„I think that it is a relatively good approximation to truth — which is much too complicated to allow anything but approximations — that mathematical ideas originate in empirics.“

"The Mathematician", in The Works of the Mind (1947) edited by R. B. Heywood, University of Chicago Press, Chicago
Contexto: I think that it is a relatively good approximation to truth — which is much too complicated to allow anything but approximations — that mathematical ideas originate in empirics. But, once they are conceived, the subject begins to live a peculiar life of its own and is … governed by almost entirely aesthetical motivations. In other words, at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration. Whenever this stage is reached the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas.

Última atualização 22 de Maio de 2020. História
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John von Neumann
1903 - 1957
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„All exact science is dominated by the idea of approximation. When a man tells you that he knows the exact truth about anything, you are safe in infering that he is an inexact man.“

—  Bertrand Russell logician, one of the first analytic philosophers and political activist 1872 - 1970

As quoted in World Unity, Vol. IX, 3rd edition (1931), p. 190
1930s

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„Force is not a fact at all, but an idea embodying what is approximately the fact.“

—  William Kingdon Clifford English mathematician and philosopher 1845 - 1879

Preface footnote, p. ix. Mr. R. Tucker searched Clifford's note books for Karl Pearson and sent him the above quote, in Clifford's handwriting.
The Common Sense of the Exact Sciences (1885)

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E. W. Hobson photo

„Much of the skill of the true mathematical physicist and of the mathematical astronomer consists in the power of adapting methods and results carried out on an exact mathematical basis to obtain approximations sufficient for the purposes of physical measurements. It might perhaps be thought that a scheme of Mathematics on a frankly approximative basis would be sufficient for all the practical purposes of application in Physics, Engineering Science, and Astronomy, and no doubt it would be possible to develop, to some extent at least, a species of Mathematics on these lines. Such a system would, however, involve an intolerable awkwardness and prolixity in the statements of results, especially in view of the fact that the degree of approximation necessary for various purposes is very different, and thus that unassigned grades of approximation would have to be provided for. Moreover, the mathematician working on these lines would be cut off from the chief sources of inspiration, the ideals of exactitude and logical rigour, as well as from one of his most indispensable guides to discovery, symmetry, and permanence of mathematical form. The history of the actual movements of mathematical thought through the centuries shows that these ideals are the very life-blood of the science, and warrants the conclusion that a constant striving toward their attainment is an absolutely essential condition of vigorous growth. These ideals have their roots in irresistible impulses and deep-seated needs of the human mind, manifested in its efforts to introduce intelligibility in certain great domains of the world of thought.“

—  E. W. Hobson British mathematician 1856 - 1933

Fonte: Presidential Address British Association for the Advancement of Science, Section A (1910), pp. 285-286; Cited in: Moritz (1914, 229): Mathematics and Science.

Claude Monet photo

„It is decidedly frightfully difficult to make something complete in all respects, and I think that there are scarcely any but those who content themselves with the approximate.“

—  Claude Monet French impressionist painter 1840 - 1926

1850 - 1870
Contexto: My dear Frédéric Bazille, I ask myself what you can be doing in Paris during fine weather, for I suppose that it must also be very fine there. Here my dear fellow, it is is charming, and I discover every day always beautiful things. It is enough to become mad [fou], so much do I have the desire to do it all, my head is cracking. Damn it, here it is the sixteenth, put aside your cliques and your claques, and come spend a couple of weeks here, it would be the best thing that you could do, because in Paris it cannot be very easy to work.
This very day, I still have a month to stay in; furthermore my sketches are becoming finished, I have even set to work additionally [remis] on some others. In sum, I am content enough with my stay here, even though my studies are very far from what I would wish. It is decidedly frightfully difficult to make something complete in all respects, and I think that there are scarcely any but those who content themselves with the approximate. Very well, my dear fellow, I want to struggle, scrape, start over again [recommencer], because one can do what one sees and understands, and it seems to me, when I see nature, that I am going to do it all, write it all out, but them go try to do it.... when one is on the job..
All this proves that one must only think about this. It is by force of observation and reflection that one finds. So let us grind away and grind away constantly. Are you making any progress? Yes, I am sure of it, but what I am sure of is that you do not work enough and not in the right way. It is not with carefree guys like your Villa and others that you will be able to work. It would be better all alone, and yet, all alone there are plenty of things that one cannot make out. In the end all of this is terrible, and it is a rough task.
... It is frightening what I see in my head.

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„If you don't think too good, don't think too much.“

—  Ted Williams American professional baseball player 1918 - 2002

As quoted in The Gigantic Book of Baseball Quotations (2007) edited by Wayne Stewart, p. 360

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„Socrates is not just expounding noble ideas in a vacuum. He is in the middle of a war between those who think truth is absolute and those who think truth is relative. He is fighting that war with everything he has.“

—  Robert M. Pirsig, livro Zen and the Art of Motorcycle Maintenance

Fonte: Zen and the Art of Motorcycle Maintenance (1974), Ch. 29
Contexto: Socrates is not just expounding noble ideas in a vacuum. He is in the middle of a war between those who think truth is absolute and those who think truth is relative. He is fighting that war with everything he has. The Sophists are the enemy.
Now Plato's hatred of the Sophists makes sense. He and Socrates are defending the Immortal Principle of the Cosmologists against what they consider to be the decadence of the Sophists. Truth. Knowledge. That which is independent of what anyone thinks about it. The ideal that Socrates died for. The ideal that Greece alone possesses for the first time in the history of the world. It is still a very fragile thing. It can disappear completely. Plato abhors and damns the Sophists without restraint, not because they are low and immoral people—there are obviously much lower and more immoral people in Greece he completely ignores. He damns them because they threaten mankind's first beginning grasp of the idea of truth. That's what it is all about.

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„The truth is too simple: one must always get there by a complicated route.“

—  George Sand French novelist and memoirist; pseudonym of Lucile Aurore Dupin 1804 - 1876

Le vrai est trop simple, il faut y arriver toujours par le compliqué.
Letter to Armand Barbès, (12 May 1867), published in Georges Lubin (ed.) Correspondance (Paris: Garnier Freres, 1964-95) vol. 20, p. 412; Bruce Kajewski Traveling with Hermes (Amherst: University of Massachusetts Press, 1992) p. 32

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