Arithmetic as Grammar

What is a number? Using material from Wittgenstein’s 1930s lectures, I argue that this question expresses a disorientation best overcome by recollecting rules that govern the number words. Why do we have the rules we do? We may be persuaded to adopt one rule rather than another by experience, when experiment shows it to be the more convenient way; we may also be persuaded by the “experience” of a new aspect. Mathematics is a “motley of techniques” for doing certain things; religion is a certain spirit meant to pervade everything we do. An important likeness is that in both instruction is essentially grammatical.