So, what does mathematics contribute to the success of science? I argue that

1. A promising way to make sense of the way in which mathematics contributes to the success of science is by distinguishing several diff erent contributions.Many philosophers seem to think that there is one thing which mathematics does. Perhaps the most influential view along these lines goes back (at least) to Wittgenstein's

*Tractatus*:

In life it is never a mathematical proposition which we need, but we use mathematical propositions only in order to infer from propositions which do not belong to mathematics to others which equally do not belong to mathematics. (6.21)But this seems too narrow. Mathematics makes any number of contributions to the success of science, and there is no straightforward way to reduce them all to a single kind.

The problems with Wittgenstein's approach are obvious. In many cases, we have no clue what the non-mathematical inputs or outputs are supposed to be. We start with mathematical descriptions and we end with equally mathematical descriptions. Either there is something defective in scientific practice, or Wittgenstein's approach is wrong. Beyond this sort of inferential or deductive contribution, there must be other kinds of contributions. But how are we to enumerate these contributions, and is there anything to be said about what they might have in common?

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