So my friend Kenny has an interesting entry over on his blog regarding Weatherson’s post on knowledge and discovery. Kenny argues that it is merely justified true belief which gives others a reason to believe a piece of knowledge counts as discovery of that piece of knowledge.

I’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 also unsure if justified true belief is even necessary for mathematical discovery or philosophical discovery. I will discuss this question and try and iron out some of the further ambiguities in the ‘discovered blah’ usage. However, the real point of this post is to muse about the relationship between JTB and knowledge in the case of the a priori. Can these two concepts really come apart in situations involving purely deductive reasoning. In particular I will argue that in the Kripke situation under consideration knowledge and JTB simply don’t come apart so while Kenny’s comments may be correct they are mostly irrelevant to this case.

Further Ambiguities of ‘Discovery’

Before we can discuss the relationship between discovery and knowledge any further we need to clarify the ways that ‘discovered blah’ is used. In the comments that Weatherson makes on Kenny’s blog he discriminates between discovering X and discovering that ‘X is true.’ In particular Weatherson suggests Fermat may have discovered Fermat’s Last Theorem (FLT) but surely didn’t discover that ‘FLT is true.’ This is indeed an important distinction but I think we need to clarify several more uses of ‘discovered’ before we can fully analyze this issue.

First though I would quibble with the idea that Fermat didn’t discover ‘FLT is true.’ Fermat didn’t have a proof of Fermat’s last theorem but quite likely he had done a fair number of calculations which gave inductive support to FLT. It would seem that if we are ever going to allow knowledge in the physical sciences when the most we can hope for is inductive support we should allow the same type of inductive inference to establish knowledge of mathematical truths. So I think it is quite plausible that Fermat did discover that ‘FLT is true.’ If we don’t accept this sort of non-proof based knowledge (or equivalently JTB in this case) this knowledge related theory of discovery would be totally inadequate to explain our usage of ‘discovered’ in cases like Ramanujan’s discoveries in number theory. Many of the discoveries made by Ramanujan were not supported by rigorous proofs but rather by intuition and calculation, i.e. by inductive means.

I think this confusion over whether Fermat discovered ‘FLT is true’ is really due to another ambiguity in our usage of ‘discover.’ In mathematics we often use ‘discovered X’ as a shorthand for ‘discovered X is provable.’ This explains our reluctance to call a purely computational observation a discovery in mathematics even though it clearly meets our standards for knowledge in the sciences and other empirical endeavors. It also comports with our willingness to call hand wavy or non-rigorous arguments in mathematics discoveries because they give us (inductive) reason to believe the statement is not only true but actually provable. Of course in some cases (some results of Ramanujan) we may be unsure whether someone really gave us reason to believe something was provable or just true explaining some of the disagreements about who discovered what occasionally seen in mathematics.

This use of ‘discovered blah’ in mathematics to mean ‘discovered blah is provable’ from some understood class of axioms is not the only specialized use of the word ‘discovered.’ In Weatherson’s original post he notes that, “One can discover lots of spiders without discovering that spiders exist.” While he uses this to suggest that there is a difference between discovering an instance of some principle and the principle itself and in particular how one might think Kripke discovered many necessary a posteriori truths without discovering that necessary a posteriori truths existed the grammar of this sentence should suggest something more complex is occurring. At the very least if we are going to take discovery to be some type of relation one can have with a piece of information we need to explain the apparently extensional usage in “discover lots of spiders.”

Just as their is a specialized sense of discovery present in mathematics I suggest that there is a special sense of discovery with respect to natural kinds totally distinct from this conception of discovery applying to pieces of information. Surely when Weatherson speaks of discovering lots of spiders he can’t mean discovering that they were spiders. Otherwise it would be impossible for someone to discover spiders before someone formulated the class arachnids or a precursor notion. It even seems wrong to suppose that someone must have discovered the fact that some creature existed with properties X, Y and Z where X, Y and Z would ultimately prove sufficient conditions to be classified as an arachnid. If the class mammal is first formulated as definitionally requiring members to have hair I still think it would be correct to speak of some alien discovering a mammal by observing a dolphin even if he did not observe the small hairs on its body. I expect some more realistic example can be given along these lines by someone who has more knowledge of taxonomy than myself.

My suspicion that this notion of discovery which grammatically appears to apply to objects not facts is related to some notion of natural kind comes from the following examples. At least to me it seems intuitively correct to speak of a color-blind man being the first person to have discovered green emeralds even if they did not know these objects were emeralds or green. It does not, however, seem right to speak of a color-blind man being the first to have discovered grue emeralds if he did not know the object he had was indeed green or an emerald. It could be that I’m just being mislead by the oddity of sentences about grue but I can’t see how we could make sense of this sense of discovery without some involvement of natural kinds. With the aid of natural kinds we can explain the discovery of spiders as discovery that there existed a creature with properties X,Y and Z and these properties would lead someone with knowledge of the natural kind in question to infer that the creature was a member of that natural kind. Thus hearing a skittering in the dark which happened to be a mouse would not constitute discovery of a mouse, though it would count as discovery of a small nocturnal creature. This seems to agree with my intuitions about usage but let me know if it strikes you differently.

I don’t think this undermines Weatherson’s point about Kripke possibly discovering that many sentences were a posteriori truths (I very much doubt he was the first person to utter those sentences). Though whether or not you find the spider analogy compelling depends on whether you think spiders and a posteriori truths are both natural kinds. However, it does call our attention to another complication of any analysis of discovery. A usage we may need to distinguish even in mathematics as I expect some kinds of Platonists will use discover in this natural kind sense.

Did Kripke Discover THAT Necessary A Posteriori Truths Exist?

While interesting this other use of the word discovered really has little impact on the question of whether Kripke discovered that there are necessary a posteriori truths. Actually I don’t think there are necessary a posteriori truths so Kripke certainly did not discover that they exist but for the moment I will assume his arguments are compelling. In this case my intuitions suggest that Kripke did discover that necessary a posteriori truths exist though someone earlier may have discovered necessary a posteriori truths in the natural kind sense.

I do not find the presence of someone merely having true belief in some fact enough to block later discovery. Imagine, for instance, that Kripke had mad the first serious philosophical mention of necessary a posteriori truths but that we later discovered the writings of some madman from the middle ages which showed he believed in necessary a posteriori truths with no justification whatsoever. Or maybe with some utterly absurd justification like, “the world would be more beautiful with necessary a posteriori truths therefore god must have made things that way.” I think it would be correct to say that Kripke discovered that necessary a posteriori truths exist.

Still I agree with Weatherson’s sentiment that one can’t go discovering philosophical truths by going through published works and fixing faulty justifications. My inclination is to dispute the idea that discovery is a nice pretty notion analyzable just in terms of temporal priority, knowledge, justification and true belief. I suspect intervening attitudes make a difference when trying to decide if some unjustified true belief counts as a discovery, or at least precludes later discovery. My solution would be to say that you can’t discover a truth that is already widely believed, though you could discover a valid justification of that truth. You can, however, discover a truth that is currently not accepted even if someone previously had an unjustified belief in it. In other words a sufficient condition for discovering something is being the first person to know it (or have JTB plus some other condition) and first knowing it at a time when it currently isn’t accepted as true. I’m unsure whether one should be regarded as discovering something when you don’t know that fact but on the basis of your discovery the truth of that fact is continuously accepted until someone does come to know that fact.

Gettier Cases?

Whatever you think the correct thing to say about these examples may be I don’t think one can maintain they constitute anything like a Gettier case. Perhaps the FLT case might be a Gettier situation if one supposes Fermat based his belief in the theorem on some false, but inductively supported, generalization about numbers. However, if Aquinas made a mistake in his supposed justification for a posteriori truths it was a deductive mistake. As Gettier cases seem to rely on an essentially inductive, or at least probabilistic, type of inference it seems likely that here we either have knowledge or don’t have JTB.

The essential aspect of all the Gettier cases I have seen is a probabilistically supported but ultimately wrong step in the inference. So long as every step in a justification is purely deductive the truth of the premises should absolutely guarantee the truth of the conclusion. It would seem, therefore, that so long as the dispute is over the deductive steps in an argument JTB and knowledge don’t come apart. Though of course it is always possible there is some other way knowledge and JTB come apart that I’m not thinking of.

At least in the case of the Aquinas situation it seems Kenny’s suggestion of identifying discovery with JTB that gives others a good reason to believe the statement will not save us from the conclusion that Aquinas did not discover that necessary a posteriori truths exist. If Aquinas did not know that necessary a posteriori truths exist it was not because some probabilistically valid inference in his justification turned out to be false. Rather it would be because he made some deductively invalid step in his reasoning preventing him from having either knowledge or JTB.

Gettier Cases and Mathematics

Just because it seems impossible to have a Gettier type case arise in deductive reasoning should not lead us to conclude they cannot happen in mathematics. I’m unsure if Kenny wants actual historical examples (who cares if it actually happened?) or just plausible candidates for Gettier cases in mathematics but the later are quite easy to create.

There are, for instance, several simple algorithms which appear to generate only primes on casual inspection. There are even some simple formulas which give the n-th prime evaluated at n for all values of n reasonably computable by hand. Someone might easily try such a formula for many values of n gain a high degree of inductive confidence in its validity and use that to prove some face about primes like the Riemmann hypothesis. Supposing they observed enough values of n to establish a high level of inductive confidence in the formula, despite the fact that it is not in fact valid, this would seem to constitute a Gettier case in mathematics.

If we want a historically more plausible situation where what is discovered is not just that something is true, but that it is actually provable we can look to the four color theorem. Suppose the computer the four color theorem was first verified on had a tiny bug which altered a calculation occurring in part of the computer aided proof. If the theorem later turned out to be true but the computer aided theorem false it would seem we would have a valid Gettier case in mathematics. However, I think we are going to need some kind of inductive or probabilistic inference any time we want to create a Gettier case like this one.