 73-74. Also collected in Mysticism and Logic: And Other Essays (1918), 60.
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Medicine is the science by which we learn the various states of the human body in health and when not in health, and the means by which health is likely to be lost and, when lost, is likely to be restored back to health. In other words, it is the art whereby health is conserved and the art whereby it is restored after being lost. While some divide medicine into a theoretical and a practical [applied] science, others may assume that it is only theoretical because they see it as a pure science. But, in truth, every science has both a theoretical and a practical side.
Avicenna
'The Definition of Medicine', in The Canon of Medicine, adapted by L. Bakhtiar (1999), 9.
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Men of science belong to two different types—the logical and the intuitive. Science owes its progress to both forms of minds. Mathematics, although a purely logical structure, nevertheless makes use of intuition. Among the mathematicians there are intuitives and logicians, analysts and geometricians. Hermite and Weierstrass were intuitives. Riemann and Bertrand, logicians. The discoveries of intuition have always to be developed by logic.
In Man the Unknown (1935), 123.
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My original decision to devote myself to science was a direct result of the discovery which has never ceased to fill me with enthusiasm since my early youth—the comprehension of the far from obvious fact that the laws of human reasoning coincide with the laws governing the sequences of the impressions we receive from the world about us; that, therefore, pure reasoning can enable man to gain an insight into the mechanism of the latter. In this connection, it is of paramount importance that the outside world is something independent from man, something absolute, and the quest for the laws which apply to this absolute appeared to me as the most sublime scientific pursuit in life.
'A Scientific Autobiography' (1948), in Scientific Autobiography and Other Papers, trans. Frank Gaynor (1950), 13.
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My view of the matter, for what it is worth, is that there is no such thing as a logical method of having new ideas, or a logical reconstruction of this process. My view may be expressed by saying that every discovery contains an “irrational element,” or “a creative intuition,” in Bergson's sense. In a similar way Einstein speaks of the “search for those highly universal laws … from which a picture of the world can be obtained by pure deduction. There is no logical path.” he says, “leading to these … laws. They can only be reached by intuition, based upon something like an intellectual love (Einfühlung) of the objects of experience.”
In The Logic of Scientific Discovery: Logik Der Forschung (1959, 2002), 8.
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Nirvana is a state of pure blissful knowledge ... It has nothing to do with the individual. The ego or its separation is an illusion. Indeed in a certain sense two ‘I’s are identical namely when one disregards all special contents–their Karma. The goal of man is to preserve his Karma and to develop it further ... when man dies his Karma lives and creates for itself another carrier.
…...
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No man can be a pure specialist without being in the strict sense an idiot.
In 'Maxims for Revolutionists: Education', in Man and Superman (1905), 230.
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No one can take from us the joy of the first becoming aware of something, the so-called discovery. But if we also demand the honor, it can be utterly spoiled for us, for we are usually not the first. What does discovery mean, and who can say that he has discovered this or that? After all it’s pure idiocy to brag about priority; for it’s simply unconscious conceit, not to admit frankly that one is a plagiarist.
Epigraph to Lancelot Law Whyte, The Unconscious before Freud (1960).
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No! What we need are not prohibitory marriage laws, but a reformed society, an educated public opinion which will teach individual duty in these matters. And it is to the women of the future that I look for the needed reformation. Educate and train women so that they are rendered independent of marriage as a means of gaining a home and a living, and you will bring about natural selection in marriage, which will operate most beneficially upon humanity. When all women are placed in a position that they are independent of marriage, I am inclined to think that large numbers will elect to remain unmarried—in some cases, for life, in others, until they encounter the man of their ideal. I want to see women the selective agents in marriage; as things are, they have practically little choice. The only basis for marriage should be a disinterested love. I believe that the unfit will be gradually eliminated from the race, and human progress secured, by giving to the pure instincts of women the selective power in marriage. You can never have that so long as women are driven to marry for a livelihood.
In 'Heredity and Pre-Natal Influences. An Interview With Dr. Alfred Russel Wallace', Humanitarian (1894), 4, 87.
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Nothing could be more admirable than the manner in which for forty years he [Joseph Black] performed this useful and dignified office. His style of lecturing was as nearly perfect as can well be conceived; for it had all the simplicity which is so entirely suited to scientific discourse, while it partook largely of the elegance which characterized all he said or did … I have heard the greatest understandings of the age giving forth their efforts in its most eloquent tongues—have heard the commanding periods of Pitt’s majestic oratory—the vehemence of Fox’s burning declamation—have followed the close-compacted chain of Grant’s pure reasoning—been carried away by the mingled fancy, epigram, and argumentation of Plunket; but I should without hesitation prefer, for mere intellectual gratification (though aware how much of it is derived from association), to be once more allowed the privilege which I in those days enjoyed of being present while the first philosopher of his age was the historian of his own discoveries, and be an eyewitness of those experiments by which he had formerly made them, once more performed with his own hands.
In 'Philosophers of the Time of George III', The Works of Henry, Lord Brougham, F.R.S. (1855), Vol. I, 19-21.
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Nothing is accidental in the universe— this is one of my Laws of Physics—except the entire universe itself, which is Pure Accident, pure divinity.
In ‘The Summing Up: Meredith Dawe’, Do What You Will, (1970). As cited in Robert Andrews, The Columbia Dictionary of Quotations (1993), 946.
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O comfortable allurement, O ravishing perswasion, to deal with a Science, whose subject is so Auncient, so pure, so excellent, so surmounting all creatures... By Numbers propertie ... we may... arise, clime, ascend, and mount up (with Speculative winges) in spirit, to behold in the Glas of creation, the Forme of Formes, the Exemplar Number of all things Numerable... Who can remaine, therefore, unpersuaded, to love, allow, and honor the excellent sciehce of Arithmatike?
John Dee
'Mathematicall Preface', in H. Billingsley, trans. The Elements of Geometry of the most Aunceint Philosopher Euclide of Megara (1570), in J. L. Hellbron, Weighing Imponderables and Other Quantitative Science around 1800 (1993), 2.
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Observation is like a piece of glass, which, as a mirror, must be very smooth, and must be very carefully polished, in order that it may reflect the image pure and undistorted.
'The Study of the Natural Sciences: An Introductory Lecture to the Course of Experimental Chemistry in the University of Munich, for the Winter Session of 1852-53,' as translated and republished in The Medical Times and Gazette (22 Jan 1853), N.S. Vol. 6, 82.
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October 9, 1863
Always, however great the height of the balloon, when I have seen the horizon it has roughly appeared to be on the level of the car though of course the dip of the horizon is a very appreciable quantity or the same height as the eye. From this one might infer that, could the earth be seen without a cloud or anything to obscure it, and the boundary line of the plane approximately the same height as the eye, the general appearance would be that of a slight concavity; but I have never seen any part of the surface of the earth other than as a plane.
Towns and cities, when viewed from the balloon are like models in motion. I shall always remember the ascent of 9th October, 1863, when we passed over London about sunset. At the time when we were 7,000 feet high, and directly over London Bridge, the scene around was one that cannot probably be equalled in the world. We were still so low as not to have lost sight of the details of the spectacle which presented itself to our eyes; and with one glance the homes of 3,000,000 people could be seen, and so distinct was the view, that every large building was easily distinguishable. In fact, the whole of London was visible, and some parts most clearly. All round, the suburbs were also very distinct, with their lines of detached villas, imbedded as it were in a mass of shrubs; beyond, the country was like a garden, its fields, well marked, becoming smaller and smaller as the eye wandered farther and farther away.
Again looking down, there was the Thames, throughout its whole length, without the slightest mist, dotted over its winding course with innumerable ships and steamboats, like moving toys. Gravesend was visible, also the mouth of the Thames, and the coast around as far as Norfolk. The southern shore of the mouth of the Thames was not so clear, but the sea beyond was seen for many miles; when at a higher elevation, I looked for the coast of France, but was unable to see it. On looking round, the eye was arrested by the garden-like appearance of the county of Kent, till again London claimed yet more careful attention.
Smoke, thin and blue, was curling from it, and slowly moving away in beautiful curves, from all except one part, south of the Thames, where it was less blue and seemed more dense, till the cause became evident; it was mixed with mist rising from the ground, the southern limit of which was bounded by an even line, doubtless indicating the meeting of the subsoils of gravel and clay. The whole scene was surmounted by a canopy of blue, everywhere free from cloud, except near the horizon, where a band of cumulus and stratus extended all round, forming a fitting boundary to such a glorious view.
As seen from the earth, the sunset this evening was described as fine, the air being clear and the shadows well defined; but, as we rose to view it and its effects, the golden hues increased in intensity; their richness decreased as the distance from the sun increased, both right and left; but still as far as 90º from the sun, rose-coloured clouds extended. The remainder of the circle was completed, for the most part, by pure white cumulus of well-rounded and symmetrical forms.
I have seen London by night. I have crossed it during the day at the height of four miles. I have often admired the splendour of sky scenery, but never have I seen anything which surpassed this spectacle. The roar of the town heard at this elevation was a deep, rich, continuous sound the voice of labour. At four miles above London, all was hushed; no sound reached our ears.
Travels in the Air (1871), 99-100.
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On coming down the stairs at dinner Tris [Trismegistus = Frankland] who walked before me seemed impressed by a mechanical impulse which impelled him along the corridor with a fervid velocity. On reaching the stair bottom I discovered the cause of the attraction. Miss Edmondson, like a pure planet, had checked his gravitating tendencies and lo! He stood radiant with smiles dropping joysparkes from his eyes as he clasped her hand. His countenance became a transparency through which the full proportions of his soul shone manifest; his blood tingled from his eyebrows to his finger ends, and wealthy with rich emotions his face became the avenue of what he felt.
Journals of John Tyndall, 18 Jan 1848. Royal Institution Archives.
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One hardly knows where, in the history of science, to look for an important movement that had its effective start in so pure and simple an accident as that which led to the building of the great Washington telescope, and went on to the discovery of the satellites of Mars.
In The Reminiscences of an Astronomer (1903), Vol. 3, 128.
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One should not wrongly reify “cause” and “effect,” as the natural scientists do (and whoever, like them, now “naturalizes” in his thinking), according to the prevailing mechanical doltishness which makes the cause press and push until it “effects” its end; one should use “cause” and “effect” only as pure concepts, that is to say, as conventional fictions for the purpose of designation and communication—not for explanation.
In Beyond Good and Evil (1886). Trans. W. Kaufmann (ed.), Basic Writings of Nietzsche (1968), 219.
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Only the applied scientist sets out to find a “useful” pot of gold. The pure scientist sets out to find nothing. Anything. Everything. The applied scientist is a prospector. The pure scientist is an explorer.
In Jacques Cousteau and Susan Schiefelbein, The Human, the Orchid, and the Octopus: Exploring and Conserving Our Natural World (2007), 181.
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Our experience up to date justifies us in feeling sure that in Nature is actualized the ideal of mathematical simplicity. It is my conviction that pure mathematical construction enables us to discover the concepts and the laws connecting them, which gives us the key to understanding nature… In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed.
In Herbert Spencer Lecture at Oxford (10 Jun 1933), 'On the Methods of Theoretical Physics'. Printed in Discovery (Jul 1933), 14, 227. Also quoted in Stefano Zambelli and Donald A. R. George, Nonlinearity, Complexity and Randomness in Economics (2012).
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Perhaps the least inadequate description of the general scope of modern Pure Mathematics—I will not call it a definition—would be to say that it deals with form, in a very general sense of the term; this would include algebraic form, functional relationship, the relations of order in any ordered set of entities such as numbers, and the analysis of the peculiarities of form of groups of operations.
In Presidential Address British Association for the Advancement of Science, Sheffield, Section A, Nature (1 Sep 1910), 84, 287.
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Physical science comes nearest to that complete system of exact knowledge which all sciences have before them as an ideal. Some fall far short of it. The physicist who inveighs against the lack of coherence and the indefiniteness of theological theories, will probably speak not much less harshly of the theories of biology and psychology. They also fail to come up to his standard of methodology. On the other side of him stands an even superior being—the pure mathematician—who has no high opinion of the methods of deduction used in physics, and does not hide his disapproval of the laxity of what is accepted as proof in physical science. And yet somehow knowledge grows in all these branches. Wherever a way opens we are impelled to seek by the only methods that can be devised for that particular opening, not over-rating the security of our finding, but conscious that in this activity of mind we are obeying the light that is in our nature.
Swarthmore Lecture (1929) at Friends’ House, London, printed in Science and the Unseen World (1929), 77-78.
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Physical scientists probably deserve the reputation they enjoy for incorruptibility and unswerving devotion to pure truth. The reason for this is that it is not worth while to bribe them.
In Science is a Sacred Cow (1950), 168-69.
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Poincaré was the last man to take practically all mathematics, pure and applied, as his province. … Few mathematicians have had the breadth of philosophic vision that Poincaré had, and none is his superior in the gift of clear exposition.
In Eric Temple Bell, Men of Mathematics (1937, 1986), 527.
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Programming is one of the most difficult branches of applied mathematics; the poorer mathematicians had better remain pure mathematicians.
…...
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Pure earth does not petrify, because the predominance of dryness over [i.e. in] the earth endows it not with coherence but rather with crumbliness. In general, stone is formed in two ways only (a) through the hardening of clay, and (b) by the congelation [of waters].
Avicenna
Congelatione et Conglutinatione Lapidium (1021-23), trans. E. J. Holmyard and D. C. Mandeville (1927), 18.
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Pure mathematics … reveals itself as nothing but symbolic or formal logic. It is concerned with implications, not applications. On the other hand, natural science, which is empirical and ultimately dependent upon observation and experiment, and therefore incapable of absolute exactness, cannot become strictly mathematical. The certainty of geometry is thus merely the certainty with which conclusions follow from non-contradictory premises. As to whether these conclusions are true of the material world or not, pure mathematics is indifferent.
In 'Non-Euclidian Geometry of the Fourth Dimension', collected in Henry Parker Manning (ed.), The Fourth Dimension Simply Explained (1910), 58.
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Pure mathematics and physics are becoming ever more closely connected, though their methods remain different. One may describe the situation by saying that the mathematician plays a game in which he himself invents the rules while the while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen. … Possibly, the two subjects will ultimately unify, every branch of pure mathematics then having its physical application, its importance in physics being proportional to its interest in mathematics.
From Lecture delivered on presentation of the James Scott prize, (6 Feb 1939), 'The Relation Between Mathematics And Physics', printed in Proceedings of the Royal Society of Edinburgh (1938-1939), 59, Part 2, 124.
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Pure mathematics consists entirely of such asseverations as that, if such and such is a proposition is true of anything, then such and such another propositions is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is of which it is supposed to be true. Both these points would belong to applied mathematics. … If our hypothesis is about anything and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the the subject in which we never know what we are talking about, not whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate.
In 'Recent Work on the Principles of Mathematics', International Monthly (1901), 4, 84.
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Pure mathematics is a collection of hypothetical, deductive theories, each consisting of a definite system of primitive, undefined, concepts or symbols and primitive, unproved, but self-consistent assumptions (commonly called axioms) together with their logically deducible consequences following by rigidly deductive processes without appeal to intuition.
In 'Non-Euclidian Geometry of the Fourth Dimension', collected in Henry Parker Manning (ed.), The Fourth Dimension Simply Explained (1910), 58.
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Pure mathematics is much more than an armoury of tools and techniques for the applied mathematician. On the other hand, the pure mathematician has ever been grateful to applied mathematics for stimulus and inspiration. From the vibrations of the violin string they have drawn enchanting harmonies of Fourier Series, and to study the triode valve they have invented a whole theory of non-linear oscillations.
In 100 Years of Mathematics: a Personal Viewpoint (1981), 3.
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Pure mathematics is not concerned with magnitude. It is merely the doctrine of notation of relatively ordered thought operations which have become mechanical.
In Schriften (1901), Zweiter Teil, 282.
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Pure Mathematics is the class of all propositions of the form “p implies q,” where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form. In addition to these, mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth.
In 'Definition of Pure Mathematics', Principles of Mathematics (1903), 3.
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Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature.
In letter (1 May 1935), Letters to the Editor, 'The Late Emmy Noether: Professor Einstein Writes in Appreciation of a Fellow-Mathematician', New York Times (4 May 1935), 12.
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Pure mathematics proves itself a royal science both through its content and form, which contains within itself the cause of its being and its methods of proof. For in complete independence mathematics creates for itself the object of which it treats, its magnitudes and laws, its formulas and symbols.
In Die Mathematik die Fackelträgerin einer neuen Zeit (1889), 94. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 11.
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Pure mathematics; may it never be of any use to anyone.
A banquet toast, quoted as “I believe that it was at a banquet of the Red Lions that he proposed the toast…”, in Alexander Macfarlane, 'Henry John Stephen Smith', Lectures on Ten British Mathematicians of the Nineteenth Century (1916), 100.
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Pure reasoning as a means of arriving at truth is like the spider who spins a web out of himself. The web is orderly and elaborate, but it is only a trap.
In Reconstruction in Philosophy (1920), 32.
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Pure truth cannot be assimilated by the crowd; it must be communicated by contagion.
Entry for 26 Oct 1875 in Amiel’s Journal: The Journal Intime of Henri-Frédéric Amiel, trans. Humphry Ward (1893), 227.
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Put glibly:
In science if you know what you are doing you should not be doing it.
In engineering if you do not know what you are doing you should not be doing it.
Of course, you seldom, if ever, see either pure state.
In The Art of Doing Science and Engineering: Learning to Learn (1975, 2005), 5.
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Real life is, to most men, a long second-best, a perpetual compromise between the ideal and the possible; but the world of pure reason ;knows no compromise, no practical limitations, no barrier to the creative activity.
Essay, 'The Study of Mathematics' (1902), collected in Philosophical Essays (1910), 73-74. Also collected in Mysticism and Logic: And Other Essays (1919), 60.
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Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos [mathematics], where pure thought can dwell in its natural home...
'The Study of Mathematics', Philosophical Essays (1910), 73-74. In J. E. Creighton (Ed.), Evander Bradley McGilvary, 'Reviews of Books', The Philosophical Review (1911), Vol 20, 422.
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Scheele, it was said, never forgot anything if it had to do with chemistry. He never forgot the look, the feel, the smell of a substance, or the way it was transformed in chemical reactions, never forgot anything he read, or was told, about the phenomena of chemistry. He seemed indifferent, or inattentive, to most things else, being wholly dedicated to his single passion, chemistry. It was this pure and passionate absorption in phenomena—noticing everything, forgetting nothing—that constituted Scheele's special strength.
Uncle Tungsten: Memories of a Chemical Boyhood (2001), 44.
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Science by itself produces a very badly deformed man who becomes rounded out into a useful creative being only with great difficulty and large expenditure of time. … It is a much smaller matter to both teach and learn pure science than it is to intelligently apply this science to the solution of problems as they arise in daily life.
As quoted in Gary W. Matkin, Technology Transfer and the University (1990), 24.
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Science is not gadgetry. The desirable adjuncts of modern living, although in many instances made possible by science, certainly do not constitute science. Basic scientific knowledge often (but not always) is a prerequisite to such developments, but technology primarily deserves the credit for having the financial courage, the ingenuity, and the driving energy to see to it that so-called ‘pure knowledge’ is in fact brought to the practical service of man. And it should also be recognized that those who have the urge to apply knowledge usefully have themselves often made significant contribution to pure knowledge and have even more often served as a stimulation to the activities of a pure researcher.
Warren Weaver (1894–1978), U.S. mathematician, scientist, educator. Science and Imagination, ch. 1, Basic Books (1967).
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Science is rooted in the will to truth. With the will to truth it stands or falls. Lower the standard even slightly and science becomes diseased at the core. Not only science, but man. The will to truth, pure and unadulterated, is among the essential conditions of his existence; if the standard is compromised he easily becomes a kind of tragic caricature of himself.
Opening statement in 'On Truth', Social Research (May 1934), 1, No. 2, 135.
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Science, at bottom, is really anti-intellectual. It always distrusts pure reason, and demands the production of objective fact.
In Minority Report: H. L. Mencken’s Notebooks (1956, 2006), 277.
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Science, though apparently transformed into pure knowledge, has yet never lost its character of being a craft; and that it is not the knowledge itself which can rightly be called science, but a special way of getting and of using knowledge. Namely, science is the getting of knowledge from experience on the assumption of uniformity in nature, and the use of such knowledge to guide the actions of men.
In 'On The Scientific Basis of Morals', Contemporary Review (Sep 1875), collected in Leslie Stephen and Frederick Pollock (eds.), Lectures and Essays: By the Late William Kingdon Clifford, F.R.S. (1886), 289.
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Scientific subjects do not progress necessarily on the lines of direct usefulness. Very many applications of the theories of pure mathematics have come many years, sometimes centuries, after the actual discoveries themselves. The weapons were at hand, but the men were not able to use them.
In Perry, Teaching of Mathematics (1902), 35.
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So the astronomer is on common ground with the physicist both in the subject and in the predicate of the conclusion, but the physicist demonstrates the predicate to belong to the subject by nature, whereas the astronomer does not care whether it belongs by nature or not. What, therefore, is the predicate for the physicist, is abstracted as the subject for the pure mathematician.
As quoted in Alistair Cameron Crombie, Robert Grosseteste and the Origins of Experimental Science, 1100-1700 (1971), 94.
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Soccer ball C60 quickly became a sort of “Rosetta Stone” leading to the discovery of a new world of geodesic structures of pure carbon built on the nanometer scale.
From Nobel Lecture (7 Dec 1996), 'Discovering the Fullerenes', collected in Ingmar Grenthe (ed.), Nobel Lectures, Chemistry 1996-2000 (2003).
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Some of my cousins who had the great advantage of University education used to tease me with arguments to prove that nothing has any existence except what we think of it. … These amusing mental acrobatics are all right to play with. They are perfectly harmless and perfectly useless. ... I always rested on the following argument. … We look up to the sky and see the sun. Our eyes are dazzled and our senses record the fact. So here is this great sun standing apparently on no better foundation than our physical senses. But happily there is a method, apart altogether from our physical senses, of testing the reality of the sun. It is by mathematics. By means of prolonged processes of mathematics, entirely separate from the senses, astronomers are able to calculate when an eclipse will occur. They predict by pure reason that a black spot will pass across the sun on a certain day. You go and look, and your sense of sight immediately tells you that their calculations are vindicated. So here you have the evidence of the senses reinforced by the entirely separate evidence of a vast independent process of mathematical reasoning. We have taken what is called in military map-making “a cross bearing.” When my metaphysical friends tell me that the data on which the astronomers made their calculations, were necessarily obtained originally through the evidence of the senses, I say, “no.” They might, in theory at any rate, be obtained by automatic calculating-machines set in motion by the light falling upon them without admixture of the human senses at any stage. When it is persisted that we should have to be told about the calculations and use our ears for that purpose, I reply that the mathematical process has a reality and virtue in itself, and that onie discovered it constitutes a new and independent factor. I am also at this point accustomed to reaffirm with emphasis my conviction that the sun is real, and also that it is hot— in fact hot as Hell, and that if the metaphysicians doubt it they should go there and see.
In My Early Life (1930).
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Success in the solution of a problem generally depends in a great measure on the selection of the most appropriate method of approaching it; many properties of conic sections (for instance) being demonstrable by a few steps of pure geometry which would involve the most laborious operations with trilinear co-ordinates, while other properties are almost self-evident under the method of trilinear co-ordinates, which it would perhaps be actually impossible to prove by the old geometry.
In Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions (1866), 154.
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Suppose then I want to give myself a little training in the art of reasoning; suppose I want to get out of the region of conjecture and probability, free myself from the difficult task of weighing evidence, and putting instances together to arrive at general propositions, and simply desire to know how to deal with my general propositions when I get them, and how to deduce right inferences from them; it is clear that I shall obtain this sort of discipline best in those departments of thought in which the first principles are unquestionably true. For in all our thinking, if we come to erroneous conclusions, we come to them either by accepting false premises to start with—in which case our reasoning, however good, will not save us from error; or by reasoning badly, in which case the data we start from may be perfectly sound, and yet our conclusions may be false. But in the mathematical or pure sciences,—geometry, arithmetic, algebra, trigonometry, the calculus of variations or of curves,— we know at least that there is not, and cannot be, error in our first principles, and we may therefore fasten our whole attention upon the processes. As mere exercises in logic, therefore, these sciences, based as they all are on primary truths relating to space and number, have always been supposed to furnish the most exact discipline. When Plato wrote over the portal of his school. “Let no one ignorant of geometry enter here,” he did not mean that questions relating to lines and surfaces would be discussed by his disciples. On the contrary, the topics to which he directed their attention were some of the deepest problems,— social, political, moral,—on which the mind could exercise itself. Plato and his followers tried to think out together conclusions respecting the being, the duty, and the destiny of man, and the relation in which he stood to the gods and to the unseen world. What had geometry to do with these things? Simply this: That a man whose mind has not undergone a rigorous training in systematic thinking, and in the art of drawing legitimate inferences from premises, was unfitted to enter on the discussion of these high topics; and that the sort of logical discipline which he needed was most likely to be obtained from geometry—the only mathematical science which in Plato’s time had been formulated and reduced to a system. And we in this country [England] have long acted on the same principle. Our future lawyers, clergy, and statesmen are expected at the University to learn a good deal about curves, and angles, and numbers and proportions; not because these subjects have the smallest relation to the needs of their lives, but because in the very act of learning them they are likely to acquire that habit of steadfast and accurate thinking, which is indispensable to success in all the pursuits of life.
In Lectures on Teaching (1906), 891-92.
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The arithmetization of mathematics … which began with Weierstrass … had for its object the separation of purely mathematical concepts, such as number and correspondence and aggregate, from intuitional ideas, which mathematics had acquired from long association with geometry and mechanics. These latter, in the opinion of the formalists, are so firmly entrenched in mathematical thought that in spite of the most careful circumspection in the choice of words, the meaning concealed behind these words, may influence our reasoning. For the trouble with human words is that they possess content, whereas the purpose of mathematics is to construct pure thought. But how can we avoid the use of human language? The … symbol. Only by using a symbolic language not yet usurped by those vague ideas of space, time, continuity which have their origin in intuition and tend to obscure pure reason—only thus may we hope to build mathematics on the solid foundation of logic.
In Tobias Dantzig and Joseph Mazur (ed.), Number: The Language of Science (1930, ed. by Joseph Mazur 2007), 99.
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The aims of pure basic science, unlike those of applied science, are neither fast-flowing nor pragmatic. The quick harvest of applied science is the useable process, the medicine, the machine. The shy fruit of pure science is understanding.
In 'The Meaning of Einstein's New Theory', Life (9 Jan 1950), 28, No. 2, 22. Einstein had just completed the mathematical formulation of the United Field Theory.
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The analytical geometry of Descartes and the calculus of Newton and Leibniz have expanded into the marvelous mathematical method—more daring than anything that the history of philosophy records—of Lobachevsky and Riemann, Gauss and Sylvester. Indeed, mathematics, the indispensable tool of the sciences, defying the senses to follow its splendid flights, is demonstrating today, as it never has been demonstrated before, the supremacy of the pure reason.
In 'What Knowledge is of Most Worth?', Presidential address to the National Education Association, Denver, Colorado (9 Jul 1895). In Educational Review (Sep 1895), 10, 109.
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The average English author [of mathematical texts] leaves one under the impression that he has made a bargain with his reader to put before him the truth, the greater part of the truth, and nothing but the truth; and that if he has put the facts of his subject into his book, however difficult it may be to unearth them, he has fulfilled his contract with his reader. This is a very much mistaken view, because effective teaching requires a great deal more than a bare recitation of facts, even if these are duly set forth in logical order—as in English books they often are not. The probable difficulties which will occur to the student, the objections which the intelligent student will naturally and necessarily raise to some statement of fact or theory—these things our authors seldom or never notice, and yet a recognition and anticipation of them by the author would be often of priceless value to the student. Again, a touch of humour (strange as the contention may seem) in mathematical works is not only possible with perfect propriety, but very helpful; and I could give instances of this even from the pure mathematics of Salmon and the physics of Clerk Maxwell.
In Perry, Teaching of Mathematics (1902), 59-61.
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The belief that mathematics, because it is abstract, because it is static and cold and gray, is detached from life, is a mistaken belief. Mathematics, even in its purest and most abstract estate, is not detached from life. It is just the ideal handling of the problems of life, as sculpture may idealize a human figure or as poetry or painting may idealize a figure or a scene. Mathematics is precisely the ideal handling of the problems of life, and the central ideas of the science, the great concepts about which its stately doctrines have been built up, are precisely the chief ideas with which life must always deal and which, as it tumbles and rolls about them through time and space, give it its interests and problems, and its order and rationality. That such is the case a few indications will suffice to show. The mathematical concepts of constant and variable are represented familiarly in life by the notions of fixedness and change. The concept of equation or that of an equational system, imposing restriction upon variability, is matched in life by the concept of natural and spiritual law, giving order to what were else chaotic change and providing partial freedom in lieu of none at all. What is known in mathematics under the name of limit is everywhere present in life in the guise of some ideal, some excellence high-dwelling among the rocks, an “ever flying perfect” as Emerson calls it, unto which we may approximate nearer and nearer, but which we can never quite attain, save in aspiration. The supreme concept of functionality finds its correlate in life in the all-pervasive sense of interdependence and mutual determination among the elements of the world. What is known in mathematics as transformation—that is, lawful transfer of attention, serving to match in orderly fashion the things of one system with those of another—is conceived in life as a process of transmutation by which, in the flux of the world, the content of the present has come out of the past and in its turn, in ceasing to be, gives birth to its successor, as the boy is father to the man and as things, in general, become what they are not. The mathematical concept of invariance and that of infinitude, especially the imposing doctrines that explain their meanings and bear their names—What are they but mathematicizations of that which has ever been the chief of life’s hopes and dreams, of that which has ever been the object of its deepest passion and of its dominant enterprise, I mean the finding of the worth that abides, the finding of permanence in the midst of change, and the discovery of a presence, in what has seemed to be a finite world, of being that is infinite? It is needless further to multiply examples of a correlation that is so abounding and complete as indeed to suggest a doubt whether it be juster to view mathematics as the abstract idealization of life than to regard life as the concrete realization of mathematics.
In 'The Humanization of Teaching of Mathematics', Science, New Series, 35, 645-46.
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The best that Gauss has given us was likewise an exclusive production. If he had not created his geometry of surfaces, which served Riemann as a basis, it is scarcely conceivable that anyone else would have discovered it. I do not hesitate to confess that to a certain extent a similar pleasure may be found by absorbing ourselves in questions of pure geometry.
Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 350.
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The climate of Colorado contains more of the essential elements which effectively promote health than that of any other country. These requisites are found in the chemical composition of the atmosphere; in the dry, pure, clean, soft, yet stimulating breezes which quicken circulation and multiply the corpuscles of the blood; in the tonic effect and exhilarating influence of the ozone; in the flood of its life-giving germ-destroying sunshine …
From F.H. Faus, 'Pike’s Peak Region Calls', Rock Island Magazine (Aug 1920), 15, No. 8, 4.
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The Commissioner of Patents may be likened to a wine merchant. He has in his office the wine of human progress of every kind and quality—wine, one may say, produced from the fermentation of the facts of the world through the yeast of human effort. Sometimes the yeast is “wild” and sometimes the “must” is poor, and while it all lies there shining with its due measure of the sparkle of divine effort, it is but occasionally that one finds a wine whose bouquet is the result of a pure culture on the true fruit of knowledge. But it is this true, pure wine of discovery that is alone of lasting significance.
In Some Chemical Problems of Today (1911), 108.
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The concept of an independent system is a pure creation of the imagination. For no material system is or can ever be perfectly isolated from the rest of the world. Nevertheless it completes the mathematician’s “blank form of a universe” without which his investigations are impossible. It enables him to introduce into his geometrical space, not only masses and configurations, but also physical structure and chemical composition. Just as Newton first conclusively showed that this is a world of masses, so Willard Gibbs first revealed it as a world of systems.
The Order of Nature: An Essay (1917), 126.
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The Darwinian process of continued interplay of a random and a selective process is not intermediate between pure chance and pure determinism, but qualitatively utterly different from either in its consequences.
In 'Comments on the Preliminary Working Papers of Eden and Waddington'. In P. Moorhead and M. Kaplan (eds.), Mathematical Challenges to the Neo-Darwinian Interpretation of Evolution (1967), 117.
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The earth holds a silver treasure, cupped between ocean bed and tenting sky. Forever the heavens spend it, in the showers that refresh our temperate lands, the torrents that sluice the tropics. Every suckling root absorbs it, the very soil drains it down; the rivers run unceasing to the sea, the mountains yield it endlessly… Yet none is lost; in vast convection our water is returned, from soil to sky, and sky to soil, and back gain, to fall as pure as blessing. There was never less; there could never be more. A mighty mercy on which life depends, for all its glittering shifts, water is constant.
In A Cup of Sky (1950), 41.
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The enthusiasm of Sylvester for his own work, which manifests itself here as always, indicates one of his characteristic qualities: a high degree of subjectivity in his productions and publications. Sylvester was so fully possessed by the matter which for the time being engaged his attention, that it appeared to him and was designated by him as the summit of all that is important, remarkable and full of future promise. It would excite his phantasy and power of imagination in even a greater measure than his power of reflection, so much so that he could never marshal the ability to master his subject-matter, much less to present it in an orderly manner.
Considering that he was also somewhat of a poet, it will be easier to overlook the poetic flights which pervade his writing, often bombastic, sometimes furnishing apt illustrations; more damaging is the complete lack of form and orderliness of his publications and their sketchlike character, … which must be accredited at least as much to lack of objectivity as to a superfluity of ideas. Again, the text is permeated with associated emotional expressions, bizarre utterances and paradoxes and is everywhere accompanied by notes, which constitute an essential part of Sylvester’s method of presentation, embodying relations, whether proximate or remote, which momentarily suggested themselves. These notes, full of inspiration and occasional flashes of genius, are the more stimulating owing to their incompleteness. But none of his works manifest a desire to penetrate the subject from all sides and to allow it to mature; each mere surmise, conceptions which arose during publication, immature thoughts and even errors were ushered into publicity at the moment of their inception, with utmost carelessness, and always with complete unfamiliarity of the literature of the subject. Nowhere is there the least trace of self-criticism. No one can be expected to read the treatises entire, for in the form in which they are available they fail to give a clear view of the matter under contemplation.
Sylvester’s was not a harmoniously gifted or well-balanced mind, but rather an instinctively active and creative mind, free from egotism. His reasoning moved in generalizations, was frequently influenced by analysis and at times was guided even by mystical numerical relations. His reasoning consists less frequently of pure intelligible conclusions than of inductions, or rather conjectures incited by individual observations and verifications. In this he was guided by an algebraic sense, developed through long occupation with processes of forms, and this led him luckily to general fundamental truths which in some instances remain veiled. His lack of system is here offset by the advantage of freedom from purely mechanical logical activity.
The exponents of his essential characteristics are an intuitive talent and a faculty of invention to which we owe a series of ideas of lasting value and bearing the germs of fruitful methods. To no one more fittingly than to Sylvester can be applied one of the mottos of the Philosophic Magazine:
“Admiratio generat quaestionem, quaestio investigationem investigatio inventionem.”
In Mathematische Annalen (1898), 50, 155-160. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 176-178.
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The fact which interests us most is the life of the naturalist. The purest science is still biographical.
In A Week on the Concord and Merrimack Rivers (1862), 382.
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The feeling of it to my lungs was not sensibly different from that of common air; but I fancied that my breast felt peculiarly light and easy for some time afterwards. Who can tell but that, in time, this pure air may become a fashionable article in luxury. Hitherto only two mice and myself have had the privilege of breathing it.
Experiments and Observations on Different Kinds of Air (1775), Vol. 2, 102.
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The first quality we know in matter is centrality,—we call it gravity,—which holds the universe together, which remains pure and indestructible in each mote, as in masses and planets, and from each atom rays out illimitable influence. To this material essence answers Truth, in the intellectual world,—Truth, whose centre is everywhere, and its circumference nowhere, whose existence we cannot disimagine,—the soundness and health of things, against which no blow can be struck but it recoils on the striker,—Truth, on whose side we always heartily are. And the first measure of a mind is its centrality, its capacity of truth, and its adhesion to it.
In 'Progress of Culture', an address read to the Phi Beta Kappa Society at Cambridge, 18 July 1867. Collected in Works of Ralph Waldo Emerson (1883), 477.
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The first successes were such that one might suppose all the difficulties of science overcome in advance, and believe that the mathematician, without being longer occupied in the elaboration of pure mathematics, could turn his thoughts exclusively to the study of natural laws.
From Preface to Traité de calcul différentiel et de calcul intégral (1864-70), i. Quoted in address to the section of Algebra and Analysis, International Congress of Arts and Sciences, St. Louis (22 Sep 1904), 'On the Development of Mathematical Analysis and its Relation to Certain Other Sciences,' as translated by M.W. Haskell in Bulletin of the American Mathematical Society (May 1905), 11, 408.
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The flights of the imagination which occur to the pure mathematician are in general so much better described in his formulas than in words, that it is not remarkable to find the subject treated by outsiders as something essentially cold and uninteresting— … the only successful attempt to invest mathematical reasoning with a halo of glory—that made in this section by Prof. Sylvester—is known to a comparative few, …
In Presidential Address British Association for the Advancement of Science (1871), Nature Vol. 4, 271,
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The future of humanity is uncertain, even in the most prosperous countries, and the quality of life deteriorates; and yet I believe that what is being discovered about the infinitely large and the infinitely small is sufficient to absolve this end of the century and millennium. What a very few are acquiring in knowledge of the physical world will perhaps cause this period not to be judged as a pure return to barbarism.
In 'News from the Sky', Other People’s Trades (1989), 23-24.
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The great beauty of Darwin’s theory of evolution is that it explains how complex, difficult to understand things could have arisen step by plausible step, from simple, easy to understand beginnings. We start our explanation from almost infinitely simple beginnings: pure hydrogen and a huge amount of energy. Our scientific, Darwinian explanations carry us through a series of well-understood gradual steps to all the spectacular beauty and complexity of life.
From speech at the Edinburgh International Science Festival (15 Apr 1992), published in the Independent newspaper. Included in excerpt in Alec Fisher, The Logic of Real Arguments (2004), 84-85. The full speech was reprinted in The Nullifidian, (Dec 1994). Transcribed online in the Richard Dawkins archive, article 89, titled: Lecture from 'The Nullifidian' (Dec 94).
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The great truths with which it [mathematics] deals, are clothed with austere grandeur, far above all purposes of immediate convenience or profit. It is in them that our limited understandings approach nearest to the conception of that absolute and infinite, towards which in most other things they aspire in vain. In the pure mathematics we contemplate absolute truths, which existed in the divine mind before the morning stars sang together, and which will continue to exist there, when the last of their radiant host shall have fallen from heaven. They existed not merely in metaphysical possibility, but in the actual contemplation of the supreme reason. The pen of inspiration, ranging all nature and life for imagery to set forth the Creator’s power and wisdom, finds them best symbolized in the skill of the surveyor. "He meted out heaven as with a span;" and an ancient sage, neither falsely nor irreverently, ventured to say, that “God is a geometer”.
In Orations and Speeches (1870), Vol. 3, 614.
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The Greeks made Space the subject-matter of a science of supreme simplicity and certainty. Out of it grew, in the mind of classical antiquity, the idea of pure science. Geometry became one of the most powerful expressions of that sovereignty of the intellect that inspired the thought of those times. At a later epoch, when the intellectual despotism of the Church, which had been maintained through the Middle Ages, had crumbled, and a wave of scepticism threatened to sweep away all that had seemed most fixed, those who believed in Truth clung to Geometry as to a rock, and it was the highest ideal of every scientist to carry on his science “more geometrico.”
In Space,Time, Matter, translated by Henry Leopold Brose (1952), 1.
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The history of acceptance of new theories frequently shows the following steps: At first the new idea is treated as pure nonsense, not worth looking at. Then comes a time when a multitude of contradictory objections are raised, such as: the new theory is too fancy, or merely a new terminology; it is not fruitful, or simply wrong. Finally a state is reached when everyone seems to claim that he had always followed this theory. This usually marks the last state before general acceptance.
In 'Field Theory and the Phase Space', collected in Melvin Herman Marx, Psychological Theory: Contemporary Readings (1951), 299.
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The ideas which these sciences, Geometry, Theoretical Arithmetic and Algebra involve extend to all objects and changes which we observe in the external world; and hence the consideration of mathematical relations forms a large portion of many of the sciences which treat of the phenomena and laws of external nature, as Astronomy, Optics, and Mechanics. Such sciences are hence often termed Mixed Mathematics, the relations of space and number being, in these branches of knowledge, combined with principles collected from special observation; while Geometry, Algebra, and the like subjects, which involve no result of experience, are called Pure Mathematics.
In The Philosophy of the Inductive Sciences (1868), Part 1, Bk. 2, chap. 1, sect. 4.
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The influence of his [Leibnitz’s] genius in forming that peculiar taste both in pure and in mixed mathematics which has prevailed in France, as well as in Germany, for a century past, will be found, upon examination, to have been incomparably greater than that of any other individual.
In Elements of the Philosophy of the Human Mind (1827), Vol. 3, Chap. 1, Sec. 3, 187.
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The laws expressing the relations between energy and matter are, however, not solely of importance in pure science. They necessarily come first in order ... in the whole record of human experience, and they control, in the last resort, the rise or fall of political systems, the freedom or bondage of nations, the movements of commerce and industry, the origin of wealth and poverty, and the general physical welfare of the race.
In Matter and Energy (1912), 10-11.
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The mathematician is perfect only in so far as he is a perfect being, in so far as he perceives the beauty of truth; only then will his work be thorough, transparent, comprehensive, pure, clear, attractive and even elegant. All this is necessary to resemble Lagrange.
In Wilhelm Meister, Wanderjahre, Zweites Buch, in 'Sprüche in Prosa' Natur, VI, 950.
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The mathematician, carried along on his flood of symbols, dealing apparently with purely formal truths, may still reach results of endless importance for our description of the physical universe.
In The Grammar of Science (1900), 505.
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The mathematicians are well acquainted with the difference between pure science, which has only to do with ideas, and the application of its laws to the use of life, in which they are constrained to submit to the imperfections of matter and the influence of accidents.
In Samuel Johnson and W. Jackson Bate (Ed.), ',The Rambler, No. 14, Saturday, 5 May 1750.' The Selected Essays from the Rambler, Adventurer, and Idler (1968), 40.
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The mind comprehends a thing the more correctly the closer the thing approaches toward pure quantity as its origin.
Letter to Mästlin (19 Apr 1597). In Gerald James Holton, Thematic Origins of Scientific Thought: Kepler to Einstein (1985), 74.
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The Modern Theory of Functions—that stateliest of all the pure creations of the human intellect.
In Lectures on Science, Philosophy and Art (1908), 16.
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The more we study Art, the less we care for Nature. What Art really reveals to us is Nature’s lack of design, her curious crudities, her extraordinary monotony, her absolutely unfinished condition. … It is fortunate for us, however, that Nature is so imperfect, as otherwise we should have had no art at all. Art is our spirited protest, our gallant attempt to teach Nature her proper place. As for the infinite variety of Nature, that is a pure myth. It is not to be found in Nature herself. It resides in the imagination, or fancy, or cultivated blindness of the man who looks at her.
In 'Decay of Lying', The Writings of Oscar Wilde: Epigrams, Phrases and Philosophies For the Use of the Young (1907), 5.
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The nineteenth century which prides itself upon the invention of steam and evolution, might have derived a more legitimate title to fame from the discovery of pure mathematics.
In International Monthly (1901), 4, 83.
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The object of pure mathematics is those relations which may be conceptually established among any conceived elements whatsoever by assuming them contained in some ordered manifold; the law of order of this manifold must be subject to our choice; the latter is the case in both of the only conceivable kinds of manifolds, in the discrete as well as in the continuous.
In Über das System der rein mathematischen Wissenschaften, Jahresbericht der Deutschen Mathematiker-Vereinigung, Bd. 1, 36. As quoted and cited in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 3.
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The object of pure Physic is the unfolding of the laws of the intelligible world; the object of pure Mathematic that of unfolding the laws of human intelligence.
In 'On a theorem connected with Newton's Rule, etc.', Collected Mathematical Papers, 3, 424.
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The order, the symmetry, the harmony enchant us ... God is pure order. He is the originator of universal harmony
…...
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The peculiar taste both in pure and in mixed nature of those relations about which it is conversant, from its simple and definite phraseology, and from the severe logic so admirably displayed in the concatenation of its innumerable theorems, are indeed immense, and well entitled to separate and ample illustration.
In Philosophy of the Human Mind (1816), Vol. 2, Chap. 2, Sec. 3, 157.
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The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example, no mathematical truths apply merely to fish, or merely to stones, or merely to colours. So long as you are dealing with pure mathematics, you are in the realm of complete and absolute abstraction. … Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about.
In Science and the Modern World: Lowell Lectures, 1925 (1925), 31.
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The progress of science requires more than new data; it needs novel frameworks and contexts. And where do these fundamentally new views of the world arise? They are not simply discovered by pure observation; they require new modes of thought. And where can we find them, if old modes do not even include the right metaphors? The nature of true genius must lie in the elusive capacity to construct these new modes from apparent darkness. The basic chanciness and unpredictability of science must also reside in the inherent difficulty of such a task.
…...
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The pure culture is the foundation for all research on infectious disease.
'Zur Untersuchungen von Pathologen Organismen', Mittheilungen aus dem Kaiserlichen Gesundheitsamte (1881), 1, 1-48. Quoted in English in Thomas D. Brock, Robert Koch: A Life in Medicine and Bacteriology (1988), 94.
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The pure mathematician starts with an unknown and ends with an unknown.
Martin H. Fischer, Howard Fabing (ed.) and Ray Marr (ed.), Fischerisms (1944), 40.
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The pure mathematician, like the musician, is a free creator of his world of ordered beauty.
In A History of Western Philosophy (1945), 33.
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The pure scientist discovers the universe. The applied scientist exploits existing scientific discoveries to create a usable product.
In Jacques Cousteau and Susan Schiefelbein, The Human, the Orchid, and the Octopus: Exploring and Conserving Our Natural World (2007), 181.
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The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience.
In Immanuel Kant and F. Max Müller (trans.), 'Method of Transcendentalism', Critique of Pure Reason (1881), Vol. 2, 610.
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The science of pure mathematics … may claim to be the most original creation of the human spirit.
In Science and the Modern World: Lowell Lectures, 1925 (1925), 29.
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The sky seems to be a pure, a cooler blue, the trees a deeper green. The whole world is charged with the glory of God and I feel fire and music under my feet.
On reading the scriptures. Quoted in Kim Lim (ed.), 1,001 Pearls of Spiritual Wisdom: Words to Enrich, Inspire, and Guide Your Life (2014), 166
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The steam-engine in its manifold applications, the crime-decreasing gas-lamp, the lightning conductor, the electric telegraph, the law of storms and rules for the mariner's guidance in them, the power of rendering surgical operations painless, the measures for preserving public health, and for preventing or mitigating epidemics,—such are among the more important practical results of pure scientific research, with which mankind have been blessed and States enriched.
President's Address to the British Association, Leeds (1858). In Charles W. Vincent and James Mason (eds.), The Year-book of Facts in Science and Art (1859), title page.
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The study of economics does not seem to require any specialised gifts of an unusually high order. Is it not, intellectually regarded, a very easy subject compared with the higher branches of philosophy and pure science? Yet good, or even competent, economists are the rarest of birds. An easy subject, at which very few excel! The paradox finds its explanation, perhaps, in that the master-economist must possess a rare combination of gifts. He must reach a high standard in several different directions and must combine talents not often found together. He must be mathematician, historian, statesman, philosopher—in some degree. He must understand symbols and speak in words. He must contemplate the particular in terms of the general, and touch abstract and concrete in the same flight of thought. He must study the present in the light of the past for the purposes of the future. No part of man's nature or his institutions must lie entirely outside his regard. He must be purposeful and disinterested in a simultaneous mood; as aloof and incorruptible as an artist, yet sometimes as near the earth as a politician.
'Alfred Marshall: 1842-1924' (1924). In Geoffrey Keynes (ed.), Essays in Biography (1933), 170.
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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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The tendency of the sciences has long been an increasing proclivity of separation and dismemberment … The mathematician turns away from the chemist; the chemist from the naturalist; the mathematician, left to himself divides himself into a pure mathematician and a mixed mathematician, who soon part company … And thus science, even mere physical science, loses all traces of unity. A curious illustration of this result may be observed in the want of any name by which we can designate the students of the knowledge of the material world collectively. We are informed that this difficulty was felt very oppressively by the members of the British Association for the Advancement of Science, at their meetings at York, Oxford and Cambridge, in the last three summers. There was no general term by which these gentlemen could describe themselves with reference to their pursuits … some ingenious gentleman [William Whewell] proposed that, by analogy with artist, they might form Scientist, and added that there could be no scruple … when we have words such as sciolist, economist, and atheist—but this was not generally palatable.
In Review of Mrs Somerville, 'On the Connexion of the Physical Sciences', The Quarterly Review (1834), 51, 58-61.
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The theoretical side of physical chemistry is and will probably remain the dominant one; it is by this peculiarity that it has exerted such a great influence upon the neighboring sciences, pure and applied, and on this ground physical chemistry may be regarded as an excellent school of exact reasoning for all students of the natural sciences.
In Theories of Solutions (1912), xx.
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The theory of ramification is one of pure colligation, for it takes no account of magnitude or position; geometrical lines are used, but these have no more real bearing on the matter than those employed in genealogical tables have in explaining the laws of procreation.
From 'On Recent Discoveries in Mechanical Conversion of Motion', Proceedings of the Royal Institution of Great Britain (1873-75), 7, 179-198, reprinted in The Collected Mathematical Papers of James Joseph Sylvester: (1870-1883) (1909), Vol. 3, 23.
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The things I want to express are so beautiful and pure.
As quoted, without citation, on the mcescher.com website.
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The totality of our so-called knowledge or beliefs, from the most casual matters of geography and history to the profoundest laws of atomic physics or even of pure mathematics and logic, is a man-made fabric which impinges on experience only along the edges. Or, to change the figure, total science is like a field of force whose boundary conditions are experience. A conflict with experience at the periphery occasions readjustments in the interior of the field. Truth values have to be redistributed over some of our statements. Reevaluation of some statements entails reevaluation of others, because of their logical interconnections—the logical laws being in turn simply certain further statements of the system, certain further elements of the field.
'Two Dogmas of Experience,' in Philosophical Review (1951). Reprinted in From a Logical Point of View (1953), 42.
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The Universe can be pictured, although still very imperfectly and inadequately, as consisting of pure thought, the thought of what for want of a wider word, we must describe as a mathematical thinker.
In The Mysterious Universe (1930, Rev. Ed. 1942), 168.
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The value of fundamental research does not lie only in the ideas it produces. There is more to it. It affects the whole intellectual life of a nation by determining its way of thinking and the standards by which actions and intellectual production are judged. If science is highly regarded and if the importance of being concerned with the most up-to-date problems of fundamental research is recognized, then a spiritual climate is created which influences the other activities. An atmosphere of creativity is established which penetrates every cultural frontier. Applied sciences and technology are forced to adjust themselves to the highest intellectual standards which are developed in the basic sciences. This influence works in many ways: some fundamental students go into industry; the techniques which are applied to meet the stringent requirements of fundamental research serve to create new technological methods. The style, the scale, and the level of scientific and technical work are determined in pure research; that is what attracts productive people and what brings scientists to those countries where science is at the highest level. Fundamental research sets the standards of modern scientific thought; it creates the intellectual climate in which our modern civilization flourishes. It pumps the lifeblood of idea and inventiveness not only into the technological laboratories and factories, but into every cultural activity of our time. The case for generous support for pure and fundamental science is as simple as that.
In 'Why Pure Science?' in Bulletin of the Atomic Scientists, 1965.
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The visible figures by which principles are illustrated should, so far as possible, have no accessories. They should be magnitudes pure and simple, so that the thought of the pupil may not be distracted, and that he may know what features of the thing represented he is to pay attention to.
In National Education Association of the United States, Report of the Committee of Ten on Secondary School Subjects, (1894), 109.
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The way of pure research is opposed to all the copy-book maxims concerning the virtues of industry and a fixed purpose, and the evils of guessing, but it is damned useful when it comes off. It is the diametrical opposite of Edison’s reputed method of trying every conceivable expedient until he hit the right one. It requires, not diligence, but experience, information, and a good nose for the essence of a problem.
Letter to Paul de Kruif (3 Aug 1933), as quoted in Nathan Reingold, Science in America: A Documentary History 1900-1939 (1981), 409.
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The whole subject of the X rays is opening out wonderfully, Bragg has of course got in ahead of us, and so the credit all belongs to him, but that does not make it less interesting. We find that an X ray bulb with a platinum target gives out a sharp line spectrum of five wavelengths which the crystal separates out as if it were a diffraction grating. In this way one can get pure monochromatic X rays. Tomorrow we search for the spectra of other elements. There is here a whole new branch of spectroscopy, which is sure to tell one much about the nature of an atom.
Letter to his mother (18 May 1913). In J. L. Heilbron (ed.), H. G. J. Moseley: The Life and Letters of an English Physicist 1887-1915 (1974), 205.
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The world of mathematics, which you condemn, is really a beautiful world; it has nothing to do with life and death and human sordidness, but is eternal, cold and passionless. To me, pure, mathematics is one of the highest forms of art; it has a sublimity quite special to itself, and an immense dignity derived, from the fact that its world is exempt I, from change and time. I am quite serious in this. The only difficulty is that none but mathematicians can enter this enchanted region, and they hardly ever have a sense of beauty. And mathematics is the only thing we know of that is capable of perfection; in thinking about it we become Gods.
Letter to Helen Thomas (30 Dec 1901). Quoted in Nicholas Griffin (ed.), The Selected Letters of Bertrand Russell (1992), Vol. 1, 224.
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There are but few saints among scientists, as among other men, but truth itself is a goal comparable to sanctity. As the Pythagoreans had already understood it more than twenty-four centuries ago, there is sanctity in pure knowledge, as there is in pure beauty, and the disinterested quest of truth is perhaps the greatest purification.
In The History of Science and the New Humanism (1987), 46-47.
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There are those who say that the human kidney was created to keep the blood pure, or more precisely, to keep our internal environment in an ideal balanced state. This I must deny. I grant that the human kidney is a marvelous organ, but I cannot grant that it was purposefully designed to excrete urine or to regulate the composition of the blood or to subserve the physiological welfare of Homo sapiens in any sense. Rather I contend that the human kidney manufactures the kind of urine that it does, and it maintains the blood in the composition which that fluid has, because this kidney has a certain functional architecture; and it owes that architecture not to design or foresight or to any plan, but to the fact that the earth is an unstable sphere with a fragile crust, to the geologic revolutions that for six hundred million years have raised and lowered continents and seas, to the predacious enemies, and heat and cold, and storms and droughts; to the unending succession of vicissitudes that have driven the mutant vertebrates from sea into fresh water, into desiccated swamps, out upon the dry land, from one habitation to another, perpetually in search of the free and independent life, perpetually failing, for one reason or another, to find it.
From Fish to Philosopher (1953), 210-1.
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There are, at present, fundamental problems in theoretical physics … the solution of which … will presumably require a more drastic revision of our fundmental concepts than any that have gone before. Quite likely, these changes will be so great that it will be beyond the power of human intelligence to get the necessary new ideas by direct attempts to formulate the experimental data in mathematical terms. The theoretical worker in the future will, therefore, have to proceed in a more direct way. The most powerful method of advance that can be suggested at present is to employ all the resources of pure mathematics in attempts to perfect and generalize the mathematical formalism that forms the existing basis of theoretical physics, and after each success in this direction, to try to interpret the new mathematical features in terms of physical entities.
At age 28.
Proceedings of the Royal Society (1931), A133, 60. In A. Pais, 'Playing With Equations, the Dirac Way'. Behram N. Kursunoglu (Ed.) and Eugene Paul Wigner (Ed.), Paul Adrien Maurice Dirac: Reminiscences about a Great Physicist (1990), 109.
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There is no “pure” science itself divorced from human values. The importance of science to the humanities and the humanities to science in their complementary contribution to the variety of human life grows daily. The need for men familiar with both is imperative.
In 'Abstract' The Impurity of Science (19 Apr 1962), the printed version of the Robbins Lecture (27 Feb 1962) given at Pomona College, Claremont, California, as published by Ernest O. Lawrence Radiation Laboratory, University of California.
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There is no drink like pure water, provided one realizes that it is alcohol that is the purifying agent.
Aphorism as given by the fictional character Dezhnev Senior, in Fantastic Voyage II: Destination Brain (1987), 220.
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There is no way to guarantee in advance what pure mathematics will later find application. We can only let the process of curiosity and abstraction take place, let mathematicians obsessively take results to their logical extremes, leaving relevance far behind, and wait to see which topics turn out to be extremely useful. If not, when the challenges of the future arrive, we won’t have the right piece of seemingly pointless mathematics to hand.
In 'The Unplanned Impact of Mathematics', Nature (14 Jul 2011), 475, No. 7355, 167.
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There is not a “pure” science. By this I mean that physics impinges on astronomy, on the one hand, and chemistry and biology on the other. And not only does each support neighbors, but derives sustenance from them. The same can be said of chemistry. Biology is, perhaps, the example par excellence today of an “impure” science.
In 'Abstract' The Impurity of Science (19 Apr 1962), the printed version of the Robbins Lecture (27 Feb 1962) given at, Pomona College, Claremont, California, as published by Ernest O. Lawrence Radiation Laboratory, University of California.
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There is only one proved method of assisting the advancement of pure science—that of picking men of genius, backing them heavily, and leaving them to direct themselves.
In a letter to The New York Times, August 13, 1945.
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These Disciplines [mathematics] serve to inure and corroborate the Mind to a constant Diligence in Study; to undergo the Trouble of an attentive Meditation, and cheerfully contend with such Difficulties as lie in the Way. They wholly deliver us from a credulous Simplicity, most strongly fortify us against the Vanity of Scepticism, effectually restrain from a rash Presumption, most easily incline us to a due Assent, perfectly subject us to the Government of right Reason, and inspire us with Resolution to wrestle against the unjust Tyranny of false Prejudices. If the Fancy be unstable and fluctuating, it is to be poized by this Ballast, and steadied by this Anchor, if the Wit be blunt it is sharpened upon this Whetstone; if luxuriant it is pared by this Knife; if headstrong it is restrained by this Bridle; and if dull it is rouzed by this Spur. The Steps are guided by no Lamp more clearly through the dark Mazes of Nature, by no Thread more surely through the intricate Labyrinths of Philosophy, nor lastly is the Bottom of Truth sounded more happily by any other Line. I will not mention how plentiful a Stock of Knowledge the Mind is furnished from these, with what wholesome Food it is nourished, and what sincere Pleasure it enjoys. But if I speak farther, I shall neither be the only Person, nor the first, who affirms it; that while the Mind is abstracted and elevated from sensible Matter, distinctly views pure Forms, conceives the Beauty of Ideas, and investigates the Harmony of Proportions; the Manners themselves are sensibly corrected and improved, the Affections composed and rectified, the Fancy calmed and settled, and the Understanding raised and excited to more divine Contemplations. All which I might defend by Authority, and confirm by the Suffrages of the greatest Philosophers.
Prefatory Oration in Mathematical Lectures (1734), xxxi.
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These machines [used in the defense of the Syracusans against the Romans under Marcellus] he [Archimedes] had designed and contrived, not as matters of any importance, but as mere amusements in geometry; in compliance with king Hiero’s desire and request, some time before, that he should reduce to practice some part of his admirable speculation in science, and by accommodating the theoretic truth to sensation and ordinary use, bring it more within the appreciation of people in general. Eudoxus and Archytas had been the first originators of this far-famed and highly-prized art of mechanics, which they employed as an elegant illustration of geometrical truths, and as means of sustaining experimentally, to the satisfaction of the senses, conclusions too intricate for proof by words and diagrams. As, for example, to solve the problem, so often required in constructing geometrical figures, given the two extremes, to find the two mean lines of a proportion, both these mathematicians had recourse to the aid of instruments, adapting to their purpose certain curves and sections of lines. But what with Plato’s indignation at it, and his invectives against it as the mere corruption and annihilation of the one good of geometry,—which was thus shamefully turning its back upon the unembodied objects of pure intelligence to recur to sensation, and to ask help (not to be obtained without base supervisions and depravation) from matter; so it was that mechanics came to be separated from geometry, and, repudiated and neglected by philosophers, took its place as a military art.
Plutarch
In John Dryden (trans.), Life of Marcellus.
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These specimens, which I could easily multiply, may suffice to justify a profound distrust of Auguste Comte, wherever he may venture to speak as a mathematician. But his vast general ability, and that personal intimacy with the great Fourier, which I most willingly take his own word for having enjoyed, must always give an interest to his views on any subject of pure or applied mathematics.
In R. Graves, Life of W. R. Hamilton (1882-89), Vol. 3, 475.
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This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus!
Ignorabimus as used here, means “we will not know” (which is slightly different from ignoramus meaning present ignorance, “we do not know”). In Lecture (1900), 'Mathematische Probleme' (Mathematical Problems), to the International Congress of Mathematicians, Paris. From the original German reprinted in David Hilbert: Gesammelte Abhandlungen (Collected Treatises, 1970), Vol. 3, 298, “Diese Überzeugung von der Lösbarkeit eines jeden mathematischer Problems ist uns ein kräftiger Ansporn während der Arbeit ; wir hören in uns den steten Zuruf: Da ist das Problem, suche die Lösung. Du kannst sie durch reines Denken finden; denn in der Mathematik gibt es kein Ignorabimus. English version as translated by Dr. Maby Winton Newson for Bulletin of the American Mathematical Society (1902), 8, 437-479. The address was first published in Göttinger Nachrichten is Nachrichten von der Königl. Gesellschaft der Wiss. zu Göttingen (1900), 253-297; and Archiv der Mathematik und Physik (1901), 3, No. 1, 44-63.
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This example illustrates the differences in the effects which may be produced by research in pure or applied science. A research on the lines of applied science would doubtless have led to improvement and development of the older methods—the research in pure science has given us an entirely new and much more powerful method. In fact, research in applied science leads to reforms, research in pure science leads to revolutions, and revolutions, whether political or industrial, are exceedingly profitable things if you are on the winning side.
In Lord Rayleigh, The Life of Sir J. J. Thomson (1943), 199
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This interpretation of the atomic number [as the number of orbital electrons] may be said to signify an important step toward the solution of the boldest dreams of natural science, namely to build up an understanding of the regularities of nature upon the consideration of pure number.
Atomic Theory and the Description of Nature (1934), 103-104Cited in Gerald James Holton, Thematic Origins of Scientific Thought: Kepler to Einstein (1985), 74.
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This pure species of air [oxygen] has the property of combining with the blood and … this combination constitutes its red colour.
From 'Expériences sur la respiration des animaux, et sur les changemens qui arrivent à l’air en passant par leur poumon', Histoire de l’Académie Royale des Sciences for 1777 (1780) as translated by Thomas Henry in 'Experiments on the Respiration of Animals on the Changes effected on the Air passing through their Lungs', Essays, on the Effects Produced by Various Processes on Atmospheric Air, etc. (1783), 13-14. Also in John F. Fulton, Selected Readings in the History of Physiology (1930), 125.
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This trend [emphasizing applied mathematics over pure mathematics] will make the queen of the sciences into the quean of the sciences.
As given, without citation, in Howard W. Eves Mathematical Circles Squared (1972), 158, which attributes it (via Dirk J. Struik) to a memorandum in which Passano wrote of the trend in the Dept. of Mathematics at M.I.T. Webmaster has as yet been unable to identify a primary source. (Can you help?) [Note: “quean” is an archaic word for: a disreputable woman; specifically : prostitute.—Merriam-Webster. “Thus the semantic spread between queen and quean could not be greater: from a woman of the highest repute to one of the lowest.” —alphadictionary.com]
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To appreciate a work of art we need bring with us nothing from life, no knowledge of its ideas and affairs, no familiarity with its emotions. Art transports us from the world of man’s activity to a world of æsthetic exaltation. For a moment we are shut off from human interests; our anticipations and memories are arrested; we are lifted above the stream of life. The pure mathematician rapt in his studies knows a state of mind which I take to be similar, if not identical. He feels an emotion for his speculations which arises from no perceived relation between them and the lives of men, but springs, inhuman or super-human, from the heart of an abstract science. I wonder, sometimes, whether the appreciators of art and of mathematical solutions are not even more closely allied.
In Art (1913), 25.
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To characterize the import of pure geometry, we might use the standard form of a movie-disclaimer: No portrayal of the characteristics of geometrical figures or of the spatial properties of relationships of actual bodies is intended, and any similarities between the primitive concepts and their customary geometrical connotations are purely coincidental.
From 'Geometry and Empirical Science', collected in Carl Hempel and James H. Fetzer (ed.), The Philosophy of Carl G. Hempel: Studies in Science, Explanation, and Rationality (2001), Chap. 2, 24. Also Carl Hempel, 'Geometry and Empirical Science', collected in J.R. Newman (ed.), The World of Mathematics (1956), Vol. 3, 1641.
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To emphasize this opinion that mathematicians would be unwise to accept practical issues as the sole guide or the chief guide in the current of their investigations, ... let me take one more instance, by choosing a subject in which the purely mathematical interest is deemed supreme, the theory of functions of a complex variable. That at least is a theory in pure mathematics, initiated in that region, and developed in that region; it is built up in scores of papers, and its plan certainly has not been, and is not now, dominated or guided by considerations of applicability to natural phenomena. Yet what has turned out to be its relation to practical issues? The investigations of Lagrange and others upon the construction of maps appear as a portion of the general property of conformal representation; which is merely the general geometrical method of regarding functional relations in that theory. Again, the interesting and important investigations upon discontinuous two-dimensional fluid motion in hydrodynamics, made in the last twenty years, can all be, and now are all, I believe, deduced from similar considerations by interpreting functional relations between complex variables. In the dynamics of a rotating heavy body, the only substantial extension of our knowledge since the time of Lagrange has accrued from associating the general properties of functions with the discussion of the equations of motion. Further, under the title of conjugate functions, the theory has been applied to various questions in electrostatics, particularly in connection with condensers and electrometers. And, lastly, in the domain of physical astronomy, some of the most conspicuous advances made in the last few years have been achieved by introducing into the discussion the ideas, the principles, the methods, and the results of the theory of functions. … the refined and extremely difficult work of Poincare and others in physical astronomy has been possible only by the use of the most elaborate developments of some purely mathematical subjects, developments which were made without a thought of such applications.
In Presidential Address British Association for the Advancement of Science, Section A, (1897), Nature, 56, 377.
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To judge in this [utilitarian] way demonstrates … how small, narrow and indolent our minds are; it shows a disposition always to calculate the reward before the work, a cold heart and a lack of feeling for everything that is great and honours mankind. Unfortunately one cannot deny that such a mode of thinking is common in our age, and I am convinced that this is closely connected with the catastrophes which have befallen many countries in recent times; do not mistake me, I do not talk of the general lack of concern for science, but of the source from which all this has come, of the tendency to look out everywhere for one’s advantage and to relate everything to one’s physical well being, of indifference towards great ideas, of aversion to any effort which derives from pure enthusiasm.
Epigraph before title page, T. W. Körner, The Pleasures of Counting (1996).
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To prove that tuberculosis is caused by the invasion of bacilli, and that it is a parasitic disease primarily caused by the growth and multiplication of bacilli, it is necessary to isolate the bacilli from the body, to grow them in pure culture until they are freed from every disease product of the animal organism, and, by introducing isolated bacilli into animals, to reproduce the same morbid condition that is known to follow from inoculation with spontaneously developed tuberculous material.
'The Etiology of Tuberculosis' (1882), Essays of Robert Koch (1987), trans. K. Codell Carter, 87.
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To the pure geometer the radius of curvature is an incidental characteristic—like the grin of the Cheshire cat. To the physicist it is an indispensable characteristic. It would be going too far to say that to the physicist the cat is merely incidental to the grin. Physics is concerned with interrelatedness such as the interrelatedness of cats and grins. In this case the “cat without a grin” and the “grin without a cat” are equally set aside as purely mathematical phantasies.
In 'The Universe and the Atom' The Expanding Universe. (1933), 103.
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Today, when so much depends on our informed action, we as voters and taxpayers can no longer afford to confuse science and technology, to confound “pure” science and “applied” science.
In Jacques Cousteau and Susan Schiefelbein, The Human, the Orchid, and the Octopus: Exploring and Conserving Our Natural World (2007), 181.
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Twin sister of natural and revealed religion, and of heavenly birth, science will never belie her celestial origin, nor cease to sympathize with all that emanates from the same pure home. Human ignorance and prejudice may for a time seem to have divorced what God has joined together; but human ignorance and prejudice shall at length pass away, and then science and religion shall be seen blending their particolored rays into one beautiful bow of light, linking heaven to earth and earth to heaven.
…...
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Two extreme views have always been held as to the use of mathematics. To some, mathematics is only measuring and calculating instruments, and their interest ceases as soon as discussions arise which cannot benefit those who use the instruments for the purposes of application in mechanics, astronomy, physics, statistics, and other sciences. At the other extreme we have those who are animated exclusively by the love of pure science. To them pure mathematics, with the theory of numbers at the head, is the only real and genuine science, and the applications have only an interest in so far as they contain or suggest problems in pure mathematics.
Of the two greatest mathematicians of modern tunes, Newton and Gauss, the former can be considered as a representative of the first, the latter of the second class; neither of them was exclusively so, and Newton’s inventions in the science of pure mathematics were probably equal to Gauss’s work in applied mathematics. Newton’s reluctance to publish the method of fluxions invented and used by him may perhaps be attributed to the fact that he was not satisfied with the logical foundations of the Calculus; and Gauss is known to have abandoned his electro-dynamic speculations, as he could not find a satisfying physical basis. …
Newton’s greatest work, the Principia, laid the foundation of mathematical physics; Gauss’s greatest work, the Disquisitiones Arithmeticae, that of higher arithmetic as distinguished from algebra. Both works, written in the synthetic style of the ancients, are difficult, if not deterrent, in their form, neither of them leading the reader by easy steps to the results. It took twenty or more years before either of these works received due recognition; neither found favour at once before that great tribunal of mathematical thought, the Paris Academy of Sciences. …
The country of Newton is still pre-eminent for its culture of mathematical physics, that of Gauss for the most abstract work in mathematics.
In History of European Thought in the Nineteenth Century (1903), 630.
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We are concerned to understand the motivation for the development of pure mathematics, and it will not do simply to point to aesthetic qualities in the subject and leave it at that. It must be remembered that there is far more excitement to be had from creating something than from appreciating it after it has been created. Let there be no mistake about it, the fact that the mathematician is bound down by the rules of logic can no more prevent him from being creative than the properties of paint can prevent the artist. … We must remember that the mathematician not only finds the solutions to his problems, he creates the problems themselves.
In A Signpost to Mathematics (1951), 19. As quoted and cited in William L. Schaaf, 'Memorabilia Mathematica', The Mathematics Teacher (Mar 1957), 50, No. 3, 230. Note that this paper incorrectly attributes “A.H. Head”.
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We have already considered with disfavour the possibility of the universe having been planned by a biologist or an engineer; from the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician.
The Mysterious Universe (1930), 134.
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We may see how unexpectedly recondite parts of pure mathematics may bear upon physical science, by calling to mind the circumstance that Fresnel obtained one of the most curious confirmations of the theory (the laws of Circular Polarization by reflection) through an interpretation of an algebraical expression, which, according to the original conventional meaning of the symbols, involved an impossible quantity.
In History of Scientific Ideas, Bk. 2, chap. 14, sect. 8.
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We must not forget that when radium was discovered no one knew that it would prove useful in hospitals. The work was one of pure science. And this is a proof that scientific work must not be considered from the point of view of the direct usefulness of it. It must be done for itself, for the beauty of science, and then there is always the chance that a scientific discovery may become like the radium a benefit for humanity.
Lecture at Vassar College, Poughkeepsie, New York (14 May 1921). In Cambridge Editorial Partnership, Speeches that Changed the World, 53.
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We think of Euclid as of fine ice; we admire Newton as we admire the peak of Teneriffe. Even the intensest labors, the most remote triumphs of the abstract intellect, seem to carry us into a region different from our own—to be in a terra incognita of pure reasoning, to cast a chill on human glory.
In Estimates of Some Englishmen and Scotchmen (1856), 411-412
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We think the heavens enjoy their spherical
Their round proportion, embracing all;
But yet their various and perplexed course,
Observed in divers ages, doth enforce
Men to find out so many eccentric parts,
Such diverse downright lines, such overthwarts,
As disproportion that pure form.
From poem, 'An Anatomy of the World: The First Anniversary', lines 251-257, as collected in The Poems of John Donne (1896), Vol. 2, 113.
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What is peculiar and new to the [19th] century, differentiating it from all its predecessors, is its technology. It was not merely the introduction of some great isolated inventions. It is impossible not to feel that something more than that was involved. … The process of change was slow, unconscious, and unexpected. In the nineteeth century, the process became quick, conscious, and expected. … The whole change has arisen from the new scientific information. Science, conceived not so much in its principles as in its results, is an obvious storehouse of ideas for utilisation. … Also, it is a great mistake to think that the bare scientific idea is the required invention, so that it has only to be picked up and used. An intense period of imaginative design lies between. One element in the new method is just the discovery of how to set about bridging the gap between the scientific ideas, and the ultimate product. It is a process of disciplined attack upon one difficulty after another This discipline of knowledge applies beyond technology to pure science, and beyond science to general scholarship. It represents the change from amateurs to professionals. … But the full self-conscious realisation of the power of professionalism in knowledge in all its departments, and of the way to produce the professionals, and of the importance of knowledge to the advance of technology, and of the methods by which abstract knowledge can be connected with technology, and of the boundless possibilities of technological advance,—the realisation of all these things was first completely attained in the nineteeth century.
In Science and the Modern World (1925, 1997), 96.
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What is this subject, which may be called indifferently either mathematics or logic? Is there any way in which we can define it? Certain characteristics of the subject are clear. To begin with, we do not, in this subject, deal with particular things or particular properties: we deal formally with what can be said about any thing or any property. We are prepared to say that one and one are two, but not that Socrates and Plato are two, because, in our capacity of logicians or pure mathematicians, we have never heard of Socrates or Plato. A world in which there were no such individuals would still be a world in which one and one are two. It is not open to us, as pure mathematicians or logicians, to mention anything at all, because, if we do so we introduce something irrelevant and not formal.
In Introduction to Mathematical Philosophy (1920), 196-197.
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What led me to my science and what fascinated me from a young age was the, by no means self-evident, fact that our laws of thought agree with the regularities found in the succession of impressions we receive from the external world, that it is thus possible for the human being to gain enlightenment regarding these regularities by means of pure thought
Max Planck and Ch. Scriba (ed), Wissenschaftliche Selbstbiographie (1990), 9. Quoted in Erhard Scheibe and Brigitte Falkenburg (ed), Between Rationalism and Empiricism: Selected Papers in the Philosophy of Physics (2001), 69
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Whatever advantage can be attributed to logic in directing and strengthening the action of the understanding is found in a higher degree in mathematical study, with the immense added advantage of a determinate subject, distinctly circumscribed, admitting of the utmost precision, and free from the danger which is inherent in all abstract logic—of leading to useless and puerile rules, or to vain ontological speculations. The positive method, being everywhere identical, is as much at home in the art of reasoning as anywhere else: and this is why no science, whether biology or any other, can offer any kind of reasoning, of which mathematics does not supply a simpler and purer counterpart. Thus, we are enabled to eliminate the only remaining portion of the old philosophy which could even appear to offer any real utility; the logical part, the value of which is irrevocably absorbed by mathematical science.
In Auguste Comte and Harriet Martineau (trans.), Positive Philosophy (1858), Vol. 1, 326-327.
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When the logician has resolved each demonstration into a host of elementary operations, all of them correct, he will not yet be in possession of the whole reality, that indefinable something that constitutes the unity ... Now pure logic cannot give us this view of the whole; it is to intuition that we must look for it.
Science and Method (1914 edition, reprint 2003), 126.
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When we are motivated by goals that have deep meaning, by dreams that need completion, by pure love that needs expressing, then we truly live life.
…...
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When we say “science” we can either mean any manipulation of the inventive and organizing power of the human intellect: or we can mean such an extremely different thing as the religion of science, the vulgarized derivative from this pure activity manipulated by a sort of priestcraft into a great religious and political weapon.
'The Art of Being Ruled'. Revolution and Progress (1926), 4.
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While, on the one hand, the end of scientific investigation is the discovery of laws, on the other, science will have reached its highest goal when it shall have reduced ultimate laws to one or two, the necessity of which lies outside the sphere of our cognition. These ultimate laws—in the domain of physical science at least—will be the dynamical laws of the relations of matter to number, space, and time. The ultimate data will be number, matter, space, and time themselves. When these relations shall be known, all physical phenomena will be a branch of pure mathematics.
'Address to the section of Mathematical and Physical Science', Reports of the British Association for the Advancement of Science (1895), 595.
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Why do I call [Isaac Newton] a magician? Because he looked on the whole universe and all that is in it as a riddle, as a secret which could be read by applying pure thought to certain evidence, certain mystic clues which God had laid about the world to allow a sort of philosopher's treasure hunt.
In 'Newton, the Man' (1946). In Geoffrey Keynes (ed.), Essays in Biography, 2nd edition (1951), 313.
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Without my attempts in natural science, I should never have learned to know mankind such as it is. In nothing else can we so closely approach pure contemplation and thought, so closely observe the errors of the senses and of the understanding, the weak and strong points of character.
Fri 13 Feb 1829. Johann Peter Eckermann, Conversations with Goethe, ed. J. K. Moorhead and trans. J. Oxenford (1971), 293.
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Carl Sagan Thumbnail In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. (1987) -- Carl Sagan
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- 90 -
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- 80 -
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