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Who said: “A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.”
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Home > Category Index for Science Quotations > Category Index D > Category: Demonstration

Demonstration Quotes (81 quotes)

La determination de la relation & de la dépendance mutuelle de ces données dans certains cas particuliers, doit être le premier but du Physicien; & pour cet effet, il falloit one mesure exacte qui indiquât d’une manière invariable & égale dans tous les lieux de la terre, le degré de l'électricité au moyen duquel les expéiences ont été faites… Aussi, l'histoire de l'électricité prouve une vérité suffisamment reconnue; c'est que le Physicien sans mesure ne fait que jouer, & qu'il ne diffère en cela des enfans, que par la nature de son jeu & la construction de ses jouets.
The determination of the relationship and mutual dependence of the facts in particular cases must be the first goal of the Physicist; and for this purpose he requires that an exact measurement may be taken in an equally invariable manner anywhere in the world… Also, the history of electricity yields a well-known truth—that the physicist shirking measurement only plays, different from children only in the nature of his game and the construction of his toys.
'Mémoire sur la mesure de force de l'électricité', Journal de Physique (1782), 21, 191. English version by Google Translate tweaked by Webmaster.
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After having a wash I proceeded to the bar where—believe it or not—there was a white-coated barman who was not only serving drinks but also cigarettes! I hastened forward and rather timidly said ‘Can I have some cigarettes?’
‘What’s your rank?’ was the slightly unexpected reply.
‘I am afraid I haven’t got one,’ I answered.
‘Nonsense—everyone who comes here has a rank.’
‘I’m sorry but I just don’t have one.’
‘Now that puts me in a spot,’ said the barman, ‘for orders about cigarettes in this camp are clear—twenty for officers and ten for other ranks. Tell me what exactly are you?’
Now I really wanted those cigarettes so I drew myself up and said ‘I am the Professor of Chemistry at Manchester University.’
The barman contemplated me for about thirty seconds and then said ‘I’ll give you five.’
Since that day I have had few illusions about the importance of professors!
In A Time to Remember: The Autobiography of a Chemist (1983), 59. This event took place after a visit to the Defence Research Establishment at Porton to observe a demonstration of a new chemical anti-tank weapon (1941).
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Any time you wish to demonstrate something, the number of faults is proportional to the number of viewers.
Anonymous
Bye's First Law of Model Railroading. In Paul Dickson, The Official Rules, (1978), 23.
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Archimedes … had stated that given the force, any given weight might be moved, and even boasted, we are told, relying on the strength of demonstration, that if there were another earth, by going into it he could remove this. Hiero being struck with amazement at this, and entreating him to make good this problem by actual experiment, and show some great weight moved by a small engine, he fixed accordingly upon a ship of burden out of the king’s arsenal, which could not be drawn out of the dock without great labor and many men; and, loading her with many passengers and a full freight, sitting himself the while far off with no great endeavor, but only holding the head of the pulley in his hand and drawing the cords by degrees, he drew the ship in a straight line, as smoothly and evenly, as if she had been in the sea. The king, astonished at this, and convinced of the power of the art, prevailed upon Archimedes to make him engines accommodated to all the purposes, offensive and defensive, of a siege. … the apparatus was, in most opportune time, ready at hand for the Syracusans, and with it also the engineer himself.
Plutarch
In John Dryden (trans.), Life of Marcellus.
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Chlorine is a poisonous gas. In case I should fall over unconscious in the following demonstration involving chlorine, please pick me up and carry me into the open air. Should this happen, the lecture for the day will be concluded.
Quoted in Ralph Oesper, The Human Side of Scientists (1975), 192.
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Considering that, among all those who up to this time made discoveries in the sciences, it was the mathematicians alone who had been able to arrive at demonstrations—that is to say, at proofs certain and evident—I did not doubt that I should begin with the same truths that they have investigated, although I had looked for no other advantage from them than to accustom my mind to nourish itself upon truths and not to be satisfied with false reasons.
In Discourse upon Method, Part 2, in Henry A. Torrey (ed., trans. )Philosophy of Descartes in Extracts from His Writings , (1892), 47-48.
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Copernicus, the most learned man whom we are able to name other than Atlas and Ptolemy, even though he taught in a most learned manner the demonstrations and causes of motion based on observation, nevertheless fled from the job of constructing tables, so that if anyone computes from his tables, the computation is not even in agreement with his observations on which the foundation of the work rests. Therefore first I have compared the observations of Copernicus with those of Ptolemy and others as to which are the most accurate, but besides the bare observations, I have taken from Copernicus nothing other than traces of demonstrations. As for the tables of mean motion, and of prosthaphaereses and all the rest, I have constructed these anew, following absolutely no other reasoning than that which I have judged to be of maximum harmony.
Dedication to the Duke of Prussia, Prutenicae Tabulae (1551), 1585 edition, as quoted in Owen Gingerich, The Eye of Heaven: Ptolemy, Copernicus, Kepler (1993), 227.
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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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Euler was a believer in God, downright and straightforward. The following story is told by Thiebault, in his Souvenirs de vingt ans de séjour à Berlin, … Thiebault says that he has no personal knowledge of the truth of the story, but that it was believed throughout the whole of the north of Europe. Diderot paid a visit to the Russian Court at the invitation of the Empress. He conversed very freely, and gave the younger members of the Court circle a good deal of lively atheism. The Empress was much amused, but some of her counsellors suggested that it might be desirable to check these expositions of doctrine. The Empress did not like to put a direct muzzle on her guest’s tongue, so the following plot was contrived. Diderot was informed that a learned mathematician was in possession of an algebraical demonstration of the existence of God, and would give it him before all the Court, if he desired to hear it. Diderot gladly consented: though the name of the mathematician is not given, it was Euler. He advanced toward Diderot, and said gravely, and in a tone of perfect conviction:
Monsieur, (a + bn) / n = x, donc Dieu existe; repondez!

Diderot, to whom algebra was Hebrew, was embarrassed and disconcerted; while peals of laughter rose on all sides. He asked permission to return to France at once, which was granted.
In Budget of Paradoxes (1878), 251. [The declaration in French expresses, “therefore God exists; please answer!” This Euler-Diderot anecdote, as embellished by De Morgan, is generally regarded as entirely fictional. Diderot before he became an encyclopedist was an accomplished mathematician and fully capable of recognizing—and responding to—the absurdity of an algebraic expression in proving the existence of God. See B.H. Brown, 'The Euler-Diderot Anecdote', The American Mathematical Monthly (May 1942), 49, No. 5, 392-303. —Webmaster.]
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Experiments on ornamental plants undertaken in previous years had proven that, as a rule, hybrids do not represent the form exactly intermediate between the parental strains. Although the intermediate form of some of the more striking traits, such as those relating to shape and size of leaves, pubescence of individual parts, and so forth, is indeed nearly always seen, in other cases one of the two parental traits is so preponderant that it is difficult or quite impossible, to detect the other in the hybrid. The same is true for Pisum hybrids. Each of the seven hybrid traits either resembles so closely one of the two parental traits that the other escapes detection, or is so similar to it that no certain distinction can be made. This is of great importance to the definition and classification of the forms in which the offspring of hybrids appear. In the following discussion those traits that pass into hybrid association entirely or almost entirely unchanged, thus themselves representing the traits of the hybrid, are termed dominating and those that become latent in the association, recessive. The word 'recessive' was chosen because the traits so designated recede or disappear entirely in the hybrids, but reappear unchanged in their progeny, as will be demonstrated later.
'Experiments on Plant Hybrids' (1865). In Curt Stern and Eva R. Sherwood (eds.), The Origin of Genetics: A Mendel Source Book (1966), 9.
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For many parts of Nature can neither be invented with sufficient subtlety, nor demonstrated with sufficient perspicuity, nor accommodated to use with sufficient dexterity, without the aid and intervention of Mathematic: of which sort are Perspective, Music, Astronomy, cosmography, Architecture, Machinery, and some others.
In De Augmentis, Bk. 3; The Advancement of Learning (1605), Book 3. As translated in Francis Bacon, ‎James Spedding and ‎Robert Leslie Ellis, 'Of the great Appendix of Natural Philosophy, both Speculative and Operative, namely Mathematic; and that it ought rather to be placed among Appendices than among Substantive Sciences. Division of Mathematic into Pure and Mixed', The Works of Francis Bacon (1858), Vol. 4, Chap. 6, 371.
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For the saving the long progression of the thoughts to remote and first principles in every case, the mind should provide itself several stages; that is to say, intermediate principles, which it might have recourse to in the examining those positions that come in its way. These, though they are not self-evident principles, yet, if they have been made out from them by a wary and unquestionable deduction, may be depended on as certain and infallible truths, and serve as unquestionable truths to prove other points depending upon them, by a nearer and shorter view than remote and general maxims. … And thus mathematicians do, who do not in every new problem run it back to the first axioms through all the whole train of intermediate propositions. Certain theorems that they have settled to themselves upon sure demonstration, serve to resolve to them multitudes of propositions which depend on them, and are as firmly made out from thence as if the mind went afresh over every link of the whole chain that tie them to first self-evident principles.
In The Conduct of the Understanding, Sect. 21.
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For, Mathematical Demonstrations being built upon the impregnable Foundations of Geometry and Arithmetick, are the only Truths, that can sink into the Mind of Man, void of all Uncertainty; and all other Discourses participate more or less of Truth, according as their Subjects are more or less capable of Mathematical Demonstration.
Inaugural lecture of Christopher Wren in his chair of astronomy at Gresham College (1657). From Parentelia (1741, 1951), 200-201.
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God forbid that Truth should be confined to Mathematical Demonstration! He who does not know truth at sight is unworthy of Her Notice.
Marginal note (c. 1808) written in his copy of 'Discourse VII', The Works of Sir Joshua Reynolds (1798), beside “…as true as mathematical demonstration…”. As given in William Blake, Edwin John Ellis (ed.) and William Butler Yeats (ed.), The Works of William Blake (1893), Vol. 2, 340.
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Hurrah for positive science! long live exact demonstration!
In Leaves of Grass (1855), 28.
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I am the most travelled of all my contemporaries; I have extended my field of enquiry wider than anybody else, I have seen more countries and climes, and have heard more speeches of learned men. No one has surpassed me in the composition of lines, according to demonstration, not even the Egyptian knotters of ropes, or geometers.
In Alan L. Mackay, A Dictionary of Scientific Quotations (1992, 1994), 71.
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I have mentioned mathematics as a way to settle in the mind a habit of reasoning closely and in train; not that I think it necessary that all men should be deep mathematicians, but that, having got the way of reasoning which that study necessarily brings the mind to, they might be able to transfer it to other parts of knowledge, as they shall have occasion. For in all sorts of reasoning, every single argument should be managed as a mathematical demonstration; the connection and dependence of ideas should be followed till the mind is brought to the source on which it bottoms, and observes the coherence all along; …
In The Conduct of the Understanding, Sect. 7.
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I have often noticed that when people come to understand a mathematical proposition in some other way than that of the ordinary demonstration, they promptly say, “Oh, I see. That’s how it must be.” This is a sign that they explain it to themselves from within their own system.
Lichtenberg: A Doctrine of Scattered Occasions: Reconstructed From: Reconstructed From His Aphorisms and Reflections (1959), 291.
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I once knew an otherwise excellent teacher who compelled his students to perform all their demonstrations with incorrect figures, on the theory that it was the logical connection of the concepts, not the figure, that was essential.
In Ernst Mach and Thomas Joseph McCormack, Space and Geometry (1906), 93.
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I should object to any experimentation which can justly be called painful, for the purpose of elementary instruction ... [but I regret] a condition of the law which permits a boy to troll for pike, or set lines with live frog bait, for idle amusement; and, at the same time, lays the teacher of that boy open to the penalty of fine and imprisonment, if he uses the same animal for the purpose of exhibiting one of the most beautiful and instructive of physiological spectacles, the circulation in the web of the foot. ... [Maybe the frog is] inconvenienced by being wrapped up in a wet rag, and having his toes tied out ... But you must not inflict the least pain on a vertebrated animal for scientific purposes (though you may do a good deal in that way for gain or for sport) without due licence of the Secretary of State for the Home Department, granted under the authority of the Vivisection Act.
... [Yet, in] 1877, two persons may be charged with cruelty to animals. One has impaled a frog, and suffered the creature to writhe about in that condition for hours; the other has pained the animal no more than one of us would be pained by tying strings round his fingers, and keeping him in the position of a hydropathic patient. The first offender says, 'I did it because I find fishing very amusing,' and the magistrate bids him depart in peace; nay, probably wishes him good sport. The second pleads, 'I wanted to impress a scientific truth, with a distinctness attainable in no other way, on the minds of my scholars,' and the magistrate fines him five pounds.
I cannot but think that this is an anomalous and not wholly creditable state of things.
'On Elementary Instruction in Physiology'. Science and Culture (1882), 92.
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I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.
In 'Reflections on my Eightieth Birthday', Portraits from Memory (1956), 54.
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I would beg the wise and learned fathers (of the church) to consider with all diligence the difference which exists between matters of mere opinion and matters of demonstration. ... [I]t is not in the power of professors of the demonstrative sciences to alter their opinions at will, so as to be now of one way of thinking and now of another. ... [D]emonstrated conclusions about things in nature of the heavens, do not admit of being altered with the same ease as opinions to what is permissible or not, under a contract, mortgage, or bill of exchange.
Letter to Cristina di Lorena, Grand Duchess of Tuscany (the mother of his patron Cosmo), 1615. Quoted in Sedley Taylor, 'Galileo and Papal Infallibility' (Dec 1873), in Macmillan's Magazine: November 1873 to April 1874 (1874) Vol 29, 94.
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If I had been taught from my youth all the truths of which I have since sought out demonstrations, and had thus learned them without labour, I should never, perhaps, have known any beyond these; at least, I should never have acquired the habit and the facility which I think I possess in always discovering new truths in proportion as I give myself to the search.
In Discours de la Méthode (1637). In English from John Veitch (trans.), A Discourse on Method (1912), 57.
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If it were always necessary to reduce everything to intuitive knowledge, demonstration would often be insufferably prolix. This is why mathematicians have had the cleverness to divide the difficulties and to demonstrate separately the intervening propositions. And there is art also in this; for as the mediate truths (which are called lemmas, since they appear to be a digression) may be assigned in many ways, it is well, in order to aid the understanding and memory, to choose of them those which greatly shorten the process, and appear memorable and worthy in themselves of being demonstrated. But there is another obstacle, viz.: that it is not easy to demonstrate all the axioms, and to reduce demonstrations wholly to intuitive knowledge. And if we had chosen to wait for that, perhaps we should not yet have the science of geometry.
In Gottfried Wilhelm Leibnitz and Alfred Gideon Langley (trans.), New Essays Concerning Human Understanding (1896), 413-414.
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If one be bird-witted, that is easily distracted and unable to keep his attention as long as he should, mathematics provides a remedy; for in them if the mind be caught away but a moment, the demonstration has to be commenced anew.
In De Augmentis, Bk. 6; Advancement of Learning, Bk. 2.
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If there were some deep principle that drove organic systems towards living systems, the operation of the principle should easily be demonstrable in a test tube in half a morning. Needless to say, no such demonstration has ever been given. Nothing happens when organic materials are subjected to the usual prescription of showers of electrical sparks or drenched in ultraviolet light, except the eventual production of a tarry sludge.
…...
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In 1906 I indulged my temper by hurling invectives at Neo-Darwinians in the following terms. “I really do not wish to be abusive [to Neo-Darwinians]; but when I think of these poor little dullards, with their precarious hold of just that corner of evolution that a blackbeetle can understand—with their retinue of twopenny-halfpenny Torquemadas wallowing in the infamies of the vivisector’s laboratory, and solemnly offering us as epoch-making discoveries their demonstrations that dogs get weaker and die if you give them no food; that intense pain makes mice sweat; and that if you cut off a dog’s leg the three-legged dog will have a four-legged puppy, I ask myself what spell has fallen on intelligent and humane men that they allow themselves to be imposed on by this rabble of dolts, blackguards, imposters, quacks, liars, and, worst of all, credulous conscientious fools.”
In Back to Methuselah: A Metabiological Pentateuch (1921), lxi
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In fact, Gentlemen, no geometry without arithmetic, no mechanics without geometry... you cannot count upon success, if your mind is not sufficiently exercised on the forms and demonstrations of geometry, on the theories and calculations of arithmetic ... In a word, the theory of proportions is for industrial teaching, what algebra is for the most elevated mathematical teaching.
... a l'ouverture du cours de mechanique industrielle á Metz (1827), 2-3, trans. Ivor Grattan-Guinness.
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In fact, no opinion should be with fervour. No one holds with fervour that seven times eight is fifty-six, because it can be shown to be the case. Fervour is only necessary in commending an opinion which is doubtful or demonstrably false.
In Institut et Musée Voltaire, Studies on Voltaire and the Eighteenth Century (1994), 314. Also quoted in Max Perutz, Is Science Necessary? (1991), 196.
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In order to discover Truth in this manner by observation and reason, it is requisite we should fix on some principles whose certainty and effects are demonstrable to our senses, which may serve to explain the phenomena of natural bodies and account for the accidents that arise in them; such only are those which are purely material in the human body with mechanical and physical experiments … a physician may and ought to furnish himself with, and reason from, such things as are demonstrated to be true in anatomy, chemistry, and mechanics, with natural and experimental philosophy, provided he confines his reasoning within the bounds of truth and simple experiment.
As quoted in selection from the writings of Herman Boerhaave, collected in Oliver Joseph Thatcher (ed.), The Ideas that Have Influenced Civilization, in the Original Documents (1800), Vol. 6, 242.
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In other branches of science, where quick publication seems to be so much desired, there may possibly be some excuse for giving to the world slovenly or ill-digested work, but there is no such excuse in mathematics. The form ought to be as perfect as the substance, and the demonstrations as rigorous as those of Euclid. The mathematician has to deal with the most exact facts of Nature, and he should spare no effort to render his interpretation worthy of his subject, and to give to his work its highest degree of perfection. “Pauca sed matura” was Gauss’s motto.
In Presidential Address British Association for the Advancement of Science, Section A, (1890), Nature, 42, 467. [The Latin motto translates as “Few, but ripe”. —Webmaster]
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It has been said that astronomy is a humbling and character building experience. There is perhaps no better demonstration of the folly of human conceits than this distant image of our tiny world. To me, it underscores our responsibility to deal more kindly with one another, and to preserve and cherish the pale blue dot, the only home we’ve ever known.
…...
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It is admitted by all that a finished or even a competent reasoner is not the work of nature alone; the experience of every day makes it evident that education develops faculties which would otherwise never have manifested their existence. It is, therefore, as necessary to learn to reason before we can expect to be able to reason, as it is to learn to swim or fence, in order to attain either of those arts. Now, something must be reasoned upon, it matters not much what it is, provided it can be reasoned upon with certainty. The properties of mind or matter, or the study of languages, mathematics, or natural history, may be chosen for this purpose. Now of all these, it is desirable to choose the one which admits of the reasoning being verified, that is, in which we can find out by other means, such as measurement and ocular demonstration of all sorts, whether the results are true or not. When the guiding property of the loadstone was first ascertained, and it was necessary to learn how to use this new discovery, and to find out how far it might be relied on, it would have been thought advisable to make many passages between ports that were well known before attempting a voyage of discovery. So it is with our reasoning faculties: it is desirable that their powers should be exerted upon objects of such a nature, that we can tell by other means whether the results which we obtain are true or false, and this before it is safe to trust entirely to reason. Now the mathematics are peculiarly well adapted for this purpose, on the following grounds:
1. Every term is distinctly explained, and has but one meaning, and it is rarely that two words are employed to mean the same thing.
2. The first principles are self-evident, and, though derived from observation, do not require more of it than has been made by children in general.
3. The demonstration is strictly logical, taking nothing for granted except self-evident first principles, resting nothing upon probability, and entirely independent of authority and opinion.
4. When the conclusion is obtained by reasoning, its truth or falsehood can be ascertained, in geometry by actual measurement, in algebra by common arithmetical calculation. This gives confidence, and is absolutely necessary, if, as was said before, reason is not to be the instructor, but the pupil.
5. There are no words whose meanings are so much alike that the ideas which they stand for may be confounded. Between the meaning of terms there is no distinction, except a total distinction, and all adjectives and adverbs expressing difference of degrees are avoided.
In On the Study and Difficulties of Mathematics (1898), chap. 1.
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It is probably no exaggeration to suppose that in order to improve such an organ as the eye at all, it must be improved in ten different ways at once. And the improbability of any complex organ being produced and brought to perfection in any such way is an improbability of the same kind and degree as that of producing a poem or a mathematical demonstration by throwing letters at random on a table.
[Expressing his reservations about Darwin's proposed evolution of the eye by natural selection.]
Opening address to the Belfast Natural History Society, as given in the 'Belfast Northern Whig,' (19 Nov 1866). As cited by Charles Darwin in The Variation of Animals & Plants Under Domestication (1868), 222.
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It seems to me, that the only Objects of the abstract Sciences or of Demonstration is Quantity and Number, and that all Attempts to extend this more perfect Species of Knowledge beyond these Bounds are mere Sophistry and Illusion.
An Enquiry Concerning Human Understanding (1748), 252.
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It was a dark and stormy night, so R. H. Bing volunteered to drive some stranded mathematicians from the fogged-in Madison airport to Chicago. Freezing rain pelted the windscreen and iced the roadway as Bing drove on—concentrating deeply on the mathematical theorem he was explaining. Soon the windshield was fogged from the energetic explanation. The passengers too had beaded brows, but their sweat arose from fear. As the mathematical description got brighter, the visibility got dimmer. Finally, the conferees felt a trace of hope for their survival when Bing reached forward—apparently to wipe off the moisture from the windshield. Their hope turned to horror when, instead, Bing drew a figure with his finger on the foggy pane and continued his proof—embellishing the illustration with arrows and helpful labels as needed for the demonstration.
In 'R. H. Bing', Biographical Memoirs: National Academy of Sciences (2002), 49. Anecdote based on the recollections of Bing's colleagues, Steve Armentrout and C. E. Burgess. The narrative was given in a memorial tribute at the University of Texas at Austin.
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It would appear that Deductive and Demonstrative Sciences are all, without exception, Inductive Sciences: that their evidence is that of experience, but that they are also, in virtue of the peculiar character of one indispensable portion of the general formulae according to which their inductions are made, Hypothetical Sciences. Their conclusions are true only upon certain suppositions, which are, or ought to be, approximations to the truth, but are seldom, if ever, exactly true; and to this hypothetical character is to be ascribed the peculiar certainty, which is supposed to be inherent in demonstration.
In System of Logic, Bk. 2, chap. 6, 1.
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It would indeed be a great delusion, if we stated that those sports of Nature [we find] enclosed in rocks are there by chance or by some vague creative power. Ah, that would be superficial indeed! In reality, those shells, which once were alive in water and are now dead and decomposed, were made thus by time not Nature; and what we now find as very hard, figured stone, was once soft mud and which received the impression of the shape of a shell, as I have frequently demonstrated.
La vana speculazione disingannata del senso (1670), trans. Ezio Vaccari, 83-4.
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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 gives the young man a clear idea of demonstration and habituates him to form long trains of thought and reasoning methodically connected and sustained by the final certainty of the result; and it has the further advantage, from a purely moral point of view, of inspiring an absolute and fanatical respect for truth. In addition to all this, mathematics, and chiefly algebra and infinitesimal calculus, excite to a high degree the conception of the signs and symbols—necessary instruments to extend the power and reach of the human mind by summarizing an aggregate of relations in a condensed form and in a kind of mechanical way. These auxiliaries are of special value in mathematics because they are there adequate to their definitions, a characteristic which they do not possess to the same degree in the physical and mathematical [natural?] sciences.
There are, in fact, a mass of mental and moral faculties that can be put in full play only by instruction in mathematics; and they would be made still more available if the teaching was directed so as to leave free play to the personal work of the student.
In 'Science as an Instrument of Education', Popular Science Monthly (1897), 253.
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Mathematics is of two kinds, Rigorous and Physical. The former is Narrow: the latter Bold and Broad. To have to stop to formulate rigorous demonstrations would put a stop to most physico-mathematical inquiries. Am I to refuse to eat because I do not fully understand the mechanism of digestion?
As quoted by Charles Melbourne Focken in Dimensional Methods and Their Applications (1953), 17.
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Mathematics, including not merely Arithmetic, Algebra, Geometry, and the higher Calculus, but also the applied Mathematics of Natural Philosophy, has a marked and peculiar method or character; it is by preeminence deductive or demonstrative, and exhibits in a nearly perfect form all the machinery belonging to this mode of obtaining truth. Laying down a very small number of first principles, either self-evident or requiring very little effort to prove them, it evolves a vast number of deductive truths and applications, by a procedure in the highest degree mathematical and systematic.
In Education as a Science (1879), 148.
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Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of Fermat, left without proof by their author, which resisted all efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when turned over in the blow-pipe flame of Gauss’s transcendent genius; or the doctrine of double periodicity, which resulted from the observation of Jacobi of a purely analytical fact of transformation; or Legendre’s law of reciprocity; or Sturm’s theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected (if my memory serves me right) with the motion of compound pendulums; or Huyghen’s method of continued fractions, characterized by Lagrange as one of the principal discoveries of that great mathematician, and to which he appears to have been led by the construction of his Planetary Automaton; or the new algebra, speaking of which one of my predecessors (Mr. Spottiswoode) has said, not without just reason and authority, from this chair, “that it reaches out and indissolubly connects itself each year with fresh branches of mathematics, that the theory of equations has become almost new through it, algebraic geometry transfigured in its light, that the calculus of variations, molecular physics, and mechanics” (he might, if speaking at the present moment, go on to add the theory of elasticity and the development of the integral calculus) “have all felt its influence”.
In 'A Plea for the Mathematician', Nature, 1, 238 in Collected Mathematical Papers, Vol. 2, 655-56.
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No matter how we twist and turn we shall always come back to the cell. The eternal merit of Schwann does not lie in his cell theory that has occupied the foreground for so long, and perhaps will soon be given up, but in his description of the development of the various tissues, and in his demonstration that this development (hence all physiological activity) is in the end traceable back to the cell. Now if pathology is nothing but physiology with obstacles, and diseased life nothing but healthy life interfered with by all manner of external and internal influences then pathology too must be referred back to the cell.
In 'Cellular-Pathologie', Archiv für pathologische Anatomie und Physiologie und fur klinische Medizin (1855), 8, 13-14, as translated in LellandJ. Rather, 'Cellular Pathology', Disease, Life, and Man: Selected Essays by Rudolf Virchow (1958), 81.
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No. I have been teaching it all my life, and I do not want to have my ideas upset.
Reply when James Clerk Maxwell’s asked if he would like to see an experimental demonstration of conical refraction. As given in James G. Crowther, The Cavendish Laboratory, 1874-1974 (1974), 9.
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Nothing afflicted Marcellus so much as the death of Archimedes, who was then, as fate would have it, intent upon working out some problem by a diagram, and having fixed his mind alike and his eyes upon the subject of his speculation, he never noticed the incursion of the Romans, nor that the city was taken. In this transport of study and contemplation, a soldier, unexpectedly coming up to him, commanded him to follow to Marcellus, which he declined to do before he had worked out his problem to a demonstration; the soldier, enraged, drew his sword and ran him through. Others write, that a Roman soldier, running upon him with a drawn sword, offered to kill him; and that Archimedes, looking back, earnestly besought him to hold his hand a little while, that he might not leave what he was at work upon inconclusive and imperfect; but the soldier, nothing moved by his entreaty, instantly killed him. Others again relate, that as Archimedes was carrying to Marcellus mathematical instruments, dials, spheres, and angles, by which the magnitude of the sun might be measured to the sight, some soldiers seeing him, and thinking that he carried gold in a vessel, slew him. Certain it is, that his death was very afflicting to Marcellus; and that Marcellus ever after regarded him that killed him as a murderer; and that he sought for his kindred and honoured them with signal favours.
Plutarch
In John Dryden (trans.), Life of Marcellus.
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O. Hahn and F. Strassmann have discovered a new type of nuclear reaction, the splitting into two smaller nuclei of the nuclei of uranium and thorium under neutron bombardment. Thus they demonstrated the production of nuclei of barium, lanthanum, strontium, yttrium, and, more recently, of xenon and caesium. It can be shown by simple considerations that this type of nuclear reaction may be described in an essentially classical way like the fission of a liquid drop, and that the fission products must fly apart with kinetic energies of the order of hundred million electron-volts each.
'Products of the Fission of the Urarium Nucleus', Nature (1939), 143, 471.
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Post-operatively the transplanted kidney functioned immediately with a dramatic improvement in the patient’s renal and cardiopulmonary status. This spectacular success was a clear demonstration that organ transplantation could be life-saving. In a way, it was spying into the future because we had achieved our long-term goal by bypassing, but not solving, the issue of biological incompatibility.
Referring to the pioneering first kidney transplant. It was well-matched since it was between twins. In Nobel Lecture (8 Dec 1990). Printed in Tore Frängsmyr and Jan Lindsten (eds.), Nobel Lectures, Physiology or Medicine 1981-1990 (1993).
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Samoa culture demonstrates how much the tragic or the easy solution of the Oedipus situation depends upon the inter-relationship between parents and children, and is not created out of whole cloth by the young child’s biological impulses.
Male and Female: A Study of the Sexes in a Changing World (1949), 119.
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Secondly, the study of mathematics would show them the necessity there is in reasoning, to separate all the distinct ideas, and to see the habitudes that all those concerned in the present inquiry have to one another, and to lay by those which relate not to the proposition in hand, and wholly to leave them out of the reckoning. This is that which, in other respects besides quantity is absolutely requisite to just reasoning, though in them it is not so easily observed and so carefully practised. In those parts of knowledge where it is thought demonstration has nothing to do, men reason as it were in a lump; and if upon a summary and confused view, or upon a partial consideration, they can raise the appearance of a probability, they usually rest content; especially if it be in a dispute where every little straw is laid hold on, and everything that can but be drawn in any way to give color to the argument is advanced with ostentation. But that mind is not in a posture to find truth that does not distinctly take all the parts asunder, and, omitting what is not at all to the point, draws a conclusion from the result of all the particulars which in any way influence it.
In Conduct of the Understanding, Sect. 7.
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Suppose we loosely define a religion as any discipline whose foundations rest on an element of faith, irrespective of any element of reason which may be present. Quantum mechanics for example would be a religion under this definition. But mathematics would hold the unique position of being the only branch of theology possessing a rigorous demonstration of the fact that it should be so classified.
Concluding remark in 'Consistency and Completeness—A Résumé', The American Mathematical Monthly (May 1956), 63, No.5, 305.
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That mathematics “do not cultivate the power of generalization,”; … will be admitted by no person of competent knowledge, except in a very qualified sense. The generalizations of mathematics, are, no doubt, a different thing from the generalizations of physical science; but in the difficulty of seizing them, and the mental tension they require, they are no contemptible preparation for the most arduous efforts of the scientific mind. Even the fundamental notions of the higher mathematics, from those of the differential calculus upwards are products of a very high abstraction. … To perceive the mathematical laws common to the results of many mathematical operations, even in so simple a case as that of the binomial theorem, involves a vigorous exercise of the same faculty which gave us Kepler’s laws, and rose through those laws to the theory of universal gravitation. Every process of what has been called Universal Geometry—the great creation of Descartes and his successors, in which a single train of reasoning solves whole classes of problems at once, and others common to large groups of them—is a practical lesson in the management of wide generalizations, and abstraction of the points of agreement from those of difference among objects of great and confusing diversity, to which the purely inductive sciences cannot furnish many superior. Even so elementary an operation as that of abstracting from the particular configuration of the triangles or other figures, and the relative situation of the particular lines or points, in the diagram which aids the apprehension of a common geometrical demonstration, is a very useful, and far from being always an easy, exercise of the faculty of generalization so strangely imagined to have no place or part in the processes of mathematics.
In An Examination of Sir William Hamilton’s Philosophy (1878), 612-13.
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The artist does not illustrate science; … [but] he frequently responds to the same interests that a scientist does, and expresses by a visual synthesis what the scientist converts into analytical formulae or experimental demonstrations.
'The Arts', in Charles Austin Beard, Whither Mankind: a Panorama of Modern Civilization (1928, 1971), 296.
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The attempted synthesis of paleontology and genetics, an essential part of the present study, may be particularly surprising and possibly hazardous. Not long ago, paleontologists felt that a geneticist was a person who shut himself in a room, pulled down the shades, watched small flies disporting themselves in milk bottles, and thought that he was studying nature. A pursuit so removed from the realities of life, they said, had no significance for the true biologist. On the other hand, the geneticists said that paleontology had no further contributions to make to biology, that its only point had been the completed demonstration of the truth of evolution, and that it was a subject too purely descriptive to merit the name 'science'. The paleontologist, they believed, is like a man who undertakes to study the principles of the internal combustion engine by standing on a street corner and watching the motor cars whiz by.
Tempo and Mode in Evolution (1944), 1.
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The credulous … advance the authority of hearsay in place of reasons for possible success or facts that can be demonstrated.
In De La Pirotechnia (1540). As translated in Pirotechnia (1959), 36. Biringuccio rejected the lore of alchemy, and believed in his own practical observation instead of the writing of ancient philosophers.
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The demonstration that no possible combination of known substances, known forms of machinery and known forms of force, can be united in a practical machine by which men shall fly along distances through the air, seems to the writer as complete as it is possible for the demonstration to be.
Side-lights on Astronomy and Kindred Fields of Popular Science (1906), 345.
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The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes.
From letter (Jan 1828) to his former teacher Berndt Holmböe. In Morris Kline, Mathematics: The Loss of Certainty (1982), 170.
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The golden age of mathematics—that was not the age of Euclid, it is ours. Ours is the age when no less than six international congresses have been held in the course of nine years. It is in our day that more than a dozen mathematical societies contain a growing membership of more than two thousand men representing the centers of scientific light throughout the great culture nations of the world. It is in our time that over five hundred scientific journals are each devoted in part, while more than two score others are devoted exclusively, to the publication of mathematics. It is in our time that the Jahrbuch über die Fortschritte der Mathematik, though admitting only condensed abstracts with titles, and not reporting on all the journals, has, nevertheless, grown to nearly forty huge volumes in as many years. It is in our time that as many as two thousand books and memoirs drop from the mathematical press of the world in a single year, the estimated number mounting up to fifty thousand in the last generation. Finally, to adduce yet another evidence of a similar kind, it requires not less than seven ponderous tomes of the forthcoming Encyclopaedie der Mathematischen Wissenschaften to contain, not expositions, not demonstrations, but merely compact reports and bibliographic notices sketching developments that have taken place since the beginning of the nineteenth century.
In Lectures on Science, Philosophy and Art (1908), 8.
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The man of science dissects the statement, verifies the facts, and demonstrates connection even where he cannot its purpose.
From 'The Great Lawsuit. Man versus Men. Woman versus Women,' in the Boston Dial (Jul 1843), 4, No. 1, 3, which she expanded (padded) to publish as Woman in the Nineteenth Century (1844).
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The pre-Darwinian age had come to be regarded as a Dark Age in which men still believed that the book of Genesis was a standard scientific treatise, and that the only additions to it were Galileo's demonstration of Leonardo da Vinci’s simple remark that the earth is a moon of the sun, Newton’s theory of gravitation, Sir Humphry Davy's invention of the safety-lamp, the discovery of electricity, the application of steam to industrial purposes, and the penny post.
Back to Methuselah: a Metabiological Pentateuch (1921), viii.
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The successful launching of the Sputnik was a demonstration of one of the highest scientific and technological achievements of man—a tantalizing invitation both to the militarist in search of ever more devastating means of destruction and to the astronomer searching for new means of carrying his instruments away from their earthbound environment.
In BBC Reith Lecture (9 Nov 1958), 'Astronomy Breaks Free', published as The Individual and the Universe (1959, 1961), 72.
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Then one day Lagrange took out of his pocket a paper which he read at the Académe, and which contained a demonstration of the famous Postulatum of Euclid, relative to the theory of parallels. This demonstration rested on an obvious paralogism, which appeared as such to everybody; and probably Lagrange also recognised it such during his lecture. For, when he had finished, he put the paper back in his pocket, and spoke no more of it. A moment of universal silence followed, and one passed immediately to other concerns.
Quoting Lagrange at a meeting of the class of mathematical and physical sciences at the Institut de France (3 Feb 1806) in Journal des Savants (1837), 84, trans. Ivor Grattan-Guinness.
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There is a great deal of emotional satisfaction in the elegant demonstration, in the elegant ordering of facts into theories, and in the still more satisfactory, still more emotionally exciting discovery that the theory is not quite right and has to be worked over again, very much as any other work of art—a painting, a sculpture has to be worked over in the interests of aesthetic perfection. So there is no scientist who is not to some extent worthy of being described as artist or poet.
'Scientist and Citizen', Speech to the Empire Club of Canada (29 Jan 1948), The Empire Club of Canada Speeches (29 Jan 1948), 209-221.
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There is no kind of material, no body, and no thing that can be produced or conceived of, which is not made up of elementary particles; and nature does not admit of a truthful exploration in accordance with the doctrines of the physicists without an accurate demonstration of the primary causes of things, showing how and why they are as they are.
Vitruvius
In De Architectura, Book 2, Chap 1, Sec. 9. As translated in Morris Hicky Morgan (trans.), Vitruvius: The Ten Books on Architecture (1914), 41.
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These duplicates in those parts of the body, without which a man might have very well subsisted, though not so well as with them, are a plain demonstration of an all-wise Contriver, as those more numerous copyings which are found among the vessels of the same body are evident demonstrations that they could not be the work of chance. This argument receives additional strength if we apply it to every animal and insect within our knowledge, as well as to those numberless living creatures that are objects too minute for a human eye: and if we consider how the several species in this whole world of life resemble one another in very many particulars, so far as is convenient for their respective states of existence, it is much more probable that a hundred millions of dice should be casually thrown a hundred millions of times in the same number than that the body of any single animal should be produced by the fortuitous concourse of matter.
In The Spectator (22 Nov 1712), No. 543, as collected in Vol. 4 (1721, 10th ed.), 48.
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These results demonstrate that there is a new polymerase inside the virions of RNA tumour viruses. It is not present in supernatents of normal cells but is present in virions of avian sarcoma and leukemia RNA tumour viruses. The polymerase seems to catalyse the incorporation of deoxyrinonucleotide triphosphates into DNA from an RNA template. Work is being performed to characterize further the reaction and the product. If the present results and Baltimore's results with Rauscher leukemia virus are upheld, they will constitute strong evidence that the DNA proviruses have a DNA genome when they are in virions. This result would have strong implications for theories of viral carcinogenesis and, possibly, for theories of information transfer in other biological systems. [Co-author with American virologist Satoshi Mizutani]
'RNA-dependent DNA Polymerase in Virions of Rous Sarcoma Virus', Nature (1970), 226, 1213.
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This formula [for computing Bernoulli’s numbers] was first given by James Bernoulli…. He gave no general demonstration; but was quite aware of the importance of his theorem, for he boasts that by means of it he calculated intra semi-quadrantem horæ! the sum of the 10th powers of the first thousand integers, and found it to be
91,409,924,241,424,243,424,241,924,242,500.

In 'Bernoulli’s Expression for ΣNr', Algebra, Vol. 2 (1879, 1889), 209. The ellipsis is for the reference (Ars Conjectandi (1713), 97). [The Latin phrase, “intra semi-quadrantem horæ!” refers to within a fraction of an hour. —Webmaster]
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This [the opening of the Vatican City radio station built by Marconi earlier in 1931] was a new demonstration of the harmony between science and religion that each fresh conquest of science ever more luminously confirms, so that one may say that those who speak of the incompatibility of science and religion either make science say that which it never said or make religion say that which it never taught.
Address to Pontifical Academy of Sciences (20 Dec 1931).In Associated Press, 'Pope Sees Harmony in Faith and Science', New York Times (21 Dec 1931), p.9. The pontiff said the opening of the radio station was “crowned by the publication of a radiophonic newspaper.”
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Those intervening ideas, which serve to show the agreement of any two others, are called proofs; and where the agreement or disagreement is by this means plainly and clearly perceived, it is called demonstration; it being shown to the understanding, and the mind made to see that it is so. A quickness in the mind to find out these intermediate ideas, (that shall discover the agreement or disagreement of any other) and to apply them right, is, I suppose, that which is called sagacity.
In An Essay concerning Human Understanding, Bk. 6, chaps. 2, 3.
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Those who assert that the mathematical sciences make no affirmation about what is fair or good make a false assertion; for they do speak of these and frame demonstrations of them in the most eminent sense of the word. For if they do not actually employ these names, they do not exhibit even the results and the reasons of these, and therefore can be hardly said to make any assertion about them. Of what is fair, however, the most important species are order and symmetry, and that which is definite, which the mathematical sciences make manifest in a most eminent degree. And since, at least, these appear to be the causes of many things—now, I mean, for example, order, and that which is a definite thing, it is evident that they would assert, also, the existence of a cause of this description, and its subsistence after the same manner as that which is fair subsists in.
Aristotle
In Metaphysics [MacMahon] Bk. 12, chap. 3.
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Those who consider James Watt only as a great practical mechanic form a very erroneous idea of his character: he was equally distinguished as a natural philosopher and a chemist, and his inventions demonstrate his profound knowledge of those sciences, and that peculiar characteristic of genius, the union of them for practical application.
As reported in Proceedings of the Public Meeting held at Preemasons' Hall, on the 18th June, 1824, for Erecting a Monument to the Late James Watt (1824), 8.
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Till Algebra, that great Instrument and Instance of Humane Sagacity, was discovered, Men, with amazement, looked on several of the Demonstrations of ancient Mathematicians, and could scarce forbear to think the finding some of those Proofs, more than humane.
In 'Of Reason', Essay Concerning Humane Understanding (1690), Book 4, Ch. 17, Sec. 11, 345. Note: humane (obsolete) = human.
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To demonstrate experimentally that a microscopic organism actually is the cause of a disease and the agent of contagion, I know no other way, in the present state of Science, than to subject the microbe (the new and happy term introduced by M. Sédillot) to the method of cultivation out of the body.
Paper read to the French Academy of Sciences (29 Apr 1878), published in Comptes Rendus de l'Academie des Sciences, 86, 1037-43, as translated by H.C.Ernst. Collected in Charles W. Eliot (ed.) The Harvard Classics, Vol. 38; Scientific Papers: Physiology, Medicine, Surgery, Geology (1910), 364.
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We see it [the as-yet unseen, probable new planet, Neptune] as Columbus saw America from the coast of Spain. Its movements have been felt, trembling along the far-reaching line of our analysis with a certainty hardly inferior to that of ocular demonstration.
Address to the British Association, Southampton (10 Sep 1845). Quoted in Memoirs of the Royal Astronomical Society (1847), 16, 400.
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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.
From 'L’Avenir des Mathématiques', Science et Méthode (1908, 1920), Livre Premier, Chap. 2, 25. English as in Henri Poincaré and George Bruce Halsted (trans.), 'The Future of Mathematics', Science and Method collected in The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method (1913), 372. From the French, “Qu’est-ce qui nous donne en effet dans une solution, dans une démonstration, le sentiment de l’élégance? C’est l’harmonie des diverses parties, leur symétrie, leur heureux balancement; c’est en un mot tout ce qui y met de l'ordre, tout ce qui leur donne de l’unité, ce qui nous permet par conséquent d’y voir clair et d’en comprendre l’ensemble en même temps que les détails”.
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Whatever things are not derived from objects themselves, whether by the external senses or by the sensation of internal thoughts, are to be taken as hypotheses…. Those things which follow from the phenomena neither by demonstration nor by the argument of induction, I hold as hypotheses.
As quoted in Ernan McMullin, 'The Principia: Significance For Empiricism', collected in Margaret J. Osler and Paul Lawrence Farber (eds.), Religion, Science, and Worldview: Essays in Honor of Richard S. Westfall (2002), 38.
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When I read some forty years ago that shells from Syria were found in the Alpes, I said, I admit, in a rather joking way, that these shells had apparently been carried by pilgrims on their return from Jerusalem. Mr. Buffon reprimanded me rather sharply in his Theory of the Earth, p. 28 I. I did not want to lose his friendship for peanuts; however, I am still of the same opinion because the impossibility of the formation of mountains by the sea is demonstrated to me.
Les Singularites de la Nature (1768), in Oeuvres Completes de Voltaire (1877-1885), Vol. 26, 408. Trans. Albert V. and Marguerite Carozzi.
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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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Where a cell arises, there a cell must have previously existed (omnis cellula e cellula), just as an animal can spring only from an animal, a plant only from a plant. In this manner, although there are still a few spots in the body where absolute demonstration has not yet been afforded, the principle is nevertheless established, that in the whole series of living things, whether they be entire plants or animal organisms, or essential constituents of the same, an eternal law of continuous development prevails.
In Lecture II 'Physiological Tissues' (17 Feb 1858), as translated by Frank Chance in Cellular Pathology: As Based Upon Physiological and Pathological Histology. Twenty Lectures Delivered in the Pathological Institute of Berlin During the Months of February, March and April, 1858 (1860), 27-28.
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Yet I also appreciate that we cannot win this battle to save species and environments without forging an emotional bond between ourselves and nature as well–for we will not fight to save what we do not love (but only appreciate in some abstract sense). So let them all continue–the films, the books, the television programs, the zoos, the little half acre of ecological preserve in any community, the primary school lessons, the museum demonstrations, even ... the 6:00 A.M. bird walks. Let them continue and expand because we must have visceral contact in order to love. We really must make room for nature in our hearts.
…...
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… There can be no doubt about faith and not reason being the ultima ratio. Even Euclid, who has laid himself as little open to the charge of credulity as any writer who ever lived, cannot get beyond this. He has no demonstrable first premise. He requires postulates and axioms which transcend demonstration, and without which he can do nothing. His superstructure indeed is demonstration, but his ground his faith. Nor again can he get further than telling a man he is a fool if he persists in differing from him. He says “which is absurd,” and declines to discuss the matter further. Faith and authority, therefore, prove to be as necessary for him as for anyone else.
In The Way of All Flesh (1917), 319-320.
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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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