The Psychological Scaffolding of Arithmetic

Where does arithmetic come from, and why are addition and multiplication its fundamental operations? Although we know that arithmetic is true, no explanation that meets standards of scientific rigor is available from philosophy, mathematical logic, or the cognitive sciences. We propose a new approach based on the assumption that arithmetic has a biological origin: Many examples of adaptive behavior such as spatial navigation suggest that organisms can perform arithmetic-like operations on represented magnitudes. If so, these operations-nonsymbolic precursors of addition and multiplication-might be optimal due to evolution and thus identifiable according to an appropriate criterion. We frame this as a metamathematical question, and using an order-theoretic criterion, prove that four qualitative conditions-monotonicity, convexity, continuity, and isomorphism-are sufficient to identify addition and multiplication over the real numbers uniquely from the uncountably infinite class of possible operations. Our results show that numbers and algebraic structure emerge from purely qualitative conditions, and as a construction of arithmetic, provide a rigorous explanation for why addition and multiplication are its fundamental operations. We argue that these conditions are preverbal psychological intuitions or principles of perceptual organization that are biologically based and shape how humans and nonhumans alike perceive the world. This is a Kantian view and suggests that arithmetic need not be regarded as an immutable truth of the universe but rather as a natural consequence of our perception. Algebraic structure may be inherent in the representations of the world formed by our perceptual system.