[Show abstract][Hide abstract] ABSTRACT: Mathematics is a system for representing and reasoning about quantities, with arithmetic as its foundation. Its deep interest for our understanding of the psychological foundations of scientific thought comes from what Eugene Wigner called the unreasonable efficacy of mathematics in the natural sciences. From a formalist perspective, arithmetic is a symbolic game, like tic-tac-toe. Its rules are more complicated, but not a great deal more complicated. Mathematics is the study of the properties of this game and of the systems that may be constructed on the foundation that it provides. Why should this symbolic game be so powerful and resourceful when it comes to building models of the physical world? And on what psychological foundations does the human mastery of this game rest? The first question is metaphysical—why is the world the way it is? We do not treat it, because it lies beyond the realm of experimental behavioral science. We review the answers to the second question that experimental research on human and non-human animal cognition suggests. The general nature of the answer is that the foundations of mathematical cognition appear does not lie in language and the language faculty. The ability to estimate quantities and to reason arithmetically with those estimates exists in the brains of animals that have no language. The same or very similar non-verbal mechanisms appear to operate in parallel with verbal estimation and reasoning in adult humans. And, they operate to some extent before children learn to speak and before they have had any tutoring in the elements of arithmetic. These findings suggest that the verbal expression of number and of arithmetic thinking is based on a non-verbal system for estimating and reasoning about discrete and continuous quantity, which we share with many non-verbal animals. A reasonable supposition is that the neural substrate for this system arose far back in the evolution of brains precisely because of the puzzle that Wigner called attention to: arithmetic reasoning captures deeply important properties of the world, which the animal brain must represent in order to act effectively in it.