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Abstract

Frege’sBegriffsschrift revealed that contrary to Kant, ^cpLogic is informative. In our last chapter, we showed that to demonstrate its informativity, Frege’s Begriffsschrift analyzed the notion of the ancestral relation and proved some theorems for induction. The ancestral is the key to the notion of mathematical induction—something Kant thought required nonlogical intuitions of consecutive order. We saw as well that Frege could not have failed to notice that his Begriffsschrift theory of ^cpLogic embodies a conception of numeric second-level concepts. But in that work he did not know how to understand numbers as objects and he did not know how to characterize the notion of one

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natural number (as an object) com­ing immediately after another in a consecutive series. In his manuscript “Boole’s Logical Calculus and the Concept-script” (1881) we find Frege assuming that natural numbers are objects. For instance, he haswhich says that 4 belongs to the series beginning with 0, in which the immediate successor of any member is obtained by adding 1. Frege clearly thought that numbers are objects. But the presentation of a theory of numbers as logical objects awaited his Grundlagen and then the mature work of hisGrundgesetze. In this chapter we aim to demonstrate that Frege’s theory of numeric second-level concepts played a central role in his conception of numbers as objects.