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The principal and most significant change from the middle to later writings on mathematics is Wittgenstein's (re- introduction of an extre- introductionapplicatiof criterion, which is used to distinguish mere "sign-ganes" from mathematical languagextra-mathematical
The remainder of this article adopts the second interpretation, explicating Wittgenstein's later Philosophy of Mathematics as largely continuous with his intermediate views, except for the important introduction of an extra-mathematical application criterion.
Given his syntactical conception of mathematics, even with the extra-mathematical application criterion, he would simply say that P, qua expression syntactically independent of PM, is not a proposition of PM, and if it is syntactically independent of all existent mathematical language-games, it is not a mathematical proposition.
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