A Quote by Felix Klein
Thus, in a sense, mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs.
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The exact sciences, which would be considered a priori as little adapted to women, for example mathematics, astronomy and physics, are exactly those in which thus far they have most distinguished themselves. This contains a warning against too precipitate conclusions about the intellectual life of woman.
Mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It's the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation.
Thus metaphysics and mathematics are, among all the sciences that belong to reason, those in which imagination has the greatest role. I beg pardon of those delicate spirits who are detractors of mathematics for saying this . . . . The imagination in a mathematician who creates makes no less difference than in a poet who invents. . . . Of all the great men of antiquity, Archimedes may be the one who most deserves to be placed beside Homer.
God and nature have thrown all human fortunes into the midst of mankind; and they are thus attainable rather by rapine than by industry, by wicked actions rather than by good. Hence it is that men feed upon each other, and those who cannot defend themselves must be worried.
If you ask ... the man in the street ... the human significance of mathematics, the answer of the world will be, that mathematics has given mankind a metrical and computatory art essential to the effective conduct of daily life, that mathematics admits of countless applications in engineering and the natural sciences, and finally that mathematics is a most excellent instrumentality for giving mental discipline... [A mathematician will add] that mathematics is the exact science, the science of exact thought or of rigorous thinking.
An entirely honest man, in the severe sense of the word, exists no more than an entirely dishonest knave: the best and the worst are only approximations of those qualities. Who are those that never contradict themselves? yet honesty never contradicts itself: Who are those that always contradict themselves? yet knavery is mere self-contradiction. Thus the knowledge of man determines not the things themselves, but their proportions, the quan?tum of congruities and incongruities.
Those who have racked their brains to discover new proofs have perhaps been induced to do so by a compulsion they could not quite explain to themselves. Instead of giving us their new proofs they should have explained to us the motivation that constrained them to search for them.
At times God's best pupils experience the most rigorous and continuous courses. Eventually those who prove to be men of Christ will thereby become distinguished alumni of life's school of affliction, graduating with honors.
Mathematics is of two kinds, Rigorous and Physical. The former is Narrow: the latter Bold and Broad. To have to stop to formulate rigorous demonstrations would put a stop to most physico-mathematical inquiries. Am I to refuse to eat because I do not fully understand the mechanism of digestion?
Medicine is a science which hath been (as we have said) more professed than laboured, and yet more laboured than advanced: the labour having been, in my judgment, rather in circle than in progression. For I find much iteration, but small addition. It considereth causes of diseases, with the occasions or impulsions; the diseases themselves, with the accidents; and the cures, with the preservation.
In those same decades, most UFO sightings were made in the daytime and frequently at close range, when shapes and surface features could be distinguished, thus making positive identification of normal sights easier and the descriptions of unusual sights more detailed. When all normal explanations had been eliminated, the witnesses could concentrate on those aspects of the experience which were most abnormal.
The countries the most famous and the most respected of antiquity are those which distinguished themselves by promoting and patronizing science, and on the contrary those which neglected or discouraged it are universally denominated rude and barbarous.
Mathematics has two faces: it is the rigorous science of Euclid, but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself.
Mathematics and logic have been proved to be one; a fact from which it seems to follow that mathematics may successfully deal with non-quantitative problems in a much broader sense than was suspected to be possible.
[I]f we revert to history, we shall find that the women who have distinguished themselves have neither been the most beautiful nor the most gentle of their sex.
Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems-general and specific statements-can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences.
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