Discovery in Mathematics

26 10 2005

Just as I looked for Gettier cases in mathematics a while ago, it looks like Brian Weatherson has some good candidates for Gettier cases in philosophy. Except that he’s looking for them to show that to discover P is not the same as to be the first person to know P. He uses the example of the existence of necessary a posteriori propositions (which he says Aquinas had justified true belief in, while Kripke was the first to properly know) to show that we might come to know something with no one discovering it. Aquinas didn’t discover it because he didn’t know, and Kripke didn’t discover it because he was just properly justifying a result he already knew of. (Even if that’s not how it actually went, it’s how it could have gone, and the moral should be the same for the relation between discovery and knowledge.)

However, I’m not quite convinced that we should deny Aquinas the discovery just because he didn’t know P. I claim that being the first person whose justified true belief in P gives others a good reason to believe P is what it takes to discover P. If Aquinas’ belief was such, then he should be credited with the discovery, and otherwise it seems that Kripke should.

In the comments on Brian’s post, Stefan Ionescu cites Fermat’s Last Theorem as a mathematical example that I think follows Brian’s example fairly closely – FLT is conjectured, Shimura and Taniyama make a (seemingly unrelated) conjecture, Ken Ribet shows S-T implies FLT, and then Andrew Wiles proves S-T. Fermat never knew it to be true, so on an account where discovery implies knowledge, he can’t be the discoverer. Wiles certainly didn’t discover FLT, because the statement was well-known and already widely believed to be true – he just definitively proved something (also widely believed) that was already known to imply FLT. Ribet also seems to be a bad candidate for the discoverer – all he did was relate two already widely believed statements.

So did Fermat discover it? Or was no one the discoverer, because Fermat never knew it to be true? (I think I’ll leave aside the question of whether knowledge of mathematical claims like FLT can come from sources other than proof.)

It certainly seems clear that Fermat discovered something – whether that something is FLT, or just “FLT is a good approximation to the truth” may be controversial. But just as Columbus discovered (from a non-Viking European context) America without knowing that’s what it was, Fermat may have discovered his theorem without having known that it was true. One might say the same thing about Michelson and Morley being the discoverers of the lack of an ether (or maybe some other related fact about light), though they never came up with a theory that properly predicted or explained it. These examples (built on Brian’s model for a mathematical parallel of his example) suggest that discovery may not require knowledge, but merely some strong enough sort of justified true belief. Just what sort of justification is involved, who knows – having mistaken a statement of an open problem for a homework assignment doesn’t seem sufficient justification to count as the discoverer.

Perhaps one way to measure whether A’s justified true belief counts as a discovery is whether B (knowing the incompleteness of A’s justification) could consider A’s belief a reason to believe the proposition. Ribet, Wiles, and the others all knew that Fermat had very good insight into the nature of the natural numbers, so even though they were convinced he didn’t have a proof, they could use his belief as a decent reason to believe that FLT was true. I don’t know enough about Aquinas’ arguments to say whether his belief provided Kripke a reason to believe that there are necessary a posteriori truths, but if it did, then perhaps he could in fact be credited with the discovery. If not, then perhaps Kripke could be, because he didn’t have an antecedent reason to believe before working out his theory.

If this account is right, then perhaps every piece of knowledge does have a discoverer (except maybe the odd open problem solved by a computer science student thinking it’s a homework assignment – apparently, since George Dantzig got his start this way and then went on to discover the Simplex Algorithm for linear programming, computer science professors often put open problems in homework sets). Probably some details will need to be worked out (especially with the “reason to believe” bit), but this seems a bit more plausible to me than the line that discovery requires knowledge. We credit explorers with all sorts of geographic discoveries they don’t properly understand or know the significance of – we should probably do the same for intellectual explorers.




4 responses

27 10 2005
Brian Weatherson

Two quick points.

First, I think the location “Fermat discovered Fermat’s Last Theorem” is ambiguous, and the ambiguity is crucial here. It’s ambiguous between a ‘discovered+NP’ reading, in the way Columbus discovered America, and a ‘discovered that’ reading, i.e. Fermat discovered that x^n + y^n = z^n has no solutions for n > 2. I think the first reading is plausibly true, but it’s the second I’m not so sure about.

Second, I think the really interesting case here concerns the case where Aquinas has an unjustified true belief. Aquinas’s proof of God’s necessity, at least in the Third Way, is really bad after all, so it’s not obvious that he did have a JTB. But I think there’s some intuitive force to the claim that his true belief stops others counting as the discoverer, even if it’s completely unjustified.

27 10 2005

It might be even more perspicuous to contrast “Fermat discovered FLT” and “Fermat discovered that FLT is true”. Then we see that I was accidentally doing some sort of semantic ascent or descent. This also deals nicely with the scientific case, where Michelson and Morley discover a phenomenon, and Einstein discovers the truth of some related proposition. But I’ll think more about whether it might be plausible that Fermat discovered that FLT is true.

I think we can’t take Aquinas’ claim to be completely unjustified though. If Kant had said “there are no necessary a posteriori truths” and someone had formed the belief that there are necessary a posteriori truths by reading this and missing the word “no”, I think that person would not block Kripke’s claim to discovery. Clearly, if no one else ever heard about this person’s belief it wouldn’t block anything. But I think even if Kripke knew this person had such a belief, Kripke could still count as discovering, if he realized that this person’s belief was totally unjustified. Just as the modern nuclear physicists discovered that it is possible to transform lead into gold, even though they knew that many people had had this belief in the past.

In Aquinas’ case, maybe Kripke thought Aquinas’ argument was really bad, but thought that it was at least remotely plausible, and though Aquinas found at least a promising candidate to consider for such a sentence. Maybe something like this is necessary and/or sufficient to block other potential discoverers. But you’re right that it really doesn’t seem sufficient to count as a discovery, even if Fermat’s conjecture does.

28 10 2005

Another interesting use of “discovery” in mathematics:

Wolfram refers incessantly to his “discovery” that simple rules can produce complex results. Now, the word “discovery” here is legitimate, but only in a special sense. When I took pre-calculus in high school, I came up with a method for solving systems of linear equations, independent of my textbook and my teacher: I discovered it. My teacher, more patient than I would be with adolescent arrogance, gently informed me that it was a standard technique, in any book on linear algebra, called “reduction to Jordan normal form”, after the man who discovered it in the 1800s. Wolfram discovered simple rules producing complexity in just the same way that I discovered Jordan normal form.

From Cosma Shalizi’s review of Wolfram’s A New Kind of Science.

6 11 2005
Computational Truth

Justified True Belief and the A Priori

‘m skeptical of Weatherson’s analysis of the problem for slightly different reasons than those Kenny emphasizes. I’m skeptical primarily because I don’t think prior true belief is enough to block discovery (a fact Kenny seems to doubt as well). I’m a…

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