Essays on the Foundations of Mathematics by Moritz Pasch

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Stephen Pollard
Springer Science & Business Media, 3 ago. 2010 - 248 páginas
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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.
 

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Índice

Translators Introduction
1
1 Fundamental Questions of Geometry
44
2 The Decidability Requirement
51
3 The Origin of the Concept of Number
55
4 Implicit Definition and the Proper Grounding of Mathematics
94
5 Rigid Bodies in Geometry
109
The Essential Ideas
117
7 Physical and Mathematical Geometry
139
9 The Concept of the Differential
153
10 Reflections on the Proper Grounding of Mathematics I
174
11 Concepts and Proofs in Mathematics
183
12 Dimension and Space in Mathematics
205
13 Reflections on the Proper Grounding of Mathematics II
214
14 The Axiomatic Method in Modern Mathematics
221
Index
243
Página de créditos

8 Natural Geometry
148

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Sobre el autor (2010)

Stephen Pollard is currently Editor of the Jewish Chronicle and has written columns for several publications including The Times and the Daily Mail and maintained a lively Spectator blog. He is also the author of the controversial 2004 biography of David Blunkett, and co-authored A Class Act: The Myth of Britain's Classless Society, which was shortlisted for the Orwell Prize.

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