Implicit Definitions

Part of the LEP Library of Exact Philosophy book series (LEP, volume 11)


Although logic from the beginning was able to perceive the above-mentioned problem, the impetus for its definitive solution came from another quarter. It came from research in a particular science, to whose needs logic, in this instance as in most others, did not adapt itself until later. In the nature of the case, the only science that could forge ahead to a rigorous formulation of our problem was one so constituted that absolute certainty had to be guaranteed for its every step. This science was mathematics. The remaining sciences, not only because of inadequate definitions but on other grounds as well, were unable to raise such lofty claims to rigor; hence they had no occasion to formulate their problems in so basic a manner. Nevertheless, the significance of the studies we are about to report is by no means confined to mathematics. On the contrary, they are in principle just as valid for scientific concepts generally as they are for those of mathematics. It is simply a matter of convenience that we take mathematical concepts as a paradigm on which to base our considerations.


Mutual Relation Intuitive Idea Absolute Certainty Intuitive Meaning Rigorous Formulation 
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  1. 7.
    Special mentions should be made of M. Pasch’s Vorlesungen über neuere Geometrie, 1882.Google Scholar
  2. 8.
    D. Hilbert, Grundlagen der Geometrie, 4th edition, 1913. (English translation of first edition by E. J. Townsend, 1902: The Foundations of Geometry.)Google Scholar

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© Springer-Verlag/Wien 1974

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