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Auguste Comte

The Philosophy of Mathematics

  • Michel Isidroцитирует6 лет назад
    since they are not established by any reasoning, must be founded on observation alone, and which form the necessary basis of all the deductions.
  • Michel Isidroцитирует6 лет назад
    Thus the calculus of Taylor never has offered, and never can offer, in any question of geometry or of mechanics, that powerful general aid which we have seen to result necessarily from the analysis of Leibnitz
  • Michel Isidroцитирует6 лет назад
    Clairaut, who first had occasion to remark their existence, saw in them a paradox of the integral calculus, since these solutions have the peculiarity of satisfying the differential equations without being comprised in the corresponding general integrals.
  • Michel Isidroцитирует6 лет назад
    That of Lagrange. This perfect unity of analysis, and this purely abstract character of its fundamental notions, are found in the highest degree in the conception of Lagrange, and are found there alone; it is, for this reason, the most rational and the most philosophical of all.
  • Michel Isidroцитирует6 лет назад
    Even if we adopt the ingenious idea of the compensation of errors, as above explained, this involves the radical inconvenience of being obliged to distinguish in mathematics two classes of reasonings, those which are perfectly rigorous, and those in which we designedly commit errors which subsequently have to be compensated. A conception which leads to such strange consequences is undoubtedly very unsatisfactory in a logical point of view.
  • Michel Isidroцитирует6 лет назад
    In this respect it must be regarded as the necessary complement of the great fundamental idea of Descartes on the general analytical representation of natural phenomena: an idea which did not begin to be worthily appreciated and suitably employed till after the formation of the infinitesimal analysis, without which it could not produce, even in geometry, very important results.
  • Michel Isidroцитирует6 лет назад
    This circumstance occurs, among other occasions, in the case of a radius vector in geometry, and diverging forces in mechanics.
  • Michel Isidroцитирует6 лет назад
    It presents, in fact, this serious inconvenience of obliging us to repeat the whole series of operations for the slightest change which may take place in a single one of the quantities considered, although their relations to one another remain unchanged; the calculations made for one case not enabling us to dispense with any of those which relate to a case very slightly different. This happens because of our inability to abstract and treat separately that purely algebraic part of the question which is common to all the cases which result from the mere variation of the given numbers.
  • Michel Isidroцитирует6 лет назад
    By suitably employing the simple and general method so happily invented by analysts, and which consists in referring all the other unknown quantities to one of them, the difficulty would always disappear if we knew how to obtain the algebraic resolution of the equations under consideration, while the numerical solution would then be perfectly useless.
  • Michel Isidroцитирует6 лет назад
    The general equation of the fifth degree itself has thus far resisted all attacks.
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