*Philip A Ebert and Marcus Rossberg (eds)*

- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780199645268
- eISBN:
- 9780191755330
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199645268.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

The collection contains an extensive introduction and 16 original papers on the philosophical and mathematical aspects of Abstractionism—a position in the philosophy of mathematics which is a ...
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The collection contains an extensive introduction and 16 original papers on the philosophical and mathematical aspects of Abstractionism—a position in the philosophy of mathematics which is a development of Frege’s original Logicism. The collection is structured as follows: After an extensive editors’ introduction to the topic of abstractionism, part II contains five contributions that deal with semantics and metaontology of Abstractionism, as well as the so-called Caesar Problem. Part III collects four contributions that discuss abstractionist epistemology, focusing on the idea of implicit definitions and non-evidential warrants (entitlements) to account for a priori mathematical knowledge. Four papers in part IV concern the mathematics of Abstractionism, in particular the issue of impredicativity, the Bad Company objection, and the question of abstractionist set theory. The last section contains three contributions that discuss Frege’s application constraint within an abstractionist setting.Less

The collection contains an extensive introduction and 16 original papers on the philosophical and mathematical aspects of Abstractionism—a position in the philosophy of mathematics which is a development of Frege’s original Logicism. The collection is structured as follows: After an extensive editors’ introduction to the topic of abstractionism, part II contains five contributions that deal with semantics and metaontology of Abstractionism, as well as the so-called Caesar Problem. Part III collects four contributions that discuss abstractionist epistemology, focusing on the idea of implicit definitions and non-evidential warrants (entitlements) to account for a priori mathematical knowledge. Four papers in part IV concern the mathematics of Abstractionism, in particular the issue of impredicativity, the Bad Company objection, and the question of abstractionist set theory. The last section contains three contributions that discuss Frege’s application constraint within an abstractionist setting.

*Philip A. Ebert and Marcus Rossberg*

- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780199645268
- eISBN:
- 9780191755330
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199645268.003.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

We offer a general introduction to Abstractionism by outlining its history and by presenting the core philosophical and mathematical tenets of the abstractionist projects. We then locate the 16 ...
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We offer a general introduction to Abstractionism by outlining its history and by presenting the core philosophical and mathematical tenets of the abstractionist projects. We then locate the 16 contributions to our volume within the current debate of Abstractionism.Less

We offer a general introduction to Abstractionism by outlining its history and by presenting the core philosophical and mathematical tenets of the abstractionist projects. We then locate the 16 contributions to our volume within the current debate of Abstractionism.

*Crispin Wright*

- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780199645268
- eISBN:
- 9780191755330
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199645268.003.0008
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

The abstractionist program of foundations for classical mathematical theories is, like its traditional logicist ancestors, first and foremost an epistemological project. Its official aim is to ...
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The abstractionist program of foundations for classical mathematical theories is, like its traditional logicist ancestors, first and foremost an epistemological project. Its official aim is to demonstrate the possibility of a certain uniform mode of a priori knowledge of the basic laws of arithmetic, real and complex analysis, and set theory. Traditional logicism aimed to show that mathematical knowledge could be accomplished using only analytic definitions and theses of pure logic and hence is not merely a priori if logic is but is effectively a proper part of logic. Abstractionism, however, adds abstraction principles to the base materials employed in the traditional logicist project—principles that, at least in the central, interesting cases, are neither pure analytic definitions nor theses of pure logic as conventionally understood. Thus, whatever significance they may carry for the prospects for logicism, the epistemological significance of technically successful abstractionist projects must turn on the epistemological status of the abstraction principles used, with any demonstration of a priority in particular being dependent on whether those principles can themselves rank as knowable a priori. My primary focus here will be on this natural thought.Less

The abstractionist program of foundations for classical mathematical theories is, like its traditional logicist ancestors, first and foremost an *epistemological* project. Its official aim is to demonstrate the possibility of a certain uniform mode of a priori knowledge of the basic laws of arithmetic, real and complex analysis, and set theory. Traditional logicism aimed to show that mathematical knowledge could be accomplished using only analytic definitions and theses of pure logic and hence is not merely a priori if logic is but is effectively a *proper part* of logic. Abstractionism, however, adds abstraction principles to the base materials employed in the traditional logicist project—principles that, at least in the central, interesting cases, are neither pure analytic definitions nor theses of pure logic as conventionally understood. Thus, whatever significance they may carry for the prospects for logicism, the epistemological significance of technically successful abstractionist projects must turn on the epistemological status of the abstraction principles used, with any demonstration of a priority in particular being dependent on whether those principles can themselves rank as knowable a priori. My primary focus here will be on this natural thought.