 On Frege's view, principles like these are true, and so they commit us to the existence of numbers, lines, and shapes. Now, of course, we have already gone through a platonistic argument — indeed, a Fregean argument — for the existence of numbers. Moreover, the standard platonist view is that the argument for the existence of mathematical objects is entirely general, covering all branches of mathematics, including geometry, so that on this view, we already have reason to believe in lines and shapes, as well as numbers. But it is worth noting that in contrast to most contemporary platonists, Frege thought of numbers, lines, and shapes as logical objects, because on his view, these things can be identified with extensions of concepts. What is the extension of a concept? Well, simplifying a bit, it is just the set of things falling under the given concept. Thus, for instance, the extension of the concept white is just the set of white  things.[13]  And so the idea here is that since logic is centrally concerned with predicates and their corresponding concepts, and since extensions are tied to concepts, we can think of extensions as logical objects. Thus, since Frege thinks that numbers, lines, and shapes can be identified with extensions, on his view, we can think of these things as logical objects.