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Essays on the Foundations of Mathematics by Moritz Pasch

  • Book
  • © 2010

Overview

Editors:
  1. Stephen Pollard

    1. Dept. Philosophy & Religion, Truman State University, Kirksville, USA

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  • Gives English speaker access to an important body of work from a turbulent and pivotal period in the history of mathematics
  • Helps us look beyone the familiar triad of formalism, instuitionism and logicism
  • Explains how we can see with the help of a guide determined to present fundamental mathematical ideas in ways that match our human capacities

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

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Table of contents (15 chapters)

  1. Front Matter

    Pages i-xi
    Download chapter PDF

  2. Translator’s Introduction

    • Stephen Pollard
    Pages 1-43

  3. Fundamental Questions of Geometry

    • Stephen Pollard
    Pages 45-49

  4. The Decidability Requirement

    • Stephen Pollard
    Pages 51-54

  5. The Origin of the Concept of Number

    • Stephen Pollard
    Pages 55-93

  6. Implicit Definition and the Proper Grounding of Mathematics

    • Stephen Pollard
    Pages 95-107

  7. Rigid Bodies in Geometry

    • Stephen Pollard
    Pages 109-116

  8. Prelude to Geometry: The Essential Ideas

    • Stephen Pollard
    Pages 117-138

  9. Physical and Mathematical Geometry

    • Stephen Pollard
    Pages 139-147

  10. Natural Geometry

    • Stephen Pollard
    Pages 149-151

  11. The Concept of the Differential

    • Stephen Pollard
    Pages 153-173

  12. Reflections on the Proper Grounding of Mathematics I

    • Stephen Pollard
    Pages 175-182

  13. Concepts and Proofs in Mathematics

    • Stephen Pollard
    Pages 183-203

  14. Dimension and Space in Mathematics

    • Stephen Pollard
    Pages 205-213

  15. Reflections on the Proper Grounding of Mathematics II

    • Stephen Pollard
    Pages 215-220

  16. The Axiomatic Method in Modern Mathematics

    • Stephen Pollard
    Pages 221-242

  17. Back Matter

    Pages 243-245
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Keywords

About this book

Moritz Pasch (1843-1930) is justly celebrated as a key figure in the history of axiomatic geometry. Less well known are his contributions to other areas of foundational research. This volume features English translations of 14 papers Pasch published in the decade 1917-1926. In them, Pasch argues that geometry and, more surprisingly, number theory are branches of empirical science; he provides axioms for the combinatorial reasoning essential to Hilbert’s program of consistency proofs; he explores "implicit definition" (a generalization of definition by abstraction) and indicates how this technique yields an "empiricist" reconstruction of set theory; he argues that we cannot fully understand the logical structure of mathematics without clearly distinguishing between decidable and undecidable properties; he offers a rare glimpse into the mind of a master of axiomatics, surveying in detail the thought experiments he employed as he struggled to identify fundamental mathematical principles; and much more. This volume will: Give English speakers access to an important body of work from a turbulent and pivotal period in the history of mathematics, help us look beyond the familiar triad of formalism, intuitionism, and logicism, show how deeply we can see with the help of a guide determined to present fundamental mathematical ideas in ways that match our human capacities, will be of interest to graduate students and researchers in logic and the foundations of mathematics.

Editors and Affiliations

  • Dept. Philosophy & Religion, Truman State University, Kirksville, USA

    Stephen Pollard

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