The Axiomatic Method in Modern Mathematics

Chapter
Part of the The Western Ontario Series in Philosophy of Science book series (WONS, volume 83)

Abstract

English translation of Moritz Pasch,“Die axiomatische Methode in der neueren Mathematik,” Annalen der Philosophie und philosophischen Kritik 5 (1926), pp. 241–274. Pasch explores the relation of immediate consequence. In a direct, unabbreviated proof , each step is an immediate consequence of prior steps. Though it is not decidable whether an arbitrary conclusion follows from arbitrary premises, it is decidable whether the conclusion immediately follows. Two sentences express the same statement if and only if each is an immediate consequence of the other. Two sentences contradict one another if and only if the denial of one follows immediately from the other. Whether one sentence is an immediate consequence of another depends entirely on the structural elements (rather than the content words) occurring in those sentences. Anyone who understands those structural elements will be able to determine whether one sentence follows immediately from the other.

Keywords

Content Word Curve Segment Straight Segment Indirect Proof Proof Step 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Dept. Philosophy & ReligionTruman State UniversityKirksvilleUSA

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