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FOR BETTER AND FOR WORSE. ABSTRACTIONISM, GOOD COMPANY, AND PLURALISM

Part of: General and miscellaneous specific topics Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  22 March 2021

ANDREA SERENI ,
MARIA PAOLA SFORZA FOGLIANI  and
LUCA ZANETTI
ANDREA SERENI*
Affiliation:
SCUOLA UNIVERSITARIA SUPERIORE IUSS PAVIA DEPARTMENT OF HUMANITIES AND LIFE SCIENCES NEtS CENTER, PAVIA, ITALY E-mail: mariapaola.sforzafogliani@iusspavia.it E-mail: luca.zanetti@iusspavia.it
MARIA PAOLA SFORZA FOGLIANI
Affiliation:
SCUOLA UNIVERSITARIA SUPERIORE IUSS PAVIA DEPARTMENT OF HUMANITIES AND LIFE SCIENCES NEtS CENTER, PAVIA, ITALY E-mail: mariapaola.sforzafogliani@iusspavia.it E-mail: luca.zanetti@iusspavia.it
LUCA ZANETTI
Affiliation:
SCUOLA UNIVERSITARIA SUPERIORE IUSS PAVIA DEPARTMENT OF HUMANITIES AND LIFE SCIENCES NEtS CENTER, PAVIA, ITALY E-mail: mariapaola.sforzafogliani@iusspavia.it E-mail: luca.zanetti@iusspavia.it
*
E-mail: andrea.sereni@iusspavia.it
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Abstract

A thriving literature has developed over logical and mathematical pluralism – i.e. the views that several rival logical and mathematical theories can be equally correct. These have unfortunately grown separate; instead, they both could gain a great deal by a closer interaction. Our aim is thus to present some novel forms of abstractionist mathematical pluralism which can be modeled on parallel ways of substantiating logical pluralism (also in connection with logical anti-exceptionalism). To do this, we start by discussing the Good Company Problem for neo-logicists recently raised by Paolo Mancosu (2016), concerning the existence of rival abstractive definitions of cardinal number which are nonetheless equally able to reconstruct Peano Arithmetic. We survey Mancosu’s envisaged possible replies to this predicament, and suggest as a further path the adoption of some form of mathematical pluralism concerning abstraction principles. We then explore three possible ways of substantiating such pluralism—Conceptual Pluralism, Domain Pluralism, Pluralism about Criteria—showing how each of them can be related to analogous proposals in the philosophy of logic. We conclude by considering advantages, concerns, and theoretical ramifications for these varieties of mathematical pluralism.

Keywords

definitions by abstractionneo-logicismGood Company Problemlogical pluralismmathematical pluralism

MSC classification

Primary: 00A30: Philosophy of mathematics
Secondary: 03A05: Philosophical and critical
Type
Research Article
Information
The Review of Symbolic Logic , Volume 16 , Issue 1 , March 2023 , pp. 268 - 297
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

BIBLIOGRAPHY

Balaguer, M. (1998a). Non-uniqueness as a non-problem. Philosophia Mathematica, 6(1), 63–84.CrossRefGoogle Scholar
Balaguer, M. (1998b). Platonism and Anti-Platonism in Mathematics. Oxford: Oxford University Press.Google Scholar
Balaguer, M. (2017). Mathematical pluralism and platonism. Journal of Indian Council of Philosophical Research, 34(2), 379–398.CrossRefGoogle Scholar
Beall, J. C. & Restall, G. (2000). Logical pluralism. Australasian Journal of Philosophy, 78(4), 475–493.CrossRefGoogle Scholar
Beall, J. C. & Restall, G. (2001). Defending logical pluralism. In Woods, J. and Brown, B., editors. Logical Consequence: Rival Approaches. Proceedings of the 1999 Conference of the Society of Exact Philosophy. Stanmore: Hermes, pp. 1–22.Google Scholar
Beall, J. C. & Restall, G. (2006). Logical Pluralism. Oxford: Oxford University Press.Google Scholar
Benacerraf, P. (1965). What numbers could not be. Philosophical Review, 74(1), 47–73.CrossRefGoogle Scholar
Black, R. (2000). Nothing matters too much, or Wright is wrong. Analysis, 60(3), 229–237.CrossRefGoogle Scholar
Boolos, G. (1990). The standard of equality of numbers. In Meaning and Method: Essays in Honor of Hilary Putnam. Cambridge: Cambridge University Press, pp. 261–277.Google Scholar
Black, R. (1997). Is Hume’s Principle analytic? In Heck, R. Jr., editor. Language, Thought and Logic. Oxford: Clarendon Press, pp. 245–261.Google Scholar
Carnap, R. (1928). Der Logische Aufbau der Welt. Hamburg: Meiner Verlag.Google Scholar
Cook, R. T. (2012). Conservativeness, stability, and abstraction. British Journal for the Philosophy of Science, 63(3), 673–696.CrossRefGoogle Scholar
Cook, R. T. & Ebert, P. A. (2005). Abstraction and identity. Dialectica, 59(2), 121–139.CrossRefGoogle Scholar
Cook, R. T. & Linnebo, Ø. (2018). Cardinality and acceptable abstraction. Notre Dame Journal of Formal Logic, 59(1), 61–74.CrossRefGoogle Scholar
Costa, N. D. & Becker Arenhart, J. R. (2018). Full-blooded anti-exceptionalism about logic. Australasian Journal of Logic, 15(2), 362–380.CrossRefGoogle Scholar
Dummett, M. (1991). Frege. Philosophy of Mathematics. Cambridge, MA: Harvard University Press.Google Scholar
Dummett, M. (1998). Neo-Fregeans: in bad company? In Schirn, M., editor. The Philosophy of Mathematics Today. Oxford: Clarendon Press.Google Scholar
Ebels-Duggan, S. C. (2015). The Nuisance Principle in infinite settings. Thought: A Journal of Philosophy, 4(4), 263–268.CrossRefGoogle Scholar
Fine, K. (2002). The Limits of Abstraction. Oxford: Oxford University Press.Google Scholar
Friend, M. (2014). Pluralism in Mathematics: A New Position in Philosophy of Mathematics. Dordrecht: Springer.Google Scholar
Hale, B. (2018). Review of Paolo Mancosu: Abstraction and Infinity . Journal of Philosophy, 115(3), 158–166.CrossRefGoogle Scholar
Hale, B. & Wright, C. (2001). Reason’s Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics. Oxford: Oxford University Press.CrossRefGoogle Scholar
Hale, B. & Wright, C. (2009). Focus restored: comments on John MacFarlane. Synthese, 170(3), 457–482.CrossRefGoogle Scholar
Heck, R. K. (1997). Finitude and Hume’s Principle. Journal of Philosophical Logic, 26(6), 589–617.CrossRefGoogle Scholar
Heck, R. K. (2011). Frege’s Theorem. Oxford: Oxford University Press.Google Scholar
Hjortland, O. (2017). Anti-exceptionalism about logic. Philosophical Studies, 174(3), 631–658.CrossRefGoogle Scholar
Lakatos, I. (1976). A renaissance of empiricism in the recent philosophy of mathematics. British Journal for the Philosophy of Science, 27(3), 201–223.CrossRefGoogle Scholar
Linnebo, Ø., editor. (2009). The Bad Company problem. Synthese, 170(Special Issue 3).CrossRefGoogle Scholar
Linnebo, Ø. (2011). Some criteria for acceptable abstraction. Notre Dame Journal of Formal Logic, 52(3), 331–338.CrossRefGoogle Scholar
Linnebo, Ø. & Uzquiano, G. (2009). Which abstraction principles are acceptable? Some limitative results. British Journal for the Philosophy of Science, 60(2), 239–252.CrossRefGoogle Scholar
MacBride, F. (2000). On Finite Hume. Philosophia Mathematica, 8(2), 150–159.CrossRefGoogle Scholar
MacBride, F. (2002). Could nothing matter? Analysis, 62(2), 125–135.CrossRefGoogle Scholar
MacFarlane, J. (2000). What Does it Mean to Say that Logic is Formal? PhD Thesis, University of Pittsburgh.Google Scholar
Maddy, P. (2002). A naturalistic look at logic. Proceedings and Addresses of the American Philosophical Association, 76(2), 61–90.CrossRefGoogle Scholar
Mancosu, P. (2016). Abstraction and Infinity. Oxford: Oxford University Press.CrossRefGoogle Scholar
Priest, G. (1979). The logic of paradox. Journal of Philosophical Logic, 8(1), 219–241.CrossRefGoogle Scholar
Priest, G. (2013). Mathematical pluralism. Logic Journal of the IGPL, 21(1), 4–13.CrossRefGoogle Scholar
Priest, G. (2014). Revising logic. In Rush, P., editor. The Metaphysics of Logic. Cambridge: Cambridge University Press, pp. 211–223.CrossRefGoogle Scholar
Priest, G. (2016). Logical disputes and the a priori. Logique et Analyse, 59(236), pp. 29–57.Google Scholar
Quine, W. V. O. (1951). Two dogmas of empiricism. Philosophical Review, 60(1), 20–43.CrossRefGoogle Scholar
Quine, W. V. O. (1969). Epistemology naturalized. In Ontological Relativity and Other Essays. New York: Columbia University Press.CrossRefGoogle Scholar
Rabin, G. (2007). Full-blooded reference. Philosophia Mathematica, 15(3), 357–365.CrossRefGoogle Scholar
Rayo, A. (2013). The Construction of Logical Space. Oxford: Oxford University Press.CrossRefGoogle Scholar
Read, S. (2019). Anti-exceptionalism about logic. Australasian Journal of Logic, 16(7), 298–318.CrossRefGoogle Scholar
Russell, G. (2015). The justification of the basic laws of logic. Journal of Philosophical Logic, 44(6), 793–803.CrossRefGoogle Scholar
Sereni, A. & Sforza Fogliani, M. P. (2020). How to water a thousand flowers. On the logic of logical pluralism. Inquiry, 63(3–4), 347–370, first published online 2017.CrossRefGoogle Scholar
Shapiro, S. (2011). The company kept by Cut Abstraction (and its relatives). Philosophia Mathematica, 19(2), 107–138.CrossRefGoogle Scholar
Shapiro, S. (2014). Varieties of Logic. Oxford: Oxford University Press.CrossRefGoogle Scholar
Shapiro, S. & Weir, A. (1999). New V, ZF and abstraction. Philosophia Mathematica, 7(3), 293–321.CrossRefGoogle Scholar
Studd, J. P. (2016). Abstraction reconceived. British Journal for the Philosophy of Science, 67(2), 579–615.CrossRefGoogle Scholar
Uzquiano, G. (2009). Bad Company generalized. Synthese, 170(3), 331–347.CrossRefGoogle Scholar
Varzi, A. C. (2002). On logical relativity. Philosophical Issues, 12(1), 197–219.CrossRefGoogle Scholar
Weir, A. (2003). Neo-Fregeanism: an embarrassment of riches. Notre Dame Journal of Formal Logic, 44(1), 13–48.CrossRefGoogle Scholar
Williamson, T. (2017). Semantic paradoxes and abductive methodology. In Armour-Garb, B., editor. Reflections on the Liar. Oxford: Oxford University Press, pp. 325–346.Google Scholar
Woods, J. (2019). Logical partisanhood. Philosophical Studies, 176(5), 1203–1224.CrossRefGoogle Scholar
Wright, C. (1983). Frege’s Conception of Numbers as Objects. Aberdeen: Aberdeen University Press.Google Scholar
Wright, C. (1997). On the philosophical significance of Frege’s Theorem. In Heck, R. K., editor. Language, Thought, and Logic: Essays in Honour of Michael Dummett. Oxford: Oxford University Press, pp. 201–244. Reprinted in Hale & Wright (2001), pp. 272–306.Google Scholar
Wright, C. (1998). On the harmless impredicativity of N= (‘Hume’s Principle’). In Schirn, M., editor. The Philosophy of Mathematics Today. Oxford: Clarendon Press, pp. 339–68. Reprinted in Hale & Wright (2001), pp. 229–255.Google Scholar
Wright, C. (1999). Is Hume’s Principle analytic? Notre Dame Journal of Formal Logic, 40(1), 6–30. Reprinted in Hale & Wright (2001), pp. 307–332.CrossRefGoogle Scholar
Wright, C. (2000). Neo-Fregean foundations for real analysis: some reflections on Frege’s Constraint. Notre Dame Journal of Formal Logic, 41(4), 317–334.CrossRefGoogle Scholar
Wyatt, N. & Payette, G. (2018). How do logics explain? Australasian Journal of Philosophy, 96(1), 157–167.Google Scholar