Structural-Abstraction Principles

Philosophia Mathematica:nkv033 (2015)
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Abstract

In this paper, I present a class of ‘structural’ abstraction principles, and describe how they are suggested by some features of Cantor's and Dedekind's approach to abstraction. Structural abstraction is a promising source of mathematically tractable new axioms for the neo-logicist. I illustrate this by showing, first, how a theorem of Shelah gives a sufficient condition for consistency in the structural setting, solving what neo-logicists call the ‘bad company’ problem for structural abstraction. Second, I show how, in the structural setting, we can measure the logical strength of abstraction principles using categories of interpretations between theories.

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Graham Leach-Krouse
Kansas State University

Citations of this work

The Structuralist Thesis Reconsidered.Georg Schiemer & John Wigglesworth - 2019 - British Journal for the Philosophy of Science 70 (4):1201-1226.
The Structuralist Thesis Reconsidered.Georg Schiemer & John Wigglesworth - 2017 - British Journal for the Philosophy of Science:axy004.

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References found in this work

The foundations of arithmetic.Gottlob Frege - 1884/1950 - Evanston, Ill.,: Northwestern University Press.
Philosophy of mathematics: structure and ontology.Stewart Shapiro - 1997 - New York: Oxford University Press.
The foundations of arithmetic: a logico-mathematical enquiry into the concept of number.Gottlob Frege - 1959 - Evanston, Ill.: Northwestern University Press. Edited by J. L. Austin.
Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 2000 - Philosophical Quarterly 50 (198):120-123.
Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 2002 - Philosophy and Phenomenological Research 65 (2):467-475.

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