On 30 July 1947 Wittgenstein penned a series of remarks that have become well-known to those interested in his writings on mathematics.  It begins with the remark “Turings ‘machines’: these machines are humans who calculate.  And one might express what he says also in the form of games”.  Though most of the extant literature interprets the remark as a criticism of Turing's philosophy of mind (that is, a criticism of forms of computationalist or functionalist behaviorism, reductionism and/or mechanism often associated with Turing), its content spells out part of Turing’s own philosophical sensibility, and applies directly to the foundations of mathematics.  For immediately after mentioning Turing, Wittgenstein frames what he calls a “variant” of Cantor’s diagonal proof.  We present and assess Wittgenstein's variant, contending that it forms a distinctive form of proof, and an elaboration rather than a rejection of Turing or Cantor.