 Overall, then, Wittgenstein in the Tractatus gives us no account of mathematics in general other than for a fragment of arithmetic, basically positive identities involving only addition. And there he denies that the sentences express propositions with truth values. Of course the book was written under extraordinarily difficult circumstances. Perhaps his account could have been developed further and more plausibly, despite the difficulties noticed above—but for some scepticism on that front, see Landini, 2007. Certainly Wittgenstein made no attempt to do so whilst engaged with F.P. Ramsey and the Vienna Circle in the 1920s. If Wittgenstein’s standpoint was not capable of much further development, we have the choice of either abandoning all of mathematics except, at most, a fragment of the arithmetic of addition; or else of rejecting the Tractarian account. One does not need to be slavishly uncritical of contemporary mathematics to see what the reasonable option is here. Admittedly, rejecting the Tractatus account is also the option Wittgenstein himself seems to adopt at the end of the book; here then we enter the issue of what the point is of taking us through such a bizarre and unconvincing theory in order to throw it away at the end. (For a more positive appraisal of Wittgenstein’s Tractarian position, see Floyd (2002). 