We’ve now had three seminars on Wittgenstein’s remarks on the foundations of mathematics in the Big Typescript, taking things very slowly. The first week, I talked about Sec. 108 (the contrast between arithmetic and a game). One of our final year undergrads gave an admirable presentation in the second week on Secs 109-11. This week, we battled with Secs 112 and 114 (leaving the discussion on Ramsey and identity in Sec. 113 till next week).
Now, a few pages ago, in Sec. 108, it seemed that it is the use or application of arithmetic that is supposed to distinguish it as mathematics from a mere game. What kind of applicability is in question? “It is mathematics, I should think, when it is used for the transition from one proposition to another.” (Sec. 108, p. 372e). So there, at any rate, Wittgenstein offers the beginnings of a story about applicability. But now, in Sec. 112, we have some distinctly odd remarks about applications. For example, “Arithmetic is its own application.” (p. 382e, repeated p. 385). What does that mean? I think it’s fair to report that we were left baffled.
To be sure, we presumably do want a story about the difference between using an empirical theory to take us from one proposition to another and using arithmetic. And Wittgenstein in effect remarks that if we use arithmetic and get empirically the wrong answer, we don’t blame arithmetic. “It might look as though the mathematical computation entitled us to make a prediction [e.g. about how many apples each a group of people will have, if you divide the pile of twelve apples between four]. But that isn’t so. What justifies us in making this prediction is a hypothesis of physics, which lies outside the calculation. The calculation is only an examination of logical forms, of structures, and of itself can’t yield anything new.” [p. 383e]. But just what does that last sentence mean? And let’s suppose for the sake of argument that, as part of our overall practice, we have the rule that we do not revise arithmetical propositions in the light of empirical results. That doesn’t make it any less the case that arithmetic is being applied to apples or whatever, or make it appropriate to say instead that “arithmetic is its own application”.
Any pointers to helpful discussions in the literature that makes sense of what is going on here will be very gratefully received!