The Innateness Hypothesis and Mathematical Concepts

In historical claims for nativism, mathematics is a paradigmatic example of innate knowledge. Claims by contemporary developmental psychologists of elementary mathematical skills in human infants are a legacy of this. However, the connection between these skills and more formal mathematical concepts and methods remains unclear. This paper assesses the current debates surrounding nativism and mathematical knowledge by teasing them apart into two distinct claims. First, in what way does the experimental evidence from infants, nonhuman animals and neuropsychology support the nativist hypothesis? Second, granting that infants have some elementary mathematical skills, does this mean that such skills play an important role in the development of mathematical knowledge? 1. Nativism and mathematical knowledge Until the late 18th century, mathematical knowledge was a paradigmatic example of innate knowledge. It often served a dual purpose, providing both a persuasive reason why innate knowledge exists, and furnishing us with an explanation of how humans can have knowledge of mathematical objects. Due to the increasing importance of foundational issues and the explicit rejection of psychologism (i.e., the view that mathematics lies within the purview of psychology) by Frege and others, nativism has lost its importance in recent philosophy of mathematics. By contrast, contemporary developmental psychologists (e.g., Spelke, Kinzler, 2007; Carey, 2004; Feigenson et al., 2004) propose that innate ideas are a naturalistic source of mathematical knowledge. They have uncovered evidence for earlydeveloped mathematical skills in infants and young children, such as the ability to estimate and discriminate numerosities (Jordan, Brannon, 2006), the capacity to detect ordinal relationships between numerosities (Brannon, 2002), and the ability to reason about spatial relationships in a Euclidean framework (see De Cruz, 2009, for a review). This article examines the relevance of nativism for the philosophy of mathematics. After providing a brief overview of historical claims for innate mathematical knowledge, we focus on arithmetic as an example of innate mathematical knowledge in current developmental psychology. We examine in detail the claim that infants are capable of predicting the outcomes of simple