 The locus classicus of game formalism is not a defence of the position by a convinced advocate, but a demolition job by a great philosopher, Gottlob Frege. Not that he was attacking a straw man position: the highly influential diatribe by Frege in volume II of his Grundgesetze Der Arithmetik (Frege, 1903) is an attack on the work of two real mathematicians, H. E. Heine and Johannes Thomae. Moreover, philosophers of mathematics are wont to claim that the position is still widely adopted by mathematicians. It must be emphasised, however, that “formalism” in this sense—the Heine/Thomae position as interpreted by Frege, and its descendants—is to be distinguished from a more sophisticated position (it is claimed), namely Hilbertian formalism. For more information on the latter see Detlefsen (1993) or consult the entries on  Hilbert’s program  and  the Frege-Hilbert controversy.)  Detlefsen (2005) also provides a detailed historical treatment of formalist themes in thinkers from the ancient Greeks up to the period of Frege and Hilbert and beyond which is the focus here. 