Essays on the Foundations of Mathematics by Moritz PaschStephen Pollard 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. |
Contents
| 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 |
| 243 | |
8 Natural Geometry | 148 |
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Common terms and phrases
analytic geometry applies arithmetic axiomatic axioms B-thing B.G. Teubner block chain collective name combinatorial reasoner common name concept of number conform congruent consider consistency proof consisting content words coordinates core concepts core propositions cyclotomic integer decidability requirement derived discussion distinct earlier empiricist endpoints example expression finite ordinal Foundations of Analysis Foundations of Mathematics grasp Hilbert Hugo Dingler immediate successor implicit definition inferences introduce last member later Lectures on modern Leipzig length logical mathematical geometry mathematical proof mathematicians meaning modern geometry Moritz Pasch natural numbers neighbor-line notation number theory paced pair pair-line planar surface position possible precedes q predicate prime divisor procedure projective geometry proof proper name real numbers rigid body satisfy segment connecting sequence specify things Springer Science+Business Media stem propositions straight line straight segment Suppose theorem Variable and Function viable zero