According to Frege...

According to Frege, we apprehend numbers by understanding the sense of sentences in which number terms appear, an understanding that is grounded (and hence shown to have a logical source) by our grasp of sentences such as (Na) together with our recognition of the equivalence captured in the Cantor-Hume Principle.[^16] What we have in the case of the Cantor-Hume Principle is what is often called an ‘abstraction principle’, and Frege’s and Russell’s different conceptions of analysis clearly lead to different views of the use of such principles.

In fact, it is significant in this respect that Frege himself never called it an ‘abstraction principle’, a phrase which itself suggests that one of the two sentences involved is on a different and ‘higher’ (i.e., more abstract) level to the other - numbers being ‘abstracted’ from the relation of one-one correlation obtaining between concepts.

Indeed, from Russell’s diametrically opposed perspective, the use of the phrase is also misleading, since it seems to grant that numbers are objects, just ‘higher’ or more abstract objects.…