Quantity, Number, and Mathematics

Mathematics is one of the most sophisticated, precise, and abstract conceptual edifices humankind has ever created, supporting science, technology, economics, and deep philosophical investigations. Yet, mathematics is a remarkably recent invention in the history of Homo sapiens. What made it possible? One fundamental building block is “number,” whose simplest forms, conventional wisdom holds, are said to be biologically endowed and shared by many species. What is shared, however, are general mechanisms for perception of quantities, not number. Number —i.e., exact symbolic quantification, as in our familiar “seven” or “8”— is complex and abstract, and although ubiquitous in the industrialized world, is far from being present in all human cultures. While there are biologically evolved preconditions for quantity discrimination shared by many species, the presence of number (and arithmetic) proper is a cultural, not a biological trait. Exact quantification was brought to being via relatively recent (only thousands of years) cultural preoccupations and practices that were crucially supported by language-mediated cognitive tools (e.g., symbolic reference, analogical/metaphorical reasoning) and material technology (e.g., writing, abacus)— essential dimensions that lie largely outside natural selection. Language thus is a necessary—but not a sufficient— condition for number. I’ll briefly review some relevant empirical data and illustrate the power of cognitive tools with the passage from “number” to the “number line.” This 17th century move (which required conceiving numbers metaphorically as locations in space) quickly led to analytic geometry, infinitesimal calculus, differential equations, and more. The path from quantity perception—shared by many species— to number, to mathematics is a story of human “biological enculturation.”