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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 R > Category: Reasoning

Reasoning Quotes (207 quotes)

Dilbert: And we know mass creates gravity because more dense planets have more gravity.
Dogbert: How do we know which planets are more dense?
Dilbert:They have more gravity.
Dogbert: That's circular reasoning.
Dilbert: I prefer to think of it as having no loose ends.
Dilbert cartoon strip (1 Mar 1999).
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He who doth with the greatest exactness imaginable, weigh every individual thing that shall or hath hapned to his Patient, and may be known from the Observations of his own, or of others, and who afterwards compareth all these with one another, and puts them in an opposite view to such Things as happen in a healthy State; and lastly, from all this with the nicest and severest bridle upon his reasoning faculty riseth to the knowledge of the very first Cause of the Disease, and of the Remedies fit to remove them; He, and only He deserveth the Name of a true Physician.
Aphorism No. 13 in Boerhaave’s Aphorisms: Concerning The Knowledge and Cure of Diseases (1715), 3.
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Je me rends parfaitement compte du desagreable effet que produit sur la majorite de l'humanité, tout ce qui se rapporte, même au plus faible dègré, á des calculs ou raisonnements mathematiques.
I am well aware of the disagreeable effect produced on the majority of humanity, by whatever relates, even at the slightest degree to calculations or mathematical reasonings.
From 'French Reply to Baron Czyllak' concerning the game at Monte Carlo, Monte Carlo Facts and Fallacies (1904), 290, originally published in L'Écho de la Mediterranée as a response to an earlier open letter by the Baron in the same magazine. Maxim defended his prior mathematical calculations about gambling games. At the end of his paper giving a cautionary mathematical analysis of 'The Gambler's Ruin', < a href="http://todayinsci.com/C/Coolidge_Julian/CoolidgeJulian-Quotations.htm">Julian Coolidge referenced this quotation, saying “it gives the best explanation which I have seen for the fact that the people continue to gamble.”
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Les mathématiciens parviennent à la solution d’un problême par le simple arrangement des données, & en réduisant le raisonnement à des opérations si simples, à des jugemens si courts, qu’ils ne perdent jamais de vue l’évidence qui leur sert de guide.
Mathematicians come to the solution of a problem by the simple arrangement of the data, and reducing the reasoning to such simple operations, to judgments so brief, that they never lose sight of the evidence that serves as their guide.
From a paper read to the Académie Royales des Sciences (18 Apr 1787), printed in Méthode de Nomenclature Chimique (1787), 12. Translation from the French by Webmaster.
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Longtemps les objets dont s'occupent les mathématiciens étaient our la pluspart mal définis; on croyait les connaître, parce qu'on se les représentatit avec le sens ou l'imagination; mais on n'en avait qu'une image grossière et non une idée précise sure laquelle le raisonment pût avoir prise.
For a long time the objects that mathematicians dealt with were mostly ill-defined; one believed one knew them, but one represented them with the senses and imagination; but one had but a rough picture and not a precise idea on which reasoning could take hold.
La valeur de la science. In Anton Bovier, Statistical Mechanics of Disordered Systems (2006), 97.
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Qui est de nous & qui seul peut nous égarer; à le mettre continuellement à épreuve de l'expérience; à ne conserver que les faits qui ne font que des données de la nature , & qui ne peuvent nous tromper; à ne chercher la vérité que dans l'enchaînement naturel des expériences & des observations
We must trust to nothing but facts: These are presented to us by Nature, and cannot deceive. We ought, in every instance, to submit our reasoning to the test of experiment, and never to search for truth but by the natural road of experiment and observation.
From the original French in Traité élémentaire de chimie (1789, 1793), discours préliminaire, x; and from edition translated into English by Robert Kerr, as Elements of Chemistry (1790), Preface, xviii.
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Une idée anticipée ou une hypothèse est donc le point de départ nécessaire de tout raisonnement expérimental. Sans cela on ne saurait faire aucune investigation ni s’instruire ; on ne pourrait qu’entasser des observations stériles. Si l’on expérimentait sans idée préconçue, on irait à l’aventure; mais d’un autre côté, ainsi que nous l’avons dit ailleurs, si l’on observait avec des idées préconçues, on ferait de mauvaises observations.
An anticipative idea or an hypothesis is, then, the necessary starting point for all experimental reasoning. Without it, we could not make any investigation at all nor learn anything; we could only pile up sterile observations. If we experimented without a preconceived idea, we should move at random.
[Also seen translated as:] A hypothesis is … the obligatory starting point of all experimental reasoning. Without it no investigation would be possible, and one would learn nothing: one could only pile up barren observations. To experiment without a preconceived idea is to wander aimlessly.
Original work in French, Introduction à l'Étude de la Médecine Expérimentale (1865). English translation by Henry Copley Green in An Introduction to the Study of Experimental Medicine (1927, 1957), 32. Alternate translation in Peter Medawar, 'Hypothesis and Imagination', collected in The Strange Case of the Spotted Mice and Other Classic Essays on Science (1974), 30.
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Where faith commences, science ends. Both these arts of the human mind must be strictly kept apart from each other. Faith has its origin in the poetic imagination; knowledge, on the other hand, originates in the reasoning intelligence of man. Science has to pluck the blessed fruits from the tree of knowledge, unconcerned whether these conquests trench upon the poetical imaginings of faith or not.
In Ernst Haeckel and E. Ray Lankester (trans.), The History of Creation (1880), Vol. 1, 9.
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A primâ facie argument in favour of the efficacy of prayer is therefore to be drawn from the very general use of it. The greater part of mankind, during all the historic ages, have been accustomed to pray for temporal advantages. How vain, it may be urged, must be the reasoning that ventures to oppose this mighty consensus of belief! Not so. The argument of universality either proves too much, or else it is suicidal.
'Statistical Inquiries into the Efficacy of Prayer', Fortnightly Review, 1872, 12, 126.
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A mathematical argument is, after all, only organized common sense, and it is well that men of science should not always expound their work to the few behind a veil of technical language, but should from time to time explain to a larger public the reasoning which lies behind their mathematical notation.
In The Tides and Kindred Phenomena in the Solar System: The Substance of Lectures Delivered in 1897 at the Lowell Institute, Boston, Massachusetts (1898), Preface, v. Preface
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A superficial knowledge of mathematics may lead to the belief that this subject can be taught incidentally, and that exercises akin to counting the petals of flowers or the legs of a grasshopper are mathematical. Such work ignores the fundamental idea out of which quantitative reasoning grows—the equality of magnitudes. It leaves the pupil unaware of that relativity which is the essence of mathematical science. Numerical statements are frequently required in the study of natural history, but to repeat these as a drill upon numbers will scarcely lend charm to these studies, and certainly will not result in mathematical knowledge.
In Primary Arithmetic: First Year, for the Use of Teachers (1897), 26-27.
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A superficial knowledge of mathematics may lead to the belief that this subject can be taught incidentally, and that exercises akin to counting the petals of flowers or the legs of a grasshopper are mathematical. Such work ignores the fundamental idea out of which quantitative reasoning grows—the equality of magnitudes. It leaves the pupil unaware of that relativity which is the essence of mathematical science. Numerical statements are frequently required in the study of natural history, but to repeat these as a drill upon numbers will scarcely lend charm to these studies, and certainly will not result in mathematical knowledge.
In Primary Arithmetic: First Year, for the Use of Teachers (1897), 26-27.
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About medications that are drunk or applied to wounds it is worth learning from everyone; for people do not discover these by reasoning but by chance, and experts not more than laymen.
Affections, in Hippocrates, trans. P. Potter (1988), Vol. 5, 69. Littré VI, 254.
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All palaetiological sciences, all speculations which attempt to ascend from the present to the remote past, by the chain of causation, do also, by an inevitable consequence, urge us to look for the beginning of the state of things which we thus contemplate; but in none of these cases have men been able, by the aid of science, to arrive at a beginning which is homogeneous with the known course of events. The first origin of language, of civilization, of law and government, cannot be clearly made out by reasoning and research; and just as little, we may expect, will a knowledge of the origin of the existing and extinct species of plants and animals, be the result of physiological and geological investigation.
In History of the Inductive Sciences (1837), Vol. 3, 581.
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Among all the liberal arts, the first is logic, and specifically that part of logic which gives initial instruction about words. … [T]he word “logic” has a broad meaning, and is not restricted exclusively to the science of argumentative reasoning. [It includes] Grammar [which] is “the science of speaking and writing correctly—the starting point of all liberal studies.”
In John of Salisbury and Daniel D. McGarry (trans.), 'Whence grammar gets its name', The Metalogicon (2009), 37. It is footnoted: Isidore, Etym., i, 5, §1.
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Among the minor, yet striking characteristics of mathematics, may be mentioned the fleshless and skeletal build of its propositions; the peculiar difficulty, complication, and stress of its reasonings; the perfect exactitude of its results; their broad universality; their practical infallibility.
In Charles S. Peirce, ‎Charles Hartshorne (ed.), ‎Paul Weiss (ed.), Collected Papers of Charles Sanders Peirce (1931), Vol. 4, 197.
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Anyone who has had actual contact with the making of the inventions that built the radio art knows that these inventions have been the product of experiment and work based on physical reasoning, rather than on the mathematicians' calculations and formulae. Precisely the opposite impression is obtained from many of our present day text books and publications.
Attributed.
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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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Biot, who assisted Laplace in revising it [The Mécanique Céleste] for the press, says that Laplace himself was frequently unable to recover the details in the chain of reasoning, and if satisfied that the conclusions were correct, he was content to insert the constantly recurring formula, “Il est àisé a voir” [it is easy to see].
In History of Mathematics (3rd Ed., 1901), 427.
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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 no process of sound reasoning can a conclusion drawn from limited data have more than a limited application.
In Higher Mathematics for Students of Chemistry and Physics (1902), 2.
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By what process of reasoning should the State of Texas be more concerned in the conviction of the guilty than in the acquittal of the innocent? [Urging the use of scopolamine “truth serum” to determine innocence.]
From paper read at the Section on State Medicine and Public Hygiene of the State Medical Association of Texas at El Paso (11 May 1922), 'The Use Of Scopolamine In Criminology', published in Texas State Journal of Medicine (Sep 1922). Reprinted in The American Journal of Police Science (Jul-Aug 1931), 2, No. 4, 328.
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Casting off the dark fog of verbal philosophy and vulgar medicine, which inculcate names alone ... I tried a series of experiments to explain more clearly many phenomena, particularly those of physiology. In order that I might subject as far as possible the reasonings of the Galenists and Peripatetics to sensory criteria, I began, after trying experiments, to write dialogues in which a Galenist adduced the better-known and stronger reasons and arguments; these a mechanist surgeon refuted by citing to the contrary the experiments I had tried, and a third, neutral interlocutor weighed the reasons advanced by both and provided an opportunity for further progress.
'Malpighi at Pisa 1656-1659', in H. B. Adelmann (ed.), Marcello Malpighi and the Evolution of Embryology (1966), Vol. 1, 155-6.
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Chemists have made of phlogiston a vague principle which is not at all rigorously defined, and which, in consequence, adapts itself to all explanations in which it is wished it shall enter; sometimes it is free fire, sometimes it is fire combined with the earthy element; sometimes it passes through the pores of vessels, sometimes they are impenetrable to it; it explains both the causticity and non-causticity, transparency and opacity, colours and absence of colours. It is a veritable Proteus which changes its form every instant. It is time to conduct chemistry to a more rigorous mode of reasoning ... to distinguish fact and observation from what is systematic and hypothetical.
'Réflexions sur le phlogistique', Mémoires de l'Académie des Sciences, 1783, 505-38. Reprinted in Oeuvres de Lavoisier (1864), Vol. 2, 640, trans. M. P. Crosland.
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Confined to its true domain, mathematical reasoning is admirably adapted to perform the universal office of sound logic: to induce in order to deduce, in order to construct. … It contents itself to furnish, in the most favorable domain, a model of clearness, of precision, and consistency, the close contemplation of which is alone able to prepare the mind to render other conceptions also as perfect as their nature permits. Its general reaction, more negative than positive, must consist, above all, in inspiring us everywhere with an invincible aversion for vagueness, inconsistency, and obscurity, which may always be really avoided in any reasoning whatsoever, if we make sufficient effort.
In Synthèse Subjective (1856), 98. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 202-203. From the original French, “Bornée à son vrai domaine, la raison mathématique y peut admirablement remplir l’office universel de la saine logique: induire pour déduire, afin de construire. … Elle se contente de former, dans le domaine le plus favorable, un type de clarté, de précision, et de consistance, dont la contemplation familière peut seule disposer l’esprit à rendre les autres conceptions aussi parfaites que le comporte leur nature. Sa réaction générale, plus négative que positive, doit surtout consister à nous inspirer partout une invincible répugnance pour le vague, l’incohérence, et l’obscurité, que nous pouvons réellement éviter envers des pensées quelconques, si nous y faisons assez d’efforts.”
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Considerable obstacles generally present themselves to the beginner, in studying the elements of Solid Geometry, from the practice which has hitherto uniformly prevailed in this country, of never submitting to the eye of the student, the figures on whose properties he is reasoning, but of drawing perspective representations of them upon a plane. ...I hope that I shall never be obliged to have recourse to a perspective drawing of any figure whose parts are not in the same plane.
Quoted in Adrian Rice, 'What Makes a Great Mathematics Teacher?' The American Mathematical Monthly, (June-July 1999), 540.
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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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Definition of Mathematics.—It has now become apparent that the traditional field of mathematics in the province of discrete and continuous number can only be separated from the general abstract theory of classes and relations by a wavering and indeterminate line. Of course a discussion as to the mere application of a word easily degenerates into the most fruitless logomachy. It is open to any one to use any word in any sense. But on the assumption that “mathematics” is to denote a science well marked out by its subject matter and its methods from other topics of thought, and that at least it is to include all topics habitually assigned to it, there is now no option but to employ “mathematics” in the general sense of the “science concerned with the logical deduction of consequences from the general premisses of all reasoning.”
In article 'Mathematics', Encyclopedia Britannica (1911, 11th ed.), Vol. 17, 880. In the 2006 DVD edition of the encyclopedia, the definition of mathematics is given as “The science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects.” [Premiss is a variant form of “premise”. —Webmaster]
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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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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 eory 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.
Variety of Men (1966), 100-1.
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Every philosophy is tinged with the colouring of some secret imaginative background, which never emerges explicitly into its train of reasoning.
In Science and the Modern World (1925), 7.
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Every work of science great enough to be well remembered for a few generations affords some exemplification of the defective state of the art of reasoning of the time when it was written; and each chief step in science has been a lesson in logic.
'The Fixation of Belief (1877). In Justus Buchler, The Philosophy of Pierce (1940), 6.
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Facts are certainly the solid and true foundation of all sectors of nature study ... Reasoning must never find itself contradicting definite facts; but reasoning must allow us to distinguish, among facts that have been reported, those that we can fully believe, those that are questionable, and those that are false. It will not allow us to lend faith to those that are directly contrary to others whose certainty is known to us; it will not allow us to accept as true those that fly in the face of unquestionable principles.
Memoires pour Servir a l'Histoire des Insectes (1736), Vol. 2, xxxiv. Quoted in Jacques Roger, The Life Sciences in Eighteenth-Century French Thought, ed. Keith R. Benson and trans. Robert Ellrich (1997), 165.
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For chemistry is no science form’d à priori; ’tis no production of the human mind, framed by reasoning and deduction: it took its rise from a number of experiments casually made, without any expectation of what follow’d; and was only reduced into an art or system, by collecting and comparing the effects of such unpremeditated experiments, and observing the uniform tendency thereof. So far, then, as a number of experimenters agree to establish any undoubted truth; so far they may be consider'd as constituting the theory of chemistry.
From 'The Author's Preface', in A New Method of Chemistry (1727), vi.
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For hundreds of pages the closely-reasoned arguments unroll, axioms and theorems interlock. And what remains with us in the end? A general sense that the world can be expressed in closely-reasoned arguments, in interlocking axioms and theorems.
In Constructions. 1974, 278.
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For there are two modes of acquiring knowledge, namely, by reasoning and experience. Reasoning draws a conclusion and makes us grant the conclusion, but does not make the conclusion certain, nor does it remove doubt so that the mind may rest on the intuition of truth, unless the mind discovers it by the path of experience; since many have the arguments relating to what can be known, but because they lack experience they neglect the arguments, and neither avoid what is harmful nor follow what is good. For if a man who has never seen fire should prove by adequate reasoning that fire burns and injures things and destroys them, his mind would not be satisfied thereby, nor would he avoid fire, until he placed his hand or some combustible substance in the fire, so that he might prove by experience that which reasoning taught. But when he has had actual experience of combustion his mind is made certain and rests in the full light of truth. Therefore reasoning does not suffice, but experience does.
Opus Majus [1266-1268], Part VI, chapter I, trans. R. B. Burke, The Opus Majus of Roger Bacon (1928), Vol. 2, 583.
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For these two years I have been gravitating towards your doctrines, and since the publication of your primula paper with accelerated velocity. By about this time next year I expect to have shot past you, and to find you pitching into me for being more Darwinian than yourself. However, you have set me going, and must just take the consequences, for I warn you I will stop at no point so long as clear reasoning will take me further.
Thomas Henry Huxley, Leonard Huxley, Life and Letters of Thomas Henry Huxley (1901), 211.
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Genius and science have burst the limits of space, and few observations, explained by just reasoning, have unveiled the mechanism of the universe. Would it not also be glorious for man to burst the limits of time, and, by a few observations, to ascertain the history of this world, and the series of events which preceded the birth of the human race?
'Preliminary discourse', to Recherches sur les Ossemens Fossiles (1812), trans. R. Kerr Essay on the Theory of the Earth (1813), 3-4.
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Geometrical reasoning, and arithmetical process, have each its own office: to mix the two in elementary instruction, is injurious to the proper acquisition of both.
In Trigonometry and Double Algebra (1849), 92.
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Geometry enlightens the intellect and sets one’s mind right. All of its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence.
In Ibn Khaldûn, Franz Rosenthal (trans.) and N.J. Dawood (ed.), The Muqaddimah: An Introduction to History (1967, 1969), Vol. 1, 378.
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He that desires to learn Truth should teach himself by Facts and Experiments; by which means he will learn more in a Year than by abstract reasoning in an Age.
In Academical Lectures on the Theory of Physic (1751), Vol. 1. As quoted in Thomas Steele Hall, A Source Book in Animal Biology (1951), 485.
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Higher Mathematics is the art of reasoning about numerical relations between natural phenomena; and the several sections of Higher Mathematics are different modes of viewing these relations.
In Higher Mathematics for Students of Chemistry and Physics (1902), Prologue, xvii.
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I believe in evidence. I believe in observation, measurement, and reasoning, confirmed by independent observers. I’ll believe anything, no matter how wild and ridiculous, if there is evidence for it. The wilder and more ridiculous something is, however, the firmer and more solid the evidence will have to be.
After stating he did definitely not believe in flying saucers, ancient astronauts, Bermuda Triangle or life after death, he explained what he would believe in. From editorial, 'Don’t You Believe?', Isaac Asimov's Science Fiction Magazine (18 Jan 1982), 6, No. 1, 6. Collected in The Roving Mind (1983), 43.
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I believe myself to possess a most singular combination of qualities exactly fitted to make me pre-eminently a discoverer of the hidden realities of nature… the belief has been forced upon me…
Firstly: Owing to some peculiarity in my nervous system, I have perceptions of some things, which no one else has… and intuitive perception of… things hidden from eyes, ears, & ordinary senses…
Secondly: my sense reasoning faculties;
Thirdly: my concentration faculty, by which I mean the power not only of throwing my whole energy & existence into whatever I choose, but also of bringing to bear on anyone subject or idea, a vast apparatus from all sorts of apparently irrelevant & extraneous sources…
Well, here I have written what most people would call a remarkably mad letter; & yet certainly one of the most logical, sober-minded, cool, pieces of composition, (I believe), that I ever framed.
Lovelace Papers, Bodleian Library, Oxford University, 42, folio 12 (6 Feb 1841). As quoted and cited in Dorothy Stein (ed.), 'This First Child of Mine', Ada: A Life and a Legacy (1985), 86.
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I can see him [Sylvester] now, with his white beard and few locks of gray hair, his forehead wrinkled o’er with thoughts, writing rapidly his figures and formulae on the board, sometimes explaining as he wrote, while we, his listeners, caught the reflected sounds from the board. But stop, something is not right, he pauses, his hand goes to his forehead to help his thought, he goes over the work again, emphasizes the leading points, and finally discovers his difficulty. Perhaps it is some error in his figures, perhaps an oversight in the reasoning. Sometimes, however, the difficulty is not elucidated, and then there is not much to the rest of the lecture. But at the next lecture we would hear of some new discovery that was the outcome of that difficulty, and of some article for the Journal, which he had begun. If a text-book had been taken up at the beginning, with the intention of following it, that text-book was most likely doomed to oblivion for the rest of the term, or until the class had been made listeners to every new thought and principle that had sprung from the laboratory of his mind, in consequence of that first difficulty. Other difficulties would soon appear, so that no text-book could last more than half of the term. In this way his class listened to almost all of the work that subsequently appeared in the Journal. It seemed to be the quality of his mind that he must adhere to one subject. He would think about it, talk about it to his class, and finally write about it for the Journal. The merest accident might start him, but once started, every moment, every thought was given to it, and, as much as possible, he read what others had done in the same direction; but this last seemed to be his real point; he could not read without finding difficulties in the way of understanding the author. Thus, often his own work reproduced what had been done by others, and he did not find it out until too late.
A notable example of this is in his theory of cyclotomic functions, which he had reproduced in several foreign journals, only to find that he had been greatly anticipated by foreign authors. It was manifest, one of the critics said, that the learned professor had not read Rummer’s elementary results in the theory of ideal primes. Yet Professor Smith’s report on the theory of numbers, which contained a full synopsis of Kummer’s theory, was Professor Sylvester’s constant companion.
This weakness of Professor Sylvester, in not being able to read what others had done, is perhaps a concomitant of his peculiar genius. Other minds could pass over little difficulties and not be troubled by them, and so go on to a final understanding of the results of the author. But not so with him. A difficulty, however small, worried him, and he was sure to have difficulties until the subject had been worked over in his own way, to correspond with his own mode of thought. To read the work of others, meant therefore to him an almost independent development of it. Like the man whose pleasure in life is to pioneer the way for society into the forests, his rugged mind could derive satisfaction only in hewing out its own paths; and only when his efforts brought him into the uncleared fields of mathematics did he find his place in the Universe.
In Florian Cajori, Teaching and History of Mathematics in the United States (1890), 266-267.
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I could not help laughing at the ease with which he explained his process of deduction. “When I hear you give your reasons,” I remarked, “the thing always appears to me to be so ridiculously simple that I could easily do it myself, though at each successive instance of your reasoning I am baffled, until you explain your process. And yet I believe that my eyes are as good as yours.”
“Quite so,” he answered, lighting a cigarette, and throwing himself down into an arm-chair. “You see, but you do not observe. The distinction is clear. For example, you have frequently seen the steps which lead up from the hall to this room.”
“Frequently.”
“How often?”
“'Well, some hundreds of times.”
“Then how many are there?”
“How many! I don't know.”
“Quite so! You have not observed. And yet you have seen. That is just my point. Now, I know that there seventeen steps, because I have both seen and observed.”
From 'Adventure I.—A Scandal in Bohemia', Adventures of Sherlock Holmes, in The Strand Magazine: An Illustrated Monthly (Jul 1891), Vol. 2, 62.
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I do not intend to go deeply into the question how far mathematical studies, as the representatives of conscious logical reasoning, should take a more important place in school education. But it is, in reality, one of the questions of the day. In proportion as the range of science extends, its system and organization must be improved, and it must inevitably come about that individual students will find themselves compelled to go through a stricter course of training than grammar is in a position to supply. What strikes me in my own experience with students who pass from our classical schools to scientific and medical studies, is first, a certain laxity in the application of strictly universal laws. The grammatical rules, in which they have been exercised, are for the most part followed by long lists of exceptions; accordingly they are not in the habit of relying implicitly on the certainty of a legitimate deduction from a strictly universal law. Secondly, I find them for the most part too much inclined to trust to authority, even in cases where they might form an independent judgment. In fact, in philological studies, inasmuch as it is seldom possible to take in the whole of the premises at a glance, and inasmuch as the decision of disputed questions often depends on an aesthetic feeling for beauty of expression, or for the genius of the language, attainable only by long training, it must often happen that the student is referred to authorities even by the best teachers. Both faults are traceable to certain indolence and vagueness of thought, the sad effects of which are not confined to subsequent scientific studies. But certainly the best remedy for both is to be found in mathematics, where there is absolute certainty in the reasoning, and no authority is recognized but that of one’s own intelligence.
In 'On the Relation of Natural Science to Science in general', Popular Lectures on Scientific Subjects, translated by E. Atkinson (1900), 25-26.
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I have hardly known a mathematician who was capable of reasoning.
Plato
The Republic. In Anton Bovier, Statistical Mechanics of Disordered Systems (2006), 159.
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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 like to look at mathematics almost more as an art than as a science; for the activity of the mathematician, constantly creating as he is, guided though not controlled by the external world of the senses, bears a resemblance, not fanciful I believe but real, to the activity of an artist, of a painter let us say. Rigorous deductive reasoning on the part of the mathematician may be likened here to technical skill in drawing on the part of the painter. Just as no one can become a good painter without a certain amount of skill, so no one can become a mathematician without the power to reason accurately up to a certain point. Yet these qualities, fundamental though they are, do not make a painter or mathematician worthy of the name, nor indeed are they the most important factors in the case. Other qualities of a far more subtle sort, chief among which in both cases is imagination, go to the making of a good artist or good mathematician.
From 'Fundamental Conceptions and Methods in Mathematics', Bulletin American Mathematical Society (1904), 9, 133. As cited in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 182.
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I presume that few who have paid any attention to the history of the Mathematical Analysis, will doubt that it has been developed in a certain order, or that that order has been, to a great extent, necessary—being determined, either by steps of logical deduction, or by the successive introduction of new ideas and conceptions, when the time for their evolution had arrived. And these are the causes that operate in perfect harmony. Each new scientific conception gives occasion to new applications of deductive reasoning; but those applications may be only possible through the methods and the processes which belong to an earlier stage.
Explaining his choice for the exposition in historical order of the topics in A Treatise on Differential Equations (1859), Preface, v-vi.
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I was pretty good in science. But again, because of the small budget, in science class we couldn’t do experiments in order to prove theories. We just believed everything. Actually I think that class was call Religion. Religion was always an easy class. All you had to do was suspend the logic and reasoning you were taught in all the other classes.
In autobiography, Brain Droppings (1998), 227.
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If scientific reasoning were limited to the logical processes of arithmetic, we should not get very far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability.
Endless Horizons (1946), 27.
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If the universe is measurably curved today, cosmologists must accept the miraculous fact that this is so for the first time in the 1010-year history of the universe; if it had been measurably non-flat at much earlier times, it would be much more obviously curved today than it is. This line of reasoning suggests that the observable universe is essentially exactly flat: that it contains precisely the critical density of mass.
(1986). As quoted in Isaac Asimov's Book of Science and Nature Quotations.
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If this “critical openminded attitude” … is wanted, the question at once arises, Is it science that should be studied in order to achieve it? Why not study law? A judge has to do everything that a scientist is exhorted to do in the way of withholding judgment until all the facts are in, and then judging impartially on the merits of the case as well as he can. … Why not a course in Sherlock Holmes? The detectives, or at least the detective-story writers, join with the scientists in excoriating “dogmatic prejudice, lying, falsification of facts, and data, and willful fallacious reasoning.”
In Science is a Sacred Cow (1950), 191.
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If we take in our hand any Volume; of Divinity or School Metaphysics, for Instance; let us ask, Does it contain any abstract Reasoning concerning Quantity or Number? No. Does it contain any experimental Reasoning concerning Matter of Fact and Existence? No. Commit it then to the Flames: For it can contain nothing but Sophistry and Illusion.
An Enquiry Concerning Human Understanding (1748), 256.
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Imagination is more robust in proportion as reasoning power is weak.
In The New Science (3rd ed., 1744), Book 1, Para. 185, as translated by Thomas Goddard Bergin and Max Harold Fisch, The New Science of Giambattista Vico (1948), 63.
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In every case the awakening touch has been the mathematical spirit, the attempt to count, to measure, or to calculate. What to the poet or the seer may appear to be the very death of all his poetry and all his visions—the cold touch of the calculating mind,—this has proved to be the spell by which knowledge has been born, by which new sciences have been created, and hundreds of definite problems put before the minds and into the hands of diligent students. It is the geometrical figure, the dry algebraical formula, which transforms the vague reasoning of the philosopher into a tangible and manageable conception; which represents, though it does not fully describe, which corresponds to, though it does not explain, the things and processes of nature: this clothes the fruitful, but otherwise indefinite, ideas in such a form that the strict logical methods of thought can be applied, that the human mind can in its inner chamber evolve a train of reasoning the result of which corresponds to the phenomena of the outer world.
In A History of European Thought in the Nineteenth Century (1896), Vol. 1, 314.
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In every enterprise … the mind is always reasoning, and, even when we seem to act without a motive, an instinctive logic still directs the mind. Only we are not aware of it, because we begin by reasoning before we know or say that we are reasoning, just as we begin by speaking before we observe that we are speaking, and just as we begin by seeing and hearing before we know what we see or what we hear.
From An Introduction to the Study of Experimental Medicine (1865), as translated by Henry Copley Greene (1957), 146.
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In Institutions of a lower grade [secondary schools], it [geology] receives far less attention than its merits deserve. Why should not a science, whose facts possess a thrilling interest; whose reasonings are admirably adapted for mental discipline, and often severely tax the strongest powers; and whose results are, many of them, as grand and ennobling as those of Astronomy itself; … why should not such a science be thought as essential in education as the kindred branches of Chemistry and Astronomy?
In 'Preface', Elementary Geology (1840, 1841), vi.
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In mathematics it [sophistry] had no place from the beginning: Mathematicians having had the wisdom to define accurately the terms they use, and to lay down, as axioms, the first principles on which their reasoning is grounded. Accordingly we find no parties among mathematicians, and hardly any disputes.
In Essays on the Intellectual Powers of Man, Essay 1, chap. 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 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 questions of science the authority of a thousand is not worth the humble reasoning of a single individual.
(1632). Attributed by F. Arago.
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In scientific study, or, as I prefer to phrase it, in creative scholarship, the truth is the single end sought; all yields to that. The truth is supreme, not only in the vague mystical sense in which that expression has come to be a platitude, but in a special, definite, concrete sense. Facts and the immediate and necessary inductions from facts displace all pre-conceptions, all deductions from general principles, all favourite theories. Previous mental constructions are bowled over as childish play-structures by facts as they come rolling into the mind. The dearest doctrines, the most fascinating hypotheses, the most cherished creations of the reason and of the imagination perish from a mind thoroughly inspired with the scientific spirit in the presence of incompatible facts. Previous intellectual affections are crushed without hesitation and without remorse. Facts are placed before reasonings and before ideals, even though the reasonings and the ideals be more beautiful, be seemingly more lofty, be seemingly better, be seemingly truer. The seemingly absurd and the seemingly impossible are sometimes true. The scientific disposition is to accept facts upon evidence, however absurd they may appear to our pre-conceptions.
The Ethical Functions of Scientific Study: An Address Delivered at the Annual Commencement of the University of Michigan, 28 June 1888, 7-8.
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In the final, the positive, state, the mind has given over the vain search after absolute notions, the origin and destination of the universe, and the causes of phenomena, and applies itself to the study of their laws—that is, their invariable relations of succession and resemblance. Reasoning and observation, duly combined, are the means of this knowledge. What is now understood when we speak of an explanation of facts is simply the establishment of a connection between single phenomena and some general facts.
The Positive Philosophy, trans. Harriet Martineau (1853), Vol. 1, 2.
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In the mountains of Parma and Piacenza, multitudes of shells and corals filled with worm-holes may be seen still adhering to the rocks, and when I was making the great horse at Milan a large sack of those which had been found in these parts was brought to my workshop by some peasants... The red stone of the mountains of Verona is found with shells all intermingled, which have become part of this stone... And if you should say that these shells have been and still constantly are being created in such places as these by the nature of the locality or by potency of the heavens in these spots, such an opinion cannot exist in brains possessed of any extensive powers of reasoning because the years of their growth are numbered upon the outer coverings of their shells; and both small and large ones may be seen; and these would not have grown without feeding, or fed without movement, and here [embedded in rock] they would not have been able to move... The peaks of the Apennines once stood up in a sea, in the form of islands surrounded by salt water... and above the plains of Italy where flocks of birds are flying today, fishes were once moving in large shoals.
'Physical Geography', in The Notebooks of Leonardo da Vinci, trans. E. MacCurdy (1938), Vol. 1, 355-6, 359.
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In the study of ideas, it is necessary to remember that insistence on hard-headed clarity issues from sentimental feeling, as it were a mist, cloaking the perplexities of fact. Insistence on clarity at all costs is based on sheer superstition as to the mode in which human intelligence functions. Our reasonings grasp at straws for premises and float on gossamers for deductions.
In Adventure of Ideas (1933), 91.
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In the whole history of the world there was never a race with less liking for abstract reasoning than the Anglo-Saxon. … Common-sense and compromise are believed in, logical deductions from philosophical principles are looked upon with suspicion, not only by legislators, but by all our most learned professional men.
In Teaching of Mathematics (1902), 20-21.
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In this manner the whole substance of our geometry is reduced to the definitions and axioms which we employ in our elementary reasonings; and in like manner we reduce the demonstrative truths of any other science to the definitions and axioms which we there employ.
In The Philosophy of the Inductive Sciences: Founded Upon Their History (1840), Vol. 1, 67.
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Indeed, the aim of teaching [mathematics] should be rather to strengthen his [the pupil’s] faculties, and to supply a method of reasoning applicable to other subjects, than to furnish him with an instrument for solving practical problems.
In John Perry (ed.), Discussion on the Teaching of Mathematics (1901), 84. The discussion took place on 14 Sep 1901 at the British Association at Glasgow, during a joint meeting of the mathematics and physics sections with the education section. The proceedings began with an address by John Perry. Magnus spoke in the Discussion that followed.
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Inductive reasoning is, of course, good guessing, not sound reasoning, but the finest results in science have been obtained in this way. Calling the guess a “working hypothesis,” its consequences are tested by experiment in every conceivable way.
In Higher Mathematics for Students of Chemistry and Physics (1902), 2.
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It frequently happens that two persons, reasoning right on a mechanical subject, think alike and invent the same thing without any communication with each other.
As quoted by Coleman Sellers, Jr., in his Lecture (20 Nov 1885) delivered at the Franklin Institute. Printed in Coleman Sellers, Jr., 'Oliver Evans and his Inventions', Journal of the Franklin Institute (Jul 1886), 122, No. 1, 15.
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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 has hitherto been a serious impediment to the progress of knowledge, that is in investigating the origin or causes of natural productions, recourse has generally been had to the examination, both by experiment and reasoning, of what might be rather than what is. The laws or processes of nature we have every reason to believe invariable. Their results from time to time vary, according to the combinations of influential circumstances; but the process remains the same. Like the poet or the painter, the chemist may, and no doubt often' does, create combinations which nature never produced; and the possibility of such and such processes giving rise to such and such results, is no proof whatever that they were ever in natural operation.
Considerations on Volcanoes (1825), 243.
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It has long been a complaint against mathematicians that they are hard to convince: but it is a far greater disqualification both for philosophy, and for the affairs of life, to be too easily convinced; to have too low a standard of proof. The only sound intellects are those which, in the first instance, set their standards of proof high. Practice in concrete affairs soon teaches them to make the necessary abatement: but they retain the consciousness, without which there is no sound practical reasoning, that in accepting inferior evidence because there is no better to be had, they do not by that acceptance raise it to completeness.
In An Examination of Sir William Hamilton’s Philosophy (1878), 611.
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It hath been an old remark, that Geometry is an excellent Logic. And it must be owned that when the definitions are clear; when the postulata cannot be refused, nor the axioms denied; when from the distinct contemplation and comparison of figures, their properties are derived, by a perpetual well-connected chain of consequences, the objects being still kept in view, and the attention ever fixed upon them; there is acquired a habit of reasoning, close and exact and methodical; which habit strengthens and sharpens the mind, and being transferred to other subjects is of general use in the inquiry after truth.
In 'The Analyst', in The Works of George Berkeley (1898), Vol. 3, 10.
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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 evidently equally foolish to accept probable reasoning from a mathematician and to demand from a rhetorician demonstrative proofs.
Aristotle
Nicomachean Ethics, 1094b, 25-7. In Jonathan Barnes (ed.), The Complete Works of Aristotle (1984), Vol. 2, 1730.
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It is his intuition, his mystical insight into the nature of things, rather than his reasoning which makes a great scientist.
In The Open Society and Its Enemies (1945).
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It is not therefore the business of philosophy, in our present situation in the universe, to attempt to take in at once, in one view, the whole scheme of nature; but to extend, with great care and circumspection, our knowledge, by just steps, from sensible things, as far as our observations or reasonings from them will carry us, in our enquiries concerning either the greater motions and operations of nature, or her more subtile and hidden works. In this way Sir Isaac Newton proceeded in his discoveries.
An Account of Sir Isaac Newton's Philosophical Discoveries, in Four Books (1748), 19.
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It is rigid dogma that destroys truth; and, please notice, my emphasis is not on the dogma, but on the rigidity. When men say of any question, “This is all there is to be known or said of the subject; investigation ends here,” that is death. It may be that the mischief comes not from the thinker but for the use made of his thinking by late-comers. Aristotle, for example, gave us our scientific technique … yet his logical propositions, his instruction in sound reasoning which was bequeathed to Europe, are valid only within the limited framework of formal logic, and, as used in Europe, they stultified the minds of whole generations of mediaeval Schoolmen. Aristotle invented science, but destroyed philosophy.
Dialogues of Alfred North Whitehead, as recorded by Lucien Price (1954, 2001), 165.
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It often happens that men, even of the best understandings and greatest circumspection, are guilty of that fault in reasoning which the writers on logick call the insufficient, or imperfect enumeration of parts, or cases: insomuch that I will venture to assert, that this is the chief, and almost the only, source of the vast number of erroneous opinions, and those too very often in matters of great importance, which we are apt to form on all the subjects we reflect upon, whether they relate to the knowledge of nature, or the merits and motives of human actions. It must therefore be acknowledged, that the art which affords a cure to this weakness, or defect, of our understandings, and teaches us to enumerate all the possible ways in which a given number of things may be mixed and combined together, that we may be certain that we have not omitted anyone arrangement of them that can lead to the object of our inquiry, deserves to be considered as most eminently useful and worthy of our highest esteem and attention. And this is the business of the art, or doctrine of combinations ... It proceeds indeed upon mathematical principles in calculating the number of the combinations of the things proposed: but by the conclusions that are obtained by it, the sagacity of the natural philosopher, the exactness of the historian, the skill and judgement of the physician, and the prudence and foresight of the politician, may be assisted; because the business of all these important professions is but to form reasonable conjectures concerning the several objects which engage their attention, and all wise conjectures are the results of a just and careful examination of the several different effects that may possibly arise from the causes that are capable of producing them.
Ars conjectandi (1713). In F. Maseres, The Doctrine of Permutations and Combinations (1795), 36.
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It was his [Leibnitz’s] love of method and order, and the conviction that such order and harmony existed in the real world, and that our success in understanding it depended upon the degree and order which we could attain in our own thoughts, that originally was probably nothing more than a habit which by degrees grew into a formal rule. This habit was acquired by early occupation with legal and mathematical questions. We have seen how the theory of combinations and arrangements of elements had a special interest for him. We also saw how mathematical calculations served him as a type and model of clear and orderly reasoning, and how he tried to introduce method and system into logical discussions, by reducing to a small number of terms the multitude of compound notions he had to deal with. This tendency increased in strength, and even in those early years he elaborated the idea of a general arithmetic, with a universal language of symbols, or a characteristic which would be applicable to all reasoning processes, and reduce philosophical investigations to that simplicity and certainty which the use of algebraic symbols had introduced into mathematics.
A mental attitude such as this is always highly favorable for mathematical as well as for philosophical investigations. Wherever progress depends upon precision and clearness of thought, and wherever such can be gained by reducing a variety of investigations to a general method, by bringing a multitude of notions under a common term or symbol, it proves inestimable. It necessarily imports the special qualities of number—viz., their continuity, infinity and infinite divisibility—like mathematical quantities—and destroys the notion that irreconcilable contrasts exist in nature, or gaps which cannot be bridged over. Thus, in his letter to Arnaud, Leibnitz expresses it as his opinion that geometry, or the philosophy of space, forms a step to the philosophy of motion—i.e., of corporeal things—and the philosophy of motion a step to the philosophy of mind.
In Leibnitz (1884), 44-45. [The first sentence is reworded to better introduce the quotation. —Webmaster]
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It [analysis] lacks at this point such plan and unity that it is really amazing that it can be studied by so many people. The worst is that it has not at all been treated with rigor. There are only a few propositions in higher analysis that have been demonstrated with complete rigor. Everywhere one finds the unfortunate manner of reasoning from the particular to the general, and it is very unusual that with such a method one finds, in spite of everything, only a few of what many be called paradoxes. It is really very interesting to seek the reason.
In my opinion that arises from the fact that the functions with which analysis has until now been occupied can, for the most part, be expressed by means of powers. As soon as others appear, something that, it is true, does not often happen, this no longer works and from false conclusions there flow a mass of incorrect propositions.
From a letter to his professor Hansteen in Christiania, Oslo in Correspondence (1902), 23 . In Umberto Bottazzini and Warren Van Egmond, The Higher Calculus (1986), 87-88.
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Like Molière’s M. Jourdain, who spoke prose all his life without knowing it, mathematicians have been reasoning for at least two millennia without being aware of all the principles underlying what they were doing. The real nature of the tools of their craft has become evident only within recent times A renaissance of logical studies in modern times begins with the publication in 1847 of George Boole’s The Mathematical Analysis of Logic.
Co-authored with James R. Newman in Gödel's Proof (1986, 2005), 30.
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LOGIC, n. The art of thinking and reasoning in strict accordance with the limitations and incapacities of the human misunderstanding. The basic of logic is the syllogism, consisting of a major and a minor premise and a conclusion—thus:
Major Premise: Sixty men can do a piece of work sixty times as quickly as one man.
Minor Premise: One man can dig a post-hole in sixty seconds; therefore—
Conclusion: Sixty men can dig a post-hole in one second.
This may be called the syllogism arithmetical, in which, by combining logic and mathematics, we obtain a double certainty and are twice blessed.
The Collected Works of Ambrose Bierce (1911), Vol. 7, The Devil's Dictionary,  196.
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Man does not limit himself to seeing; he thinks and insists on learning the meaning of phenomena whose existence has been revealed to him by observation. So he reasons, compares facts, puts questions to them, and by the answers which he extracts, tests one by another. This sort of control, by means of reasoning and facts, is what constitutes experiment, properly speaking; and it is the only process that we have for teaching ourselves about the nature of things outside us.
In Claude Bernard and Henry Copley Greene (trans.), An Introduction to the Study of Experimental Medicine (1927, 1957), 5.
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Man is made for science; he reasons from effects to causes, and from causes to effects; but he does not always reason without error. In reasoning, therefore, from appearances which are particular, care must be taken how we generalize; we should be cautious not to attribute to nature, laws which may perhaps be only of our own invention.
'Theory of the Earth', Transactions of the Royal Society of Edinburgh, 1788, 1, 273.
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Man is the Reasoning Animal. Such is the claim. I think it is open to dispute. Indeed, my experiments have proven to me that he is the Unreasoning Animal. Note his history, as sketched above. It seems plain to me that whatever he is he is not a reasoning animal. His record is the fantastic record of a maniac. I consider that the strongest count against his intelligence is the fact that with that record back of him he blandly sets himself up as the head animal of the lot: whereas by his own standards he is the bottom one.
In truth, man is incurably foolish. Simple things which the other animals easily learn, he is incapable of learning. Among my experiments was this. In an hour I taught a cat and a dog to be friends. I put them in a cage. In another hour I taught them to be friends with a rabbit. In the course of two days I was able to add a fox, a goose, a squirrel and some doves. Finally a monkey. They lived together in peace; even affectionately.
Next, in another cage I confined an Irish Catholic from Tipperary, and as soon as he seemed tame I added a Scotch Presbyterian from Aberdeen. Next a Turk from Constantinople; a Greek Christian from Crete; an Armenian; a Methodist from the wilds of Arkansas; a Buddhist from China; a Brahman from Benares. Finally, a Salvation Army Colonel from Wapping. Then I stayed away two whole days. When I came back to note results, the cage of Higher Animals was all right, but in the other there was but a chaos of gory odds and ends of turbans and fezzes and plaids and bones and flesh—not a specimen left alive. These Reasoning Animals had disagreed on a theological detail and carried the matter to a Higher Court.
In Letters from the Earth: Uncensored Writings (),
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Mathematical economics is old enough to be respectable, but not all economists respect it. It has powerful supporters and impressive testimonials, yet many capable economists deny that mathematics, except as a shorthand or expository device, can be applied to economic reasoning. There have even been rumors that mathematics is used in economics (and in other social sciences) either for the deliberate purpose of mystification or to confer dignity upon commonplaces as French was once used in diplomatic communications. …. To be sure, mathematics can be extended to any branch of knowledge, including economics, provided the concepts are so clearly defined as to permit accurate symbolic representation. That is only another way of saying that in some branches of discourse it is desirable to know what you are talking about.
In J.R. Newman (ed.), Commentary on Cournot, Jevons and the Mathematics of Money', The World of Mathematics (1956), Vol. 2, 1200.
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Mathematical proofs, like diamonds, are hard and clear, and will be touched with nothing but strict reasoning.
In 'Mr Locke’s Reply to the Bishop of Worcester’s Answer to his Second Letter', collected in The Works of John Locke (1824), Vol. 3, 428.
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Mathematical reasoning is deductive in the sense that it is based upon definitions which, as far as the validity of the reasoning is concerned (apart from any existential import), needs only the test of self-consistency. Thus no external verification of definitions is required in mathematics, as long as it is considered merely as mathematics.
In Universal Algebra (1898), Preface, vi.
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Mathematicians are only dealing with the structure of reasoning, and they do not really care what they are talking about. They do not even need to know what they are talking about … But the physicist has meaning to all his phrases. … In physics, you have to have an understanding of the connection of words with the real world.
In The Character of Physical Law (1965), 55.
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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 in its widest signification is the development of all types of formal, necessary, deductive reasoning.
In Universal Algebra (1898), Preface, vi.
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Mathematics is a form of poetry which transcends poetry in that it proclaims a truth; a form of reasoning which transcends reasoning in that it wants to bring about the truth it proclaims; a form of action, of ritual behavior, which does not find fulfilment in the act but must proclaim and elaborate a poetic form of truth.
'Why Mathematics Grows', Journal of the History of Ideas (Jan-Mar 1965), 26, No. 1, 3. In Salomon Bochner and Robert Clifford Gunning (ed.) Collected Papers of Salomon Bochner (1992), Vol. 4, 191. Footnoted as restating about Mathematics what was written about Myth by Henri Frankfort, et al., in The Intellectual Adventures of Ancient Man (1946), 8.
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Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems—general and specific statements—can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences.
Essay,'Private Games', in Lewis Wolpert, Alison Richards (eds.), A Passion for Science (1988), 53.
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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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More dependence must be placed on facts than on reasonings, which must agree with facts.
Aristotle
Cited as De Gener. Animal., III., 10, 760. 639 in George Henry Lewes, Aristotle: A Chapter from the History of Science including analyses of Aristotle's scientific writings (1864),
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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 religion consists of a humble admiration of the illimitable superior spirit who reveals himself in the slight details we are able to perceive with our frail and feeble minds. That deeply emotional conviction of the presence of a superior reasoning power, which is revealed in the incomprehensible universe, forms my idea of God.
As quoted in obituary, 'Einstein Noted as an Iconoclast In Research, Politics and Religion', The New York Times (19 Apr 1955), 25.
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Nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical reasoning involved and contrast it with the complex and far-reaching physical consequences, a deep sense of respect for the power of the symmetry laws never fails to develop.
Nobel Lecture (11 Dec 1957). In Nobel Lectures: Physics, 1981-1990) (1998), 394-395.
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No process of sound reasoning can establish a result not contained in the premises.
In Higher Mathematics for Students of Chemistry and Physics (1902), 2.
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Nor do I know any study which can compete with mathematics in general in furnishing matter for severe and continued thought. Metaphysical problems may be even more difficult; but then they are far less definite, and, as they rarely lead to any precise conclusion, we miss the power of checking our own operations, and of discovering whether we are thinking and reasoning or merely fancying and dreaming.
In Conflict of Studies (1873), 13.
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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 less applicable to life than mathematical reasoning. A proposition in mathematics is decidedly false or true. Everywhere else the true is mingled with the false.
Quoted, without citation, in David Eugene Smith, The Teaching of Elementary Mathematics (1904), 170.
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Now, what I want is, Facts. Teach these boys and girls nothing but Facts. Facts alone are wanted in life. Plant nothing else. And root out everything else. You can only form the minds of reasoning animals upon Facts: nothing else will ever be of any service to them. This is the principle on which I bring up my own children, and this is the principle on which I bring up these children. Stick to Facts, sir! ... In this life, we want nothing but Facts, sir: nothing but Facts!
Spoken by fictional character Thomas Gringrind, first paragraph, chap. 1, Hard Times, published in Household Words (1 Apr 1854), Vol. 36, 1.
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One of the most conspicuous and distinctive features of mathematical thought in the nineteenth century is its critical spirit. Beginning with the calculus, it soon permeates all analysis, and toward the close of the century it overhauls and recasts the foundations of geometry and aspires to further conquests in mechanics and in the immense domains of mathematical physics. … A searching examination of the foundations of arithmetic and the calculus has brought to light the insufficiency of much of the reasoning formerly considered as conclusive.
In History of Mathematics in the Nineteenth Century', Congress of Arts and Sciences (1906), Vol. 1, 482. As quoted and cited in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 113-114.
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One rarely hears of the mathematical recitation as a preparation for public speaking. Yet mathematics shares with these studies [foreign languages, drawing and natural science] their advantages, and has another in a higher degree than either of them.
Most readers will agree that a prime requisite for healthful experience in public speaking is that the attention of the speaker and hearers alike be drawn wholly away from the speaker and concentrated upon the thought. In perhaps no other classroom is this so easy as in the mathematical, where the close reasoning, the rigorous demonstration, the tracing of necessary conclusions from given hypotheses, commands and secures the entire mental power of the student who is explaining, and of his classmates. In what other circumstances do students feel so instinctively that manner counts for so little and mind for so much? In what other circumstances, therefore, is a simple, unaffected, easy, graceful manner so naturally and so healthfully cultivated? Mannerisms that are mere affectation or the result of bad literary habit recede to the background and finally disappear, while those peculiarities that are the expression of personality and are inseparable from its activity continually develop, where the student frequently presents, to an audience of his intellectual peers, a connected train of reasoning. …
One would almost wish that our institutions of the science and art of public speaking would put over their doors the motto that Plato had over the entrance to his school of philosophy: “Let no one who is unacquainted with geometry enter here.”
In A Scrap-book of Elementary Mathematics: Notes, Recreations, Essays (1908), 210-211.
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One would have to have been brought up in the “spirit of militarism” to understand the difference between Hiroshima and Nagasaki on the one hand, and Auschwitz and Belsen on the other. The usual reasoning is the following: the former case is one of warfare, the latter of cold-blooded slaughter. But the plain truth is that the people involved are in both instances nonparticipants, defenseless old people, women, and children, whose annihilation is supposed to achieve some political or military objective.… I am certain that the human race is doomed, unless its instinctive detestation of atrocities gains the upper hand over the artificially constructed judgment of reason.
Max Born
In The Born-Einstein Letters: Correspondence Between Albert Einstein and Max Born (1971), 205. Born’s commentary (at age 86) added for the book, printed after letter to Albert Einstein, 8 Nov 1953.
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One-story intellects, two-story intellects, three-story intellects with skylights. All fact-collectors, who have no aim beyond their facts, are one-story men. Two-story men compare, reason, generalize, using the labors of the fact-collectors as well as their own. Three-story men idealize, imagine, predict; their best illumination comes from above, through the skylight. There are minds with large ground-floors, that can store an infinite amount of knowledge; some librarians, for instance, who know enough of books to help other people, without being able to make much other use of their knowledge, have intellects of this class. Your great working lawyer has two spacious stories; his mind is clear, because his mental floors are large, and he has room to arrange his thoughts so that lie can get at them,—facts below, principles above, and all in ordered series; poets are often narrow below, incapable of clear statement, and with small power of consecutive reasoning, but full of light, if sometimes rather bare of furniture, in the attics.
The Poet at the Breakfast Table (1883), 50.
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Our reasonings are grounded upon two great principles, that of contradiction, in virtue of which we judge false that which involves a contradiction, and true that which is opposed or contradictory to the false.
The Monadology and Other Philosophical Writings (1714), trans. Robert Latta (1898) 235.
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Philosophy would long ago have reached a high level if our predecessors and fathers had put this into practice; and we would not waste time on the primary difficulties, which appear now as severe as in the first centuries which noticed them. We would have the experience of assured phenomena, which would serve as principles for a solid reasoning; truth would not be so deeply sunken; nature would have taken off most of her envelopes; one would see the marvels she contains in all her individuals. ...
Les Préludes de l'Harmonie Universelle (1634), 135-139. In Charles Coulston Gillispie (ed.), Dictionary of Scientific Biography (1974), Vol. 9, 316.
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Proper Experiments have always Truth to defend them; also Reasoning join’d with Mathematical Evidence, and founded upon Experiment, will hold equally true; but should it be true, without those Supports it must be altogether useless.
In Academical Lectures on the Theory of Physic (1751), Vol. 1. As quoted in Thomas Steele Hall, A Source Book in Animal Biology (1951), 485.
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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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Relations between authors and referees are, of course, almost always strained. Authors are convinced that the malicious stupidity of the referee is alone preventing them from laying their discoveries before an admiring world. Referees are convinced that authors are too arrogant and obtuse to recognize blatant fallacies in their own reasoning, even when these have been called to their attention with crystalline lucidity. All physicists know this, because all physicists are both authors and referees, but it does no good. The ability of one person to hold both views is an example of what Bohr called complementarity.
In Boojums All the Way Through: Communicating Science in a Prosaic Age (1990), 19-20.
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Science gives us the grounds of premises from which religious truths are to be inferred; but it does not set about inferring them, much less does it reach the inference; that is not its province. It brings before us phenomena, and it leaves us, if we will, to call them works of design, wisdom, or benevolence; and further still, if we will, to proceed to confess an Intelligent Creator. We have to take its facts, and to give them a meaning, and to draw our own conclusions from them. First comes Knowledge, then a view, then reasoning, then belief. This is why Science has so little of a religious tendency; deductions have no power of persuasion. The heart is commonly reached, not through the reason, but through the imagination, by means of direct impressions, by the testimony of facts and events, by history, by description. Persons influence us, voices melt us, looks subdue us, deeds inflame us. Many a man will live and die upon a dogma; no man will be a martyr for a conclusion.
Letter collected in Tamworth Reading Room: Letters on an Address Delivered by Sir Robert Peel, Bart., M.P. on the Establishment of a Reading Room at Tamworth (1841), 32. Excerpted in John Henry Newman, An Essay in Aid of a Grammar of Assent (1870), 89 & 94 footnote.
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Science is not a system of certain, or -established, statements; nor is it a system which steadily advances towards a state of finality... And our guesses are guided by the unscientific, the metaphysical (though biologically explicable) faith in laws, in regularities which we can uncover—discover. Like Bacon, we might describe our own contemporary science—'the method of reasoning which men now ordinarily apply to nature'—as consisting of 'anticipations, rash and premature' and as 'prejudices'.
The Logic of Scientific Discovery (1959), 278.
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Science is organized knowledge; and before knowledge can be organized, some of it must first be possessed. Every study, therefore, should have a purely experimental introduction; and only after an ample fund of observations has been accumulated, should reasoning begin.
In essay 'The Art of Education', The North British Review (May 1854), 137.
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Science quickens and cultivates directly the faculty of observation, which in very many persons lies almost dormant through life, the power of accurate and rapid generalizations, and the mental habit of method and arrangement; it accustoms young persons to trace the sequence of cause and effect; it familiarizes then with a kind of reasoning which interests them, and which they can promptly comprehend; and it is perhaps the best corrective for that indolence which is the vice of half-awakened minds, and which shrinks from any exertion that is not, like an effort of memory, merely mechanical.
Anonymous
Report of the Royal Commission on Education (1861), Parliamentary Papers (1864), Vol 20, 32-33, as cited in Paul White, Thomas Huxley: Making the "Man of Science" (2003), 77, footnote. Also quoted in John Lubbock, The Pleasures of Life (1887, 2007), 63.
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Scientific reasoning is a kind of dialogue between the possible and the actual, between what might be and what is in fact the case.
Induction and Intuition in Scientific Thought (1969), 48.
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Scientists are convinced that they, as scientists, possess a number of very admirable human qualities, such as accuracy, observation, reasoning power, intellectual curiosity, tolerance, and even humility.
In Science is a Sacred Cow (1950), 15-16.
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Scientists are not robotic inducing machines that infer structures of explanation only from regularities observed in natural phenomena (assuming, as I doubt, that such a style of reasoning could ever achieve success in principle). Scientists are human beings, immersed in culture, and struggling with all the curious tools of inference that mind permits ... Culture can potentiate as well as constrain–as Darwin’s translation of Adam Smith’s laissez-faire economic models into biology as the theory of natural selection. In any case, objective minds do not exist outside culture, so we must make the best of our ineluctable embedding.
…...
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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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See skulking Truth to her old cavern fled,
Mountains of Casuistry heap’d o’er her head!
Philosophy, that lean’d on Heav’n before,
Shrinks to her second cause, and is no more.
Physic of Metaphysic begs defence,
And Metaphysic calls for aid on Sense!
See Mystery to Mathematics fly!
In The Dunciad, collected in The Poetical Works of Alexander Pope (1828), Vol. 3, 211.
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Simple as the law of gravity now appears, and beautifully in accordance with all the observations of past and of present times, consider what it has cost of intellectual study. Copernicus, Galileo, Kepler, Euler, Lagrange, Laplace, all the great names which have exalted the character of man, by carrying out trains of reasoning unparalleled in every other science; these, and a host of others, each of whom might have been the Newton of another field, have all labored to work out, the consequences which resulted from that single law which he discovered. All that the human mind has produced—the brightest in genius, the most persevering in application, has been lavished on the details of the law of gravity.
in The Ninth Bridgewater Treatise: A Fragment (1838), 57.
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Since the world is what it is, it is clear that valid reasoning from sound principles cannot lead to error; but a principle may be so nearly true as to deserve theoretical respect, and yet may lead to practical consequences which we feel to be absurd. There is therefore a justification for common sense in philosophy, but only as showing that our theoretical principles cannot be quite correct so long as their consequences are condemned by an appeal to common sense which we feel to be irresistible.
In A History of Western Philosophy, (1945, 1996), 553.
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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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Speaking concretely, when we say “making experiments or making observations,” we mean that we devote ourselves to investigation and to research, that we make attempts and trials in order to gain facts from which the mind, through reasoning, may draw knowledge or instruction.
Speaking in the abstract, when we say “relying on observation and gaining experience,” we mean that observation is the mind's support in reasoning, and experience the mind's support in deciding, or still better, the fruit of exact reasoning applied to the interpretation of facts. It follows from this that we can gain experience without making experiments, solely by reasoning appropriately about well- established facts, just as we can make experiments and observations without gaining experience, if we limit ourselves to noting facts.
Observation, then, is what shows facts; experiment is what teaches about facts and gives experience in relation to anything.
From An Introduction to the Study of Experimental Medicine (1865), as translated by Henry Copley Greene (1957), 11.
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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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That deeply emotional conviction of the presence of a superior reasoning power, which is revealed in the incomprehensible universe, forms my idea of God.
…...
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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 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 Mathematics are Friends to Religion, inasmuch as they charm the Passions, restrain the Impetuosity of the Imagination, and purge the Mind from Error and Prejudice. Vice is Error, Confusion, and false Reasoning; and all Truth is more or less opposite to it. Besides, Mathematical Studies may serve for a pleasant Entertainment for those Hours which young Men are apt to throw away upon their Vices; the Delightfulness of them being such as to make Solitude not only easy, but desirable.
In An Essay On the Usefulness of Mathematical Learning, (1701) 8-9.
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The “British Association for the Promotion of Science,” … is almost necessary for the purposes of science. The periodical assemblage of persons, pursuing the same or différent branches of knowledge, always produces an excitement which is favourable to the development of new ideas; whilst the long period of repose which succeeds, is advantageous for the prosecution of the reasonings or the experiments then suggested; and the récurrence of the meeting in the succeeding year, will stimulate the activity of the inquirer, by the hope of being then enabled to produce the successful result of his labours.
In 'Future Prospects', On the Economy of Machinery and Manufactures (1st ed., 1832), chap. 32, 274. Note: The British Association for the Advancement of Science held its first meeting at York in 1831, the year before the first publication of this book in 1832.
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The ancients devoted a lifetime to the study of arithmetic; it required days to extract a square root or to multiply two numbers together. Is there any harm in skipping all that, in letting the school boy learn multiplication sums, and in starting his more abstract reasoning at a more advanced point? Where would be the harm in letting the boy assume the truth of many propositions of the first four books of Euclid, letting him assume their truth partly by faith, partly by trial? Giving him the whole fifth book of Euclid by simple algebra? Letting him assume the sixth as axiomatic? Letting him, in fact, begin his severer studies where he is now in the habit of leaving off? We do much less orthodox things. Every here and there in one’s mathematical studies one makes exceedingly large assumptions, because the methodical study would be ridiculous even in the eyes of the most pedantic of teachers. I can imagine a whole year devoted to the philosophical study of many things that a student now takes in his stride without trouble. The present method of training the mind of a mathematical teacher causes it to strain at gnats and to swallow camels. Such gnats are most of the propositions of the sixth book of Euclid; propositions generally about incommensurables; the use of arithmetic in geometry; the parallelogram of forces, etc., decimals.
In Teaching of Mathematics (1904), 12.
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The art of reasoning consists in getting hold of the subject at the right end, of seizing on the few general ideas that illuminate the whole, and of persistently organizing all subsidiary facts round them.
In 'Presidential Address to the London Branch of the Mathematical Association', Mathematical Gazette (Mar 1913), 7, No. 104, 92.
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The basis of the discovery is imagination, careful reasoning and experimentation where the use of knowledge created by those who came before is an important component.
Nobel Banquet speech (10 Dec 1982). In Wilhelm Odelberg (ed.), Les Prix Nobel. The Nobel Prizes 1982 (1983)
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The cause of the six-sided shape of a snowflake is none other than that of the ordered shapes of plants and of numerical constants; and since in them nothing occurs without supreme reason—not, to be sure, such as discursive reasoning discovers, but such as existed from the first in the Creators's design and is preserved from that origin to this day in the wonderful nature of animal faculties, I do not believe that even in a snowflake this ordered pattern exists at random.
Di Nive Sexangula, On the Six-Cornered Snowflake (1611), K18, 1. 6-12. Trans. and ed. Colin Hardie (1966), 33.
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The conception of correspondence plays a great part in modern mathematics. It is the fundamental notion in the science of order as distinguished from the science of magnitude. If the older mathematics were mostly dominated by the needs of mensuration, modern mathematics are dominated by the conception of order and arrangement. It may be that this tendency of thought or direction of reasoning goes hand in hand with the modern discovery in physics, that the changes in nature depend not only or not so much on the quantity of mass and energy as on their distribution or arrangement.
In History of European Thought in the Nineteenth Century (1903), Vol. 2, 736.
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The difference between science and the fuzzy subjects is that science requires reasoning while those other subjects merely require scholarship.
In Time Enough For Love: the Lives of Lazarus Long (1973), 366.
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The discovery of the laws of definite proportions is one of the most important and wonderful among the great and brilliant achievements of modern chemistry. It is sufficient of itself to convince any reasoning mind, that order and system pervade the universe, and that the minutest atoms of matter, and the vast orbs that move round the heavens are equally under the control of the invariable laws of the creator.
Elements of Chemistry (1845), 84.
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The distinctive Western character begins with the Greeks, who invented the habit of deductive reasoning and the science of geometry.
In 'Western Civilization', collected in In Praise of Idleness and Other Essays (1935), 161.
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The doctrine that logical reasoning produces no new truths, but only unfolds and brings into view those truths which were, in effect, contained in the first principles of the reasoning, is assented to by almost all who, in modern times, have attended to the science of logic.
In The Philosophy of the Inductive Sciences: Founded Upon Their History (1840), Vol. 1, 67.
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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 errors which arise from the absence of facts are far more numerous and more durable than those which result from unsound reasoning respecting true data.
In 'Of Price as Measured by Money', On the Economy of Machinery and Manufactures (1st ed., 1832), chap. 17, 112.
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The first step in all physical investigations, even in those which admit of the application of mathematical reasoning and the deductive method afterwards, is the observation of natural phenomena; and the smallest error in such observation in the beginning is sufficient to vitiate the whole investigation afterwards. The necessity of strict and minute observation, then, is the first thing which the student of the physical sciences has to learn; and it is easy to see with what great advantage the habit thus acquired may be carried into everything else afterwards.
Presidential Address to Anniversary meeting of the Royal Society (30 Nov 1859), Proceedings of the Royal Society of London (1860), 10, 164-165.
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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 great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible.
In History of Mathematics (3rd Ed., 1901), 468.
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The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible. … [I]t is hard to believe that our reasoning power was brought, by Darwin’s process of natural selection, to the perfection which it seems to possess.
In 'The Unreasonable Effectiveness of Mathematics in the Natural Sciences,' Communications in Pure and Applied Mathematics (Feb 1960), 13, No. 1 (February 1960). Collected in Eugene Paul Wigner, A.S. Wightman (ed.), Jagdish Mehra (ed.), The Collected Works of Eugene Paul Wigner (1955), Vol. 6, 536.
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The greatest enemy, however, to true arithmetic work is found in so-called practical or illustrative problems, which are freely given to our pupils, of a degree of difficulty and complexity altogether unsuited to their age and mental development. … I am, myself, no bad mathematician, and all the reasoning powers with which nature endowed me have long been as fully developed as they are ever likely to be; but I have, not infrequently, been puzzled, and at times foiled, by the subtle logical difficulty running through one of these problems, given to my own children. The head-master of one of our Boston high schools confessed to me that he had sometimes been unable to unravel one of these tangled skeins, in trying to help his own daughter through her evening’s work. During this summer, Dr. Fairbairn, the distinguished head of one of the colleges of Oxford, England, told me that not only had he himself encountered a similar difficulty, in the case of his own children, but that, on one occasion, having as his guest one of the first mathematicians of England, the two together had been completely puzzled by one of these arithmetical conundrums.
Address before the Grammar-School Section of the Massachusetts Teachers’ Association (25 Nov 1887), 'The Teaching of Arithmetic in the Boston Schools', printed The Academy (Jan 1888). Collected in Francis Amasa Walker, Discussions in Education (1899), 253.
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The ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every province of thought, or of external experience, in which the succession of thoughts, or of events can be definitely ascertained and precisely stated. So that all serious thought which is not philosophy, or inductive reasoning, or imaginative literature, shall be mathematics developed by means of a calculus.
In Universal Algebra (1898), Preface.
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The influence of electricity in producing decompositions, although of inestimable value as an instrument of discovery in chemical inquiries, can hardly be said to have been applied to the practical purposes of life, until the same powerful genius [Davy] which detected the principle, applied it, by a singular felicity of reasoning, to arrest the corrosion of the copper-sheathing of vessels. … this was regarded as by Laplace as the greatest of Sir Humphry's discoveries.
Reflections on the Decline of Science in England (1830), 16.
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The iron labor of conscious logical reasoning demands great perseverance and great caution; it moves on but slowly, and is rarely illuminated by brilliant flashes of genius. It knows little of that facility with which the most varied instances come thronging into the memory of the philologist or historian. Rather is it an essential condition of the methodical progress of mathematical reasoning that the mind should remain concentrated on a single point, undisturbed alike by collateral ideas on the one hand, and by wishes and hopes on the other, and moving on steadily in the direction it has deliberately chosen.
In Ueber das Verhältniss der Naturwissenschaften zur Gesammtheit der Wissenschaft, Vorträge und Reden (1896), Bd. 1, 178.
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The knowledge of Natural-History, being Observation of Matters of Fact, is more certain than most others, and in my slender Opinion, less subject to Mistakes than Reasonings, Hypotheses, and Deductions are; ... These are things we are sure of, so far as our Senses are not fallible; and which, in probability, have been ever since the Creation, and will remain to the End of the World, in the same Condition we now find them.
A Voyage to the Islands Madera, Barbados, Nieves, S. Christophers and Jamaica: With the Natural History of the Herbs and Trees, Four-footed Beasts, Fishes, Birds, Insects, Reptiles, &c. of the Last of those Islands (1707), Vol. 1, 1.
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The language of experiment is more authoritative than any reasoning: facts can destroy our ratiocination—not vice versa.
In Marcello Pera, The Ambiguous Frog: The Galvani-Volta Controversy on Animal Electricity (1992). Cited in Patrick F. Dunn, Measurement and Data Analysis for Engineering and Science (2010), 15.
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The love of experiment was very strong in him [Charles Darwin], and I can remember the way he would say, “I shan't be easy till I have tried it,” as if an outside force were driving him. He enjoyed experimenting much more than work which only entailed reasoning, and when he was engaged on one of his books which required argument and the marshalling of facts, he felt experimental work to be a rest or holiday.
In Charles Darwin: His Life Told in an Autobiographical Chapter, and in a Selected Series of his Published Letters (1908), 95.
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The mathematician pays not the least regard either to testimony or conjecture, but deduces everything by demonstrative reasoning, from his definitions and axioms. Indeed, whatever is built upon conjecture, is improperly called science; for conjecture may beget opinion, but cannot produce knowledge.
In Essays on the Intellectual Powers of Man, Essay 1, chap. 3.
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The mind God is looking for in man is a doubting, questioning mind, not a dogmatic mind; dogmatic reasoning is wrong reasoning. Dogmatic reason ties a huge rock to a man’s foot and stops him forever from advancing.
From the play Galileo Galilei (2001) .
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The modern, and to my mind true, theory is that mathematics is the abstract form of the natural sciences; and that it is valuable as a training of the reasoning powers not because it is abstract, but because it is a representation of actual things.
From 'Introduction', Mathematical Teaching and its Modern Methods (1886), 9-10.
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The moment of truth, the sudden emergence of new insight, is an act of intuition. Such intuitions give the appearance of miraculous flashes, or short circuits of reasoning. In fact they may be likened to an immersed chain, of which only the beginning and the end are visible above the surface of consciousness. The diver vanishes at one end of the chain and comes up at the other end, guided by invisible links.
In The Act of Creation (1964).
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The moving power of mathematical invention is not reasoning but imagination.
Quoted in Robert Perceval Graves, Life of Sir W. R. Hamilton, Vol. 3 (1889), 219.
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The one who stays in my mind as the ideal man of science is, not Huxley or Tyndall, Hooker or Lubbock, still less my friend, philosopher and guide Herbert Spencer, but Francis Galton, whom I used to observe and listen to—I regret to add, without the least reciprocity—with rapt attention. Even to-day. I can conjure up, from memory’s misty deep, that tall figure with its attitude of perfect physical and mental poise; the clean-shaven face, the thin, compressed mouth with its enigmatical smile; the long upper lip and firm chin, and, as if presiding over the whole personality of the man, the prominent dark eyebrows from beneath which gleamed, with penetrating humour, contemplative grey eyes. Fascinating to me was Francis Galton’s all-embracing but apparently impersonal beneficence. But, to a recent and enthusiastic convert to the scientific method, the most relevant of Galton’s many gifts was the unique contribution of three separate and distinct processes of the intellect; a continuous curiosity about, and rapid apprehension of individual facts, whether common or uncommon; the faculty for ingenious trains of reasoning; and, more admirable than either of these, because the talent was wholly beyond my reach, the capacity for correcting and verifying his own hypotheses, by the statistical handling of masses of data, whether collected by himself or supplied by other students of the problem.
In My Apprenticeship (1926), 134-135.
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The philosopher may very justly be delighted with the extent of his views, the artificer with the readiness of his hands, but let the one remember that without mechanical performance, refined speculation is an empty dream, and the other that without theoretical reasoning, dexterity is little more than brute instinct.
In 'The Rambler' (17 Apr 1750), No. 9. Collected in The Rambler (1763), Vol. 1, 48.
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The physicist, in his study of natural phenomena, has two methods of making progress: (1) the method of experiment and observation, and (2) the method of mathematical reasoning. The former is just the collection of selected data; the latter enables one to infer results about experiments that have not been performed. There is no logical reason why the second method should be possible at all, but one has found in practice that it does work and meets with reasonable success.
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, 122.
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The problems of the infinite have challenged man’s mind and have fired his imagination as no other single problem in the history of thought. The infinite appears both strange and familiar, at times beyond our grasp, at times easy and natural to understand. In conquering it, man broke the fetters that bound him to earth. All his faculties were required for this conquest—his reasoning powers, his poetic fancy, his desire to know.
With co-author James R Newman, in 'Beyond the Google', Mathematics and the Imagination (1940), 35.
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The Reader may here observe the Force of Numbers, which can be successfully applied, even to those things, which one would imagine are subject to no Rules. There are very few things which we know, which are not capable of being reduc’d to a Mathematical Reasoning, and when they cannot, it’s a sign our Knowledge of them is very small and confus’d; and where a mathematical reasoning can be had, it’s as great folly to make use of any other, as to grope for a thing in the dark when you have a Candle standing by you.
Of the Laws of Chance, or, a Method of the Hazards of Game (1692), Preface.
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The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure.
From the original French, “On ne trouvera point de Figures dans set Ouvrage. Les méthodes que j’y expose ne demandent ni constructions, ni raisonnements géométriqus ou méchaniques, mais seulement des opérations algébriques, assujetties à une march régulière et uniforme.” In 'Avertissement', Mécanique Analytique (1788, 1811), Vol. 1, i. English version as given in Cornelius Lanczos, The Variational Principles of Mechanics (1966), Vol. 1, 347.
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The reasoning of mathematicians is founded on certain and infallible principles. Every word they use conveys a determinate idea, and by accurate definitions they excite the same ideas in the mind of the reader that were in the mind of the writer. When they have defined the terms they intend to make use of, they premise a few axioms, or self-evident principles, that every one must assent to as soon as proposed. They then take for granted certain postulates, that no one can deny them, such as, that a right line may be drawn from any given point to another, and from these plain, simple principles they have raised most astonishing speculations, and proved the extent of the human mind to be more spacious and capacious than any other science.
In Diary, Works (1850), Vol. 2, 21.
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The reasoning of mathematics is a type of perfect reasoning.
In Common Sense in Education and Teaching (1905), 222.
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The required techniques of effective reasoning are pretty formal, but as long as programming is done by people that don’t master them, the software crisis will remain with us and will be considered an incurable disease. And you know what incurable diseases do: they invite the quacks and charlatans in, who in this case take the form of Software Engineering gurus.
…...
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The results of systematic symbolical reasoning must always express general truths, by their nature; and do not, for their justification, require each of the steps of the process to represent some definite operation upon quantity. The absolute universality of the interpretation of symbols is the fundamental principle of their use.
In 'The Foundations of Higher Mathematics', The Philosophy of the Inductive Sciences (1847), Part I, Bk. 2, 149.
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The scientific spirit is of more value than its products, and irrationally held truths may be more harmful than reasoned errors.
'The Coming of Age of the Origin of Species' (1880). In Collected Essays (1893), Vol. 2, 229.
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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 traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand. He prefers to face a blackboard and to turn his back on the class. He writes a, he says b, he means c, but it should be d. Some of his sayings are handed down from generation to generation:
“In order to solve this differential equation you look at it till a solution occurs to you.”
“This principle is so perfectly general that no particular application of it is possible.”
“Geometry is the science of correct reasoning on incorrect figures.”
“My method to overcome a difficulty is to go round it.”
“What is the difference between method and device? A method is a device which you used twice.”
In How to Solve It: A New Aspect of Mathematical Method (2004), 208.
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The whole of Mathematics consists in the organization of a series of aids to the imagination in the process of reasoning.
In Universal Algebra (1898), 12.
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There are also two kinds of truths, those of reasoning and those of fact. Truths of reasoning are necessary and their opposite is impossible: truths of fact are contingent and their opposite is possible. When a truth is necessary, reason can be found by analysis, resolving it into more simple ideas and truths, until we come to those which are primary.
The Monadology and Other Philosophical Writings (1714), trans. Robert Latta (1898), 235-6.
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There can be no scientific foundation of religion, and belief must always remain the foundation of religion, while that of science is logical reasoning from facts, that is, sense perceptions; and all that we can say is, that the two, science and religion, are not necessarily incompatible, but are different and unrelated activities of the human mind.
In 'Religion and Modern Science', The Christian Register (16 Nov 1922), 101, 1089. The article is introduced as “the substance of an address to the Laymen’s League in All Soul’s Church (5 Nov 1922).
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There is an integration of the present impressions with such past ones as they resemble, and a differentiation of them from such past ones as they do not resemble; and this comparison of present with past impressions, dependent on memory, implies classification, and is the germ of what we call Perception and Reasoning.
In Outlines of Cosmic Philosophy: Based on the Doctrine of Evolution (1874), Vol. 2, 155-156.
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There is no substitute for honest, thorough, scientific effort to get correct data (no matter how much of it clashes with preconceived ideas). There is no substitute for actually reaching a correct claim of reasoning. Poor data and good reasoning give poor results. Good data and poor reasoning give poor results. Poor data and poor reasoning give rotten results.
In 'Right Answers—A Short Guide for Obtaining Them,' Computers and Automation, September 1969.
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There is probably no other science which presents such different appearances to one who cultivates it and to one who does not, as mathematics. To this person it is ancient, venerable, and complete; a body of dry, irrefutable, unambiguous reasoning. To the mathematician, on the other hand, his science is yet in the purple bloom of vigorous youth, everywhere stretching out after the “attainable but unattained” and full of the excitement of nascent thoughts; its logic is beset with ambiguities, and its analytic processes, like Bunyan’s road, have a quagmire on one side and a deep ditch on the other and branch off into innumerable by-paths that end in a wilderness.
In 'The Theory of Transformation Groups', (A review of Erster Abschnitt, Theorie der Transformationsgruppen (1888)), Bulletin New York Mathematical Society (1893), 2 (First series), 61.
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This science [experimental science] alone, therefore, knows how to test perfectly what can be done by nature, what by the effort of art, what by trickery, what the incantations, conjurations, invocations, deprecations, sacrifices that belong to magic mean and dream of, and what is in them, so that all falsity may be removed and the truth alone of art and nature may be retained. This science alone teaches us how to view the mad acts of magicians, that they may be not ratified but shunned, just as logic considers sophistical reasoning.
In Opus Majus (1267).
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Those who are accustomed to judge by feeling do not understand the process of reasoning, because they want to comprehend at a glance and are not used to seeking for first principles. Those, on the other hand, who are accustomed to reason from first principles do not understand matters of feeling at all, because they look for first principles and are unable to comprehend at a glance.
In Pensées (1670), Section 7, No. 33. As translated in W.H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms (1966), 351. Also translated as “Those who are accustomed to judge by feeling do not understand the process of reasoning, for they would understand at first sight, and are not used to seek for principles. And others, on the contrary, who are accustomed to reason from principles, do not at all understand matters of feeling, seeking principles, and being unable to see at a glance,” in Blaise Pascal and W.F. Trotter (trans.), 'Thoughts', No. 3, collected in Charles W. Eliot (ed.), The Harvard Classics (1910), Vol. 48, 9. From the original French, “Ceux qui sont accoutumés à juger par le sentiment ne comprennent rien aux choses de raisonnement, car ils veulent d’abord pénétrer d’une vue et ne sont point accoutumés à chercher les principes. Et les autres, au contraire, qui sont accoutumés à raisonner par principes, ne comprennent rien aux choses de sentiment, y cherchant des principes et ne pouvant voir d’une vue,” in Ernest Havet (ed.), Pensées de Pascal (1892), 224.
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Though we must not without further consideration condemn a body of reasoning merely because it is easy, nevertheless we must not allow ourselves to be lured on merely by easiness; and we should take care that every problem which we choose for attack, whether it be easy or difficult, shall have a useful purpose, that it shall contribute in some measure to the up-building of the great edifice.
From 'On Some Recent Tendencies in Geometric Investigation', Rivista di Matematica (1891), 63. In Bulletin American Mathematical Society (1904), 465.
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Tis evident that all reasonings concerning matter of fact are founded on the relation of cause and effect, and that we can never infer the existence of one object from another, unless they be connected together, either mediately or immediately... Here is a billiard ball lying on the table, and another ball moving toward it with rapidity. They strike; and the ball which was formerly at rest now acquires a motion. This is as perfect an instance of the relation of cause and effect as any which we know, either by sensation or reflection.
An Abstract of A Treatise on Human Nature (1740), ed. John Maynard Keynes and Piero Sraffa (1938), 11.
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To arrive at the simplest truth, as Newton knew and practiced, requires years of contemplation. Not activity Not reasoning. Not calculating. Not busy behaviour of any kind. Not reading. Not talking. Not making an effort. Not thinking. Simply bearing in mind what it is one needs to know. And yet those with the courage to tread this path to real discovery are not only offered practically no guidance on how to do so, they are actively discouraged and have to set about it in secret, pretending meanwhile to be diligently engaged in the frantic diversions and to conform with the deadening personal opinions which are continually being thrust upon them.
In 'Appendix 1', The Laws of Form (1969), 110.
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To be worthy of the name, an experimenter must be at once theorist and practitioner. While he must completely master the art of establishing experimental facts, which are the materials of science, he must also clearly understand the scientific principles which guide his reasoning through the varied experimental study of natural phenomena. We cannot separate these two things: head and hand. An able hand, without a head to direct it, is a blind tool; the head is powerless without its executive hand.
In Claude Bernard and Henry Copley Greene (trans.), An Introduction to the Study of Experimental Medicine (1927, 1957), 3.
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To behold is not necessa