"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)
"Knowledge is only real and can only be set forth fully in the form of science, in the form of system." (G W Friedrich Hegel, "The Phenomenology of Mind", 1807)
"The Combinatorial Analysis is a branch of mathematics which teaches us to ascertain and exhibit all the possible ways in which a given number of things may be associated and mixed together; so that we may be certain that we have not missed any collection or arrangement of these things, that has not been enumerated. [...] Besides its uses in mathematical investigations, it not only enables us to form our ideas of the elegant compositions of design, but to contemplate the prodigious variety which constitutes the beauties of nature, and which arises from the combinations of objects, by their number, forms, color, and positions. It has a relation to every species of useful knowledge upon which the mind of man can be employed." (Peter Nicholson, "Essays on the Combinatorial Analysis", 1818)
"In the same way as philosophy loses sight of its true object and appropriate matter, when either it passes into and merges in theology, or meddles with external politics, so also does it mar its proper form when it attempts to mimic the rigorous method of mathematics." (K W Friedrich von Schlegel, "Philosophy of Life", 1828)
"In music the dignity of art seems to find supreme expression. There is no subject matter to be discounted. It is all form and significant content. It elevates and ennobles whatever it expresses."(Johann Wolfgang von Goethe, "Maxims and Reflections", 1829)
"The function of theory is to put all this in systematic order, clearly and comprehensively, and to trace each action to an adequate, compelling cause. […] Theory should cast a steady light on all phenomena so that we can more easily recognize and eliminate the weeds that always spring from ignorance; it should show how one thing is related to another, and keep the important and the unimportant separate. If concepts combine of their own accord to form that nucleus of truth we call a principle, if they spontaneously compose a pattern that becomes a rule, it is the task of the theorist to make this clear." (Carl von Clausewitz, "On War", 1832)
"Geometry is that of mathematical science which is devoted to consideration of form and size, and may be said to be the best and surest guide to study of all sciences in which ideas of dimension or space are involved. Almost all the knowledge required by navigators, architects, surveyors, engineers, and opticians, in their respective occupations, is deduced from geometry and branches of mathematics. All works of art are constructed according to the rules which geometry involves; and we find the same laws observed in the works of nature. The study of mathematics, generally, is also of great importance in cultivating habits of exact reasoning; and in this respect it forms a useful auxiliary to logic." (William Chambers & Robert Chambers, "Chambers's Information for the People" Vol. 2, 1835)
"A limit is a peculiar and fundamental conception, the use of which in proving the propositions of Higher Geometry cannot be superseded by any combination of other hypotheses and definitions. The axiom just noted that what is true up to the limit is true at the limit, is involved in the very conception of a limit: and this principle, with its consequences, leads to all the results which form the subject of the higher mathematics, whether proved by the consideration of evanescent triangles, by the processes of the Differential Calculus, or in any other way." (William Whewell, "The Philosophy of the Inductive Sciences", 1840)
"This principle, which is thus made the foundation of the operations and results of Symbolical Algebra, has been called 'The principle of the permanence of equivalent forms', and may be stated as follows: Whatever algebraical forms are equivalent, when the symbols are general in form but specific in value, will be equivalent likewise when the symbols are general in value as well as in form." (George Peacock, "A Treatise on Algebra", 1842)
"There are, undoubtedly, the most ample reasons for stating both the principles and theorems [of geometry] in their general form […] But, that an unpractised learner, even in making use of one theorem to demonstrate another, reasons rather from particular to particular than from the general proposition, is manifest from the difficulty he finds in applying a theorem to a case in which the configuration of the diagram is extremely unlike that of the diagram by which the original theorem was demonstrated. A difficulty which, except in cases of unusual mental powers, long practice can alone remove, and removes chiefly by rendering us familiar with all the configurations consistent with the general conditions of the theorem." (John S Mill, "A System of Logic", 1843)
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