In the paper the problem of mathematical explanations in science is discussed. The focus is on the programming account, according to which mathematical theorems impose some modal constraints on the physical world (so to say – they are “programming it”).

After presenting some examples (like P=NP, Con(PA), Con(ZFC)), the problem of the necessary background assumptions is discussed. In particular, it might turn out, that in some cases strong (and non-standard) assumptions are necessary (so that – so to say – a quite “abstract metaphysics” is programming the world). There is also an important empirical aspect: the “computational resources” of the universe should be taken into account.

The talk also exhibits the links between the programming account and the discussion concerning the explanatory character of mathematical theorems – in particular the potential explanatory role of proofs. The discussion is important for the realism-antirealism debate, in particular in the context of the recently much discussed Enhanced Indispensability Argument (EIA), where the explanatory virtues of mathematics are of primary importance.