As noted before, I think we get (mature, human) mathematical knowledge by benefiting from caual interactions with the world that lead us to find "coherent" combinations of mathematical statements obvious, and that our acceptance of these coherent stipulations helps determine the meaning of our words in such a way that these stipulations express truths in our language.
But this suggests a question. (Or at least, related views suggested a question to Shapiro and Ebert) Suppose someone accepts ZF and just guesses some elaborate provable truth T, and then stipulates {ZF+T}. Do they count as knowing that T? Doesnt my view commit me to thinking that they do?
The combination of ZF+T is indeed coherent, so I think that people who naturally found T just as obvious as people with mainstream mathematical intuitions find ZF would count as expressing mathematical truths, and indeed knowing that T. (see my paper the Doctoroids for more on this, though I wrote it before seeing the Shapiro).
But what about someone who feels uncertain about whether T, but tries to just stipulate it?
In general, I think, such a person won't count as having knowledge, because they are taking what is (relative to their current state of knowledge) an excessive epistemic risk - and hence they lack justification for their true beliefs. If their current mathematical faculties and other experiences do not give sufficient reason think that adding T to their beliefs would lead to a logically consistent system, they also lack sufficient reason to think that adding T would lead to a system of axioms that correctly describe some realm of mathematical reality. Thus, they are being epistemically irresponcible in adding this axiom.
However, if their current mathematical and other reasoning does suggest (though not prove) that adding T would be consistent, they can be justified in adding T as an axiom (although they may not be justified in assuming that once they have e.g. stipulated the axiom of choice to be true, they are talking about the same mathematical structure as they originally were).
Do they count as knowing that T?
ReplyDeleteWell tell me how you define as "knowing that T" and specify exactly how they guess and I'll tell you the answer.
I mean sure there are some situations where I feel comfortable saying "They know blah" and others where I feel comfortable saying "They don't know blah." I can even give some heuristics that would let some (english speaking) alien who has never been exposed to the concept of knowledge predict in most cases which sentence I would assent to if pressed.
However, the same could be said about my determinations about who is a hot girl, the heuristics are just a bit more messy. In both cases I'd be willing to assert general principles I take to hold of the concepts, e.g., "If I know x then x is true" or "If x is a hot girl then x is not ugly." But in neither case would it be sensible to assume that there are clear cut rules to determine the application of the word in every case or even that different people will agree on the application in peculiar cases.
Now if you just mean something like, "tell me how most people would answer if presented with this question," I can do a survey and in many cases there will be a robust result but sometimes the result will depend on subtle factors of how the survey is given. This problem only becomes worse as you pile on more and more constraints on the survey, e.g., "after reflection, presented with the best arguments, unlimited time to think etc.." so these attempts to handwave away the issue of whether there is a well definied question at all don't succeed.
I mean we use words. There are empirical facts about how we use words. That's it. Now sometimes it's useful to reason about how we use words but we shouldn't confuse the useful *fiction* that there are objective facts about whether something is knowledge with reality.
Are there objective facts about whether there are any heaps of sand in Hawaii, or any bald men in Canada?
ReplyDeleteThe fact that a term is vague, doesn't mean that there can't be objective facts about whether it does or doesn't apply in various clear cases.