Tom_Breton comments on A Priori - Less Wrong
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In a comment on "How to convince we that 2+2=3", I pointed out that the study of neccessary truths is not the same as the possession of neccessary truths (credit to David Deutsch for that important insight). Unfortunately, the discussion here seems to have gotten hung up on a philosophical formulation that blurs that important distinction, a priori. Eliezer's quotative paragraph illustrates the problem:
All of these definitions seem to assume there is no distinction between the existence of neccessary truths and knowing neccessary truths (more correctly, justifiably assigning extremely high probability to them). But there are neccessary truths that are not knowable by any means we have or expect to have. Eg, the digits of Gregory Chaitin's Omega constant, beyond the first few. Omega is the probability that a random Turing machine will halt. Whatever value it has, it neccessarily has.
(One might say more charitably that these definitions are only categorizing knowledge and say nothing about non-knowledge. If so, they mislead, and also make a subtler mistake. Neccessary truths are not a special type of knowledge, they are topic of knowledge)
One can understand why the mistake is made. Epistemology, the branch of philosophy about how we know what we know, is not looking for a way to assign untouchable status to what seems its most certain knowledge.