David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Philosophical Quarterly 58 (230):59-79 (2008)
This paper has two goals. The ﬁrst goal is to show that the structuralists’ claims about dependence are more signiﬁcant to their view than is generally recognized. I argue that these dependence claims play an essential role in the most interesting and plausible characterization of this brand of structuralism. The second goal is to defend a compromise view concerning the dependence relations that obtain between mathematical objects. Two extreme views have tended to dominate the debate, namely the view that all mathematical objects depend on the structures to which they belong and the view that none do. I present counterexamples to each of these extreme views. I defend instead a compromise view according to which the structuralists are right about many kinds of mathematical objects (roughly, the algebraic ones), whereas the anti-structuralists are right about others (in particular, the sets). I end with some remarks about how to understand the crucial notion of dependence, which despite being at the heart of the debate is rarely examined in any detail.
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References found in this work BETA
George Boolos (1971). The Iterative Conception of Set. Journal of Philosophy 68 (8):215-231.
John P. Burgess (1999). Book Review: Stewart Shapiro. Philosophy of Mathematics: Structure and Ontology. [REVIEW] Notre Dame Journal of Formal Logic 40 (2):283-291.
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Citations of this work BETA
Julian C. Cole (2010). Mathematical Structuralism Today. Philosophy Compass 5 (8):689-699.
Luca Incurvati (2012). How to Be a Minimalist About Sets. Philosophical Studies 159 (1):69-87.
Uri Nodelman & Edward N. Zalta (2014). Foundations for Mathematical Structuralism. Mind 123 (489):39-78.
Steven French & Michela Massimi (2013). Philosophy of Science A Personal Peek Into the Future. Metaphilosophy 44 (3):230-240.
Georg Schiemer (2014). Invariants and Mathematical Structuralism. Philosophia Mathematica 22 (1):70-107.
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