Hundreds of years ago, calculus, while still very useful, lacked rigor. It was later made rigorous and put on a sound basis by Dedekind and others. Another example is intuitive set theory, which was made rigorous by Zermelo. Are there any present day examples of mathematics that lacks a solid foundation, but which could possibly be made rigorous in the future?
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1Physics gives examples. I think there are calculations in quantum field theory that provide a lot of insight into the physics but haven't yet been elevated to the status of theorems. – Jalex Stark May 02 '18 at 22:41
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I'm not sure this counts, but you might be interested in this article on issues in symplectic geometry. – Noah Schweber May 02 '18 at 23:26
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There's no agreed upon definition of weak n-category, but there's a lot of theory about how they "should" behave. – Oscar Cunningham May 04 '18 at 20:31