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!

The Cardinal Collective(00:30:26) :If You Liked Philosophy of ScienceIf you liked your philosophy of science class (don’t tell me you didn’t take one), you’ll love Kenny’s post on Antimeta about Gettier cases in mathematics….

peter(15:35:22) :I don’t know enough phil of math to understand the jargon you use, but as someone trained in pure math I think there are many cases of mathematicians “knowing” something which they cannot prove to the satisfaction of later generations of mathematicians. Indeed, so common is this that a common saying of pure mathematicians is that a great mathematician can be recognized by the number of true theorems he/she publishes with invalid or flawed proofs. (In other words, later mathematicans have to come and clean up the mess.)

Moreover, this distinction (knowing something and proving it rigorously) is what arguably distinguishes applied mathematics from pure mathematics. Applied mathematicians, of all stripes, have all manner of tricks and rubrics and heuristics which “work”, but which require (or may not yet have) hard, rigorous, messy technical work to prove. It’s one reason pure mathematicians commonly look their noses down on their applied brethren.

Your example of infinitesimals is a good one. Newton and Leibniz had the correct intuitions about the differential calculus, but not a rigorous theory to support them. It took Cauchy and Weierstrauss, two centuries years later, to fix this with an operational semantics in the form of epsilon-delta arguments. Then, another century on, Robinson found a rigorous way to talk about infinitesimals.

But the intuitions of great mathematicians are not always correct. Ramanujan supposedly intuited both valid and invalid results.

Kenny(14:39:19) :I suppose at this point, it just becomes a question of intuitions. If you think that Newton and Frege did

knowtheir results, then perhaps justified true belief really is all that is necessary for knowledge in mathematics. I’m just trying to come up with examples in which someone has justified true belief anddoesn’tqualify as knowing, like someone who sees a hologram of an apple on a table and therefore justifiably believes there is an apple on the table, but doesn’t know this fact because he has no connection to the actual apple that is hidden behind a screen on the other end of the table.wo(01:20:22) :How about this:

Fred is in a maths lecture and follows a complicated proof of some obscure ‘theorem’ P. In fact, the proof is subtly fallacious and P is actually false. Not realizing this, Fred comes to believe P, and infers “P or Q”, where Q is some arbitrary mathematical truth Fred has no reason at all to believe, nor does he believe it.

If we are ever justified in believing a mathematical proposition without having gone through a minutely detailed and correct deductive proof, Fred is justified in believing (the falsehood) P. So he is also justified in believing (the truth) “P or Q”. But he doesn’t know “P or Q”.

wo's weblog(09:34:05) :Gettier Cases in Mathematics and MetaphysicsI once believed that in non-contingent matters, knowledge is true,

justified belief. I guess my reasoning went like this: