This first of a two-volume book discusses the process of conceptual recasting of mathematical contents under a new and more convenient form, complying with the ideal of epistemic economy (expressing the same content by appealing to fewer or minimal conceptual resources). It accounts for this ideal and discusses its philosophical significance, also in opposition to the most frequently considered ideal of ontological parsimony, in the context of an original approach to mathematical knowledge. If it is admitted that mathematical objects have no external existence, the former ideal is much more sensible and more plausible to be pursued, despite lacking any clear and specifically dedicated account. To illustrate it, this first volume considers many examples of second-order definitions of natural numbers, by comparing Peano Arithmetic with Frege one, and showing the greater epistemic economy of the latter over the former. This ensures the book is of interest to scholars and students in the philosophy of mathematics and logic.