In the run-up to my qual, I was discussing Benacerraf’s epistemological objections to platonism and the way it used a causal theory of knowledge. John MacFarlane challenged me to come up with a Gettier-type example that could possibly push one to adopt something like a causal theory for mathematical knowledge as well as physical knowledge. At the time the best I could come up with was a lot of the work of Newton and Euler in calculus, which came up with correct theorems, but used a lot of shady dealings with infinitesimals and non-convergent infinite series. By our standards, we shouldn’t be able to say that Newton and Euler knew these results any more than we’d say that a first-year calculus student knows the answer to a homework problem “solved” by plugging in x=0 and then canceling zeros from the numerator and denominator of some fraction. However, I think it’s status as justified true belief may thus also be questionable.
I think I may have come up with a better example recently though. Consider any of Cantor’s results in set theory. He was working with a fairly intuitive theory that may not have been consistent, but managed to establish many important results that were later verified with the machinery of ZFC set theory. Now consider Frege, who may well have formalized some of Cantor’s proofs of these results in his notation, and proved them using unrestricted comprehension. However, if the only instances of comprehension that he needed to use are the ones that appear as the pairing, powerset, union, replacement, separation, and empty set axioms of ZFC, then it seems more plausible to say that he actually was justified in his beliefs. Every step in his proofs was an instance of a schema we now generally recognize as valid. However, he believed these schemas were valid because they were all instances of the larger schema of full comprehension, which is inconsistent, and thus not valid.
So Frege may well have had JTB but not knowledge of various results from ZF (putting aside temporarily the question of whether any mathematical claims are “true”).
I’m beginning to worry now that this example isn’t much better, because on reviewing my problem sets that I wrote up for my set theory students at Mathcamp last summer, I couldn’t find any consequences of the comprehension fragment of ZFC that weren’t themselves instances of the comprehension schema. So it would be exceedingly implausible for Frege to have justified any of these results on the basis of the particular instances that are also instances of ZF schemas, rather than “justifying” the results directly as instances of comprehension. So if anyone can come up with a good particular example, let me know!