 A central theme of the Grundlagen is a detailed criticism of Mill’s philosophy of mathematics. Frege argues that mathematical truths are not empirical truths and that numbers cannot be properties of aggregates of objects. First, Frege denies Mill’s claim that mathematical statements are about matters of fact. Frege’s objection is that there is no physical fact of the matter to which the numbers 0 or 777864 refer. Moreover, someone who learns how to calculate does not thereby gain any new empirical knowledge (1884, 9–11). Second, Frege insists that there is no general inductive law from which all mathematical sentences can be said to follow (1884, 10). Third, while Frege grants Mill that some empirical knowledge may well be necessary for us to learn mathematics, he points out that empirical knowledge cannot justify mathematical truths (1884, 12). Fourth, Frege counters Mill’s claim according to which numbers are properties of aggregates of objects with the observations that aggregates do not have — in and of themselves — characteristic manners in which they can be divided. Frege also points out that the numbers 0 and 1 are not aggregates at all. And finally, Frege accuses Mill of overlooking that numbers can be predicated of both concrete and abstract objects (1884, 31).