Abstract: Mathematical methods and results are widely used in contemporary philosophy, playing crucial roles in many philosophical arguments. One obvious kind of mathematical involvement is representational, when an argument employs mathematical language to represent philosophically salient notions: modelling preferences as linear orders, utilities as real numbers, or possible worlds as assignments of truth values to atomic propositions. However, there is a second kind of mathematical involvement which some philosophical arguments exhibit, namely reliance on substantial mathematical theorems as implicit or explicit premises. Examples of this phenomenon include Dutch book theorems in arguments for epistemic norms, completeness theorems in squeezing arguments for the adequacy of consequence relations, and impossibility theorems in arguments concerning the possibility of democratic social decision-making.
The aim of this talk is to explore this phenomenon using a framework I will refer to as ‘reverse philosophy’. By formalising the mathematical premises of philosophical arguments in second-order arithmetic, we can use the proof-theoretic hierarchy studied in reverse mathematics to measure their strength. One result of this analysis is to show that although many paradigmatic applications of mathematics in philosophy do not rely on higher set theory, they do imply the existence of non-computable sets. This is significant because it focuses attention away from traditional concerns about the cardinality (or more generally the ontology) of the sets which must exist in order to sustain a given argument, and towards concerns about their computational or descriptive complexity. Moreover, because this analysis can provide quite fine-grained information in terms of hierarchies familiar from computability theory such as the Turing degrees or the arithmetical hierarchy, we can then use these results to assess the consequences of mathematical involvement for debates in which the argument is embedded—for example by making explicit suppressed premises, highlighting idealisations, or suggesting novel avenues of reply.
Please note: A notice will be sent out by the convenors for each talk with the Zoom link.
Philosophy of Mathematics Seminar Convenors: Daniel Isaacson and Beau Mount
