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 Hilbertas program of consistency proofs; he explores qimplicit definitionq (a generalization of definition by abstraction) and indicates how this technique yields an qempiricistq 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.introduction to the field can dispense with many of the fundamental ideas of geometry: most notably, the Euclidean notion of parallelism a a ... As I already mentioned, projective geometry employs, from the start, only a few basic concepts.

Title | : | Essays on the Foundations of Mathematics by Moritz Pasch |

Author | : | Stephen Pollard |

Publisher | : | Springer Science & Business Media - 2010-08-03 |

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