probabilistic knowledge of mathematical truth
Abram Demski
The following post about probabilistic knowledge of mathematics was interesting:
http://antimeta.wordpress.com/2009/04/22/probabilistic-proofs-of-undecidable-sentences/
The author concludes that we cannot gain probabilistic knowledge of
claims that are independent of our axioms.
The argument, however, appears to specifically allow us to be
justified in ascribing arbitrarily high probability without "knowing".
The example of lottery tickets is given: according to their definition
of knowledge, we do not "know" that a lottery ticket is a losing
ticket, even though we should rationally ascribe an extremely low
probability to the ticket being a winner. I'd tend to think that the
question of what probability should be ascribed to mathematical
statements based on evidence is more important. So, although their
conclusion is to admit probabilistic proofs only in very limited
situations, it seems that restriction doesn't apply to using such
methods to ascribe probabilities, only to saying we *know* things.
So, I still tend to think that we can have probabilistically justified
beliefs for statements not decided by our axioms, in the sense of
ascribing probabilities to such statements.
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Abram Demski
http://dragonlogic-ai.blogspot.com/