Is "Do We Have A Priori Knowledge?" an Empirical Question?

In his article "A Priori Knowledge" in the Oxford Handbook of Epistemology (2002), Albert Casullo canvases the array of accounts of a priori knowledge (what it means -- how we want the term to function).

One approach, which he calls non-epistemic, is to give a reductive analysis of a priori knowledge in terms of metaphysical notions (P is knowable a priori iff P is a necessary truth; P is knowable a priori iff P is an analytic truth). Such accounts, he argues, are predisposed to fail at answering the question "what about the knowledge is a priori?" And further, Kripkean examples seem to render any reduction to necessity a failure: it is knowable a priori that the meter bar (in Paris, London?), itself being what makes it true that certain lengths are 1 meter, is 1 meter long -- one would be making a mistake to measure it to be sure it was 1 meter long. But, it seems, the length of a meter could have been different, since it was just stipulated, and so `that stick there -- in Paris or where ever -- is 1 meter long' is true contingently). Also, "I exist" is not a necessary truth, since I might not have existed, yet this seems knowable via rational reflection.

Another approach, which Casullo attributes to BonJour, is this: "S is a priori justified in believing P iff S can intuitively "see" that S is a necessary truth." The same Kripkean worries apply to the last clause, but further, notice the regress: if 1+2=3 is knowable a priori, then one must be able to see the truth of "it's necessary that 1+2=3," and if one is to know the last modal proposition via reflection alone, then one must know that it is also necessary (it's necessary that it's necessary that 1+2=3), and off we go.

Another approach, which Casullo attributes to Chisholm, is an axiomatic understanding: S knows P a priori iff P is a consequence of facts that are axiomatic for S, where P is axiomatic for S iff P is true and S believes P and anyone who believes P is certain of it. Casullo objects to this approach on the grounds that is rules out it should be possible to have false a priori beliefs. For example, presumably we want to say that for thousands of years people believed a priori that one could not consistently deny the parallel postulate, but as it turns out, there are consistent non-Euclidean geometries. It seems right to say that we should understand a priori justification in a way that allows for varying degrees of confidence.

Another approach, which Casullo attributes to Putnam, is that S knows P a priori iff P could not be defeated by experience. Casullo rejects this as well, on the grounds that we should allow for testimony, an experiential source of justification, being able to revise our a priori formed beliefs. Consider a mathematician who is usually correct, but sometimes makes mistakes. She is doing a proof while her brain is hooked up to a computer which detects cognitive states which are associated with her making mistakes. She finishes a proof, thinks she's carried it out correctly, but the machine says an that an unreliable cognitive process came into play. Presumably we want to say that she should now be allowed to revise how certain she is with her result, or even choose to suspend believe in the theorem at hand.

Casullo ends the article with an absolutely striking suggestion. Perhaps we should settle the matter empirically: we should figure out what your mind would have to be doing to supply non-experiential justification to a belief, and then ask, does that ever happen to people? Talk about a plot twist! I curious as to what the readers of this blog have to say about this approach. I'm not sure what I want to say about it, other than that I think it's really interesting!