Frege was certainly arguing against formalism; he was the one responsible for taking its defenders to task. But the formalism he presented and argued against (

Frege was certainly arguing against formalism; he was the one responsible for taking its defenders to task. But the formalism he presented and argued against (and the version held by its defenders of the day) is clearly the form described in the article and the discussions he sparked later all circled around the same. Frege quotes Heine as > Suppose I am not satisfied to have nothing but positive rational numbers. I do not answer the question, What is a number?, by defining number conceptually, say by introducing irrationals as limits whose existence is presupposed. I define from the standpoint of the pure formalist and call certain tangible signs numbers. and Thomae > The formal conception of arithmetic accepts more modest limitations than does the logical conception. It does not ask what numbers are and what they do, but rather what is demanded of them in arithmetic. For the formalist, arithmetic is a game with signs, which are called empty. That means they have no other content (in the calculating game) than they are assigned by their behavior with respect to certain rules of combination (rules of the game). within the Grundgesetze der Arithmetik. And indeed Frege defines the position he is arguing against as > Arithmetic is concerned only with the rules governing the manipulation of the arithmetical signs, not, however, with the denotation of the signs These are from Max Black's translation of the Grundgesetze der Arithmetik in his three part series "Frege Against the Formalists." These quotes are from part 1 in The Philosophical Review, Vol. 59, No. 1 (Jan., 1950), pp. 79-93