In his latest essay, The Limits of Science?, Mark Rowlands holds up Kant's Critique of Pure Reason as (possible) paradigmatic (or at least exemplary) of an epistemic methodological alternative to science.
I'm not a philosophologer (I'm not concerned with having a detailed knowledge of the philosophical canon), and I read Critique of Pure Reason more than twenty years ago, before I was seriously interested in philosophy. I wasn't impressed then, and none of the commentary I've read about it since has given me much motivation to revisit the work.
Going by my general impressions from commentary and decades-old memory, I recall that Kant's overall argument rests on what we can (and, to his credit, much that we cannot) say about synthetic a priori propositions. A synthetic proposition is some proposition that states a truth about the world (e.g. "all crows are black"), as opposed to an analytic proposition, which is about the meanings of words (e.g. "all bachelors are unmarried"). An a priori proposition is some proposition that is known (or can be known) prior to experience, as opposed to an a posteriori proposition, which can be known only on the basis of (i.e. after) experience.
Kant's argument rests on our a priori knowledge of mathematics and geometry and our deep a priori intuitions about space and time. (One might also add our a priori intuitions about morality.) I think this sort of argument fails both on the specifics and in general. Kant cannot be faulted too much: He works at the birth of modern science, immediately after Newton (Kant: 1724–1804; Newton: 1643-1727). He precedes Lobachevsky (1792–1856), Riemann (1826–1866), Darwin, Gödel and Einstein.
Non-Euclidian geometry, chaos theory, Gödel's incompleteness theorems, and all the weirdness of 20th and 20st century mathematics casts considerable doubt on the a priori truths of mathematics. Furthermore, Tobias Dantzig's history of mathematics, Number, the Language of Science, shows us just how hard-earned were the mathematical concepts of even Kant's day. Dantzig argues persuasively that our "primary" mathematical intuition is very limited, and the sophistication of modern mathematics can be seen as groping for concepts that would allow us to understand practical problems; our mathematical discovery has been driven as much or more by the requirements of experience as by the exercise of pure reason.
Einstein puts paid to our ordinary, intuitive "Newtonian" intuitions about space and time. Worse yet, quantum mechanics subverts our core assumptions about reality itself. If paradigmatic synthetic concepts are actually wrong — or obviously limited to the conditions of our ordinary experience — it's suspect to maintain they are actually known a priori.
It's provably true that individual human beings have true synthetic propositional beliefs that precede their own individual experience. Someone of Kant's time would be justified in concluding that such beliefs are fundamentally a priori, the result of the exercise of pure reason uninformed by experience.
But someone of Kant's time precedes Darwin, and Darwin makes all the difference. Because, of course, it becomes a plausible counter-argument to say that our true synthetic "a priori" beliefs are actually a posteriori with respect to the experience of evolution. Our brains, far from being engines of "pure" reason, have been shaped by at least 500,000,000 years of multicellular evolution (and perhaps even by the full 4,000,000,000 years of the evolution of terrestrial life). Evolution is not only undeniably an experiential process, it is also scientific. The parallels are exact: theoretical structure = genome; creation of hypotheses = random mutation; acceptance or rejection of the hypotheses by experiment = natural selection.
I am almost certainly missing some subtlety of Kant's argument, and it's possible I'm missing it completely; like I said, it's been a long time and I'm not much of a philosophologer. But Darwin seems to rebut any argument that rests in any way on synthetic a priori propositions.