Whether or not it is ultimately defensible and this is an absolutely crucial question for Wittgenstein's Philosophy of Mathematics this strongly counter-intuitive aspect of Wittgenstein's account of algorithmic decidability, proof, and the sense of a mathematical proposition is a piece with his rejection of predeterminacy in mathematics.
Chip Hankey, 25, is a New York options trader for Group One Trading, a job that requires him to calculate complex mathematical propositions rapidly, with large sums of money on the line.
Unlike the criterion of truth for an empirical proposition, which can be known before the proposition is decided, we cannot know the criterion of truth for an undecided mathematical proposition, though we are "acquainted with criteria for the truth of similar propositions" (RFM VI, §13).
There cannot be "undecidable propositions," Wittgenstein argues (PR §173), because an expression that is not decidable in some actual calculus is simply not a mathematical proposition, since "every proposition in mathematics must belong to a calculus of mathematics" (PG 376).
Mathematicians once believed that the truth or falsity of any mathematical proposition could be decided by applying the rules of mathematical logic.
What could it mean to say that a mathematical proposition was true if there was no possibility of proving it?
In this sense of conversion, the passage from a proposition to its converse is not, in general, a valid inference; and though often a mathematical proposition and its converse may both hold, separate proofs must be given for each case.
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.
(PR, 52) {§6.5} a mathematical proof is an analysis of the mathematical proposition.
In this manner, Wittgenstein defines both a mathematical calculus and a mathematical proposition in epistemic terms.
