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Wei_Dai comments on A Request for Open Problems - Less Wrong
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Here are some papers I found recently that seem to represent the state of the art on the issue of how to deal with uncertainty in mathematics. None of them really get very far beyond "let's try to apply probability theory", but at least the problem is not being completely ignored by mathematicians and philosophers.
One could say that most of math is already about uncertainty: when you have a system and ways of refining it, it is in a way a form of applying knowledge to resolve uncertainty. For example, applying a function to a parameter, or combining morphisms. A lot of analysis is about approximation or representing systems that expect future observations. It is a very narrow sense of "dealing with uncertainty" that would require going to the fringe.
I don't understand the point of your comment. It should have been clear from the context that by "dealing with uncertainty in mathematics" I did not mean things like proving or disproving a conjecture, thus resolving its uncertainty, but rather how to make bets involving mathematical statements that we don't know how to either prove or disprove. Are you saying that the latter is not an important problem, or just that you don't like that I'm using the phrase "dealing with uncertainty in mathematics" to refer to it?
You don't have to resolve all of uncertainty in one go. For example, you could restrict a function to part of a domain, thus deciding that it is only this part that you are interested in, instead of the whole thing.
What you seem to mean is non-rigorous methods for making uncertain conclusions about mathematical structures. It is about dealing with uncertainty about mathematics on non-mathematical level of rigor. Correct?
Yes, something like that, except that "non-rigorous" seems too prejudicial. Why not just "methods for making uncertain conclusions about mathematical structures", or "dealing with uncertainty about mathematics"?