Séminaire Philmath

Resumé : There is a popular sentiment among logicians and mathematicians that Zilber's conjecture provides a sense in which canonical parts of mathematics can be recovered from logical resources alone. This suggests that a fruitful philosophical analysis of Zilber's conjecture can be given in terms of logicism, or the view that mathematics (or at least some non-trivial parts of it) is reducible to logic. The purpose of this paper is to provide such an analysis by articulating a model-theoretic approach to logicism, which consists of (1) a model-theoretic account of reduction and (2) a model-theoretic account of logicality. I then identify some of the philosophical challenges facing this model-theoretic approach to logicism and offer a preliminary defense. Even though Zilber's conjecture in its original form is false, this approach to logicism remains worthy of philosophical consideration because it provides us with a novel conception of logicality and reduction that is continuous with contemporary mathematical practice. Moreover, this approach offers an interpretive framework for appreciating the philosophical significance of Zilber's conjecture in the contexts in which it does hold.