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Philosophy of Mathematics

Geoffrey Hellman

Pages 412 - 424

There is a surprising variety of programs in the philosophy and foundations of mathematics that have found mereology a useful and, in some cases, an indispensable tool. After emphasising a number of key relevant features of mereology, we will briefly examine five such programs, including (1) Goodman and Quine’s (1947) efforts to recover the syntax of mathematical language as part of a finitist, formalist philosophy of mathematics; (2) Tarski (1929; 1956), Menger (1940), et al.’s programs to reconstruct geometry and topology from regions of a space, introducing points and any other objects of lower dimensionality only by definitions in terms of regions and relations thereof; (3) Field’s (1980) and Burgess’ (1984) ‘synthetic mechanics’ as an effort to recover nominalistically certain applications of mathematics in physics inspired by synthetic geometry; (4) Lewis’ (1991) attempt to ground set theory on a combination of mereology and plural logic, which he called ‘megethology’ (theory of size); (5) Hellman’s (1989; 1996)) modalstructuralism employing the same machinery as (4) together with modal logic, but to provide an eliminative structuralist alternative to platonist, face-value readings of abstract mathematical theories.

1Department of Philosophy, University of Minnesota

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8 Hellman, G., (1996), “Structuralism without Structures”, Philosophia Mathematica 4: 100-123.

9 Hellman, G., (2005), “Structuralism”, Ch. 17 in Shapiro, S. (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic, Oxford: Oxford University Press, 536-562.

10 Hellman, G.; Shapiro, S., (2012), “Towards a Point-free Account of the Continuous”, Iyyun: The Jerusalem Philosophical Quarterly 61: 263-287.

11 Hellman, G.; Shapiro, S., (2013), “The Classical Continuum without Points”, Review of Symbolic Logic. 6 (3): 488-512.

12 Lewis, D., (1991), Parts of Classes, Oxford: Blackwell.

13 Lewis, D., (1993), “Mathematics Is Megethology”, Philosophia Mathematica 1: 3-23.

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