Philosophical Uses of Categoricity Arguments

by Penelope Maddy, Jouko Väänänen

This Element addresses the viability of categoricity arguments in philosophy by focusing with some care on the specific conclusions that a sampling of prominent figures have attempted to draw – the same theorem might successfully support one such conclusion while failing to support another. It begins with Dedekind, Zermelo, and Kreisel, casting doubt on received readings of the latter two and highlighting the success of all three in achieving what are argued to be their actual goals. These earlier uses of categoricity arguments are then compared and contrasted with more recent work of Parsons and the co-authors Button and Walsh. Highlighting the roles of first- and second-order theorems, of external and internal theorems, the Element concludes that categoricity arguments have been more effective in historical cases that reflect philosophically on internal mathematical matters than in recent questions of pre-theoretic metaphysics.

62 pages, Kindle Edition

Published December 21, 2023

About the author

Penelope Maddy is a UCI Distinguished Professor of Logic and Philosophy of Science and of Mathematics at the University of California, Irvine. She is well-known for her influential work in the philosophy of mathematics where she has worked on realism and naturalism.
Maddy received her Ph.D. from Princeton University in 1979.
Her early work, culminating in Realism in Mathematics, tried to defend Kurt Gödel's position that mathematics is a true description of a mind-independent realm that we can access through our intuition. However, she suggested that some mathematical entities are in fact concrete, unlike, notably, Gödel, who assumed all mathematical objects are abstract. She suggested that sets can be causally efficacious, and in fact share all the causal and spatiotemporal properties of their elements. Thus, when I see the three cups on the table in front of me, I also see the set as well. She used recent work in cognitive science and psychology to support this position, pointing out that just as at a certain age we begin to see objects rather than mere sense perceptions, there is also a certain age at which we begin to see sets rather than just objects.
In the 1990s, she moved away from this position, towards a position described in Naturalism in Mathematics. Her "naturalist" position, like Quine's, suggests that since science is our most successful project so far for knowing about the world, philosophers should adopt the methods of science in their own discipline, and especially when discussing science. However, rather than a unified picture of the sciences like Quine's, she has a picture on which mathematics is separate. This way, mathematics is neither supported nor undermined by the needs and goals of science, but is allowed to obey its own criteria. This means that traditional metaphysical and epistemological concerns of the philosophy of mathematics are misplaced. Like Wittgenstein, she suggests that many of these puzzles arise merely because of the application of language outside its proper domain of significance.
Throughout her career, she has been dedicated to understanding and explaining the methods that set theorists use in agreeing on axioms, especially those that go beyond ZFC.

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