incompatible complaints?

A traditional objection to empiricism is that you couldn’t learn math empirically because math doesn’t make empirically testable predictions. On the other hand a traditional objection to formalism is that it would be surprising if a mere formal game had the scientific applications which math does. So prima facie it seems like on the one hand we are saying that the problem with empiricism is that math *doesn’t* make empirically testable predictions while on the other hand the problem with formalism is that math *does* seem to make such predictions and these predictions turn out right. So it seems like these traditional objections can’t both work.

One natural/obvious thing to say here would be that the applications problem for formalism is not that math itself makes empirical predictions (as the empiricist would like) but that it can be combined with a lot of empirical stuff to make predictions. But if the “applicability of math” just consists in the fact that mathematical statements can be usefully combined with a lot of empirically discovered stuff to make correct predictions this doesn’t seem like much of a problem for the formalist. Given two complex systems like the game of producing mathematical proofs on the one hand and the physical world on the other hand it would be more surprising if we *didn’t* find some parallels between the two. Newton et al just noticed the bits of math that seem to match certain transactions in the physical world. For all we know you *could* do just the same with any sufficiently complex formal game.

Specifically, the feature of math which (immediately seem to) pose a problem for the formalist are the *direct* applications – the claim that when there are 2Fs and 2Gs there are 4 F-or-Gs and predictions about what a computer will do so long as its electricity behaves in a certain way, or whether you will literally be able to tile a certain floor with certain physical features and certain tiles or whether a someone doing a certain kind of formal manipulations will get a certain string. It’s not surprising that some connection can be found between a complex formal game and what goes on in the external world, but here the ‘arbitrary formal manipulations’ of mathematical practice seem to call the shots and directly make predictions –and then these predictions turn out to be correct. If one allows that the formalism of math has direct predictive applications then it does indeed seem miraculous that the particular mathematical game which we ended up with makes the right predictions.

But if math does have these *direct* predictive applications (if this program halts you should expect to find a mouse in the wiring, no one will ever manage to cover a flat floor with tiles in those shapes) which turn out to be correct when tested then how can we make the other traditional objection that ‘you can’t learn math empirically because math doesn’t make empirical predictions’?