Abstract
A number of nominalist
approaches to mathematics have been developed: Field’s [1980] conservative
nominalism, Hellman’s [1989] modal-structuralism, Lewis’ [1990] megethology
(under a certain interpretation), and Azzouni’s [2004] deflationary nominalism.
In this paper, Bueno argues that all of these approaches face a serious
problem: they fail to provide an adequate nominalization strategy for set
theory. Field’s approach ultimately presupposes such a nominalization, but B.
argues that it is unable to provide one. It is part of Hellman’s approach to
offer tentatively a nominalist reformulation of set theory. However, the
approach faces difficulties with the interpretation of the second-order
quantifier and the modal operators introduced. Lewis’ proposal - under a
certain interpretation - also attempts to provide a nominalization of set
theory. However, if the nominalization were to be carried out, it would make
set theory contingent upon the size of reality, which conflicts with the way in
which the theory is typically conceived. Finally, Azzouni’s proposal faces some
tensions that emerge from the alleged ontological independence of sets. I
conclude that set theory becomes the crucial area in which nominalists should
concentrate their efforts if nominalism is to become a plausible account. I
then sketch a fictionalist alternative that does not face the difficulties
above.
About the Author
Otávio Bueno is Professor of Philosophy at the University of Miami.
He has held visiting professorships or fellowships at Princeton
University, University
of York (in the UK), University
of Leeds, and at the University of São Paulo.
His research concentrates in philosophy of science, philosophy of mathematics,
and philosophy of logic. He has published widely in journals and collections,
including: Noûs, Mind, Philosophy of Science,
Synthese, Journal of Philosophical Logic, Studies
in History and Philosophy of Science, British
Journal for the Philosophy of Science, Analysis,
Erkenntnis, History and Philosophy of Logic, and Logique et Analyse. He is the author of two books, Constructive Empiricism: A Restatement and
Defense (CLE, 1999), and Elements of Paraconsistent Set Theory (CLE, 1998; with Newton da Costa and Jean-Yves
Béziau). And he edited New Waves in
Philosophy of Mathematics (Palgrave-Macmillan, 2009; with Øystein Linnebo).
e-mail: otaviobueno@mac.com
http://web.me.com/otaviobuno/Site/Otavio_Bueno.html
|
Philosophia Contributions to Philosophy 
