Essays on the Foundations of Mathematics by Moritz Pasch

2010
ISBN: 978-94-007-3314-5
Verlag: Springer Netherlands

Buch, Englisch, Band 83, 248 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 400 g

Reihe: The Western Ontario Series in Philosophy of Science

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.

Zielgruppe

Research

Autoren/Hrsg.

Pollard, Stephen

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Philosophie der Mathematik
• Geisteswissenschaften Philosophie Philosophie der Mathematik, Philosophie der Physik
• Mathematik | Informatik Mathematik Geometrie
• Geisteswissenschaften Geschichtswissenschaft Geschichtliche Themen Wissenschafts- und Universitätsgeschichte
• Geisteswissenschaften Philosophie Wissenschaftstheorie, Wissenschaftsphilosophie
• Mathematik | Informatik Mathematik Mathematik Allgemein Geschichte der Mathematik
• Interdisziplinäres Wissenschaften Wissenschaften: Allgemeines Geschichte der Naturwissenschaften, Formalen Wissenschaften & Technik

Inhaltsverzeichnis

Translator’s Introduction.- Fundamental Questions of Geometry.- The Decidability Requirement.- The Origin of the Concept of Number.- Implicit Definition and the Proper Grounding of Mathematics.- Rigid Bodies in Geometry.- Prelude to Geometry: The Essential Ideas.- Physical and Mathematical Geometry.- Natural Geometry.- The Concept of the Differential.- Reflections on the Proper Grounding of Mathematics I.- Concepts and Proofs in Mathematics.- Dimension and Space in Mathematics.- Reflections on the Proper Grounding of Mathematics II.- The Axiomatic Method in Modern Mathematics.