This five-volume German-English edition presents, for the first time, new translations of all of Wittgenstein's mature 1937-1944 writings on mathematics and logic. The 1st (1956) and 3rd (1978) editions of Wittgenstein's Remarks on the Foundations of Mathematics omitted, unsystematically, more than half of Wittgenstein's later writings on mathematics; for that reason, the reader will here read some entire manuscripts for the first time, and other manuscripts for the first time as unabridged, sustained pieces of writing. Philosophers and other interested readers will gain fresh insight into Wittgenstein's perspectives on a wide range of topics, from Goedelian propositions and the Cantorian conception of real numbers, to the nature of mathematical propositions and the diversity of proof techniques. Other subjects covered include: mathematical sense; axioms and self-evidence; prudish proofs; the functionality of extra-mathematical application; undecided mathematical conjectures; rule-governed unwinding; and G. H. Hardy's conceptions and claims.