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Conclusion Quotes (160 quotes)
Conclusions Quotes

“The Universe repeats itself, with the possible exception of history.” Of all earthly studies history is the only one that does not repeat itself. ... Astronomy repeats itself; botany repeats itself; trigonometry repeats itself; mechanics repeats itself; compound long division repeats itself. Every sum if worked out in the same way at any time will bring out the same answer. ... A great many moderns say that history is a science; if so it occupies a solitary and splendid elevation among the sciences; it is the only science the conclusions of which are always wrong.
In 'A Much Repeated Repetition', Daily News (26 Mar 1904). Collected in G. K. Chesterton and Dale Ahlquist (ed.), In Defense of Sanity: The Best Essays of G.K. Chesterton (2011), 82.
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Dilbert: Evolution must be true because it is a logical conclusion of the scientific method.
Dogbert: But science is based on the irrational belief that because we cannot perceive reality all at once, things called “time” and “cause and effect” exist.
Dilbert: That’s what I was taught and that’s what I believe.
Dogbert: Sounds cultish.
Dilbert comic strip (8 Feb 1992).
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L'imagination au contraire qui tend à nous porter continuellement au-delà du vrai, l'amour-propre et la confiance en nous-mêmes, qu'il sait si bien nous inspirer, nous sollicitent à tirer des conséquences qui ne dérivent pas immédiatement des faits.
Imagination, on the contrary, which is ever wandering beyond the bounds of truth, joined to self-love and that self-confidence we are so apt to indulge, prompt us to draw conclusions which are not immediately derived from facts.
From the original French in Traité élémentaire de chimie (1789, 1793), discours préliminaire, ix; and from edition translated into English by Robert Kerr, as Elements of Chemistry (1790), Preface, xvii.
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A conclusion is the place where you got tired thinking.
In Charlie Walker, My Few Wise Words of Wisdom (200), 151.
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A great surgeon performs operations for stone by a single method; later he makes a statistical summary of deaths and recoveries, and he concludes from these statistics that the mortality law for this operation is two out of five. Well, I say that this ratio means literally nothing scientifically and gives us no certainty in performing the next operation; for we do not know whether the next case will be among the recoveries or the deaths. What really should be done, instead of gathering facts empirically, is to study them more accurately, each in its special determinism. We must study cases of death with great care and try to discover in them the cause of mortal accidents so as to master the cause and avoid the accidents.
From An Introduction to the Study of Experimental Medicine (1865), as translated by Henry Copley Greene (1957), 137-138. (Note that Bernard overlooks how the statistical method can be useful: a surgeon announcing a mortality rate of 40% invites comparison. A surgeon with worse outcomes should adopt this method. If a surgeon has a better results, that method should be adopted.)
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A professor … may be to produce a perfect mathematical work of art, having every axiom stated, every conclusion drawn with flawless logic, the whole syllabus covered. This sounds excellent, but in practice the result is often that the class does not have the faintest idea of what is going on. … The framework is lacking; students do not know where the subject fits in, and this has a paralyzing effect on the mind.
In A Concrete Approach to Abstract Algebra (1959), 1-2.
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After five years' work I allowed myself to speculate on the subject, and drew up some short notes; these I enlarged in 1844 into a sketch of the conclusions, which then seemed to me probable: from that period to the present day I have steadily pursued the same object. I hope that I may be excused for entering on these personal details, as I give them to show that I have not been hasty in coming to a decision.
From On the Origin of Species by Means of Natural Selection; or, The Preservation of Favoured Races in the Struggle for Life (1861), 9.
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Always be suspicious of conclusions that reinforce uncritical hope and follow comforting traditions of Western thought.
From The Flamingo's Smile (1987), 401.
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And this is the ultimate lesson that our knowledge of the mode of transmission of typhus has taught us: Man carries on his skin a parasite, the louse. Civilization rids him of it. Should man regress, should he allow himself to resemble a primitive beast, the louse begins to multiply again and treats man as he deserves, as a brute beast. This conclusion would have endeared itself to the warm heart of Alfred Nobel. My contribution to it makes me feel less unworthy of the honour which you have conferred upon me in his name.
'Investigations on Typhus', Nobel Lecture, 1928. In Nobel Lectures: Physiology or Medicine 1922-1941 (1965), 187.
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Antoine Magnan, a French zoologist, in 1934 made some very careful studies of bumblebee flight and came to the conclusion that bumblebees cannot fly at all! Fortunately, the bumblebees never heard this bit of news and so went on flying as usual.
Insects (1968, 1972), 68. Referring to Antoine Magnan Le Vol des Insectes (1934), Vol. 1 of Locomotion Chez les Animaux. Cited
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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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At this stage you must admit that whatever is seen to be sentient is nevertheless composed of atoms that are insentient. The phenomena open to our observation so not contradict this conclusion or conflict with it. Rather they lead us by the hand and compel us to believe that the animate is born, as I maintain, of the insentient.
In On the Nature of the Universe, translated by R. E. Latham (1951, 1994), 59.
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Because a fact seems strange to you, you conclude that it is not one. ... All science, however, commences by being strange. Science is successive. It goes from one wonder to another. It mounts by a ladder. The science of to-day would seem extravagant to the science of a former time. Ptolemy would believe Newton mad.
In Victor Hugo and Lorenzo O'Rourke (trans.) Victor Hugo's Intellectual Autobiography: (Postscriptum de ma vie) (1907), 322.
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Because intelligence is our own most distinctive feature, we may incline to ascribe superior intelligence to the basic primate plan, or to the basic plan of the mammals in general, but this point requires some careful consideration. There is no question at all that most mammals of today are more intelligent than most reptiles of today. I am not going to try to define intelligence or to argue with those who deny thought or consciousness to any animal except man. It seems both common and scientific sense to admit that ability to learn, modification of action according to the situation, and other observable elements of behavior in animals reflect their degrees of intelligence and permit us, if only roughly, to compare these degrees. In spite of all difficulties and all the qualifications with which the expert (quite properly) hedges his conclusions, it also seems sensible to conclude that by and large an animal is likely to be more intelligent if it has a larger brain at a given body size and especially if its brain shows greater development of those areas and structures best developed in our own brains. After all, we know we are intelligent, even though we wish we were more so.
In The Meaning of Evolution: A Study of the History of Life and of its Significance for Man (1949), 78.
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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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Both religion and science must preserve their autonomy and their distinctiveness. Religion is not founded on science nor is science an extension of religion. Each should possess its own principles, its pattern of procedures, its diversities of interpretation and its own conclusions.
In Letter (1 Jun 1988) to Father George V. Coyne, Director of the Vatican Observatory. On vatican.va website.
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But although in theory physicists realize that their conclusions are ... not certainly true, this ... does not really sink into their consciousness. Nearly all the time ... they ... act as if Science were indisputably True, and what's more, as if only science were true.... Any information obtained otherwise than by the scientific method, although it may be true, the scientists will call “unscientific,” using this word as a smear word, by bringing in the connotation from its original [Greek] meaning, to imply that the information is false, or at any rate slightly phony.
In Science is a Sacred Cow (1950), 176-77.
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But as my conclusions have lately been much misrepresented, and it has been stated that I attribute the modification of species exclusively to natural selection, I may be permitted to remark that in the first edition of this work, and subsequently, I placed in a most conspicuous position—namely, at the close of the Introduction—the following words: “I am convinced that natural selection has been the main but not the exclusive means of modification.” This has been of no avail. Great is the power of steady misrepresentation; but the history of science shows that fortunately this power does not long endure.
In The Origin of Species by Means of Natural Selection with additions and corrections from sixth and last English edition (1899), Vol. 2, 293.
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But nature is remarkably obstinate against purely logical operations; she likes not schoolmasters nor scholastic procedures. As though she took a particular satisfaction in mocking at our intelligence, she very often shows us the phantom of an apparently general law, represented by scattered fragments, which are entirely inconsistent. Logic asks for the union of these fragments; the resolute dogmatist, therefore, does not hesitate to go straight on to supply, by logical conclusions, the fragments he wants, and to flatter himself that he has mastered nature by his victorious intelligence.
'On the Principles of Animal Morphology', Proceedings of the Royal Society of Edinburgh (2 Apr 1888), 15, 289. Original as Letter to Mr John Murray, communicated to the Society by Professor Sir William Turner. Page given as in collected volume published 1889.
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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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Chemistry affords two general methods of determining the constituent principles of bodies, the method of analysis, and that of synthesis. When, for instance, by combining water with alkohol, we form the species of liquor called, in commercial language, brandy or spirit of wine, we certainly have a right to conclude, that brandy, or spirit of wine, is composed of alkohol combined with water. We can produce the same result by the analytical method; and in general it ought to be considered as a principle in chemical science, never to rest satisfied without both these species of proofs. We have this advantage in the analysis of atmospherical air, being able both to decompound it, and to form it a new in the most satisfactory manner.
Elements of Chemistry (1790), trans. R. Kerr, 33.
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Chlorine is a poisonous gas. In case I should fall over unconscious in the following demonstration involving chlorine, please pick me up and carry me into the open air. Should this happen, the lecture for the day will be concluded.
Quoted in Ralph Oesper, The Human Side of Scientists (1975), 192.
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Conclusion: Big helix in several chains, phosphates on outside, phosphate-phosphate inter-helical bonds disrupted by water. Phosphate links available to proteins.
Underlined in typewritten lecture notes, with handwritten annotations, as report (7 Feb 1952) on 'Colloquium November 1951'. As given in Anne Sayre, Rosalind Franklin and DNA (1975), 128.
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Consider the plight of a scientist of my age. I graduated from the University of California at Berkeley in 1940. In the 41 years since then the amount of biological information has increased 16 fold; during these 4 decades my capacity to absorb new information has declined at an accelerating rate and now is at least 50% less than when I was a graduate student. If one defines ignorance as the ratio of what is available to be known to what is known, there seems no alternative to the conclusion that my ignorance is at least 25 times as extensive as it was when I got my bachelor’s degree. Although I am sure that my unfortunate condition comes as no surprise to my students and younger colleagues, I personally find it somewhat depressing. My depression is tempered, however, by the fact that all biologists, young or old, developing or senescing, face the same melancholy situation because of an interlocking set of circumstances.
In 'Scientific innovation and creativity: a zoologist’s point of view', American Zoologist (1982), 22, 228.
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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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Even a wise experiment when made by a fool generally leads to a false conclusion, but that fools’ experiments conducted by a genius often prove to be leaps through the dark into great discoveries.
Commenting on Charles Darwin’s “fool’s experiments”, in 'Charles Robert Darwin', collected in C.D. Warner (ed.), Library of the World’s Best Literature Ancient and Modern (1896), Vol. 2, 4391-4392.
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Every investigator must before all things look upon himself as one who is summoned to serve on a jury. He has only to consider how far the statement of the case is complete and clearly set forth by the evidence. Then he draws his conclusion and gives his vote, whether it be that his opinion coincides with that of the foreman or not.
In The Maxims and Reflections of Goethe (1906), 190.
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First, as concerns the success of teaching mathematics. No instruction in the high schools is as difficult as that of mathematics, since the large majority of students are at first decidedly disinclined to be harnessed into the rigid framework of logical conclusions. The interest of young people is won much more easily, if sense-objects are made the starting point and the transition to abstract formulation is brought about gradually. For this reason it is psychologically quite correct to follow this course.
Not less to be recommended is this course if we inquire into the essential purpose of mathematical instruction. Formerly it was too exclusively held that this purpose is to sharpen the understanding. Surely another important end is to implant in the student the conviction that correct thinking based on true premises secures mastery over the outer world. To accomplish this the outer world must receive its share of attention from the very beginning.
Doubtless this is true but there is a danger which needs pointing out. It is as in the case of language teaching where the modern tendency is to secure in addition to grammar also an understanding of the authors. The danger lies in grammar being completely set aside leaving the subject without its indispensable solid basis. Just so in Teaching of Mathematics it is possible to accumulate interesting applications to such an extent as to stunt the essential logical development. This should in no wise be permitted, for thus the kernel of the whole matter is lost. Therefore: We do want throughout a quickening of mathematical instruction by the introduction of applications, but we do not want that the pendulum, which in former decades may have inclined too much toward the abstract side, should now swing to the other extreme; we would rather pursue the proper middle course.
In Ueber den Mathematischen Unterricht an den hoheren Schulen; Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 11, 131.
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For me, the idea of a creation is not conceivable without invoking the necessity of design. One cannot be exposed to the law and order of the universe without concluding that there must be design and purpose behind it all.
In letter to California State board of Education (14 Sep 1972).
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For undemocratic reasons and for motives not of State, they arrive at their conclusions—largely inarticulate. Being void of self-expression they confide their views to none; but sometimes in a smoking room, one learns why things were done.
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Gauss [replied], when asked how soon he expected to reach certain mathematical conclusions, “that he had them long ago, all he was worrying about was how to reach them.”
In Louis Pasteur, Free Lance of Science (1950), 365. Also excerpted in 'Mechanisms of Discovery', collected in I.S. Gordon and S. Sorkin (eds.) The Armchair Science Reader (1959), 336.
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Geology got into the hands of the theoreticians who were conditioned by the social and political history of their day more than by observations in the field. … We have allowed ourselves to be brainwashed into avoiding any interpretation of the past that involves extreme and what might be termed “catastrophic” processes. However, it seems to me that the stratigraphical record is full of examples of processes that are far from “normal” in the usual sense of the word. In particular we must conclude that sedimentation in the past has often been very rapid indeed and very spasmodic. This may be called the “Phenomenon of the Catastrophic Nature of the Stratigraphic Record.”
In The Nature of the Stratigraphical Record (3rd ed., 1993), 70.
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Geometrical axioms are neither synthetic a priori conclusions nor experimental facts. They are conventions: our choice, amongst all possible conventions, is guided by experimental facts; but it remains free, and is only limited by the necessity of avoiding all contradiction. ... In other words, axioms of geometry are only definitions in disguise.
That being so what ought one to think of this question: Is the Euclidean Geometry true?
The question is nonsense. One might as well ask whether the metric system is true and the old measures false; whether Cartesian co-ordinates are true and polar co-ordinates false.
In George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane (1982), 110.
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He who attempts to draw any conclusion whatever as to the nation's wealth or poverty from the mere fact of a favorable or unfavorable Balance of Trade, has not grasped the first fundamental principle of Political Economy.
As quoted by Richard Theodore Ely in 'Transfers of Economic Goods, Including, as Agents of Transfers, Money, Credit, and Banking', Political Economy, Political Science and Sociology: a Practical and Scientific Presentation of Social and Economic Subjects (1899), 485. Engel's statement is presented in quotation marks, but since the author, Ely, qualifies it with 'Dr. Ernst Engel, once made the following observation at a meeting of a "Seminary," of which the writer was a member at the time.' This suggests the quote is only given as a recollection, and not Engel's exact words.
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Historical science is not worse, more restricted, or less capable of achieving firm conclusions because experiment, prediction, and subsumption under invariant laws of nature do not represent its usual working methods. The sciences of history use a different mode of explanation, rooted in the comparative and observational richness in our data. We cannot see a past event directly, but science is usually based on inference, not unvarnished observation (you don’t see electrons, gravity, or black holes either).
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How can a modern anthropologist embark upon a generalization with any hope of arriving at a satisfactory conclusion? By thinking of the organizational ideas that are present in any society as a mathematical pattern.
In Rethinking Anthropology (1961), 2.
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I am turned into a sort of machine for observing facts and grinding out conclusions.
As quoted in Adrian J. Desmond and James Richard Moore, Darwin (1994), 644. Described by the authors as a remark Darwin was heard to sigh, suspecting that his largest botany book, the 600-page Movement in Plants, was as dull as ditchwater.
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I cannot anyhow be contented to view this wonderful universe, and especially the nature of man, and to conclude that everything is the result of brute force. I am inclined to look at everything as resulting from designed laws, with the details, whether good or bad, left to the working out of what we call chance. Not that this notion at all satisfies me. I feel most deeply that the whole subject is too profound for the human intellect. A dog might as well speculate on the mind of Newton. Let each man hope and believe what he can.
Letter to Asa Gray (22 May 1860). In Charles Darwin and Francis Darwin (ed.), Charles Darwin: His Life Told in an Autobiographical Chapter, and in a Selected Series of His Published Letters (1892), 236.
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I conclude that, while it is true that science cannot decide questions of value, that is because they cannot be intellectually decided at all, and lie outside the realm of truth and falsehood. Whatever knowledge is attainable, must be attained by scientific methods; and what science cannot discover, mankind cannot know.
Religion and Science (1935), 243.
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I had come to the conclusion, that the principal alimentary matters might be reduced to the three great classes, namely the saccharine, the oily and the albuminous.
'On the Ultimate Composition of Simple Alimentary Substances; with some Preliminary Remarks on the Analysis of Organised Bodies in General', Philosophical Transactions of the Royal Society (1827), 117, 357.
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I have come to the conclusion that mankind consume twice too much food. According to my computation, I have eaten and drunk, between my tenth and seventieth year, forty-four horse-wagon loads more than was good for me.
Quoted in Lydia Maria Francis Child, 'Hints About Health', Looking Toward Sunset (1891), 428.
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I have come to the conclusion that the exertion, without which a knowledge of mathematics cannot be acquired, is not materially increased by logical rigor in the method of instruction.
In Jahresbericht der Deutschen Mathematiker Vereinigung (1898), 143.
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I have therefore tried to show the tendency displayed throughout history, by the most profound investigators, to pass from the world of the senses to a world where vision becomes spiritual, where principles are elaborated, and from which the explorer emerges with conceptions and conclusions, to be approved or rejected according as they coincide with sensible things.
Heat, A Mode of Motion (1880, 1915), 6th ed., viii.
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I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives.
Attributed. Quoted in James GleickChaos (1988), 38. Contact webmaster if you know a primary print source.
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I think and think for months and years. Ninety-nine times, the conclusion is false. The hundredth time I am right.
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I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.
In 'Reflections on my Eightieth Birthday', Portraits from Memory (1956), 54.
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If ... the past may be no Rule for the future, all Experience becomes useless and can give rise to no Inferences or Conclusions.
An Enquiry Concerning Human Understanding (1748), 65.
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If all economists were laid end to end, they would not reach a conclusion.
Anonymous
Often seen misattributed (?) to George Bernard Shaw, but there appears to be no source in the writings of Shaw. However, the theme "laid end to end, they would not" has long been used humorously in various incarnations. For example, “If all the arguments in the world were placed end to end, they would not reach any conclusion”, as used by Tom Sims in 'Tom Sims Says', Brownwood Bulletin (18 Jul 1925), 4. The subject quote was attributed to Shaw by George Soule in 'Bridges to the Unknown', in The Saturday Review of Literature (20 May 1933), 9, No. 44, 601. See discussion on the quoteinvestigator.com website.
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If one looks with a cold eye at the mess man has made of history, it is difficult to avoid the conclusion that he has been afflicted by some built-in mental disorder which drives him towards self-destruction.
In The Ghost in the Machine (1967).
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If there is something very slightly wrong in our definition of the theories, then the full mathematical rigor may convert these errors into ridiculous conclusions.
Feynman Lectures on Gravitation, edited by Brian Hatfield (2002), 21.
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In every branch of Natural Science progress is now so rapid that few accepted conclusions can be regarded as more than provisional; and this is especially true of prehistoric Archaeology.
In Ancient Hunters and Their Modern Representatives (1911), Preface, viii.
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In fact, I’ve come to the conclusion that I never did know anything about it [electricity].
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In less than eight years “The Origin of Species” has produced conviction in the minds of a majority of the most eminent living men of science. New facts, new problems, new difficulties as they arise are accepted, solved, or removed by this theory; and its principles are illustrated by the progress and conclusions of every well established branch of human knowledge.
From a review of four books on the subject 'Mimicry, and Other Protective Resemblances Among Animals', in The Westminster Review (Jul 1867), 88, 1. Wallace is identified as the author in the article as reprinted in William Beebe, The Book of Naturalists: An Anthology of the Best Natural History (1988), 108.
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In mathematics, … and in natural philosophy since mathematics was applied to it, we see the noblest instance of the force of the human mind, and of the sublime heights to which it may rise by cultivation. An acquaintance with such sciences naturally leads us to think well of our faculties, and to indulge sanguine expectations concerning the improvement of other parts of knowledge. To this I may add, that, as mathematical and physical truths are perfectly uninteresting in their consequences, the understanding readily yields its assent to the evidence which is presented to it; and in this way may be expected to acquire the habit of trusting to its own conclusions, which will contribute to fortify it against the weaknesses of scepticism, in the more interesting inquiries after moral truth in which it may afterwards engage.
In Elements of the Philosophy of the Human Mind (1827), Vol. 3, Chap. 1, Sec. 3, 182.
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In reality, all Arguments from Experience are founded on the Similarity which we discover among natural Objects, and by which we are induc'd to expect effects similar to those which we have found to follow from such Objects. And tho' none but a Fool or Madman will ever pretend to dispute the Authority of Experience, or to reject that great Guide of human Life, it may surely be allow'd a Philosopher to have so much Curiosity at least as to examine the Principle of human Nature, which gives this mighty Authority to Experience, and makes us draw Advantage from that Similarity which Nature has plac'd among different Objects. From Causes which appear similar we expect similar Effects. This is the Sum of our experimental Conclusions.
An Enquiry Concerning Human Understanding (1748), 63.
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In scientific investigations it is grievously wrong to pander to the public’s impatience for results, or to let them think that for discovery it is necessary only to set up a great manufactory and a system of mass production. If in treatment team work is effective, in research it is the individual who counts first and above all. No great thought has ever sprung from anything but a single mind, suddenly conceiving. Throughout the whole world there has been too violent a forcing of the growth of ideas; too feverish a rush to perform experiments and publish conclusions. A year of vacation for calm detachment with all the individual workers thinking it all over in a desert should be proclaimed.
In Viewless Winds: Being the Recollections and Digressions of an Australian Surgeon (1939), 286.
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In the 1920s, there was a dinner at which the physicist Robert W. Wood was asked to respond to a toast … “To physics and metaphysics.” Now by metaphysics was meant something like philosophy—truths that you could get to just by thinking about them. Wood took a second, glanced about him, and answered along these lines: The physicist has an idea, he said. The more he thinks it through, the more sense it makes to him. He goes to the scientific literature, and the more he reads, the more promising the idea seems. Thus prepared, he devises an experiment to test the idea. The experiment is painstaking. Many possibilities are eliminated or taken into account; the accuracy of the measurement is refined. At the end of all this work, the experiment is completed and … the idea is shown to be worthless. The physicist then discards the idea, frees his mind (as I was saying a moment ago) from the clutter of error, and moves on to something else. The difference between physics and metaphysics, Wood concluded, is that the metaphysicist has no laboratory.
In 'Wonder and Skepticism', Skeptical Enquirer (Jan-Feb 1995), 19, No. 1.
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In the strict formulation of the law of causality—if we know the present, we can calculate the future—it is not the conclusion that is wrong but the premise.
On an implication of the uncertainty principle.
Quoted in David C. Cassidy, Beyond Uncertainty: Heisenberg, Quantum Physics, and the Bomb (2009), 162.
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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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Inexact method of observation, as I believe, is one flaw in clinical pathology to-day. Prematurity of conclusion is another, and in part follows from the first; but in chief part an unusual craving and veneration for hypothesis, which besets the minds of most medical men, is responsible. Except in those sciences which deal with the intangible or with events of long past ages, no treatises are to be found in which hypothesis figures as it does in medical writings. The purity of a science is to be judged by the paucity of its recorded hypotheses. Hypothesis has its right place, it forms a working basis; but it is an acknowledged makeshift, and, at the best, of purpose unaccomplished. Hypothesis is the heart which no man with right purpose wears willingly upon his sleeve. He who vaunts his lady love, ere yet she is won, is apt to display himself as frivolous or his lady a wanton.
The Mechanism and Graphic Registration of the Heart Beat (1920), vii.
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It has been pointed out already that no knowledge of probabilities, less in degree than certainty, helps us to know what conclusions are true, and that there is no direct relation between the truth of a proposition and its probability. Probability begins and ends with probability. That a scientific investigation pursued on account of its probability will generally lead to truth, rather than falsehood, is at the best only probable.
In A Treatise on Probability (1921), 322.
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It is a misfortune to pass at once from observation to conclusion, and to regard both as of equal value; but it befalls many a student.
In The Maxims and Reflections of Goethe (1906), 183.
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It is a strange fact, characteristic of the incomplete state of our present knowledge, that totally opposing conclusions are drawn about prehistoric conditions on our planet, depending on whether the problem is approached from the biological or the geophysical viewpoint.
In The Origins of Continents and Oceans (4th ed. 1929), trans. John Biram (1966), 5.
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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 certainly true that all physical phenomena are subject to strictly mathematical conditions, and mathematical processes are unassailable in themselves. The trouble arises from the data employed. Most phenomena are so highly complex that one can never be quite sure that he is dealing with all the factors until the experiment proves it. So that experiment is rather the criterion of mathematical conclusions and must lead the way.
In Matter, Ether, Motion (1894), 89.
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It is easier to love humanity as a whole than to love one’s neighbor. There may even be a certain antagonism between love of humanity and love of neighbor; a low capacity for getting along with those near us often goes hand in hand with a high receptivity to the idea of the brotherhood of men. About a hundred years ago a Russian landowner by the name of Petrashevsky recorded a remarkable conclusion: “Finding nothing worthy of my attachment either among women or among men, I have vowed myself to the service of mankind.” He became a follower of Fourier, and installed a phalanstery on his estate. The end of the experiment was sad, but what one might perhaps have expected: the peasants—Petrashevsky’s neighbors-burned the phalanstery.
In 'Brotherhood', The Ordeal of Change (1963), 91.
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It is hard to hide our genes completely. However devoted someone may be to the privacy of his genotype, others with enough curiosity and knowledge can draw conclusions from the phenotype he presents and from the traits of his relatives.
In The Lives to Come: the Genetic Revolution and Human Possibilities (1997), 127.
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It is not possible to find in all geometry more difficult and more intricate questions or more simple and lucid explanations [than those given by Archimedes]. Some ascribe this to his natural genius; while others think that incredible effort and toil produced these, to all appearance, easy and unlaboured results. No amount of investigation of yours would succeed in attaining the proof, and yet, once seen, you immediately believe you would have discovered it; by so smooth and so rapid a path he leads you to the conclusion required.
Plutarch
In John Dryden (trans.), Life of Marcellus.
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It is now necessary to indicate more definitely the reason why mathematics not only carries conviction in itself, but also transmits conviction to the objects to which it is applied. The reason is found, first of all, in the perfect precision with which the elementary mathematical concepts are determined; in this respect each science must look to its own salvation .... But this is not all. As soon as human thought attempts long chains of conclusions, or difficult matters generally, there arises not only the danger of error but also the suspicion of error, because since all details cannot be surveyed with clearness at the same instant one must in the end be satisfied with a belief that nothing has been overlooked from the beginning. Every one knows how much this is the case even in arithmetic, the most elementary use of mathematics. No one would imagine that the higher parts of mathematics fare better in this respect; on the contrary, in more complicated conclusions the uncertainty and suspicion of hidden errors increases in rapid progression. How does mathematics manage to rid itself of this inconvenience which attaches to it in the highest degree? By making proofs more rigorous? By giving new rules according to which the old rules shall be applied? Not in the least. A very great uncertainty continues to attach to the result of each single computation. But there are checks. In the realm of mathematics each point may be reached by a hundred different ways; and if each of a hundred ways leads to the same point, one may be sure that the right point has been reached. A calculation without a check is as good as none. Just so it is with every isolated proof in any speculative science whatever; the proof may be ever so ingenious, and ever so perfectly true and correct, it will still fail to convince permanently. He will therefore be much deceived, who, in metaphysics, or in psychology which depends on metaphysics, hopes to see his greatest care in the precise determination of the concepts and in the logical conclusions rewarded by conviction, much less by success in transmitting conviction to others. Not only must the conclusions support each other, without coercion or suspicion of subreption, but in all matters originating in experience, or judging concerning experience, the results of speculation must be verified by experience, not only superficially, but in countless special cases.
In Werke [Kehrbach] (1890), Bd. 5, 105. As quoted, cited and translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 19.
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It is still false to conclude that man is nothing but the highest animal, or the most progressive product of organic evolution. He is also a fundamentally new sort of animal and one in which, although organic evolution continues on its way, a fundamentally new sort of evolution has also appeared. The basis of this new sort of evolution is a new sort of heredity, the inheritance of learning. This sort of heredity appears modestly in other mammals and even lower in the animal kingdom, but in man it has incomparably fuller development and it combines with man's other characteristics unique in degree with a result that cannot be considered unique only in degree but must also be considered unique in kind.
In The Meaning of Evolution: A Study of the History of Life and of its Significance for Man (1949), 286.
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It may well be doubted whether, in all the range of Science, there is any field so fascinating to the explorer—so rich in hidden treasures—so fruitful in delightful surprises—as that of Pure Mathematics. The charm lies chiefly, I think, in the absolute certainty of its results: for that is what, beyond all mental treasures, the human intellect craves for. Let us only be sure of something! More light, more light … “And if our fate be death, give light and let us die” This is the cry that, through all the ages, is going up from perplexed Humanity, and Science has little else to offer, that will really meet the demands of its votaries, than the conclusions of Pure Mathematics.
Opening of 'Introduction', A New Theory of Parallels (1890), xv. As a non-fiction work, the author’s name on the title page of this book was Charles Lutwidge Dodgson. Being better known for his works of fiction as Lewis Carroll, all quotes relating to this one person, published under either name, are gathered on this single web page under his pen name.
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It was my science that drove me to the conclusion that the world is much more complicated than can be explained by science. It is only through the supernatural that I can understand the mystery of existence.
As quoted in Sharon Begley, 'Science Finds God', Newsweek (1998).
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It was noted long ago that the front row of burlesque houses was occupied predominantly by bald-headed men. In fact, such a row became known as the bald-headed row. It might be assumed from this on statistical evidence that the continued close observation of chorus girls in tights caused loss of hair from the top of the head.
[Disputing a statistical study for the American Cancer Society showing smoking to be a cancer causative.]
In Bess Furman, '2 Cite Extraction of Cigarette Tar', New York Times (26 Jul 1957), 21. The article reported on testimony before the Legal and Monetary Affairs Subcommittee of the House Government Operations Committee.
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It would appear that Deductive and Demonstrative Sciences are all, without exception, Inductive Sciences: that their evidence is that of experience, but that they are also, in virtue of the peculiar character of one indispensable portion of the general formulae according to which their inductions are made, Hypothetical Sciences. Their conclusions are true only upon certain suppositions, which are, or ought to be, approximations to the truth, but are seldom, if ever, exactly true; and to this hypothetical character is to be ascribed the peculiar certainty, which is supposed to be inherent in demonstration.
In System of Logic, Bk. 2, chap. 6, 1.
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It [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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Kant, discussing the various modes of perception by which the human mind apprehends nature, concluded that it is specially prone to see nature through mathematical spectacles. Just as a man wearing blue spectacles would see only a blue world, so Kant thought that, with our mental bias, we tend to see only a mathematical world.
In The Mysterious Universe (1930), 115.
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Let us now recapitulate all that has been said, and let us conclude that by hermetically sealing the vials, one is not always sure to prevent the birth of the animals in the infusions, boiled or done at room temperature, if the air inside has not felt the ravages of fire. If, on the contrary, this air has been powerfully heated, it will never allow the animals to be born, unless new air penetrates from outside into the vials. This means that it is indispensable for the production of the animals that they be provided with air which has not felt the action of fire. And as it would not be easy to prove that there were no tiny eggs disseminated and floating in the volume of air that the vials contain, it seems to me that suspicion regarding these eggs continues, and that trial by fire has not entirely done away with fears of their existence in the infusions. The partisans of the theory of ovaries will always have these fears and will not easily suffer anyone's undertaking to demolish them.
Nouvelles Recherches sur les Découvertes Microscopiques, et la Génération des Corps Organisés (1769), 134-5. Quoted in Jacques Roger, The Life Sciences in Eighteenth-Century French Thought, ed. Keith R. Benson and trans. Robert Ellrich (1997), 510-1.
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Man has never been a particularly modest or self-deprecatory animal, and physical theory bears witness to this no less than many other important activities. The idea that thought is the measure of all things, that there is such a thing as utter logical rigor, that conclusions can be drawn endowed with an inescapable necessity, that mathematics has an absolute validity and controls experience—these are not the ideas of a modest animal. Not only do our theories betray these somewhat bumptious traits of self-appreciation, but especially obvious through them all is the thread of incorrigible optimism so characteristic of human beings.
In The Nature of Physical Theory (1936), 135-136.
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Many persons nowadays seem to think that any conclusion must be very scientific if the arguments in favor of it are derived from twitching of frogs’ legs—especially if the frogs are decapitated—and that—on the other hand—any doctrine chiefly vouched for by the feelings of human beings—with heads on their shoulders—must be benighted and superstitious.
Pragmatism: A New Name for Old Ways of Thinking (1907)
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Mathematics has the completely false reputation of yielding infallible conclusions. Its infallibility is nothing but identity. Two times two is not four, but it is just two times two, and that is what we call four for short. But four is nothing new at all. And thus it goes on and on in its conclusions, except that in the higher formulas the identity fades out of sight.
As quoted in Richard von Mises, 'Mathematical Postulates and Human Understanding', collected in J.R. Newman (ed.) The World of Mathematics (1956), Vol. 3, 1754.
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Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional.
(1923). Quoted, without source, in E.T. Bell, Men of Mathematics (1937), Vol. 1, li.
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Mathematics is the science which draws necessary conclusions.
First line of Linear Associative Algebra: read before the National Academy of Sciences, Washington City (1870), 2.
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Mathematics—in a strict sense—is the abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations.
In New English Dictionary as quoted in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 5. This definition of Mathematics appeared in Oxford English Dictionary (1933), as “The abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations, and which includes as its main divisions geometry, arithmetic, and algebra.”
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Measurement has too often been the leitmotif of many investigations rather than the experimental examination of hypotheses. Mounds of data are collected, which are statistically decorous and methodologically unimpeachable, but conclusions are often trivial and rarely useful in decision making. This results from an overly rigorous control of an insignificant variable and a widespread deficiency in the framing of pertinent questions. Investigators seem to have settled for what is measurable instead of measuring what they would really like to know.
'Patient Care—Mystical Research or Researchable Mystique/', Clinical Research (1964), 12, no. 4, 422.
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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 perhaps has so retarded the reception of the higher conclusions of Geology among men in general, as ... [the] instinctive parsimony of the human mind in matters where time is concerned.
Proceedings of the Geological Society of London (1903), 59, lxx.
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Occasionally and frequently the exercise of the judgment ought to end in absolute reservation. It may be very distasteful, and great fatigue, to suspend a conclusion; but as we are not infallible, so we ought to be cautious; we shall eventually find our advantage, for the man who rests in his position is not so far from right as he who, proceeding in a wrong direction, is ever increasing his distance.
Lecture at the Royal Institution, 'Observations on the Education of the Judgment'. Collected in Edward Livingston Youmans (ed)., Modern Culture (1867), 219.
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Of Science generally we can remark, first, that it is the most perfect embodiment of Truth, and of the ways of getting at Truth. More than anything else does it impress the mind with the nature of Evidence, with the labour and precautions necessary to prove a thing. It is the grand corrective of the laxness of the natural man in receiving unaccredited facts and conclusions. It exemplifies the devices for establishing a fact, or a law, under every variety of circumstances; it saps the credit of everything that is affirmed without being properly attested.
In Education as a Science (1879), 147-148.
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On the whole, I cannot help saying that it appears to me not a little extraordinary, that a theory so new, and of such importance, overturning every thing that was thought to be the best established in chemistry, should rest on so very narrow and precarious a foundation, the experiments adduced in support of it being not only ambiguous or explicable on either hypothesis, but exceedingly few. I think I have recited them all, and that on which the greatest stress is laid, viz. That of the formation of water from the decomposition of the two kinds of air, has not been sufficiently repeated. Indeed it required so difficult and expensive an apparatus, and so many precautions in the use of it, that the frequent repetition of the experiment cannot be expected; and in these circumstances the practised experimenter cannot help suspecting the accuracy of the result and consequently the certainty of the conclusion.
Considerations on the Doctrine of Phlogiston (1796), 57-8.
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Ordinarily logic is divided into the examination of ideas, judgments, arguments, and methods. The two latter are generally reduced to judgments, that is, arguments are reduced to apodictic judgments that such and such conclusions follow from such and such premises, and method is reduced to judgments that prescribe the procedure that should be followed in the search for truth.
Ampére expresses how arguments have a logical structure which he expected should be applied to relate scientific theories to experimental evidence. In James R. Hofmann, André-Marie Ampère (1996), 158. Cites Académie des Sciences Ampère Archives, École Normale lecture 15 notes, box 261.
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Our emphasis on science has resulted in an alarming rise in world populations, the demand and ever-increasing emphasis of science to improve their standards and maintain their vigor. I have been forced to the conclusion that an over-emphasis of science weakens character and upsets life's essential balance.
In 'The Wisdom of Wilderness', Life (22 Dec 1967), 63, No. 25, 9.
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Proof that a given condition always precedes or accompanies a phenomenon does not warrant concluding with certainty that a given condition is the immediate cause of that phenomenon. It must still be established that when this condition is removed, the phenomen will no longer appear.
From An Introduction to the Study of Experimental Medicine (1865), as translated by Henry Copley Greene (1957), 55.
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Psychologists … have found that people making an argument or a supposedly factual claim can manipulate us by the words they choose and the way they present their case. We can’t avoid letting language do our thinking for us, but we can become more aware of how and when language is steering us toward a conclusion that, upon reflection, we might choose to reject.
As co-author with Kathleen Hall Jamieson, in unSpun: Finding Facts in a World of Disinformation (2007), 70.
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Psychology … tells us that we rarely work through reasons and evidence in a systematic way; weighing information carefully and suspending the impulse to draw conclusions. Instead, much of the time we use mental shortcuts or rules of thumb that save us mental effort. These habits often work reasonably well, but they also can lead us to conclusions we might dismiss if we applied more thought.
As co-author with Kathleen Hall Jamieson, in unSpun: Finding Facts in a World of Disinformation (2007), 70.
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Pure mathematics … reveals itself as nothing but symbolic or formal logic. It is concerned with implications, not applications. On the other hand, natural science, which is empirical and ultimately dependent upon observation and experiment, and therefore incapable of absolute exactness, cannot become strictly mathematical. The certainty of geometry is thus merely the certainty with which conclusions follow from non-contradictory premises. As to whether these conclusions are true of the material world or not, pure mathematics is indifferent.
In 'Non-Euclidian Geometry of the Fourth Dimension', collected in Henry Parker Manning (ed.), The Fourth Dimension Simply Explained (1910), 58.
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Science derives its conclusions by the laws of logic from our sense perceptions, Thus it does not deal with the real world, of which we know nothing, but with the world as it appears to our senses. … All our sense perceptions are limited by and attached to the conceptions of time and space. … Modern physics has come to the same conclusion in the relativity theory, that absolute space and absolute time have no existence, but, time and space exist only as far as things or events fill them, that is, are forms of sense perception.
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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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 often regarded as the most objective and truth-directed of human enterprises, and since direct observation is supposed to be the favored route to factuality, many people equate respectable science with visual scrutiny–just the facts ma’am, and palpably before my eyes. But science is a battery of observational and inferential methods, all directed to the testing of propositions that can, in principle, be definitely proven false ... At all scales, from smallest to largest, quickest to slowest, many well-documented conclusions of science lie beyond the strictly limited domain of direct observation. No one has ever seen an electron or a black hole, the events of a picosecond or a geological eon.
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Science is the knowledge of many, orderly and methodically digested and arranged, so as to become attainable by one. The knowledge of reasons and their conclusions constitutes abstract, that of causes and their effects, and of the laws of nature, natural science.
A Preliminary Discourse on the Study of Natural Philosophy (1830).
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Science, in the immediate, produces knowledge and, indirectly, means of action. It leads to methodical action if definite goals are set up in advance. For the function of setting up goals and passing statements of value transcends its domain. While it is true that science, to the extent of its grasp of causative connections, may reach important conclusions as to the compatibility and incompatibility of goals and evaluations, the independent and fundamental definitions regarding goals and values remain beyond science’s reach.
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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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So the astronomer is on common ground with the physicist both in the subject and in the predicate of the conclusion, but the physicist demonstrates the predicate to belong to the subject by nature, whereas the astronomer does not care whether it belongs by nature or not. What, therefore, is the predicate for the physicist, is abstracted as the subject for the pure mathematician.
As quoted in Alistair Cameron Crombie, Robert Grosseteste and the Origins of Experimental Science, 1100-1700 (1971), 94.
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Starting from statistical observations, it is possible to arrive at conclusions which not less reliable or useful than those obtained in any other exact science. It is only necessary to apply a clear and precise concept of probability to such observations.
In Probability, Statistics, and Truth (1939), 1. In the 1957 edition, this was rewritten as, “Starting from statistical observations and applying to them a clear and precise concept of probability it is possible to arrive at conclusions which are just as reliable and “truth-full” and quite as practically useful as those obtained in any other exact science.”
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Statisticians tell us that for many years the death-rate from cancer has been slowly but steadily rising: and not unnaturally, many people conclude from this that for some reason or other we are becoming more susceptible to cancer. Actually, that conclusion does not follow at all. The rise in the cancer death-rate is probably due entirely to the fact that other causes of death have been reduced. Numbers of people who, if they had been born a century earlier, would have died in their twenties of typhoid or smallpox, say, are now living on into their seventies and dying of cancer.
From 'Figures Can Lie', Science Digest (Sep 1951), 30, No. 3, 53. (As condensed from The Listener). Excerpted in Meta Riley Emberger and Marian Ross Hall, Scientific Writing (1955), 407.
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Statistics: The only science that enables different experts using the same figures to draw different conclusions.
Evan Esar
The Dictionary of Humorous Quotations (1949). In Robert Harris Shutler, Mathematics 436 - Finely Explained (2004), 3.
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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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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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Tactics used by many practitioners of pseudoscience: make a large number of vaguely scientific arguments in the hope of making the desired conclusion seem inevitable. It is essential to recognize that a disconnected assemblage of weak arguments does not create a single, strong scientific argument.
Co-author with Matt Ford, Chris Lee and Jonathan Gitlin, in 'Diluting the Scientific Method: Ars Looks at Homeopathy' (11 Sep 2007) on arstechnica.com web site.
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Talent jogs to conclusions to which genius takes giant leaps.
In 'Genius', Wellman’s Miscellany (Dec 1871), 4, No. 6, 203.
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That a free, or at least an unsaturated acid usually exists in the stomachs of animals, and is in some manner connected with the important process of digestion, seems to have been the general opinion of physiologists till the time of SPALLANZANI. This illustrious philosopher concluded, from his numerous experiments, that the gastric fluids, when in a perfectly natural state, are neither acid nor alkaline. Even SPALLANZANI, however, admitted that the contents of the stomach are very generally acid; and this accords not only with my own observation, but with that, I believe, of almost every individual who has made any experiments on the subject. ... The object of the present communication is to show, that the acid in question is the muriatic [hydrochloric] acid, and that the salts usually met with in the stomach, are the alkaline muriates.
'On the Nature of the Acid and Saline Matters Usually Existing in the Stomachs of Animals', Philosophical Transactions of the Royal Society of London (1824), 114, 45-6.
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That the great majority of those who leave school should have some idea of the kind of evidence required to substantiate given types of belief does not seem unreasonable. Nor is it absurd to expect that they should go forth with a lively interest in the ways in which knowledge is improved and a marked distaste for all conclusions reached in disharmony with the methods of scientific inquiry.
Address to Section L, Education, of the American Association for the Advancement of Science, at Boston (1909), 'Science as Subject-Matter and as Method'. Published in Science (28 Jan 1910), N.S. Vol. 31, No. 787, 126.
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The apodictic quality of mathematical thought, the certainty and correctness of its conclusions, are due, not to a special mode of ratiocination, but to the character of the concepts with which it deals. What is that distinctive characteristic? I answer: precision, sharpness, completeness,* of definition. But how comes your mathematician by such completeness? There is no mysterious trick involved; some ideas admit of such precision, others do not; and the mathematician is one who deals with those that do.
In 'The Universe and Beyond', Hibbert Journal (1904-1905), 3, 309. An editorial footnote indicates “precision, sharpness, completeness” — i.e., in terms of the absolutely clear and indefinable.
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The art of drawing conclusions from experiments and observations consists in evaluating probabilities and in estimating whether they are sufficiently great or numerous enough to constitute proofs. This kind of calculation is more complicated and more diff
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The contradictory experiments of chemists leave us at liberty to conclude what we please. My conclusion is, that art has not yet invented sufficient aids to enable such subtle bodies [air, light, &c.] to make a well-defined impression on organs as blunt as ours; that it is laudable to encourage investigation but to hold back conclusion.
Letter to Rev. James Madison (Paris, 19 Jul 1788). In Thomas Jefferson and John P. Foley (ed.), The Jeffersonian Cyclopedia (1900), 135. From H.A. Washington, The Writings of Thomas Jefferson (1853-54). Vol 2, 431.
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The evidence at present available points strongly to the conclusion that the spirals are individual galaxies, or island universes, comparable with our own galaxy in dimension and in number of component units.
[Stating his conviction on the nature of nebulae during the Shapley-Curtis debate on 26 Apr 1920 to the National Academy of Sciences.]
In Aleksandr Sergeevich Sharov and Igor Dmitrievich Novikov, Edwin Hubble: The Discoverer of the Big Bang Universe (1993), 27.
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The fool rushed to certain conclusions.
The middling man approached them warily, and with circumspection.
And the wise man, perceiving them from afar, decided to go round another way.
'Discretion' in Little Stings (1907, 1908), 140.
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The great art consists in devising décisive experiments, leaving no place to the imagination of the observer. Imagination is needed to give wings to thought at the beginning of experimental investigations on any given subject. When, however, the time has come to conclude, and to interpret the facts derived from observations, imagination must submit to the factual results of the experiments.
Speech (8 Jul 1876), to the French Academy of Medicine. As translated in René J. Dubos, Louis Pasteur, Free Lance of Science (1950, 1986), 376. Date of speech identified in Maurice B. Strauss, Familiar Medical Quotations (1968), 502.
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The great science [mathematics] occupies itself at least just as much with the power of imagination as with the power of logical conclusion.
In 'Pestalozzi's Idee eines A B C der Anschauung', Werke[Kehrbach] (1890), Bd.l, 174. As quoted, cited and translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 31.
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The history of mathematics may be instructive as well as agreeable; it may not only remind us of what we have, but may also teach us to increase our store. Says De Morgan, “The early history of the mind of men with regards to mathematics leads us to point out our own errors; and in this respect it is well to pay attention to the history of mathematics.” It warns us against hasty conclusions; it points out the importance of a good notation upon the progress of the science; it discourages excessive specialization on the part of the investigator, by showing how apparently distinct branches have been found to possess unexpected connecting links; it saves the student from wasting time and energy upon problems which were, perhaps, solved long since; it discourages him from attacking an unsolved problem by the same method which has led other mathematicians to failure; it teaches that fortifications can be taken by other ways than by direct attack, that when repulsed from a direct assault it is well to reconnoiter and occupy the surrounding ground and to discover the secret paths by which the apparently unconquerable position can be taken.
In History of Mathematics (1897), 1-2.
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The investigation of causal relations between economic phenomena presents many problems of peculiar difficulty, and offers many opportunities for fallacious conclusions. Since the statistician can seldom or never make experiments for himself, he has to accept the data of daily experience, and discuss as best he can the relations of a whole group of changes; he cannot, like the physicist, narrow down the issue to the effect of one variation at a time. The problems of statistics are in this sense far more complex than the problems of physics.
Udny Yule
In 'On the Theory of Correlation', Journal of the Royal Statistical Society (Dec 1897), 60, 812, as cited in Stephen M. Stigler, The History of Statistics: The Measurement of Uncertainty Before 1900 (1986), 348.
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The Mathematics, I say, which effectually exercises, not vainly deludes or vexatiously torments studious Minds with obscure Subtilties, perplexed Difficulties, or contentious Disquisitions; which overcomes without Opposition, triumphs without Pomp, compels without Force, and rules absolutely without Loss of Liberty; which does not privately over-reach a weak Faith, but openly assaults an armed Reason, obtains a total Victory, and puts on inevitable Chains; whose Words are so many Oracles, and Works as many Miracles; which blabs out nothing rashly, nor designs anything from the Purpose, but plainly demonstrates and readily performs all Things within its Verge; which obtrudes no false Shadow of Science, but the very Science itself, the Mind firmly adhering to it, as soon as possessed of it, and can never after desert it of its own Accord, or be deprived of it by any Force of others: Lastly the Mathematics, which depends upon Principles clear to the Mind, and agreeable to Experience; which draws certain Conclusions, instructs by profitable Rules, unfolds pleasant Questions; and produces wonderful Effects; which is the fruitful Parent of, I had almost said all, Arts, the unshaken Foundation of Sciences, and the plentiful Fountain of Advantage to human Affairs.
Address to the University of Cambridge upon being elected Lucasian Professor of Mathematics (14 Mar 1664). In Mathematical Lectures (1734), xxviii.
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The more I think of it, I find this conclusion more impressed upon me—that the greatest thing a human soul ever does in this world is to see something, and tell what it saw in a plain way.
Modern Painters: pt. 4. Of Many Things (1850), 268. books.google.com John Ruskin - 1850
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The mystic and the physicist arrive at the same conclusion; one starting from the inner realm, the other from the outer world. The harmony between their views confirms the ancient Indian wisdom that Brahman, the ultimate reality without, is identical to Atman, the reality within.
In The Tao of Physics (1975), 305.
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The one [the logician] studies the science of drawing conclusions, the other [the mathematician] the science which draws necessary conclusions.
In Charles S. Peirce, ‎Charles Hartshorne (ed.), ‎Paul Weiss (ed.), Collected Papers of Charles Sanders Peirce (1931), Vol. 4, 199.
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The only possible conclusion the social sciences can draw is: some do, some don’t.
Attributed.
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The ordinary patient goes to his doctor because he is in pain or some other discomfort and wants to be comfortable again; he is not in pursuit of the ideal of health in any direct sense. The doctor on the other hand wants to discover the pathological condition and control it if he can. The two are thus to some degree at cross purposes from the first, and unless the affair is brought to an early and happy conclusion this diversion of aims is likely to become more and more serious as the case goes on.
Address, opening of 1932-3 session of U.C.H. Medical School (4 Oct 1932), 'Art and Science in Medicine', The Collected Papers of Wilfred Trotter, FRS (1941), 98.
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The other line of argument, which leads to the opposite conclusion, arises from looking at artificial automata. Everyone knows that a machine tool is more complicated than the elements which can be made with it, and that, generally speaking, an automaton A, which can make an automaton B, must contain a complete description of B, and also rules on how to behave while effecting the synthesis. So, one gets a very strong impression that complication, or productive potentiality in an organization, is degenerative, that an organization which synthesizes something is necessarily more complicated, of a higher order, than the organization it synthesizes. This conclusion, arrived at by considering artificial automaton, is clearly opposite to our early conclusion, arrived at by considering living organisms.
From lecture series on self-replicating machines at the University of Illinois, Lecture 5 (Dec 1949), 'Re-evaluation of the Problems of Complicated Automata—Problems of Hierarchy and Evolution', Theory of Self-Reproducing Automata (1966).
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The physiological combustion theory takes as its starting point the fundamental principle that the amount of heat that arises from the combustion of a given substance is an invariable quantity–i.e., one independent of the circumstances accompanying the combustion–from which it is more specifically concluded that the chemical effect of the combustible materials undergoes no quantitative change even as a result of the vital process, or that the living organism, with all its mysteries and marvels, is not capable of generating heat out of nothing.
Bemerkungen über das mechanische Aequivalent der Wärme [Remarks on the Mechanical Equivalent of Heat] (1851), 17-9. Trans. Kenneth L. Caneva, Robert Mayer and the Conservation of Energy (1993), 240.
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The present state of the earth and of the organisms now inhabiting it, is but the last stage of a long and uninterrupted series of changes which it has undergone, and consequently, that to endeavour to explain and account for its present condition without any reference to those changes (as has frequently been done) must lead to very imperfect and erroneous conclusions.
In 'On the Law which has regulated the Introduction of New Species', The Annals and Magazine of Natural History (1855), 16, No. 93, 184.
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The regularity with which we conclude that further advances in a particular field are impossible seems equaled only by the regularity with which events prove that we are of too limited vision. And it always seems to be those who have the fullest opportunity to know who are the most limited in view. What, then, is the trouble? I think that one answer should be: we do not realize sufficiently that the unknown is absolutely infinite, and that new knowledge is always being produced.
Quoted in Guy Suits, 'Willis Rodney Whitney', National Academy of Sciences, Biographical Memoirs (1960), 357.
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The same set of statistics can produce opposite conclusions at different levels of aggregation.
'Penetrating the Rhetoric', The Vision of the Anointed (1996), 102.
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The speculative propositions of mathematics do not relate to facts; … all that we are convinced of by any demonstration in the science, is of a necessary connection subsisting between certain suppositions and certain conclusions. When we find these suppositions actually take place in a particular instance, the demonstration forces us to apply the conclusion. Thus, if I could form a triangle, the three sides of which were accurately mathematical lines, I might affirm of this individual figure, that its three angles are equal to two right angles; but, as the imperfection of my senses puts it out of my power to be, in any case, certain of the exact correspondence of the diagram which I delineate, with the definitions given in the elements of geometry, I never can apply with confidence to a particular figure, a mathematical theorem. On the other hand, it appears from the daily testimony of our senses that the speculative truths of geometry may be applied to material objects with a degree of accuracy sufficient for the purposes of life; and from such applications of them, advantages of the most important kind have been gained to society.
In Elements of the Philosophy of the Human Mind (1827), Vol. 3, Chap. 1, Sec. 3, 180.
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The strongest arguments prove nothing so long as the conclusions are not verified by experience. Experimental science is the queen of sciences and the goal of all speculation.
Opus Tertium. Translation as stated in Popular Science (Aug 1901), 337.
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The theoretical idea … does not arise apart from and independent of experience; nor can it be derived from experience by a purely logical procedure. It is produced by a creative act. Once a theoretical idea has been acquired, one does well to hold fast to it until it leads to an untenable conclusion.
'On the Generalized Theory of Gravitation', Scientific American (Apr 1950). Collected in David H. Levy (ed.), The Scientific American Book of the Cosmos (2000), 14.
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The theory that gravitational attraction is inversely proportional to the square of the distance leads by remorseless logic to the conclusion that the path of a planet should be an ellipse, … It is this logical thinking that is the real meat of the physical sciences. The social scientist keeps the skin and throws away the meat. … His theorems no more follow from his postulates than the hunches of a horse player follow logically from the latest racing news. The result is guesswork clad in long flowing robes of gobbledygook.
In Science is a Sacred Cow (1950), 149-150.
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The third [argument of motion is] to the effect that the flying arrow is at rest, which result follows from the assumption that time is composed of moments: if this assumption is not granted, the conclusion will not follow.Arrow paradox
Zeno
Aristotle, Physics, 239b, 30-1. In Jonathan Barnes (ed.), The Complete Works of Aristotle (1984), Vol. 1, 405.
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The unavoidable conclusion is that the unprecedented meekness of the majority is responsible for the increase in violence. Social stability is the product of an equilibrium between a vigorous majority and violent minorities. Disorder does not come from an increased inner pressure or from the interaction of explosive ingredients. There is no reason to believe that the nature of the violent minorities is now greatly different from what it was in the past. What has changed is the will and ability of the majority to react.
In 'Thoughts on the Present', First Things, Last Things (1971), 110.
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The velocity of light is one of the most important of the fundamental constants of Nature. Its measurement by Foucault and Fizeau gave as the result a speed greater in air than in water, thus deciding in favor of the undulatory and against the corpuscular theory. Again, the comparison of the electrostatic and the electromagnetic units gives as an experimental result a value remarkably close to the velocity of light–a result which justified Maxwell in concluding that light is the propagation of an electromagnetic disturbance. Finally, the principle of relativity gives the velocity of light a still greater importance, since one of its fundamental postulates is the constancy of this velocity under all possible conditions.
Studies in Optics (1927), 120.
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The whole strenuous intellectual work of an industrious research worker would appear, after all, in vain and hopeless, if he were not occasionally through some striking facts to find that he had, at the end of all his criss-cross journeys, at last accomplished at least one step which was conclusively nearer the truth.
Nobel Lecture (2 Jun 1920), in Nobel Lectures in Physics, 1901-1921 (1998), 407.
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There are then two kinds of intellect: the one able to penetrate acutely and deeply into the conclusions of given premises, and this is the precise intellect; the other able to comprehend a great number of premises without confusing them, and this is the mathematical intellect. The one has force and exactness, the other comprehension. Now the one quality can exist without the other; the intellect can be strong and narrow, and can also be comprehensive and weak.
In Pascal’s Pensées (1958), 3.
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There can be no ultimate statements science: there can be no statements in science which can not be tested, and therefore none which cannot in principle be refuted, by falsifying some of the conclusions which can be deduced from them.
The Logic of Scientific Discovery (1959), 47.
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There is a finite number of species of plants and animals—even of insects—upon the earth. … Moreover, the universality of the genetic code, the common character of proteins in different species, the generality of cellular structure and cellular reproduction, the basic similarity of energy metabolism in all species and of photosynthesis in green plants and bacteria, and the universal evolution of living forms through mutation and natural selection all lead inescapably to a conclusion that, although diversity may be great, the laws of life, based on similarities, are finite in number and comprehensible to us in the main even now.
Presidential Address (28 Dec 1970) to the American Association for the Advancement of Science. 'Science: Endless Horizons or Golden Age?', Science (8 Jan 1971), 171, No. 3866, 24.
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These machines [used in the defense of the Syracusans against the Romans under Marcellus] he [Archimedes] had designed and contrived, not as matters of any importance, but as mere amusements in geometry; in compliance with king Hiero’s desire and request, some time before, that he should reduce to practice some part of his admirable speculation in science, and by accommodating the theoretic truth to sensation and ordinary use, bring it more within the appreciation of people in general. Eudoxus and Archytas had been the first originators of this far-famed and highly-prized art of mechanics, which they employed as an elegant illustration of geometrical truths, and as means of sustaining experimentally, to the satisfaction of the senses, conclusions too intricate for proof by words and diagrams. As, for example, to solve the problem, so often required in constructing geometrical figures, given the two extremes, to find the two mean lines of a proportion, both these mathematicians had recourse to the aid of instruments, adapting to their purpose certain curves and sections of lines. But what with Plato’s indignation at it, and his invectives against it as the mere corruption and annihilation of the one good of geometry,—which was thus shamefully turning its back upon the unembodied objects of pure intelligence to recur to sensation, and to ask help (not to be obtained without base supervisions and depravation) from matter; so it was that mechanics came to be separated from geometry, and, repudiated and neglected by philosophers, took its place as a military art.
Plutarch
In John Dryden (trans.), Life of Marcellus.
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They [mathematicians] only take those things into consideration, of which they have clear and distinct ideas, designating them by proper, adequate, and invariable names, and premising only a few axioms which are most noted and certain to investigate their affections and draw conclusions from them, and agreeably laying down a very few hypotheses, such as are in the highest degree consonant with reason and not to be denied by anyone in his right mind. In like manner they assign generations or causes easy to be understood and readily admitted by all, they preserve a most accurate order, every proposition immediately following from what is supposed and proved before, and reject all things howsoever specious and probable which can not be inferred and deduced after the same manner.
In Mathematical Lectures (1734), 65-66.
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Those who nod sagely and quote the tragedy of the commons in relation to environmental problems from pollution of the atmosphere to poaching of national parks tend to forget that Garrett Hardin revised his conclusions many times…. He recognized, most importantly, that anarchy did not prevail on the common pastures of medieval England in the way he had described…. “A managed commons, though it may have other defects, is not automatically subject to the tragic fate of the unmanaged commons,” wrote Hardin…. At sea, where a common exists in most waters… None of Hardin’s requirements for a successfully managed common is fulfilled by high-seas fishery regimes.
In The End of the Line: How Overfishing is Changing the World and what We Eat (2004), 153-155.
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Thus we conclude, that the strata both primary and secondary, both those of ancient and those of more recent origin, have had their materials furnished from the ruins of former continents, from the dissolution of rocks, or the destruction of animal or vegetable bodies, similar, at least in some respects, to those that now occupy the surface of the earth.
Illustrations of the Huttonian Theory of the Earth (1802), 14-5.
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To conclude, therefore, let no man out of a weak conceit of sobriety, or an ill-apply'd moderation, think or maintain, that a man can search too far or be too well studied in the book of God’s word, or in the book of God’s works; divinity or philosophy; but rather let men endeavour an endless progress or proficience in both.
In Of Proficience and Advancement of Learning Divine and Human (1605), collected in The Works of Francis Bacon (1711), Vol. 2, 417. Charles Darwin placed this quote on the title page of his On the Origin of Species.
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To say that mind is a product or function of protoplasm, or of its molecular changes, is to use words to which we can attach no clear conception. You cannot have, in the whole, what does not exist in any of the parts; and those who argue thus should put forth a definite conception of matter, with clearly enunciated properties, and show, that the necessary result of a certain complex arrangement of the elements or atoms of that matter, will be the production of self-consciousness. There is no escape from this dilemma—either all matter is conscious, or consciousness is something distinct from matter, and in the latter case, its presence in material forms is a proof of the existence of conscious beings, outside of, and independent of, what we term matter. The foregoing considerations lead us to the very important conclusion, that matter is essentially force, and nothing but force; that matter, as popularly understood, does not exist, and is, in fact, philosophically inconceivable. When we touch matter, we only really experience sensations of resistance, implying repulsive force; and no other sense can give us such apparently solid proofs of the reality of matter, as touch does. This conclusion, if kept constantly present in the mind, will be found to have a most important bearing on almost every high scientific and philosophical problem, and especially on such as relate to our own conscious existence.
In 'The Limits of Natural Selection as Applied to Man', last chapter of Contributions to the Theory of Natural Selection (1870), 365-366.
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We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim and of the road; we want to understand the idea of the proof, the deeper context.
Unterrichtsblätter für Mathematik und Naturwissenschaften (1932), 38, 177-188. As translated by Abe Shenitzer, in 'Part I. Topology and Abstract Algebra as Two Roads of Mathematical Comprehension', The American Mathematical Monthly (May 1995), 102, No. 7, 453.
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We do not draw conclusions with our eyes, but with our reasoning powers, and if the whole of the rest of living nature proclaims with one accord from all sides the evolution of the world of organisms, we cannot assume that the process stopped short of Man. But it follows also that the factors which brought about the development of Man from his Simian ancestry must be the same as those which have brought about the whole of evolution.
Translation of Weismann's work in German, by John Arthur Thomson and Margaret R. Thomson, The Evolution Theory (1904), Vol. 2, 393.
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We receive it as a fact, that some minds are so constituted as absolutely to require for their nurture the severe logic of the abstract sciences; that rigorous sequence of ideas which leads from the premises to the conclusion, by a path, arduous and narrow, it may be, and which the youthful reason may find it hard to mount, but where it cannot stray; and on which, if it move at all, it must move onward and upward… . Even for intellects of a different character, whose natural aptitude is for moral evidence and those relations of ideas which are perceived and appreciated by taste, the study of the exact sciences may be recommended as the best protection against the errors into which they are most likely to fall. Although the study of language is in many respects no mean exercise in logic, yet it must be admitted that an eminently practical mind is hardly to be formed without mathematical training.
In Orations and Speeches (1870), Vol. 8, 510.
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When I examine myself and my methods of thought, I come to the conclusion that the gift of fantasy has meant more to me that my talent for absorbing positive knowledge.
…...
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When someone says his conclusions are objective, he means that they are based on prejudices which many other people share.
In The Decline and Fall of Science (1976), 3.
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When the elements are arranged in vertical columns according to increasing atomic weight, so that the horizontal lines contain analogous elements again according to increasing atomic weight, an arrangement results from which several general conclusions may be drawn.
'The Relations of the Properties to the Atomic Weights of the Elements', Zeitschrift fur Chemie, 1869.
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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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[About mathematicians’ writings] Extreme external elegance, sometimes a somewhat weak skeleton of conclusions characterizes the French; the English, above all Maxwell, are distinguished by the greatest dramatic bulk.
In Ceremonial Speech (15 Nov 1887) celebrating the 301st anniversary of the Karl-Franzens-University Graz. Published as Gustav Robert Kirchhoff: Festrede zur Feier des 301. Gründungstages der Karl-Franzens-Universität zy Graz (1888), 29, as translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 187. From the original German, “Höchste äussere Eleganz, mitunter etwas schwaches Knochengerüste der Schlüsse charakterisirt die Franzosen, die grösste dramatische Wucht die Engländer, vor Allen Maxwell.”
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[Louis Rendu, Bishop of Annecy] collects observations, makes experiments, and tries to obtain numerical results; always taking care, however, so to state his premises and qualify his conclusions that nobody shall be led to ascribe to his numbers a greater accuracy than they merit. It is impossible to read his work, and not feel that he was a man of essentially truthful mind and that science missed an ornament when he was appropriated by the Church.
In The Glaciers of the Alps (1860), 299.
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[My] numberless observations... made on the Strata... [have] made me confident of their uniformity throughout this Country & [have] led me to conclude that the same regularity... will be found to extend to every part of the Globe for Nature has done nothing by piecemeal. [T]here is no inconsistency in her productions. [T]he Horse never becomes an Ass nor the Crab an Apple by any intermixture or artificial combination whatever[. N]or will the Oak ever degenerate into an Ash or an Ash into an Elm. [H]owever varied by Soil or Climate the species will still be distinct on this ground. [T]hen I argue that what is found here may be found elsewhere[.] When proper allowances are made for such irregularities as often occur and the proper situation and natural agreement is well understood I am satisfied there will be no more difficulty in ascertaining the true quality of the Strata and the place of its possition [sic] than there is now in finding the true Class and Character of Plants by the Linean [sic] System.
Natural Order of the Strata in England and Wales Accurately Delineated and Described, unpublished manuscript, Department of Geology, University of Oxford, 1801, f. 7v.
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[The object of education is] to train the mind to ascertain the sequence of a particular conclusion from certain premises, to detect a fallacy, to correct undue generalisation, to prevent the growth of mistakes in reasoning. Everything in these must depend on the spirit and the manner in which the instruction itself is conveyed and honoured. If you teach scientific knowledge without honouring scientific knowledge as it is applied, you do more harm than good. I do think that the study of natural science is so glorious a school for the mind, that with the laws impressed on all these things by the Creator, and the wonderful unity and stability of matter, and the forces of matter, there cannot be a better school for the education of the mind.
Giving Evidence (18 Nov 1862) to the Public Schools Commission. As quoted in John L. Lewis, 125 Years: The Physical Society & The Institute of Physics (1999), 168-169.
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… on these expanded membranes [butterfly wings] Nature writes, as on a tablet, the story of the modifications of species, so truly do all changes of the organisation register themselves thereon. Moreover, the same colour-patterns of the wings generally show, with great regularity, the degrees of blood-relationship of the species. As the laws of nature must be the same for all beings, the conclusions furnished by this group of insects must be applicable to the whole world.
From The Naturalist on the River Amazons: A record of Adventures, Habits of Animals, Sketches of Brazilian and Indian life, and Aspects of Nature under the Equator, During Eleven Years of Travel (1864), 413.
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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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- 70 -
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- 60 -
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- 50 -
Stephen Hawking
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- 40 -
Pierre Fermat
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- 30 -
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- 20 -
Carl Sagan
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- 10 -
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