 As we shall see, Wittgenstein’s Philosophy of Mathematics begins in a rudimentary way in the Tractatus, develops into a finitistic constructivism in the middle period (Philosophical Remarks (1929–30) and Philosophical Grammar (1931–33), respectively; hereafter PR and PG, respectively), and is further developed in new and old directions in the manuscripts used for Remarks on the Foundations of Mathematics (1937–44; hereafter RFM). As Wittgenstein’s substantive views on mathematics evolve from 1918 through 1944, his writing and philosophical styles evolve from the assertoric, aphoristic style of the Tractatus to a clearer, argumentative style in the middle period, to a dialectical, interlocutory style in RFM and the Philosophical Investigations (hereafter PI).