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Indicate Quotes (18 quotes)
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A great department of thought must have its own inner life, however transcendent may be the importance of its relations to the outside. No department of science, least of all one requiring so high a degree of mental concentration as Mathematics, can be developed entirely, or even mainly, with a view to applications outside its own range. The increased complexity and specialisation of all branches of knowledge makes it true in the present, however it may have been in former times, that important advances in such a department as Mathematics can be expected only from men who are interested in the subject for its own sake, and who, whilst keeping an open mind for suggestions from outside, allow their thought to range freely in those lines of advance which are indicated by the present state of their subject, untrammelled by any preoccupation as to applications to other departments of science. Even with a view to applications, if Mathematics is to be adequately equipped for the purpose of coping with the intricate problems which will be presented to it in the future by Physics, Chemistry and other branches of physical science, many of these problems probably of a character which we cannot at present forecast, it is essential that Mathematics should be allowed to develop freely on its own lines.
In Presidential Address British Association for the Advancement of Science, Section A, (1910), Nature, 84, 286.
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Few will deny that even in the first scientific instruction in mathematics the most rigorous method is to be given preference over all others. Especially will every teacher prefer a consistent proof to one which is based on fallacies or proceeds in a vicious circle, indeed it will be morally impossible for the teacher to present a proof of the latter kind consciously and thus in a sense deceive his pupils. Notwithstanding these objectionable so-called proofs, so far as the foundation and the development of the system is concerned, predominate in our textbooks to the present time. Perhaps it will be answered, that rigorous proof is found too difficult for the pupil’s power of comprehension. Should this be anywhere the case,—which would only indicate some defect in the plan or treatment of the whole,—the only remedy would be to merely state the theorem in a historic way, and forego a proof with the frank confession that no proof has been found which could be comprehended by the pupil; a remedy which is ever doubtful and should only be applied in the case of extreme necessity. But this remedy is to be preferred to a proof which is no proof, and is therefore either wholly unintelligible to the pupil, or deceives him with an appearance of knowledge which opens the door to all superficiality and lack of scientific method.
In 'Stücke aus dem Lehrbuche der Arithmetik', Werke, Bd. 2 (1904), 296.
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Genius is the whistle of the locomotive, which with steaming shrieks indicates its progress, but common sense is the driving-wheel which moves the train.
In Igerne and Other Writings of Arthur Handly Marks (1897), 348.
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I have now reached the point where I may indicate briefly what to me constitutes the essence of the crisis of our time. It concerns the relationship of the individual to society. The individual has become more conscious than ever of his dependence upon society. But he does not experience this dependence as a positive asset, as an organic tie, as a protective force, but rather as a threat to his natural rights, or even to his economic existence. Moreover, his position in society is such that the egotistical drives of his make-up are constantly being accentuated, while his social drives, which are by nature weaker, progressively deteriorate. All human beings, whatever their position in society, are suffering from this process of deterioration. Unknowingly prisoners of their own egotism, they feel insecure, lonely, and deprived of the naive, simple, and unsophisticated enjoyment of life. Man can find meaning in life, short and perilous as it is, only through devoting himself to society.
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In Euclid each proposition stands by itself; its connection with others is never indicated; the leading ideas contained in its proof are not stated; general principles do not exist. In modern methods, on the other hand, the greatest importance is attached to the leading thoughts which pervade the whole; and general principles, which bring whole groups of theorems under one aspect, are given rather than separate propositions. The whole tendency is toward generalization. A straight line is considered as given in its entirety, extending both ways to infinity, while Euclid is very careful never to admit anything but finite quantities. The treatment of the infinite is in fact another fundamental difference between the two methods. Euclid avoids it, in modern mathematics it is systematically introduced, for only thus is generality obtained.
In 'Geometry', Encyclopedia Britannica (9th edition).
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In this physical world there is no real chaos; all is in fact orderly; all is ordered by the physical principles. Chaos is but unperceived order- it is a word indicating the limitations of the human mind and the paucity of observational facts. The words “chaos,” “accidental,” “chance,” “unpredictable," are conveniences behind which we hide our ignorance.
From Of Stars and Men: The Human Response to an Expanding Universe (1958 Rev. Ed. 1964), Foreword.
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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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Logic does not pretend to teach the surgeon what are the symptoms which indicate a violent death. This he must learn from his own experience and observation, or from that of others, his predecessors in his peculiar science. But logic sits in judgment on the sufficiency of that observation and experience to justify his rules, and on the sufficiency of his rules to justify his conduct. It does not give him proofs, but teaches him what makes them proofs, and how he is to judge of them.
In A System of Logic, Ratiocinative and Inductive: Being a Connected View of the Principles of Evidence, and the Methods of Scientific Investigation (1843), Vol. 1, 11.
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Make it compulsory for a doctor using a brass plate to have inscribed on it, in addition to the letters indicating his qualifications, the words “Remember that I too am mortal.”
In 'Preface on Doctors', The Doctor’s Dilemma (1909, 1911), xci.
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Once the hatch was opened, I turned the lock handle and bright rays of sunlight burst through it. I opened the hatch and dust from the station flew in like little sparklets, looking like tiny snowflakes on a frosty day. Space, like a giant vacuum cleaner, began to suck everything out. Flying out together with the dust were some little washers and nuts that dad got stuck somewhere; a pencil flew by.
My first impression when I opened the hatch was of a huge Earth and of the sense of unreality concerning everything that was going on. Space is very beautiful. There was the dark velvet of the sky, the blue halo of the Earth and fast-moving lakes, rivers, fields and clouds clusters. It was dead silence all around, nothing whatever to indicate the velocity of the flight… no wind whistling in your ears, no pressure on you. The panorama was very serene and majestic.
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The aid which we feel impelled to give to the helpless is mainly an incidental result of the instinct of sympathy, which was originally acquired as part of the social instincts, but subsequently rendered, in the manner previously indicated, more tender and more widely diffused. Nor could we check our sympathy, even at the urging of hard reason, without deterioration in the noblest part of our nature.
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The aim of science is to apprehend this purely intelligible world as a thing in itself, an object which is what it is independently of all thinking, and thus antithetical to the sensible world.... The world of thought is the universal, the timeless and spaceless, the absolutely necessary, whereas the world of sense is the contingent, the changing and moving appearance which somehow indicates or symbolizes it.
'Outlines of a Philosophy of Art,' Essays in the Philosophy of Art, Indiana University Press (1964).
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The concepts of ‘soul’ or ‘life’ do not occur in atomic physics, and they could not, even indirectly, be derived as complicated consequences of some natural law. Their existence certainly does not indicate the presence of any fundamental substance other than energy, but it shows only the action of other kinds of forms which we cannot match with the mathematical forms of modern atomic physics ... If we want to describe living or mental processes, we shall have to broaden these structures. It may be that we shall have to introduce yet other concepts.
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The evidence indicates that woman is, on the whole, biologically superior to man.
In The Natural Superiority of Women (1952, 1999), 91.
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The information we have so far from the exploration of the planets seems to indicate that the earth is probably the only place in this solar system where there is life.
In Space World (1985), 5, 25.
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The theory of evolution has often been perverted so as to indicate that what is merely animal and brutal must gain the ascendancy. The contrary seems to me to be the case, for in man it is the spirit, and not the body, which is the deciding factor.
In Proceedings of Linnean Society of London (1913), 36.
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Unfortunately what is little recognized is that the most worthwhile scientific books are those in which the author clearly indicates what he does not know; for an author most hurts his readers by concealing difficulties.
As quoted in Nicholas J. Rose, Mathematical Maxims and Minims (1988). Also used as an epigraph without citation in Morris Kline, Mathematical Thought From Ancient to Modern Times (1990), Vol. 2, 752. If you know the primary source, perhaps in French, please contact Webmaster.
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When a conjecture inspires new hopes or creates new fears, action is indicated. There is an important asymmetry between hope, which leads to actions that will test its basis, and fear, which leads to restriction of options frequently restricting testing of the basis for the fear. As we know only too well, many of our hopes do not survive their tests. However, fears accumulate untested. Our inventory of untested fears has always made humanity disastrously vulnerable to thought control. While science was independent of politics, its greatest triumph was the reduction of that vulnerability.
Dartmouth College (1994)
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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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Bible
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- 70 -
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- 60 -
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- 50 -
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- 40 -
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- 30 -
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- 20 -
Carl Sagan
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- 10 -
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