 To illustrate each type, consider a domain where differences between these three lines of inquiry are especially clear: arithmetic. A descriptive theory of arithmetic might concern the psychology of arithmetical reasoning, a model of approximate numeracy in animals, or an algorithm for implementing arbitrary-precision arithmetic on a digital computer. The normative standard of full arithmetic is Peano’s axiomatization of arithmetic, which distills natural number arithmetic down to a function for one number succeeding another and mathematical induction. But one might also consider Robinson’s induction-free fragment of Peano arithmetic (Tarski, Mostowski, &amp; Robinson 1953) or axioms for some system of cardinal arithmetic in the hierarchy for large cardinals. A prescriptive theory for arithmetic will reference both a fixed normative standard and relevant facts about the arithmetical capabilities of the organism or machine performing arithmetic. A curriculum for improving the arithmetical performance of elementary school children will differ from one designed to improve the performance of adults. Even though the normative standard of Peano arithmetic is the same for both children and adults, stable psychological differences in these two populations may warrant prescribing different approaches for improving their arithmetic. Continuing, even though Peano’s axioms are the normative standard for full arithmetic, nobody would prescribe Peano’s axioms for the purpose of improving anyone’s sums. There is no mistaking Peano’s axioms for a descriptive theory of arithmetical reasoning, either. Even so, a descriptive theory of arithmetic will presuppose the Peano axioms as the normative standard for full arithmetic, even if only implicitly. In describing how people sum two numbers, after all, one presumes that they are attempting to sum two numbers rather than concatenate them, count out in sequence, or send a message in code.