"Well, since paradoxes are at hand, let us see how it might be demonstrated that in a finite continuous extension it is not impossible for infinitely many voids to be found." (Galileo Galilei, "Dialogue Concerning the Two Chief World Systems", 1632
"It will seem not a little paradoxical to ascribe a great importance to observations even in that part of the mathematical sciences which is usually called Pure Mathematics, since the current opinion is that observations are restricted to physical objects that make impression on the senses. As we must refer the numbers to the pure intellect alone, we can hardly understand how observations and quasi-experiments can be of use in investigating the nature of the numbers. Yet, in fact, as I shall show here with very good reasons, the properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are even many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations; they will lead us continually to new properties which we shall endeavor to prove afterwards. The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such a discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful. (Leonhard Euler, "Specimen de usu observationum in mathesi pura" , Novi Commentarii academiae scientiarum Petropolitanae 6, 1756/57)
"It is to be desired, that the charges of paradox and mystery, said to be introduced into algebra by negative and impossible quantities, should be proposed distinctly, in a precise form, fit to be apprehended and made the subject of discussion." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"Mathematical science has been at times embarrassed with contradictions and paradoxes; yet they are not to be imputed to imaginary symbols, rather than to any other symbols invented for the purpose of rendering demonstration compendious, and expeditious. It may; however, be justly remarked, that mathematicians, neglecting to exercise mental superintendance, are too prone to trust to mechanical dexterity; and that some, instead of establishing the truth of conclusions on antecedent reasons, have endeavoured to prop it by imperfect analogies or mere algebraic forms. On the other hand, there are mathematicians, whose zeal for just reasoning has been alarmed at a verbal absurdity and, from a name improperly applied, or a definition incautiously given, l have been hurried to the precipitate conclusion, that operations with symbols of which the mind can form no idea, must necessarily be doubtful and unintelligible." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"[...] all knowledge, and especially the weightiest knowledge of the truth, to which only a brief triumph is allotted between the two long periods in which it is condemned as paradoxical or disparaged as trivial." (Arthur Schopenhauer, "The World as Will and Representation", 1819)
"To concentrate on a craft is the best procedure. For the person of inferior gifts it will always remain a craft. The more gifted person will raise it to an art. And as for the man of highest endowment, in doing one thing he does all things; or, to put it less paradoxically, in the one thing that he does properly, he sees a symbol of all things that are done right. " (Johann Wolfgang von Goethe, 1829)
"There is no study which presents so simple a beginning as that of geometry, there is none in which difficulties grow more rapidly as we proceed, and what may appear at first rather paradoxical, the more acute the student the more serious will the impediments in the way of his progress appear." (Augustus De Morgan, "On the Study and Difficulties of Mathematics", 1830)
"Divergent series are in general very mischievous affairs, and it is shameful that any one should have founded a demonstration upon them. You can demonstrate anything you please by employing them, and it is they who have caused so much misfortune, and given birth to so many paradoxes." (Martin Ohm, "The Spirit of Mathematical Analysis and its Relation to a Logical System", 1842)
"One should not think slightly of the paradoxical; for the paradox is the source of the thinker's passion, and the thinker without a paradox is like a lover without feeling: a paltry mediocrity." (Søren Kierkegaard, "Philosophical Fragments: Or, A Fragment of Philosophy" 1844)
"It is the duty of the human understanding to understand that there are things which it cannot understand, and what those things are. Human understanding has vulgarly occupied itself with nothing but understanding, but if it would only take the trouble to understand itself at the same time it would simply have to posit the paradox." (Søren Kierkegaard, [The Journals of Søren Kierkegaard] 1847)
"Take away the paradox from a thinker and you have a professor." (Søren Kierkegaard,
[journal entry] 1849)
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