- Series Information
- The Oxford Handbook of Philosophy of Mathematics and Logic
- Notes on the Contributors
- Philosophy of Mathematics and Its Logic: Introduction
- A Priority and Application: Philosophy of Mathematics in the Modern Period
- Later Empiricism and Logical Positivism
- Wittgenstein on Philosophy of Logic and Mathematics
- The Logicism of Frege, Dedekind, and Russell
- Logicism in the Twenty‐first Century
- Logicism Reconsidered
- Intuitionism and Philosophy
- Intuitionism in Mathematics
- Intuitionism Reconsidered
- Quine and the Web of Belief
- Three Forms of Naturalism
- Naturalism Reconsidered
- Nominalism Reconsidered
- Structuralism Reconsidered
- Mathematics—Application and Applicability
- Logical Consequence, Proof Theory, and Model Theory
- Logical Consequence From a Constructivist View
- Relevance in Reasoning
- No Requirement of Relevance
- Higher‐order Logic
- Higher‐order Logic Reconsidered
Abstract and Keywords
This article focuses on issues which neo-Fregeanism must address, even if the scope of its leading claims is restricted to elementary arithmetic. Many of these concern the capacity of abstraction principles—centrally, but not only, Hume's Principle itself—to discharge the implicitly definitional role in which the neo-Fregean casts them, and thereby to subserve a satisfactory apriorist epistemology for (at least part of) mathematics. Others concern the other main assumption that undergirds the specifically logicist aspect of the neo-Fregean project (and equally, of course, Frege's original project): that the logic to which abstraction principles are to be adjoined may legitimately be taken to include higher-order—at the very least, second-order—logic without compromise of the epistemological purposes of the project.
Bob Hale is Professor of Metaphysical Philosophy at the University of Glasgow.
Crispin Wright is Bishop Wardlaw Professor at the University of St. Andrews, Global Distinguished Professor at New York University, and Director of the Research Centre, Arché. His writings in the philosophy of mathematics include Wittgenstein on the Foundations of Mathematics (1980); Frege's Conception of Numbers as Objects (1983); and, with Bob Hale, The Reason's Proper Study (Oxford University Press, 2001). His most recent books, Rails to Infinity (2001) and Saving the Differences (2003), respectively collect his writings on central themes of Wittgenstein's Philosophical Investigations and those further developing themes of his Truth and Objectivity (1992).
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