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othercriteria comments on Strong intutions. Weak arguments. What to do? - Less Wrong
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On the other hand, working on topology for a while gives one the meta-intuition that one should check reasonable sounding statements on the long line, the topologists's sine curve, the Cantor set, etc.
Or better, one's idea of what constitutes a "reasonable-sounding statement" in the first place changes, to better accommodate what is actually true.
(Checking those examples is good; but even better would be not to need to, due to having an appropriate feeling for how abstract a topological space is.)
Completely agreed. Part of this might look like a shift in definitions/vocabulary over time. Coming to topology from analysis, sequence felt like a natural way to interrogate limiting behavior. After a while though, it sort of became clear that thinking sequentially requires putting first-countability assumptions everywhere. Introducing nets did away with the need for this assumption and better captured what convergence ought to mean in a general topological spaces.
Sure. But we don't have much in the way of actual AI to check our intuitions against in the same way.