Titlebook: Essays on the Foundations of Mathematics by Moritz Pasch; Stephen Pollard Book 2010 Springer Science+Business Media B.V. 2010 Arithmetic.A

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Concepts and Proofs in Mathematics,endent essays. In “Proof and Definition in Mathematics,” Pasch analyzes the notion of direct proof. In “Equality in Mathematics,” he explains why “equals” in mathematics is best understood as “is identical to.” In “The Decidability Requirement in Mathematics,” he offers examples of decidable and und

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Dimension and Space in Mathematics,f his “Prelude to Geometry” (Chapter 6 above) by characterizing bodies of 0, 1, 2, and 3 dimensions. He then discusses degrees (or “dimensions”) of polynomials and concludes by reviewing the fundamental structures of synthetic geometry and their counterparts in analytic geometry.

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Reflections on the Proper Grounding of Mathematics II,orrectness of his earlier paper on “The Origin of the Concept of Number,” Pasch concedes that he did not provide a foundation for number theory that was “complete in every detail.” He now undertakes to “reconsider and improve” his treatment of two topics: the distinction between restricted and unres

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The Axiomatic Method in Modern Mathematics,iate 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 i

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The Axiomatic Method in Modern Mathematics, 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.

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Fundamental Questions of Geometry,athematical proofs be presented “in a purely deductive form” and suggests that “full insight into the structure of deduction” requires that proofs be “atomized”: broken up into steps linked by inferences of a certain elementary form.

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Rigid Bodies in Geometry,eduction” by identifying axioms . supplying those axioms with content. In this paper, Pasch begins the job of supplying fundamental geometric propositions with content based on experience. His two goals are “to fix the concept of rigid body and, among such bodies, to distinguish between the ‘ extended’ and ‘ unextended’ ones.”

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