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Induction Quotes (59 quotes)

A great part of its [higher arithmetic] theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed.
Quoted in H. Eves, Mathematical Circles (1977) .
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A principle of induction would be a statement with the help of which we could put inductive inferences into a logically acceptable form. In the eyes of the upholders of inductive logic, a principle of induction is of supreme importance for scientific method: “... this principle”, says Reichenbach, “determines the truth of scientific theories. To eliminate it from science would mean nothing less than to deprive science of the power to decide the truth or falsity of its theories. Without it, clearly, science would no longer have the right to distinguish its theories from the fanciful and arbitrary creations of the poet’s mind.” Now this principle of induction cannot be a purely logical truth like a tautology or an analytic statement. Indeed, if there were such a thing as a purely logical principle of induction, there would be no problem of induction; for in this case, all inductive inferences would have to be regarded as purely logical or tautological transformations, just like inferences in inductive logic. Thus the principle of induction must be a synthetic statement; that is, a statement whose negation is not self-contradictory but logically possible. So the question arises why such a principle should be accepted at all, and how we can justify its acceptance on rational grounds.
…...
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Almost everyone... seems to be quite sure that the differences between the methodologies of history and of the natural sciences are vast. For, we are assured, it is well known that in the natural sciences we start from observation and proceed by induction to theory. And is it not obvious that in history we proceed very differently? Yes, I agree that we proceed very differently. But we do so in the natural sciences as well.
In both we start from myths—from traditional prejudices, beset with error—and from these we proceed by criticism: by the critical elimination of errors. In both the role of evidence is, in the main, to correct our mistakes, our prejudices, our tentative theories—that is, to play a part in the critical discussion, in the elimination of error. By correcting our mistakes, we raise new problems. And in order to solve these problems, we invent conjectures, that is, tentative theories, which we submit to critical discussion, directed towards the elimination of error.
The Myth of the Framework: In Defence of Science and Rationality (1993), 140.
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Analysis and natural philosophy owe their most important discoveries to this fruitful means, which is called induction. Newton was indebted to it for his theorem of the binomial and the principle of universal gravity.
In Frederick Wilson Truscott (trans.) and Frederick Lincoln Emory (trans.), A Philosophical Essay on Probabilities (1902), 176.
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As in Mathematicks, so in Natural Philosophy, the Investigation of difficult Things by the Method of Analysis, ought ever to precede the Method of Composition. This Analysis consists in making Experiments and Observations, and in drawing general Conclusions from them by Induction, and admitting of no Objections against the Conclusions, but such as are taken from Experiments, or other certain Truths. For Hypotheses are not to be regarded in experimental Philosophy.
From Opticks, (1704, 2nd ed. 1718), Book 3, Query 31, 380.
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Ask a scientist what he conceives the scientific method to be, and he will adopt an expression that is at once solemn and shifty eyed: solemn because he feels he ought to declare an opinion; shifty eyed because he is wondering how to conceal the fact that he has no opinion to declare. If taunted he would probably mumble something about “Induction” and “Establishing the Laws of Nature”, but if anyone working in a laboratory professed to be trying to establish the Laws of Nature by induction, we should think he was overdue for leave.
From a Jayne Lecture (1968), 'Induction and Intuition in Scientific Thought', printed in Memoirs of the American Philosophical Society (1969), Vol. 75. Lecture republished as Induction and Intuition in Scientific Thought (2009), 11. Also included in Peter Medawar, Pluto’s Republic (1984), 80.
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But Medicine is a demonstrative Science, and all its processes should be proved by established principles, and be based on positive inductions. That the proceedings of Medicine are not of this character, in to be attributed to the manner of its cultivation, and not to the nature of the Science itself.
Samuel Jackson, Principles of Medicine (1832). Quoted in Alva Curtis, A Fair Examination and Criticism of All the Medical Systems in Vogue (1855), 1.
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Experiment is fundamentally only induced observation.
Section title in An Introduction to the Study of Experimental Medicine (1865), as translated by Henry Copley Greene (1957), 19.
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Experiments may be of two kinds: experiments of simple fact, and experiments of quantity. ...[In the latter] the conditions will ... vary, not in quality, but quantity, and the effect will also vary in quantity, so that the result of quantitative induction is also to arrive at some mathematical expression involving the quantity of each condition, and expressing the quantity of the result. In other words, we wish to know what function the effect is of its conditions. We shall find that it is one thing to obtain the numerical results, and quite another thing to detect the law obeyed by those results, the latter being an operation of an inverse and tentative character.
Principles of Science: A Treatise on Logic and Scientific Method (1874, 1892), 439.
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Facts are not pure unsullied bits of information; culture also influences what we see and how we see it. Theories, moreover, are not inexorable inductions from facts. The most creative theories are often imaginative visions imposed upon facts; the source of imagination is also strongly cultural.
In The Mismeasure of Man (1981, 1996), 54.
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Given any domain of thought in which the fundamental objective is a knowledge that transcends mere induction or mere empiricism, it seems quite inevitable that its processes should be made to conform closely to the pattern of a system free of ambiguous terms, symbols, operations, deductions; a system whose implications and assumptions are unique and consistent; a system whose logic confounds not the necessary with the sufficient where these are distinct; a system whose materials are abstract elements interpretable as reality or unreality in any forms whatsoever provided only that these forms mirror a thought that is pure. To such a system is universally given the name MATHEMATICS.
In 'Mathematics', National Mathematics Magazine (Nov 1937), 12, No. 2, 62.
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Having discovered … by observation and comparison that certain objects agree in certain respects, we generalise the qualities in which they coincide,—that is, from a certain number of individual instances we infer a general law; we perform an act of Induction. This induction is erroneously viewed as analytic; it is purely a synthetic process.
In Lecture VI of his Biennial Course, by William Hamilton and Henry L. Mansel (ed.) and John Veitch (ed.), Metaphysics (1860), Vol. 1, 101.
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Here I shall present, without using Analysis [mathematics], the principles and general results of the Théorie, applying them to the most important questions of life, which are indeed, for the most part, only problems in probability. One may even say, strictly speaking, that almost all our knowledge is only probable; and in the small number of things that we are able to know with certainty, in the mathematical sciences themselves, the principal means of arriving at the truth—induction and analogy—are based on probabilities, so that the whole system of human knowledge is tied up with the theory set out in this essay.
Philosophical Essay on Probabilities (1814), 5th edition (1825), trans. Andrew I. Dale (1995), 1.
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I am convinced that it is impossible to expound the methods of induction in a sound manner, without resting them upon the theory of probability. Perfect knowledge alone can give certainty, and in nature perfect knowledge would be infinite knowledge, which is clearly beyond our capacities. We have, therefore, to content ourselves with partial knowledge—knowledge mingled with ignorance, producing doubt.
The Principles of Science: A Treatise on Logic and Scientific Method, 2nd edition (1877), 197.
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If men of science owe anything to us, we may learn much from them that is essential. For they can show how to test proof, how to secure fulness and soundness in induction, how to restrain and to employ with safety hypothesis and analogy.
Lecture, 'The Study of History' (11 Jun 1895) delivered at Cambridge, published as A Lecture on The Study of History (1895), 54.
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If the [Vestiges] be true, the labours of sober induction are in vain; religion is a lie; human law is a mass of folly, and a base injustice; morality is moonshine; our labours for the black people of Africa were works of madmen; and man and woman are only better beasts!
Letter to Charles Lyell (9 Apr 1845). In John Willis Clark and Thomas McKenny Hughes (eds.), The Life and Letters of the Reverend Adam Sedgwick (1890), Vol. 2, 84.
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In experimental philosophy, propositions gathered from phenomena by induction should be considered either exactly or very nearly true notwithstanding any contrary hypotheses, until yet other phenomena make such propositions either more exact or liable to exceptions.
The Principia: Mathematical Principles of Natural Philosophy (1687),3rd edition (1726), trans. I. Bernard Cohen and Anne Whitman (1999), Book 3, Rules of Reasoning in Philosophy, Rule 4, 796.
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Induction and analogy are the special characteristics of modern mathematics, in which theorems have given place to theories and no truth is regarded otherwise than as a link in an infinite chain. “Omne exit in infinitum” is their favorite motto and accepted axiom.
In 'A Plea for the Mathematician', Nature, Vol. 1, 861. [The Latin phrase “Omne exit in infinitum” means “Everything goes to infinity”.
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Induction for deduction, with a view to construction.
Attributed in John Arthur Thomson, quote at heading of chapter 'Scientific Method', Introduction to Science (1911), 57, but without further citation. Webmaster has found no primary source for confirmation. Please contact if you can help.
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Induction is the process of generalizing from our known and limited experience, and framing wider rules for the future than we have been able to test fully. At its simplest, then, an induction is a habit or an adaptation—the habit of expecting tomorrow’s weather to be like today’s, the adaptation to the unwritten conventions of community life.
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Induction may be defined, the operation of discovering and proving general propositions.
In A System of Logic, Ratiocinative and Inductive: Being a Connected View of the Principles of Evidence, and the Methods of Scientific Investigation (1843), Vol. 1, 347.
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Induction, then, is that operation of the mind by which we infer that what we know to be true in a particular case or cases, will be true in all cases which resemble the former in certain assignable respects. In other words, induction is the process by which we conclude that what is true of certain individuals of a class is true of the whole class, or that what is true at certain times will be true in similar circumstances at all times.
In A System of Logic, Ratiocinative and Inductive: Being a Connected View of the Principles of Evidence, and the Methods of Scientific Investigation (1843), Vol. 1, 352.
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Induction. The mental operation by which from a number of individual instances, we arrive at a general law. The process, according to Hamilton, is only logically valid when all the instances included in the law are enumerated. This being seldom, if ever, possible, the conclusion of an Induction is usually liable to more or less uncertainty, and Induction is therefore incapable of giving us necessary (general) truths.
Stated as narrative, not a direct quote, by his biographer W.H.S. Monck in 'Glossary of Philosophical Terms', appended in Sir William Hamilton (1881), 181.
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It is often in our Power to obtain an Analogy where we cannot have an Induction.
Observations on Man, His Frame, His Duty, and His Expectations (1749).
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It seems to me, he says, that the test of “Do we or not understand a particular subject in physics?” is, “Can we make a mechanical model of it?” I have an immense admiration for Maxwell’s model of electromagnetic induction. He makes a model that does all the wonderful things that electricity docs in inducing currents, etc., and there can be no doubt that a mechanical model of that kind is immensely instructive and is a step towards a definite mechanical theory of electromagnetism.
From stenographic report by A.S. Hathaway of the Lecture 20 Kelvin presented at Johns Hopkins University, Baltimore, on 'Molecular Dynamics and the Wave Theory of Light' (1884), 132. (Hathaway was a Mathematics fellow there.) This remark is not included in the first typeset publication—a revised version, printed twenty years later, in 1904, as Lord Kelvin’s Baltimore Lectures on Molecular Dynamics and the Wave Theory of Light. The original notes were reproduced by the “papyrograph” process. They are excerpted in Pierre Maurice Marie Duhem, Essays in the History and Philosophy of Science (1996), 54-55.
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It would appear that Deductive and Demonstrative Sciences are all, without exception, Inductive Sciences: that their evidence is that of experience, but that they are also, in virtue of the peculiar character of one indispensable portion of the general formulae according to which their inductions are made, Hypothetical Sciences. Their conclusions are true only upon certain suppositions, which are, or ought to be, approximations to the truth, but are seldom, if ever, exactly true; and to this hypothetical character is to be ascribed the peculiar certainty, which is supposed to be inherent in demonstration.
In System of Logic, Bk. 2, chap. 6, 1.
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Mathematics is not the discoverer of laws, for it is not induction; neither is it the framer of theories, for it is not hypothesis; but it is the judge over both, and it is the arbiter to which each must refer its claims; and neither law can rule nor theory explain without the sanction of mathematics.
In 'Linear Associative Algebra', American Journal of Mathematics (1881), 4, 97.
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Plots have I laid, inductions dangerous…
From Richard III (Nov 1633), Act 1, Scene 1. In The Plays of William Shakespeare (1804), Vol. 5, 2.
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Science, in its ultimate ideal, consists of a set of propositions arranged in a hierarchy, the lowest level of the hierarchy being concerned with particular facts, and the highest with some general law, governing everything in the universe. The various levels in the hierarchy have a two-fold logical connection, travelling one up, one down; the upward connection proceeds by induction, the downward by deduction.
In The Scientific Outlook (1931, 2009), 38.
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Such is the tendency of the human mind to speculation, that on the least idea of an analogy between a few phenomena, it leaps forward, as it were, to a cause or law, to the temporary neglect of all the rest; so that, in fact, almost all our principal inductions must be regarded as a series of ascents and descents, and of conclusions from a few cases, verified by trial on many.
In A Preliminary Discourse on the Study of Natural Philosophy (1830), 164-165.
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The actual evolution of mathematical theories proceeds by a process of induction strictly analogous to the method of induction employed in building up the physical sciences; observation, comparison, classification, trial, and generalisation are essential in both cases. Not only are special results, obtained independently of one another, frequently seen to be really included in some generalisation, but branches of the subject which have been developed quite independently of one another are sometimes found to have connections which enable them to be synthesised in one single body of doctrine. The essential nature of mathematical thought manifests itself in the discernment of fundamental identity in the mathematical aspects of what are superficially very different domains. A striking example of this species of immanent identity of mathematical form was exhibited by the discovery of that distinguished mathematician … Major MacMahon, that all possible Latin squares are capable of enumeration by the consideration of certain differential operators. Here we have a case in which an enumeration, which appears to be not amenable to direct treatment, can actually be carried out in a simple manner when the underlying identity of the operation is recognised with that involved in certain operations due to differential operators, the calculus of which belongs superficially to a wholly different region of thought from that relating to Latin squares.
In Presidential Address to British Association for the Advancement of Science (1910), Nature, 84, 290.
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The advance from the simple to the complex, through a process of successive differentiations, is seen alike in the earliest changes of the Universe to which we can reason our way back, and in the earliest changes which we can inductively establish; it is seen in the geologic and climatic evolution of the Earth; it is seen in the unfolding of every single organism on its surface, and in the multiplication of kinds of organisms; it is seen in the evolution of Humanity, whether contemplated in the civilized individual, or in the aggregate of races; it is seen in the evolution of Society in respect alike of its political, its religious, and its economical organization; and it is seen in the evolution of all those endless concrete and abstract products of human activity which constitute the environment of our daily life. From the remotest past which Science can fathom, up to the novelties of yesterday, that in which Progress essentially consists, is the transformation of the homogeneous into the heterogeneous.
Progress: Its Law and Cause (1857), 35.
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The functional validity of a working hypothesis is not a priori certain, because often it is initially based on intuition. However, logical deductions from such a hypothesis provide expectations (so-called prognoses) as to the circumstances under which certain phenomena will appear in nature. Such a postulate or working hypothesis can then be substantiated by additional observations ... The author calls such expectations and additional observations the prognosis-diagnosis method of research. Prognosis in science may be termed the prediction of the future finding of corroborative evidence of certain features or phenomena (diagnostic facts). This method of scientific research builds up and extends the relations between the subject and the object by means of a circuit of inductions and deductions.
In 'The Scientific Character of Geology', The Journal of Geology (Jul 1961), 69, No. 4, 454-5.
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The general mental qualification necessary for scientific advancement is that which is usually denominated “common sense,” though added to this, imagination, induction, and trained logic, either of common language or of mathematics, are important adjuncts.
From presidential address (24 Nov 1877) to the Philosophical Society of Washington. As cited by L.A. Bauer in his retiring president address (5 Dec 1908), 'The Instruments and Methods of Research', published in Philosophical Society of Washington Bulletin, 15, 103. Reprinted in William Crookes (ed.) The Chemical News and Journal of Industrial Science (30 Jul 1909), 59.
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The increasing technicality of the terminology employed is also a serious difficulty. It has become necessary to learn an extensive vocabulary before a book in even a limited department of science can be consulted with much profit. This change, of course, has its advantages for the initiated, in securing precision and concisement of statement; but it tends to narrow the field in which an investigator can labour, and it cannot fail to become, in the future, a serious impediment to wide inductive generalisations.
Year Book of Science (1892), preface, from review in Chemical News and Journal of Physical Science (14 Apr 1892), 65, 190.
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The Law of Causation, the recognition of which is the main pillar of inductive science, is but the familiar truth that invariability of succession is found by observation to obtain between every fact in nature and some other fact which has preceded it.
In 'On the Law of Universal Causation', A System of Logic, Ratiocinative and Inductive: Being a Connected View of the Principles of Evidence and the Methods of Scientific Investigation, (1846), 197.
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The Mathematician deals with two properties of objects only, number and extension, and all the inductions he wants have been formed and finished ages ago. He is now occupied with nothing but deduction and verification.
In 'On the Educational Value of the Natural History Sciences', Lay Sermons, Addresses and Reviews (1872), 87.
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The momentous laws of induction between currents and between currents and magnets were discovered by Michael Faraday in 1831-32. Faraday was asked: “What is the use of this discovery?” He answered: “What is the use of a child—it grows to be a man.” Faraday’s child has grown to be a man and is now the basis of all the modern applications of electricity.
In An Introduction to Mathematics (1911), 34-35.
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The only hope of science is genuine induction.
In Maturin Murray Ballou, Edge-Tools of Speech (1899), 440.
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The only hope [of science] ... is in genuine induction.
Aphorism 14. In Francis Bacon and Basil Montagu, The Works of Francis Bacon (1831), Vol. 14, 32.
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The question of the origin of the hypothesis belongs to a domain in which no very general rules can be given; experiment, analogy and constructive intuition play their part here. But once the correct hypothesis is formulated, the principle of mathematical induction is often sufficient to provide the proof.
As co-author with Herbert Robbins, in What Is Mathematics?: An Elementary Approach to Ideas and Methods (1941, 1996), 15.
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The science of optics, like every other physical science, has two different directions of progress, which have been called the ascending and the descending scale, the inductive and the deductive method, the way of analysis and of synthesis. In every physical science, we must ascend from facts to laws, by the way of induction and analysis; and we must descend from laws to consequences, by the deductive and synthetic way. We must gather and group appearances, until the scientific imagination discerns their hidden law, and unity arises from variety; and then from unity must reduce variety, and force the discovered law to utter its revelations of the future.
In On a General Method of Expressing the Paths of Light, & of the Planets, by the Coefficients of a Characteristic Function (1833), 7-8. [The spelling as “groupe” in the original text, has her been corrected to “group” to avoid an intrusive “sic”.]
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The scientist, if he is to be more than a plodding gatherer of bits of information, needs to exercise an active imagination. The scientists of the past whom we now recognize as great are those who were gifted with transcendental imaginative powers, and the part played by the imaginative faculty of his daily life is as least as important for the scientist as it is for the worker in any other field—much more important than for most. A good scientist thinks logically and accurately when conditions call for logical and accurate thinking—but so does any other good worker when he has a sufficient number of well-founded facts to serve as the basis for the accurate, logical induction of generalizations and the subsequent deduction of consequences.
‘Imagination in Science’, Tomorrow (Dec 1943), 38-9. Quoted In Barbara Marinacci (ed.), Linus Pauling In His Own Words: Selected Writings, Speeches, and Interviews (1995), 82.
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The validity of all the Inductive Methods depends on the assumption that every event, or the beginning of every phenomenon, must have some cause; some antecedent, upon the existence of which it is invariably and unconditionally consequent.
A System of Logic: Ratiocinative and Inductive (1843), Vol. 2, 107.
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There are no royal roads to knowledge, and we can only advance to new and important truths along the rugged path of experience, guided by cautious induction.
In 'Report of the Secretary', Annual Report of the Board of Regents of the Smithsonian Institution for 1856 (1857), 36.
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There is a tradition of opposition between adherents of induction and of deduction. In my view it would be just as sensible for the two ends of a worm to quarrel.
From address to the Mathematical and Physical Science Section of the British Association, Newcastle-on-Tyne (1916). In The Chemical News and Journal of Physical Science (22 Sep 1916), 142.114, No. 2965,
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There is in every step of an arithmetical or algebraical calculation a real induction, a real inference from facts to facts, and what disguises the induction is simply its comprehensive nature, and the consequent extreme generality of its language.
In System of Logic, Bk. 2, chap. 6, 2.
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There is no inductive method which could lead to the fundamental concepts of physics. Failure to understand this fact constituted the basic philosophical error of so many investigators of the nineteenth century.
Opening of section 4, 'The Theory of Relativity', in Physics and Reality (1936), collected in Essays in Physics (1950), 34.
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This maze of symbols, electric and magnetic potential, vector potential, electric force, current, displacement, magnetic force, and induction, have been practically reduced to two, electric and magnetic force.
Describing Heaviside’s refinement of the original Maxwell Equations. In Joseph Larmore (ed.), The Scientific Writings of the Late George Francis Fitzgerald (1902) 294.
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We are living now, not in the delicious intoxication induced by the early successes of science, but in a rather grisly morning-after, when it has become apparent that what triumphant science has done hitherto is to improve the means for achieving unimproved or actually deteriorated ends.
Ends and Means: an Inquiry into the Nature of Ideals and into Methods Employed for their Realization (1937), 310.
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We are told that “Mathematics is that study which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation.” I think no statement could have been made more opposite to the facts of the case; that mathematical analysis is constantly invoking the aid of new principles, new ideas, and new methods, not capable of being defined by any form of words, but springing direct from the inherent powers and activities of the human mind, and from continually renewed introspection of that inner world of thought of which the phenomena are as varied and require as close attention to discern as those of the outer physical world (to which the inner one in each individual man may, I think, be conceived to stand somewhat in the same relation of correspondence as a shadow to the object from which it is projected, or as the hollow palm of one hand to the closed fist which it grasps of the other), that it is unceasingly calling forth the faculties of observation and comparison, that one of its principal weapons is induction, that it has frequent recourse to experimental trial and verification, and that it affords a boundless scope for the exercise of the highest efforts of the imagination and invention.
In Presidential Address to British Association, Exeter British Association Report (1869), pp. 1-9, in Collected Mathematical Papers, Vol. 2, 654.
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We have here spoken of the prediction of facts of the same kind as those from which our rule was collected. But the evidence in favour of our induction is of a much higher and more forcible character when it enables us to explain and determine cases of a kind different from those which were contemplated in the formation of our hypothesis. The instances in which this has occurred, indeed, impress us with a conviction that the truth of our hypothesis is certain. No accident could give rise to such an extraordinary coincidence. No false supposition could, after being adjusted to one class of phenomena, so exactly represent a different class, when the agreement was unforeseen and contemplated. That rules springing from remote and unconnected quarters should thus leap to the same point, can only arise from that being where truth resides.
In The Philosophy of the Inductive Sciences (1840), Vol. 2, 230.
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We know that the probability of well-established induction is great, but, when we are asked to name its degree we cannot. Common sense tells us that some inductive arguments are stronger than others, and that some are very strong. But how much stronger or how strong we cannot express.
In A Treatise on Probability (1921), Chap. 22, 259.
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Whatever things are not derived from objects themselves, whether by the external senses or by the sensation of internal thoughts, are to be taken as hypotheses…. Those things which follow from the phenomena neither by demonstration nor by the argument of induction, I hold as hypotheses.
As quoted in Ernan McMullin, 'The Principia: Significance For Empiricism', collected in Margaret J. Osler and Paul Lawrence Farber (eds.), Religion, Science, and Worldview: Essays in Honor of Richard S. Westfall (2002), 38.
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When we have amassed a great store of such general facts, they become the objects of another and higher species of classification, and are themselves included in laws which, as they dispose of groups, not individuals have a far superior degree of generality, till at length, by continuing the process, we arrive at axioms of the highest degree of generality of which science is capable. This process is what we mean by induction.
In A Preliminary Discourse on the Study of Natural Philosophy (1830), 102.
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When you say, “The burned child dreads the fire, “ you mean that he is already a master of induction.
Epigraph in Isaac Asimov’s Book of Science and Nature Quotations (1988), 132.
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[Vestiges begins] from principles which are at variance with all sober inductive truth. The sober facts of geology shuffled, so as to play a rogue’s game; phrenology (that sinkhole of human folly and prating coxcombry); spontaneous generation; transmutation of species; and I know not what; all to be swallowed, without tasting and trying, like so much horse-physic!! Gross credulity and rank infidelity joined in unlawful marriage, and breeding a deformed progeny of unnatural conclusions!
Letter to Charles Lyell (9 Apr 1845). In John Willis Clark and Thomas McKenny Hughes (eds.), The Life and Letters of the Reverend Adam Sedgwick (1890), Vol. 2, 83.
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[Mathematics] is that [subject] which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation.
In 'The Scientific Aspects of Positivism', Fortnightly Review (1898) in Lay Sermons, Addresses and Reviews, (1872), 169.
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… just as the astronomer, the physicist, the geologist, or other student of objective science looks about in the world of sense, so, not metaphorically speaking but literally, the mind of the mathematician goes forth in the universe of logic in quest of the things that are there; exploring the heights and depths for facts—ideas, classes, relationships, implications, and the rest; observing the minute and elusive with the powerful microscope of his Infinitesimal Analysis; observing the elusive and vast with the limitless telescope of his Calculus of the Infinite; making guesses regarding the order and internal harmony of the data observed and collocated; testing the hypotheses, not merely by the complete induction peculiar to mathematics, but, like his colleagues of the outer world, resorting also to experimental tests and incomplete induction; frequently finding it necessary, in view of unforeseen disclosures, to abandon one hopeful hypothesis or to transform it by retrenchment or by enlargement:—thus, in his own domain, matching, point for point, the processes, methods and experience familiar to the devotee of natural science.
In Lectures on Science, Philosophy and Art (1908), 26
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Carl Sagan Thumbnail In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. (1987) -- Carl Sagan
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- 90 -
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- 80 -
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- 40 -
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- 30 -
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- 20 -
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