<abstract> The cultural history of the real numbers began with the positive integers. Kronecker is often quoted as saying, "God made the integers; all else is the work of man," by which he meant that the system of real numbers had been erected by mathematicians on the intuitively obvious foundation provided by the integers. Taken as a statement about the cultural history of mathematics, this is beyond dispute. But if this is taken as a claim about the psychological foundations of arithmetic reasoning, then we suggest that here, as in many other areas of psychology, introspection and intuition are poor guides to the inner workings of the mind.
We suggest that it is the system of real numbers that is the psychologically primitive system, both in the phylogenetic and the ontogenetic sense. We review evidence that a system for arithmetic reasoning with real numbers evolved before language evolved. When language evolved, it picked out from the real numbers only the integers, thereby making the integers the foundation of the cultural history of the number. Secondly, we suggest that this ancestral non-verbal real number system becomes operative in the prelinguistic child and makes possible the acquisition of language- mediated counting and language-mediated arithmetic reasoning. It is the foundation on which an individual?s language-mediated understanding of what numbers are and what may be done with them rests. </abstract>