The purpose of this question is to ask about the role of mathematical rigor in physics. In order to formulate a question that can be answered, and not just discussed, I divided this large issue into five specific questions.
Update February, 12, 2018: Since the question was put yesterday on hold as too board, I ask future to refer only to questions one and two listed below. I will ask a separate questions on item 3 and 4. Any information on question 5 can be added as a remark.
What are the most important and the oldest insights (notions, results) from physics that are still lacking rigorous mathematical formulation/proofs.
The endeavor of rigorous mathematical explanations, formulations, and proofs for notions and results from physics is mainly taken by mathematicians. What are examples that this endeavor was beneficial to physics itself.
What are examples that insisting on rigour delayed progress in physics.
What are examples that solid mathematical understanding of certain issues from physics came from further developments in physics itself. (In particular, I am interested in cases where mathematical rigorous understanding of issues from classical mechanics required quantum mechanics, and also in cases where progress in physics was crucial to rigorous mathematical solutions of questions in mathematics not originated in physics.)
The role of rigor is intensely discussed in popular books and blogs. Please supply references (or better annotated references) to academic studies of the role of mathematical rigour in modern physics.
(Of course, I will be also thankful to answers which elaborate on a single item related to a single question out of these five questions. See update)
Related Math Overflow questions:
Examples-of-non-rigorous-but-efficient-mathematical-methods-in-physics (related to question 1);