Numerical cognition and mathematical realism

Clarke-Doane (2012) has developed an evolutionary debunking argument specifically aimed at mathematical realism. It relies on intuitions elicited by the following toy example: Suppose that a lion is hiding behind bush A, and a second lion is hiding behind bush B. Human ancestor P believes that 1 + 1 = 2 and flees. Human ancestor Q believes that 1 + 1 = 0 and stays. Prima facie, we can argue that because 1 + 1 = 2, ancestor P has an advantage over Q. This arithmetical truth figures in the evolutionary explanation, therefore, our belief that 1 + 1 = 2 tracks a mathematical truth. However, Clarke-Doane (2012) proceeds with a counterfactual scenario, imagining a world where 1 + 1 really equals 0. Realistically construed, 1 + 1 = 0 speaks about numbers. Now suppose we hold the first-order logical truths constant, but change the mathematical truth to 1 + 1 = 0. In that case, ancestor Q, although she now has the correct mathematical belief, would still get eaten, as there would still be one lion and another lion waiting for her. Accordingly, numerical truths do not play a relevant role in this evolutionary scenario — if the numerical truths had been different, but the first-order properties remained the same, it would have been more adaptively advantageous to believe that 1 + 1 = 2. In the next section, I examine the psychological literature on numerical cognition to assess whether premise 1 holds for numbers. In section 4, I scrutinize Clarke-Doane’s argument in more detail. I argue that the current cognitive scientific literature on numerical thinking gives us no good reasons to believe premise 2 is true, which opens the possibility of a realist understanding of this literature. 0