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Home > Category Index for Science Quotations > Category Index I > Category: Infinite Series

Infinite Series Quotes (8 quotes)

La théorie des séries infinies en général est justqu’à présent très mal fondée. On applique aux séries infinies toutes les opérations, come si elles aient finies; mais cela est-il bien permis? Je crois que non. Où est-il démonstré qu/on ontient la différentielle dune série infinie en prenant la différentiaella de chaque terme. Rien n’est plus facile que de donner des exemples où cela n’est pas juste.
Until now the theory of infinite series in general has been very badly grounded. One applies all the operations to infinite series as if they were finite; but is that permissible? I think not. Where is it demonstrated that one obtains the differential of an infinite series by taking the differential of each term? Nothing is easier than to give instances where this is not so.
Quoted in Reinhold Remmert and Robert B. Burckel, Theory of Complex Functions: Readings in Mathematics (1991), 125.
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If it be urged that the action of the potato is chemical and mechanical only, and that it is due to the chemical and mechanical effects of light and heat, the answer would seem to lie in an enquiry whether every sensation is not chemical and mechanical in its operation? Whether those things which we deem most purely spiritual are anything but disturbances of equilibrium in an infinite series of levers, beginning with those that are too small for microscopic detection, and going up to the human arm and the appliances which it makes use of? Whether there be not a molecular action of thought, whence a dynamical theory of the passions shall be deducible?
In Erewhon, Or, Over the Range (1872), 192.
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If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words, the most important parts of mathematics stand without a foundation.
In Letter to a friend, as quoted in George Finlay Simmons, Calculus Gems (1992), 188.
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The first law of Engineering Mathematics: All infinite series converge, and moreover converge to the first term.
Anonymous
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The value the world sets upon motives is often grossly unjust and inaccurate. Consider, for example, two of them: mere insatiable curiosity and the desire to do good. The latter is put high above the former, and yet it is the former that moves some of the greatest men the human race has yet produced: the scientific investigators. What animates a great pathologist? Is it the desire to cure disease, to save life? Surely not, save perhaps as an afterthought. He is too intelligent, deep down in his soul, to see anything praiseworthy in such a desire. He knows by life-long observation that his discoveries will do quite as much harm as good, that a thousand scoundrels will profit to every honest man, that the folks who most deserve to be saved will probably be the last to be saved. No man of self-respect could devote himself to pathology on such terms. What actually moves him is his unquenchable curiosity–his boundless, almost pathological thirst to penetrate the unknown, to uncover the secret, to find out what has not been found out before. His prototype is not the liberator releasing slaves, the good Samaritan lifting up the fallen, but the dog sniffing tremendously at an infinite series of rat-holes.
In 'Types of Men: The Scientist', Prejudices (1923), 269-70.
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The [first] argument asserts the non-existence of notion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal.
Dichotomy paradox
Zeno
Aristotle, Physics, 239b10 11. In Reginald E. Allen, Greek Philosophy: Thales to Aristotle? (1991), 46.
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What animates a great pathologist? Is it the desire to cure disease, to save life? Surely not, save perhaps as an afterthought. He is too intelligent, deep in his soul, to see anything praiseworthy in such a desire. He knows from life-long observation that his discoveries will do quite as much harm as good, that a thousand scoundrels will profit to every honest man, that the folks who most deserve to be saved will probably be the last to be saved. ... What actually moves him is his unquenchable curiosity—his boundless, almost pathological thirst to penetrate the unknown, to uncover the secret, to find out what has not been found out before. ... [like] the dog sniffing tremendously at an infinite series of rat-holes. ... And yet he stands in the very front rank of the race
In 'The Scientist', Prejudices: third series (1922), 269-70.
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With the exception of the geometrical series, there does not exist in all of mathematics a single infinite series the sum of which has been rigorously determined. In other words, the things which are the most important in mathematics are also those which have the least foundation.
From letter (Jan 1828) to his former teacher Berndt Holmböe. In Morris Kline, Mathematics: The Loss of Certainty (1982), 170.
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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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