Abstractionism: Essays in Philosophy of Mathematics

ISBN : 9780199645268

Philip A. Ebert; Marcus Rossberg
368 Pages
156 x 234 mm
Pub date
Dec 2016
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Abstractionism, which is a development of Frege's original Logicism, is a recent and much debated position in the philosophy of mathematics. This volume contains 16 original papers by leading scholars on the philosophical and mathematical aspects of Abstractionism. After an extensive editors' introduction to the topic of abstractionism, five contributions deal with the semantics and meta-ontology of Abstractionism, as well as the so-called Caesar Problem. Four papers then discuss abstractionist epistemology, focusing on the idea of implicit definitions and non-evidential warrants (entitlements) to account for a priori mathematical knowledge. This is followed by four chapters concerning the mathematics of Abstractionism, in particular the issue of impredicativity, the Bad Company objection, and the question of abstractionist set theory. Finally, the last section of the book contains three contributions that discuss Frege's application constraint within an abstractionist setting.


I. Introduction
1 Philip A. Ebert and Marcus Rossberg: Introduction to Abstractionism

II. Semantics and Ontology of Abstraction
2 William Stirton: Caesar and Circularity
3 Richard G. Heck, Jr.: The Existence (and Non-existence) of Abstract Objects
4 Matti Eklund: Hale and Wright on the Metaontology of Neo-Fregeanism
5 Fraser MacBride: Neo-Fregean Ontology: Just Don't Ask Too Many Questions
6 Friederike Moltmann: The Number of Planets, a Number-Referring Term?

III. Epistemology of Abstraction
7 Philip A. Ebert: A Framework for Implicit Definitions and the A Priori
8 Crispin Wright: Abstraction and Epistemic Entitlement: On the Epistemological Status of Hume's Principle
9 Nikolaj Jang Lee Linding Pedersen: Hume's Principle and Entitlement: On the Epistemology of the Neo-Fregean Programme
10 Agustin Rayo: Neo-Fregeanism Reconsidered

IV. Mathematics of Abstraction
11 Roy T. Cook: Conservativeness, Cardinality, and Bad Company
12 Oystein Linnebo: Impredicativity in the Neo-Fregean Programme
13 Hannes Leitgeb: Abstraction Grounded: A Note on Abstraction and Truth
14 Stewart Shapiro and Gabriel Uzquiano: Ineffability within the Limits of Abstraction Alone

V. Application Constraint
15 Paul McCallion: On Frege's Applications Constraint
16 Peter Simons: Applications of Complex Numbers and Quaternions: Historical Remarks, with a Note on Clifford Algebra
17 Bob Hale: Definitions of Numbers and Their Applications

About the author: 

Philip A. Ebert received his PhD in Philosophy from the University of St Andrews in 2006 and was a Post Doctoral Fellow at the Arche Centre from 2005-2007. He is currently a Senior Lecturer at the University of Stirling. ; Marcus Rossberg received his PhD in Philosophy from the University of St Andrews in 2006 and was a Post Doctoral Fellow at the Arche Centre from 2005-2008. He is currently an Associate Professor at the University of Connecticut.

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