9781440059605
The Foundations Of Mathematics: A Contribution To The Philosophy Of Geometry (Classic Reprint) - Paul Carus
Forgotten Books (2009)
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THE SEARCH FOR THE FOUNDATIONS OF GEOMETRY: HISTORICAL SKETCH.AXIOMS AND THE AXIOM OF PARALLELS.MATHEMATICS as commonly taught in our schools is based upon axioms. These axioms so called are a few simple formulas which the beginner must take on trust.Axioms are defined to be self-evident propositions, and are claimed to be neither demonstrable nor in need of demonstration. They are statements which are said to command the assent of every one who comprehends their meaning.The word axiom1 means "honor, reputation, high rank, authority," and is used by Aristotle, almost in the modern sense of the term, as "a self-evident highest principle," or "a truth so obvious as to be in no need of proof." It is derived from the verb agtovv, "to deem worthy, to think fit, to maintain," and is cognate with a&oq, "worth" or "worthy."Euclid does not use the term "axiom." He starts with Definitions,2 which describe the meanings of point, line, surface, plane, angle, etc. He1 dtTable of Contents TABLE OF CONTENTS; THE SEARCH FOR THE FOUNDATIONS OF GEOMETRY: HISTORICAL SKETCH; PAGE; Axioms and the Axiom of Parallels i; Metageometry 5; Precursors 7; Gauss *1; Riemann >5; Lobatchevsky 20; Bolyai 22; Later Geometricians 24; Grassmann 27; Euclid Still Unimpaired 3«; THE PHILOSOPHICAL BASIS OF MATHEMATICS; The Philosophical Problem 35; Transcendentalism and Empiricism 3$; The A Priori and the Purely Formal 40; Anyness and its Universality 46; Apriority of Different Degrees 49; Space as a Spread of Motion 56; Uniqueness of Pure Space 61; Mathematical Space and Physiological Space 63; Homogeneity of Space Due to Abstraction 66; Even Boundaries as Standards of Measurement 69; The Straight Line Indispensable 72; The Superreal 76; Discrete Units and the Continuum 78; MATHEMATICS AND METAGEOMETRY; Different Geometrical Systems 82; Tridimensional^ 84; Three a Concept of Bound

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Cover Price 7,85 €
No. of Pages 154
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