"The two operations of our understanding [… are] intuition and deduction, on which alone we have said we must rely in the acquisition of knowledge." (René Descartes, "Rules for the Direction of the Mind", 1684)
"[…] for the saving the long progression of the thoughts to remote and first principles in every case, the mind should provide itself several stages; that is to say, intermediate principles, which it might have recourse to in the examining those positions that come in its way. These, though they are not self-evident principles, yet, if they have been made out from them by a wary and unquestionable deduction, may be depended on as certain and infallible truths, and serve as unquestionable truths to prove other points depending upon them, by a nearer and shorter view than remote and general maxims." (John Locke, "The Conduct of the Understanding", 1706)
"Extravagant theories, however, in those parts of philosophy, where our knowledge is yet imperfect, are not without their use; as they encourage the execution of laborious experiments, or the investigation of ingenious deductions, to conform or refute them." (Erasmus Darwin, "The botanic garden: A poem, in two parts", 1793)
"One very reprehensible mode of theory-making consists, after honest deductions from a few facts have been made, in torturing other facts to suit the end proposed, in omitting some, and in making use of any authority that may lend assistance to the object desired; while all those which militate against it are carefully put on one side or doubted." (Henry De la Beche, "Sections and Views, Illustrative of Geological Phaenomena", 1830)
"Facts [...] are not truths; they are not conclusions; they are not even premises, but in the nature and parts of premises. The truth depends on, and is only arrived at, by a legitimate deduction from all the facts which are truly material." (Samuel T Coleridge, "The Table Talk and Omniana of Samuel Taylor Coleridge", 1831)
"The deduction of effect from cause is often blocked by some insuperable extrinsic obstacle: the true causes may be quite unknown." (Carl von Clausewitz, "On War", 1832)
"Every stage of science has its train of practical applications and systematic inferences, arising both from the demands of convenience and curiosity, and from the pleasure which, as we have already said, ingenious and active-minded men feel in exercising the process of deduction."
"In the original discovery of a proposition of practical utility, by deduction from general principles and from experimental data, a complex algebraical investigation is often not merely useful, but indispensable; but in expounding such a proposition as a part of practical science, and applying it to practical purposes, simplicity is of the importance: - and […] the more thoroughly a scientific man has studied higher mathematics, the more fully does he become aware of this truth – and […] the better qualified does he become to free the exposition and application of principles from mathematical intricacy." (William J M Rankine, "On the Harmony of Theory and Practice in Mechanics", 1856)
"The principle of deduction is, that things which agree with the same thing agree with one another. The principle of induction is, that in the same circumstances and in the same substances, from the same causes the same effects will follow. The mathematical and metaphysical sciences are founded on deduction; the physical sciences rest on induction." (William Fleming, "A vocabulary of the philosophical sciences", 1857)
"If an idea presents itself to us, we must not reject it simply because it does not agree with the logical deductions of a reigning theory." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)
"Observe this: the abstraction of the philosopher is meant to keep the object itself, with its perturbing suggestions, out of sight, allowing only one quality to fill the field of vision; whereas the abstraction of the poet is meant to bring the object itself into more vivid relief, to make it visible by means of the selected qualities. In other words, the one aims at abstract symbols, the other at picturesque effects. The one can carry on his deductions by the aid of colourless signs, X or Y. The other appeals to the emotions through the symbols which will most vividly express the real objects in their relations to our sensibilities." (George H Lewes, "The Principles of Success in Literature", 1865)
"Modern discoveries have not been made by large collections of facts, with subsequent discussion, separation, and resulting deduction of a truth thus rendered perceptible. A few facts have suggested an hypothesis, which means a supposition, proper to explain them. The necessary results of this supposition are worked out, and then, and not till then, other facts are examined to see if their ulterior results are found in Nature." (Augustus de Morgan, "A Budget of Paradoxes", 1872)
"Deduction is certain and infallible, in the sense that each step in deductive reasoning will lead us to some result, as certain as the law itself. But it does not follow that deduction will lead the reasoner to every result of a law or combination of laws." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)
"Whatever lies beyond the limits of experience, and claims another origin than that of induction and deduction from established data, is illegitimate." (George H Lewes, "The Foundations of a Creed", 1875)
"To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction." (Gottlob Frege, "The Foundations of Arithmetic", 1884)
"[…] deduction consists in constructing an icon or diagram the relations of whose parts shall present a complete analogy with those of the parts of the object of reasoning, of experimenting upon this image in the imagination, and of observing the result so as to discover unnoticed and hidden relations among the parts." (Charles S Peirce, 1885)
"There is as great a distinction between mathematics and the mathematical sciences as there is between induction and the inductive sciences. Practically, few cases of induction do not involve, to a greater or less extent, deductions; so few mathematical processes do not involve some strictly logical procedure." (Charles C Everett, "The Science of Thought", 1890)
"By investigating the universe by an inductive method (endeavoring from the much which is observable to arrive at a little which may be verified and is indubitable) the new science refuses to recognise dogma as truth, but through reason, by a slow and laborious method of investigation, strives for and attains to true deductions." (Dmitri Mendeleev, "Principles of Chemistry", 1891)
"In deduction the mind is under the dominion of a habit or association by virtue of which a general idea suggests in each case a corresponding reaction." (Charles S Peirce, "The Law of Mind", 1892)
"In every science, after having analysed the ideas, expressing the more complicated by means of the more simple, one finds a certain number that cannot be reduced among them, and that one can define no further. These are the primitive ideas of the science; it is necessary to acquire them through experience, or through induction; it is impossible to explain them by deduction." (Giuseppe Peano, "Notations de Logique Mathématique", 1894)
"All deduction rests ultimately upon the data derived from experience. This is the tortoise that supports our conception of the cosmos." (Percival Lowell, "Mars", 1895)
"Deduction is that mode of reasoning which examines the state of things asserted in the premises, forms a diagram of that state of things, perceives in the parts of the diagram relations not explicitly mentioned in the premises, satisfies itself by mental experiments upon the diagram that these relations would always subsist, or at least would do so in a certain proportion of cases, and concludes their necessary, or probable, truth." (Charles S Peirce, "Kinds of Reasoning", cca. 1896)
"The discovery that all mathematics follows inevitably from a small collection of fundamental laws is one which immeasurably enhances the intellectual beauty of the whole; to those who have been oppressed by the fragmentary and incomplete nature of most existing chains of deduction this discovery comes with all the overwhelming force of a revelation; like a palace emerging from the autumn mist as the traveler ascends an Italian hill-side, the stately stories of the mathematical edifice appear in their due order and proportion, with a new perfection in every part." (Bertrand A W Russell, "Mysticism and Logic and Other Essays", 1925)
"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." (Alfred N Whitehead, "The Aims of Education & Other Essays", 1929)
"If an explanation is so vague in its inherent nature, or so unskillfully molded in its formulation, that specific deductions subject to empirical verification or refutation can not be based upon it, then it can never serve as a working hypothesis. A hypothesis with which one can not work is not a working hypothesis." (Douglas W Johnson, "Role of Analysis in Scientific Investigation", Bulletin of the Geological Society of America, 1933)
"Our reasonings grasp at straws for premises and float on gossamers for deductions." (Alfred N Whitehead, "Adventures of Ideas", 1933)
"[...] when the pioneer in science sends for the groping feelers of his thoughts, he must have a vivid intuitive imagination, for new ideas are not generated by deduction, but by an artistically creative imagination. Nevertheless, the worth of a new idea is invariably determined, not by the degree of its intuitiveness - which, incidentally, is to a major extent a matter of experience and habit - but by the scope and accuracy of the individual laws to the discovery of which it eventually leads. (Max Planck, The Meaning and Limits of Exact Science, Science Vol. 110 (2857), 1949)
"Insight is not the same as scientific deduction, but even at that it may be more reliable than statistics." (Anthony Standen, "Science Is a Sacred Cow", 1950)
"[…] the chief reason in favor of any theory on the principles of mathematics must always be inductive, i.e., it must lie in the fact that the theory in question enables us to deduce ordinary mathematics. In mathematics, the greatest degree of self-evidence is usually not to be found quite at the beginning, but at some later point; hence the early deductions, until they reach this point, give reasons rather from them, than for believing the premises because true consequences follow from them, than for believing the consequences because they follow from the premises." (Alfred N Whitehead, "Principia Mathematica", 1950)
"All followers of the axiomatic method and most mathematicians think that there is some such thing as an absolute ‘mathematical rigor’ which has to be satisfied by any deduction if it is to be valid. The history of mathematics shows that this is not the case, that, on the contrary, every generation is surpassed in rigor again and again by its successors." (Richard von Mises, "Positivism: A Study in Human Understanding", 1951)
"We are driven to conclude that science, like mathematics, is a system of axioms, assumptions, and deductions; it may start from being, but later leaves it to itself, and ends in the formation of a hypothetical reality that has nothing to do with existence; or it is the discovery of an ideal being which is, of course, present in what we call actuality, and renders it an existence for us only by being present in it." (Poolla T Raju, "Idealistic Thought of India", 1953)
"[…] the grand aim of all science […] is to cover the greatest possible number of empirical facts by logical deductions from the smallest possible number of hypotheses or axioms." (Albert Einstein, 1954)
"This method of deduction [...] is often called 'combinatory'. Its usefulness is not exhausted at this stage, but it does even at the outset lead to some valuable conclusions [...]" (John Chadwick, "The Decipherment of Linear B", 1958)
"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 or by experiments especially arranged to test details. The value of the hypothesis is strengthened if the observed facts fit the expectation within the limits of permissible error." (R Willem van Bemmelen, "The Scientific Character of Geology", The Journal of Geology Vol 69 (4), 1961)
"Intellect begins with the observation of nature, proceeds to memorize and classify the facts thus observed, and by logical deduction builds up that edifice of knowledge properly called science. But admittedly we also know by feeling, and we can combine the two faculties, and present knowledge in the guise of art." (Herbert Read, "Selected Writings: Poetry and Criticism", 1963)
"Perfect logic and faultless deduction make a pleasant theoretical structure, but it may be right or wrong; the experimenter is the only one to decide, and he is always right." (Léon Brillouin, "Scientific Uncertainty and Information", 1964)
"[…] the human reason discovers new relations between things not by deduction, but by that unpredictable blend of speculation and insight […] induction, which - like other forms of imagination - cannot be formalized." (Jacob Bronowski, "The Reach of Imagination", 1967)
"To give a causal explanation of an event means to deduce a statement which describes it, using as premises of the deduction one or more universal laws, together with certain singular statements, the initial conditions. [...] We have thus two different kinds of statement, both of which are necessary ingredients of a complete causal explanation." (Karl Popper, "The Philosophy of Karl Popper", 1974)
"The advantage of semantic networks over standard logic is that some selected set of the possible inferences can be made in a specialized and efficient way. If these correspond to the inferences that people make naturally, then the system will be able to do a more natural sort of reasoning than can be easily achieved using formal logical deduction." (Avron Barr, Natural Language Understanding, AI Magazine Vol. 1 (1), 1980)
"It is actually impossible in theory to determine exactly what the hidden mechanism is without opening the box, since there are always many different mechanisms with identical behavior. Quite apart from this, analysis is more difficult than invention in the sense in which, generally, induction takes more time to perform than deduction: in induction one has to search for the way, whereas in deduction one follows a straightforward path." (Valentino Braitenberg, "Vehicles: Experiments in Synthetic Psychology", 1984)
"It is difficult to distinguish deduction from what in other circumstances is called problem-solving. And concept learning, inference, and reasoning by analogy are all instances of inductive reasoning. (Detectives typically induce, rather than deduce.) None of these things can be done separately from each other, or from anything else. They are pseudo-categories." (Frank Smith, "To Think: In Language, Learning and Education", 1990)
"A mathematical proof is a chain of logical deductions, all stemming from a small number of initial assumptions ('axioms') and subject to the strict rules of mathematical logic. Only such a chain of deductions can establish the validity of a mathematical law, a theorem. And unless this process has been satisfactorily carried out, no relation - regardless of how often it may have been confirmed by observation - is allowed to become a law. It may be given the status of a hypothesis or a conjecture, and all kinds of tentative results may be drawn from it, but no mathematician would ever base definitive conclusions on it. (Eli Maor, "e: The Story of a Number", 1994)
"Mathematicians apparently don’t generally rely on the formal rules of deduction as they are thinking. Rather, they hold a fair bit of logical structure of a proof in their heads, breaking proofs into intermediate results so that they don’t have to hold too much logic at once. In fact, it is common for excellent mathematicians not even to know the standard formal usage of quantifiers (for all and there exists), yet all mathematicians certainly perform the reasoning that they encode." (William P Thurston, "On Proof and Progress in Mathematics", 1994)
"Model building is the art of selecting those aspects of a process that are relevant to the question being asked. As with any art, this selection is guided by taste, elegance, and metaphor; it is a matter of induction, rather than deduction. High science depends on this art." (John H Holland," Hidden Order: How Adaptation Builds Complexity", 1995)
"The methods of science include controlled experiments, classification, pattern recognition, analysis, and deduction. In the humanities we apply analogy, metaphor, criticism, and (e)valuation. In design we devise alternatives, form patterns, synthesize, use conjecture, and model solutions." (Béla H Bánáthy, "Designing Social Systems in a Changing World", 1996)
"Both induction and deduction, reasoning from the particular and the general, and back again from the universal to the specific, form the essence to scientific thinking." (Hans Christian von Baeyer, "Information, The New Language of Science", 2003)
"Paradox is the sharpest scalpel in the satchel of science. Nothing concentrates the mind as effectively, regardless of whether it pits two competing theories against each other, or theory against observation, or a compelling mathematical deduction against ordinary common sense." (Hans Christian von Baeyer, "Information, The New Language of Science", 2003)
"The concept of proof perhaps marks the true beginning of mathematics as the art of deduction rather than just numerological observation, the point at which mathematical alchemy gave way to mathematical chemistry." (Marcus du Sautoy, "The Music of the Primes", 2004)
"It seems that scientists are often attracted to beautiful theories in the way that insects are attracted to flowers - not by logical deduction, but by something like a sense of smell." (Steven Weinberg, "Physics Today", 2005)
"Human language is a vehicle of truth but also of error, deception, and nonsense. Its use, as in the present discussion, thus requires great prudence. One can improve the precision of language by explicit definition of the terms used. But this approach has its limitations: the definition of one term involves other terms, which should in turn be defined, and so on. Mathematics has found a way out of this infinite regression: it bypasses the use of definitions by postulating some logical relations (called axioms) between otherwise undefined mathematical terms. Using the mathematical terms introduced with the axioms, one can then define new terms and proceed to build mathematical theories. Mathematics need, not, in principle rely on a human language. It can use, instead, a formal presentation in which the validity of a deduction can be checked mechanically and without risk of error or deception." (David Ruelle, "The Mathematician's Brain", 2007)
"Knowledge about things beyond our immediate environment may be acquired through deduction, if the initial premises are believed to be correct." (Nayef Al-Rodhan, "Sustainable History and the Dignity of Man", 2009)
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