You might think that a satisfactory account for mathematical knowledge would have to at least prevent us from ever winding up with/keeping contradictory mathematical beliefs.

But this is wrong for two reasons:

a) OLD REASON: "We" did have contradictory beliefs about mathematics at various periods in history (naive set theory, inconsistent reasoning about infinity).

b) NEW REASON: Many people have contradictory beliefs about baldness (as brought out by the sorites paradox), but this doesn't prevent them from knowing many things about baldness, who is bald etc.

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