Wittgenstein's diagonal argument
a variation on Cantor and Turing
pp. 25-44
in: P. Dybjer, Sten Lindström, Erik Palmgren, Göran Sundholm (eds), Epistemology versus ontology, Berlin, Springer, 2012Abstract
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 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.
