David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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Synthese 170 (3):415 - 441 (2009)
This paper discusses the neo-logicist approach to the foundations of mathematics by highlighting an issue that arises from looking at the Bad Company objection from an epistemological perspective. For the most part, our issue is independent of the details of any resolution of the Bad Company objection and, as we will show, it concerns other foundational approaches in the philosophy of mathematics. In the first two sections, we give a brief overview of the "Scottish" neo-logicist school, present a generic form of the Bad Company objection and introduce an epistemic issue connected to this general problem that will be the focus of the rest of the paper. In the third section, we present an alternative approach within philosophy of mathematics, a view that emerges from Hilbert's Grundlagen der Geometrie (1899, Leipzig: Teubner; Foundations of geometry (trans.: Townsend, E.). La Salle, Illinois: Open Court, 1959.). We will argue that Bad Company-style worries, and our concomitant epistemic issue, also affects this conception and other foundationalist approaches. In the following sections, we then offer various ways to address our epistemic concern, arguing, in the end, that none resolves the issue. The final section offers our own resolution which, however, runs against the foundationalist spirit of the Scottish neo-logicist program.
|Keywords||Abstraction principles Neo-Logicism Frege Hilbert Foundationalism Implicit definitions Basic knowledge|
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References found in this work BETA
Graham Priest (2006). In Contradiction: A Study of the Transconsistent. Oxford University Press.
Crispin Wright (2004). Warrant for Nothing (and Foundations for Free)? Aristotelian Society Supplementary Volume 78 (1):167–212.
Stewart Shapiro (1997). Philosophy of Mathematics: Structure and Ontology. Oxford University Press.
Bob Hale (ed.) (2001). The Reason's Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics. Oxford University Press.
George Boolos (1998). Logic, Logic, and Logic. Harvard University Press.
Citations of this work BETA
Thomas Donaldson (2015). Platitudes in Mathematics. Synthese 192 (6):1799-1820.
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