Quantifiers, Quantifiers, and Quantifiers: Themes in Logic, Metaphysics, and Language pp 191-219 | Cite as

# Chalmers, Quantifier Variance and Mathematicians’ Freedom

## Abstract

Philosophers of mathematics have been much struck by mathematicians’ apparent freedom to introduce new kinds of mathematical objects, such as complex numbers, sets and the objects and arrows of category theory. In this paper, I explore a way of using recent work on quantifier variance to explain this apparent freedom to introduce theories about new kinds of mathematical objects. In Ontological Antirealism, David Chalmers sketches a method for describing a class of alternative quantifier senses which are more ontologically profligate than our own using appeals to set theoretic models. I suggest a modification of this method which frees it of certain arbitrary limitations on size, by replacing appeals to set theory with appeals to an (independently motivated) notion of broadly logical possibility. Once amended in this way, Chalmers’ technique allows us to flesh out a Neo-Carnapian explanation for mathematicians’ freedom to introduce new kinds of mathematical objects which avoids some major problems for existing accounts.

## Keywords

Truth Condition Mathematical Object Relation Symbol Logical Possibility Impossible World## References

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