How many multiplications can we do?

In discussions in cognitive science, philosophy of language, philosophy of mind, and linguistics, it is often taken for granted that we (as well as some machines) have certain abilities, such as the ability to do multiplications or the ability to identify grammatical sentences. Such abilities are regarded as in some sense infinitary, and they are identified with, or taken to be based upon, knowledge of the relevant rules (the rule of multiplication, or the rules of grammar). In what follows, I argue that whatever such abilities we do possess are not infinitary in any plausible sense. Therefore, the (alleged) infinitary nature of our (or a machine's) knowledge of such rules cannot be accounted for by bringing it back to infinitary abilities.
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