Plausibly, some true set theoretic axioms can be known a priori with a high degree of confidence. The Axiom of Separation is a good candidate. But other axioms, like the Axiom of Choice or the Continuum Hypothesis, we are much less confident of. Suppose that these axioms are true.
Could smarter beings than we are know them a priori? Maybe, but probably even they would not know them with complete confidence. There will be arguments for the axioms and arguments against the axioms. It seems likely that there will be axioms of set theory that both are true but that no rational being is going to have an a priori degree of confidence greater than, say, 0.99. Moreover, it is likely that there are many such independent axioms, perhaps infinitely many. When you conjoin enough such axioms, the probability of the conjunction will get smaller and smaller for a rational being, until you get to the point where the conjunction will be a priori quite incredible. Thus, there will be conjunctions of a priori knowable axioms that will themselves not be a priori knowable.
Thus the a priori knowable is not closed under conjunction. This does not bother me. As Jon Kvanvig said to me, the a priori is an epistemological category, and so we shouldn't expect closure. But I think this will generate serious problems for anybody like Chalmers who wants to put a heavy philosophical burden on the concept of the a priori.
But maybe someone could have a rational insight into set theory that yields complete certainty as to a controverted axiom, of a sort that remains no matter how many independent axioms are conjoined? For instance, theists are apt to think that God has such an insight. But God is not, I think, an a priori knower of set theory. First, I say that abstract objects are nothing but divine thoughts, and so God knows set theory by introspection, and introspection is more akin to the a posteriori. Second, even apart from such an ontology of sets, it is really hard to see if the rational insight really should count as a priori. I don't know how the rational insight would work, either for God or for a godlike knower, but rational insight into set theory is something like a vision of set theoretic reality. But that's much more like the a posteriori.
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It's not clear if Chalmers should be bothered by the failure of closure for a priori knowledge. Chalmers distinguishes between the *conclusive a priori* and the *inconclusive a priori*, where the latter corresponds more-or-less to what Hawthorne calls the "deeply contingent a priori." (Paradigm case: Suppose I know p on the basis of induction, where E is my total inductive base. Then the material conditional E-->p should be a priori. But since induction is an ampliative form of inference, certainty in E-->p would be inappropriate.) When Chalmers makes heavy philosophical use out of the notion of the a priori (e.g. in defining primary-intension truth-at-a-(centered)-world, in defining negative conceivability, in defining scrutability, etc.), he makes clear that the operative notion of the a priori is the *conclusive* a priori. He might just be able to take your arguments as showing that we can only have inconclusive a priori knowledge of (certain) mathematical truths.
All that said, he normally puts mathematical truths on the side of the conclusive a priori. And it might screw up some of his projects if he were to drop this assumption. For example, he defines primary-intension truth-at-a-centered-world (w, i) roughly as follows: Let D(w, i) be the "canonical description" of w, understood as a conjunction of D(w)-- the set of all truths about w couched in neutral (non-twin-earthable) vocabulary-- with D(i), a specification of the center of the form "I am the F," where F is a neutral, uniquely identifying description of the center. Then S is true at (w,i) iff the conditional D(w,i)-->S is a priori conclusive. Given that mathematical truths are couched solely in neutral vocabulary, it will follow that mathematical truths have a necessary primary intension. But your remarks in this post suggest that some mathematical truths are conceivably false. But then we have a "strong necessity," i.e. a sentence S which is not a priori certain but which has a necessary primary intension. This is a problem, because the prohibition on strong necessities is one of the central planks in his zombie argument (and his whole 2D system, for that matter).
Good point with regard to Chalmers.
But on the mathematics side, there is pretty much an epistemic continuum between the axioms of Peano arithmetic (which most people are pretty confident of, though there may be issues about the exact version of induction to be used) and the continuum hypothesis (say). I really doubt there is going to be a qualitative difference somewhere along that line. Moreover, judging by some MathOverflow discussions, there are mathematicians who wouldn't be very surprised to find that ZF is inconsistent, and hence that one of the seemingly uncontroversial axioms isn't true.
And do we have any reason to think there are *any* claims that are conclusively a priori, except maybe obvious tautologies? Presumably closure holds for the conclusively a priori. Yet we may be pretty confident that bachelors are unmarried and that what is known is believed, but once you conjoin a thousand such claims, our rational confidence in the conjunction should not be all that high.
Well, it seems there is a qualitative difference between axioms which are existence-entailing (entail the existence of some abstract entity, or more generally, entail the existence of anything at all) and those which do not. It seems that there is a qualitative difference, for example, between the axiom of extensionality, which merely says that any sets x and y are identical iff all the same things stand in the set-membership relation to them but does not logically entail that there are any sets, and the axiom of infinity, which logically entails the existence of sets. Generally, one might say: where S logically/conceptually entails the existence of a certain sort of thing (or anything at all), then S is not conclusive a priori. (I might know with absolute certainty, via introspection, that a thinker exists, but this is not a priori knowledge.)
This is not to say that we should be absolutely certain in the non-existence-entailing mathematical axioms. (Perhaps if it turns out that there are no sets, then it would turn out that the set-membership predicate is meaningless, and so statements involving it, like the axiom of extensionality, are without truth value.) It's just to say that it may be possible to find a principled divide in this territory.
Maybe.
But if we draw the line there, then the Peano axioms end up on the non a priori side of the line, since they assert that there is a natural number 0 and that every natural number has a successor. And even 0=0 will then be on the non a priori side, since it entails the existence of 0. (I guess one can have a weaker claim in a non-free logic: if there is a 0, then 0=0.)
All this isn't perhaps much of a problem for me since I'm a structuralist about mathematics.
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