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The aim of this Element is to provide an overview of abstractionism in the philosophy of mathematics. The authors distinguish between mathematical abstractionism, which interprets mathematical theories on the basis of abstraction principles, and philosophical abstractionism, which attributes a philosophical significance to mathematical abstractionism. They then survey the main semantic, ontological, and epistemological theses that are associated with philosophical abstractionism. Finally, the authors suggest that the most recent developments in the debate pull abstractionism in different directions.
Introduction
Mathematical Abstractionism
Introduction
Second-Order Logic
Frege’s Grundgesetze der Arithmetik
Arithmetical Abstraction
Real Number Abstraction
Set Abstraction
Invariance
The Bad Company Problem
Philosophical Abstractionism I: Semantics
Introduction
The Substantival View
Nonsubstantival Views
The Caesar Problem
Philosophical Abstractionism II: Ontology
Introduction
The Existence of Mathematical Objects
Real Definitions
Philosophical Abstractionism III: Epistemology
Introduction
Puzzle 1: Epistemic Access
Puzzle 2: Knowledge by Abstraction
Epistemic Bad Company and Related Concerns
New Directions in Philosophical Abstractionism
Abstractionist Structuralism
References
Contributions
Acknowledgements