 Superficially, at least, it seems as if Wittgenstein is offering an essentialist argument for the conclusion that real number arithmetic should not be extended in such-and-such a way. Such an essentialist account of real and irrational numbers seems to conflict with the actual freedom mathematicians have to extend and invent, with Wittgenstein’s intermediate claim (PG 334) that “[f]or [him] one calculus is as good as another”, and with Wittgenstein’s acceptance of complex and imaginary numbers. Wittgenstein’s foundationalist critic (e.g., set theorist) will undoubtedly say that we have extended the term “irrational number” to lawless and pseudo-irrationals because they are needed for the mathematical continuum and because such “conceivable numbers” are much more like rule-governed irrationals than rationals.