The first of five planned volumes, this book collects Wittgenstein’s unabridged writings on mathematics and logic composed between 1941 and 1943. In 1956, a significant but abridged portion of Ludwig Wittgenstein’s 1937–1944 writings for a continuation of what became Philosophical Investigations was published as Remarks on the Foundations of Mathematics. Now, in the five German-English facing pages volumes of Remarks on Mathematics and Logic, these writings are presented in full as Wittgenstein wrote them: in chronological order, without editorial omissions, selections, or rearrangements. Volume IV contains materials from 1941 to 1943, including one previously unpublished manuscript and three manuscripts from which selections were made for the previous abridgment. The topics covered in this volume include the philosophy of mathematics, mathematical foundationalism, surveyability in mathematics, language-games and proof-networks, Gödelian undecidability, pictures in mathematics, Cantorian diagonal proofs, applied mathematics without pure mathematics, and mathematics and chess, among others. This German-English facing pages edition presents Wittgenstein’s writings in their original context—as Wittgenstein wrote them, in chronological order, and without editorial omissions, selections, or rearrangements.