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It seems self-evident that mathematics must keep uncertainty at arms length. For example, to deal with probability mathematics establishes a set of assumptions. The assumptions themselves are not probabilistic - i.e. the "laws" of probability are assumed to be true "all the time", not randomly true or false. As another example, consider multi-valued logics. As far as I can tell when people reason about a system of "multi-valued" logic they use ordinary logie - i.e. they prove thing about it in terms of statements that are assumed to be one of "true" or "false". So the basic framework for conducting mathematics is "classical". If it wants to deal with a a type of uncertainty, it assumes there are statements about the uncertainty that are themselves certainly true.
Mathematics can (traditionally) be discussed without discussing the mathematicians who are doing it. But in physics, the physicist might acknowledge himself as a physical system. Is there any sort of paradox or inherent limitation when physicists trying to discuss uncertainty in nature? They use mathematics, so the discussion takes place as if it were implemented by a "classical" physical system.
(I suppose an abstract mathematical paradox wouldn't disturb physicists. This is more a question of whether a mathematician could get agitated about what physicists do.)
Mathematics can (traditionally) be discussed without discussing the mathematicians who are doing it. But in physics, the physicist might acknowledge himself as a physical system. Is there any sort of paradox or inherent limitation when physicists trying to discuss uncertainty in nature? They use mathematics, so the discussion takes place as if it were implemented by a "classical" physical system.
(I suppose an abstract mathematical paradox wouldn't disturb physicists. This is more a question of whether a mathematician could get agitated about what physicists do.)