**Mathematical counterfactuals**

Daniel Lassiter

(Stanford)

If 2 weren’t prime, what would the smallest prime be? If 3 + 3 were 7, what would 3 + 4 be? People who have access to the requisite mathematical concepts seem to have intuitions about the answers to these questions. These intuitions can’t be generated by reasoning about the closest possible world(s) where the antecedent is true–as in the Stalnaker/Lewis semantics–since there aren’t any. I suggest that we can make sense of people’s intuitions about such counterfactuals if we adopt a procedural model of counterfactual reasoning. On this approach, we reason about counterfactuals by intervening on the procedure used to construct instances of the scenario described by consequent, where the intervention is designed to ensure that the mutated scenario will verify the antecedent. This approach generalizes Pearl’s (2000) approach to counterfactuals based on interventions on causal models, and makes an intriguing prediction in the case of impossible antecedents: since there are generally many procedures that could generate the same mathematical object, judgments about mathematical counterfactuals may vary depending on the procedure that a given evaluator is using. Furthermore, since there is no uniquely correct procedure, there is no way to adjudicate between these diverging judgments. If this is correct, then at least some mathematical counterfactuals are irremediably subjective.