I have not heard anyone from Philosphy of sciences saying "all scientific explanations are fundamentally ad hoc". In fact, Adolf Grunbaum studied the nature of

I have not heard anyone from Philosphy of sciences saying "all scientific explanations are fundamentally ad hoc". In fact, Adolf Grunbaum studied the nature of Adhoc explanations. Even Imre Lakatos claimed that prima facie ad hoc (those from the protective belt) can end up being non ad hoc. What I have in my mind when I said mathematical models provide ad hoc suport is this. Today, it is consensus in the philosphy of sciences that there are two kinds of confirmation of a theory: confirming instance vs positive instance. Mathematical models generate "positive instances". A non-adhoc hypothesis generates confirming instances, a special type of positive instances. For more, pp. 61-62, Larry Laudan's "Science and Relativism: some key controversies in the philosophy of science. Lets look at a hypothesis that says "All swans are white". One keeps on doing field work, saying that 1001th swan is white, 1002th swan is white, etc. All these field reports are "positive" instances. Such field reports don't add much.