If mathematics were merely convention, then its success in physics would indeed be a miracle — why should arbitrary symbols line up so exactly with the predictability of nature? And if it were merely empirical, then we could never be sure it applies universally and necessarily... — Wayfarer

Science isn't committed to the reality of alethic modalities (necessity, possibility, probability) in the devout epistemological sense you seem to imply here, for they are merely tools of logic and language - the modalities do not express propositional content unless they are falsifiable, which generally isn't the case.

A nice case of the “unreasonable effectiveness” is Dirac’s prediction of anti-matter — it literally “fell out of the equations” long before there was any empirical validation of it. That shows mathematics is not just convention or generalisation, but a way of extending knowledge synthetically a priori. — Wayfarer

IMO, that is a merely an instance of an inductive argument happening to succeed. A purpose of any theory is to predict the future by appealing to induction -- but there is no evidence of inductive arguments being more right than wrong on average. Indeed, even mathematics expresses that it cannot be unreasonably effective, aka Wolpert's No Free Lunch Theorems of Statistical Learning Theory.

Humans have a very selective memory when it comes to remembering successes as opposed to failures. Untill the conjecture is tested under scrutiny, it can be dismissed.

Synthetic a priori = adds new content, but is knowable independently of experience.

That last category was Kant’s unique insight. Mathematics is built around it — “7+5=12” is not analytic, because “12” isn’t contained in “7+5,” but it’s still a priori. — Wayfarer

Let's slow down on this one. Kant doesn't speak about "content" in the [Prolegomena] (where the 7+5 example is discussed). He says that the concept of "12" is not the same as the concept of "7+5". According to him, we need an "intuition" ("perception" would probably be our way of saying it today) of the physical in order to discover "12". (He suggests that our five fingers, and then seven fingers, would do the trick.) "Hence our concept is really amplified by the proposition 7+5 = 12, and we add to the first concept a second concept not thought in it." What Kant regards as analytic here is the judgment that 7 and 5 must add up to some number -- but this does not tell us what particular number.

The place where this can be challenged, I think, is the reliance on intuition. If this is truly the case, don't we have to question whether the judgment is indeed a priori? Kant addresses this in Sec. 281: "How is it possible to intuit anything a priori?" I don't want to take this any farther, except to say that the case for math as a series of synthetic a priori judgments, even on Kant's own terms, is far from closed.

Need the year of publication, for whatever text you’re saying has Groundwork in its title. The Groundwork I’m familiar with is a treatise on moral philosophy, having nothing to do with mathematical judgements, and 7 + 5 is not discussed as far as I could determine, but that a categorical "ought" implies a synthetic a priori proposition, is.

And sec 281 doesn’t Google.

Thanks.

Oh good lord, sorry, I meant the Prolegomena. :grimace:

Quite, the Prolegomena. That’s where I read it.

A nice case of the “unreasonable effectiveness” is Dirac’s prediction of anti-matter — it literally “fell out of the equations” long before there was any empirical validation of it. That shows mathematics is not just convention or generalisation, but a way of extending knowledge synthetically a priori.
— Wayfarer

IMO, that is a merely an instance of an inductive argument happening to succeed. A purpose of any theory is to predict the future by appealing to induction -- but there is no evidence of inductive arguments being more right than wrong on average. Indeed, even mathematics expresses that it cannot be unreasonably effective, aka Wolpert's No Free Lunch Theorems of Statistical Learning Theory. — sime

But inductive arguments are a posteriori by definition. Dirac’s prediction of antimatter was not an inductive guess but a deductive consequence of the mathematics of the electron. It’s a perfect case of the synthetic a priori: by synthesising the elements of the theory, he saw that negative counterparts must exist — long before observation confirmed it.