Frege’s platonism and mathematical creation: some new perspectives

Abstract In this three-part essay, I investigate Frege’s platonist and anti-creationist position in Grundgesetze der Arithmetik and to some extent also in Die Grundlagen der Arithmetik. In Sect. 1.1, I analyze his arithmetical and logical platonism in Grundgesetze. I argue that the reference-fixing strategy for value-range names—and indirectly also for numerical singular terms—that Frege pursues in Grundgesetze I gives rise to a conflict with the supposed mind- and language-independent existence of numbers and logical objects in general. In Sect. 1.2 and 1.3, I discuss the non-creativity of Frege’s definitions in Grundgesetze and the case of what I call weakly creative definitions. In Part II of this essay, I first deal with Stolz’s and Dedekind’s (intended) creation of numbers. In what follows, I focus on Grundgesetze II, §146, where Frege considers a potential creationist charge in relation to the stipulation that he makes in Grundgesetze I, §3 with the purpose of partially fixing the references of value-range names. I place equal emphasis on the related twin stipulations that he makes in Grundgesetze I, §10. In §10, Frege identifies the truth-values with their unit classes in order to fix the references of value-range names (almost) completely. He does so in a piecemeal fashion. Although in Grundgesetze II, §146 Frege refers also to Grundgesetze I, §9 and §10 in this connection, he does not explain why he thinks that the transsortal identifications in §10 and also the stipulation that he makes in §9 regarding the value-range notation may give rise to a creationist charge in addition to or in connection with the stipulation in §3, and if so, how he would have responded to it. The two main issues that I discuss in Part II are: (a) Has Frege created value-ranges in general in Grundgesetze I, §3? (b) Has he created the unit classes of the True and the False in §10? In Part III, I discuss, inter alia, the question of whether in developing the whole wealth of objects and functions that arithmetic deals with from the primitive functions of Grundgesetze by applying the formation rules Frege creates special value-ranges and special functions. This procedure is fundamentally different from the reference-fixing strategy regarding value-range names that Frege pursues in Grundgesetze I, §3, §10–12. It is just another aspect of his anti-creationism. In Grundgesetze II, §147, Frege makes a concession to an imagined creationist opponent which might suggest that he was fully convinced neither of the defensibility of his anti-creationist position regarding the syntactic development of the subject matter of arithmetic nor of his actual defence in §146 of the non-creativity of the introduction of value-ranges via logical abstraction in Grundgesetze I, §3 and the twin stipulations in §10. I argue that not only in Grundgesetze II, §146 but also in Grundgesetze II, §147 Frege falls short of defending his anti-creationist position. I further argue that on the face of it his creationist rival gains the upper hand in the envisioned debate in more than one respect.