In the Tractatus, Wittgenstein gives a definition of number as the exponent of an operation. In something I read recently, although for the life of me I cannot figure out what it was, the author pointed out that the basic idea in the Tractatus is the same as that of Church numerals. The definition of number in TLP has seemed somewhat obscure to me in the context of the TLP, so this comparison helped clarify things. The definition of number in TLP comes at 6.02. Lets call the basic operation S and name an element x. The number 0 is x, 1 is Sx, n is Snx and n+1 is SSnx. While not in lambda notation, this is fairly close to Church’s definition for addition.

Together with some of Michael Kremer’s remarks on Tractarian views of math, it makes the TLP seem concerned with computation as opposed to just structural concerns, which one would expect of something in the broadly logicist vein. (This may not be fair to all logicists. There did not seem to be a comparable concern with computation in Frege or Russell. The distinction I’m using here is the one drawn by Jeremy Avigad between math as a theory of structure and of computation in his “Response to Questionnaire” on his website. Although, Avigad points out that before the 20th century mathematicians were concerned with computational aspects of proof more than structural.) In Russell’s preface to the TLP, he criticizes Wittgenstein’s definition of number for not being able to handle transfinite numbers. If computation is supposed to be an important theme in the TLP, then this would not be bad. The transfinite numbers would not be the sort of thing that we would be computing with recursively. An interesting historical question is whether there were any reviews written of Church’s work which pointed out that his definition only worked for finite numbers, echoing Russell’s criticism of the TLP. Since the TLP was written before Church’s, Turing’s or Goedel’s work on computability made it a more precise mathematical notion, it seems likely that it would remain implicit in the book rather than being made explicit as a main theme.

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