 Wittgenstein attempts no theory of mathematics in the Tractatus beyond arithmetic, a rather narrow fragment of arithmetic at that. The theory clearly shares the anti-platonism of the game formalist. There are no numbers, arithmetic is to be construed as a calculus in which one manipulates exponents or indices of operators. What is an operator? Wittgenstein distinguishes operator terms from function terms but commentators have struggled to explain what the distinction comes to. It is clear that Wittgenstein held that two occurrences of a function term \(f\) applied to different strings \(t\) and \(u\) have different meanings, where by ‘meaning’ Wittgenstein means referent, something like Frege’s Bedeutung. Thus ‘\(f\)’ in “\(f(t)\)” does not refer to the same entity as the outermost \(f\) in “\(f(f(t)\)”; this is supposed to be the basis of the solution to Russell’s paradox (¶3.333). In particular ‘the father of’ in “the father of John”, means something different there compared to its outermost occurrence in “the father of the father of John”. There can be no genuine iterated application of functions in other words, a cure for Russell’s paradox that many will find about as bad as the disease.