"Even with the examples of the infinite considered so far it could not escape our notice that not all infinite multitudes are to be regarded as equal to one another in respect of their plurality, but that some of them are greater (or smaller) than others, i.e. another multitude is contained as a part in one multitude (or on the contrary one multitude occurs in another as a mere part). This also is a claim which sounds to many paradoxical." (Bernard Bolzano, "Paradoxes of the Infinite", 1851)
"Disinterestedness is as great a puzzle and paradox as ever. Indeed, strictly speaking, it is a species of irrationality, or insanity, as regards the individual's self; a contradiction of the most essential nature of a sentient being, which is to move to pleasure and from pain." (Alexander Bain, "On the Study of Character, including an estimate of phrenology", 1861)
"Modern discoveries have not been made by large collections of facts, with subsequent discussion, separation, and resulting deduction of a truth thus rendered perceptible. A few facts have suggested an hypothesis, which means a supposition, proper to explain them. The necessary results of this supposition are worked out, and then, and not till then, other facts are examined to see if their ulterior results are found in Nature." (Augustus de Morgan, "A Budget of Paradoxes", 1872)
"The manner in which a paradoxer will show himself, as to sense or nonsense, will not depend upon what he maintains, but upon whether he has or has not made a sufficient knowledge of what has been done by others, especially as to the mode of doing it, a preliminary to inventing knowledge for himself." (Augustus De Morgan, "A Budget of Paradoxes", 1872)
"Thought is symbolical of Sensation as Algebra is of Arithmetic, and because it is symbolical, is very unlike what it symbolises. For one thing, sensations are always positive; in this resembling arithmetical quantities. A negative sensation is no more possible than a negative number. But ideas, like algebraic quantities, may be either positive or negative. However paradoxical the square of a negative quantity, the square root of an unknown quantity, nay, even in imaginary quantity, the student of Algebra finds these paradoxes to be valid operations. And the student of Philosophy finds analogous paradoxes in operations impossible in the sphere of Sense. Thus although it is impossible to feel non-existence, it is possible to think it; although it is impossible to frame an image of Infinity, we can, and do, form the idea, and reason on it with precision." (George H Lewes "Problems of Life and Mind", 1873)
"As lightning clears the air of impalpable vapours, so an incisive paradox frees the human intelligence from the lethargic influence of latent and unsuspected assumptions. Paradox is the slayer of Prejudice." (James J Sylvester, "The Collected Mathematical Papers of James Joseph Sylvester", 1875)
"It sounds paradoxical to say the attainment of scientific truth has been effected, to a great extent, by the help of scientific errors." (Thomas H Huxley, "The Progress of Science", 1887)
"Paradoxes have no place in science. Their removal is the substitution of true for false statements and thoughts." (William T Kelvin, "On Sun’s Heat", 1887)
"However we define error, the idea of calculating its extent may appear paradoxical. A science of errors seems a contradiction in terms." (Francis Y Edgeworth, "The Element of Chance in Competitive Examinations", Journal of the Royal Statistical Society Vol. 53, 1890)
"The folly of mistaking a paradox for a discovery, a metaphor for a proof, a torrent of verbiage for a spring of capital truths, and oneself for an oracle, is inborn in us." (Paul Valéry, "Introduction to the Method of Leonardo da Vinci", 1895)
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