„In order to see the difference which exists between… studies,—for instance, history and geometry, it will be useful to ask how we come by knowledge in each. Suppose, for example, we feel certain of a fact related in history… if we apply the notions of evidence which every-day experience justifies us in entertaining, we feel that the improbability of the contrary compels us to take refuge in the belief of the fact; and, if we allow that there is still a possibility of its falsehood, it is because this supposition does not involve absolute absurdity, but only extreme improbability.
In mathematics the case is wholly different… and the difference consists in this—that, instead of showing the contrary of the proposition asserted to be only improbable, it proves it at once to be absurd and impossible. This is done by showing that the contrary of the proposition which is asserted is in direct contradiction to some extremely evident fact, of the truth of which our eyes and hands convince us. In geometry, of the principles alluded to, those which are most commonly used are—
I. If a magnitude is divided into parts, the whole is greater than either of those parts.
II. Two straight lines cannot inclose a space.
III. Through one point only one straight line can be drawn, which never meets another straight line, or which is parallel to it.
It is on such principles as these that the whole of geometry is founded, and the demonstration of every proposition consists in proving the contrary of it to be inconsistent with one of these.“

Fuente: On the Study and Difficulties of Mathematics (1831), Ch. I.

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Augustus De Morgan8
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„But if some mind very different from ours were to look upon some property of some curved line as we do on the evenness of a straight line, he would not recognize as such the evenness of a straight line; nor would he arrange the elements of his geometry according to that very different system, and would investigate quite other relationships as I have suggested in my notes.
We fashion our geometry on the properties of a straight line because that seems to us to be the simplest of all. But really all lines that are continuous and of a uniform nature are just as simple as one another. Another kind of mind which might form an equally clear mental perception of some property of any one of these curves, as we do of the congruence of a straight line, might believe these curves to be the simplest of all, and from that property of these curves build up the elements of a very different geometry, referring all other curves to that one, just as we compare them to a straight line. Indeed, these minds, if they noticed and formed an extremely clear perception of some property of, say, the parabola, would not seek, as our geometers do, to rectify the parabola, they would endeavor, if one may coin the expression, to parabolify the straight line.“

—  Roger Joseph Boscovich Croat-Italian physicist 1711 - 1787

"Boscovich's mathematics", an article by J. F. Scott, in the book Roger Joseph Boscovich (1961) edited by Lancelot Law Whyte.
"Transient pressure analysis in composite reservoirs" (1982) by Raymond W. K. Tang and William E. Brigham.
"Non-Newtonian Calculus" (1972) by Michael Grossman and Robert Katz.

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„I tell you that if natural bodies have it from Nature to be moved by any movement, this can only be circular motion, nor is it possible that Nature has given to any of its integral bodies a propensity to be moved by straight motion. I have many confirmations of this proposition, but for the present one alone suffices, which is this. I suppose the parts of the universe to be in the best arrangement, so that none is out of its place, which is to say that Nature and God have perfectly arranged their structure. This being so, it is impossible for those parts to have it from Nature to be moved in straight, or in other than circular motion, because what moves straight changes place, and if it changes place naturally, then it was at first in a place preternatural to it, which goes against the supposition. Therefore, if the parts of the world are well ordered, straight motion is superfluous and not natural, and they can only have it when some body is forcibly removed from its natural place, to which it would then return by a straight line, for thus it appears that a part of the earth does [move] when separated from its whole. I said "it appears to us," because I am not against thinking that not even for such an effect does Nature make use of straight line motion.“

—  Galileo Galilei Italian mathematician, physicist, philosopher and astronomer 1564 - 1642

A note on this statement is included by Stillman Drake in his Galileo at Work, His Scientific Biography (1981): Galileo adhered to this position in his Dialogue at least as to the "integral bodies of the universe." by which he meant stars and planets, here called "parts of the universe." But he did not attempt to explain the planetary motions on any mechanical basis, nor does this argument from "best arrangement" have any bearing on inertial motion, which to Galileo was indifference to motion and rest and not a tendency to move, either circularly or straight.
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„You can take every one of Spinoza's propositions, and take the contrary propositions, and“

—  Baruch Spinoza Dutch philosopher 1632 - 1677

Richard Feynman, in The Pleasure of Finding Things Out (1999), Ch. 9. The Smartest Man in the World
Contexto: My son is taking a course in philosophy, and last night we were looking at something by Spinoza and there was the most childish reasoning! There were all these attributes, and Substances, and all this meaningless chewing around, and we started to laugh. Now how could we do that? Here's this great Dutch philosopher, and we're laughing at him. It's because there's no excuse for it! In the same period there was Newton, there was Harvey studying the circulation of the blood, there were people with methods of analysis by which progress was being made! You can take every one of Spinoza's propositions, and take the contrary propositions, and look at the world and you can't tell which is right.

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„Our knowledge is composed not of facts, but of the relations which facts and ideas bear to themselves and to each other; and real knowledge consists not in an acquaintance with facts, which only makes a pedant, but in the use of facts, which makes a philosopher.“

—  Henry Thomas Buckle English historian 1821 - 1862

" The Influence Of Women On The Progress Of Knowledge http://www.public.coe.edu/~theller/soj/u-rel/buckle.html". Lecture given at the Royal Institution 19 March 1858. In: The Miscellaneous and Posthumous Works of Henry Thomas Buckle (1872)

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„We use power, of course, in the international fields in a way which is the exact contrary to the way in which we use it within the state.“

—  Norman Angell British politician 1872 - 1967

Peace and the Public Mind (1935)
Contexto: We use power, of course, in the international fields in a way which is the exact contrary to the way in which we use it within the state. In the international field, force is the instrument of the rival litigants, each attempting to impose his judgment upon the other. Within the state, force is the instrument of the community, the law, primarily used to prevent either of the litigants imposing by force his view upon the other. The normal purpose of police — to prevent the litigant taking the law into his own hands, being his own judge — is the precise contrary of the normal purpose in the past of armies and navies, which has been to enable the litigant to be his own judge of his own rights when in conflict about them with another.

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„See now the power of truth; the same experiment which at first glance seemed to show one thing, when more carefully examined, assures us of the contrary.“

—  Galileo Galilei Italian mathematician, physicist, philosopher and astronomer 1564 - 1642

Discourses and Mathematical Demonstrations Relating to Two New Sciences (1638); Discorsi e dimostrazioni matematiche, intorno à due nuove scienze, as translated by Henry Crew and Alfonso de Salvio (1914)
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Fuente: Discorsi E Dimostrazioni Matematiche: Intorno a Due Nuoue Scienze, Attenenti Alla Mecanica & I Movimenti Locali

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