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This is a bit of a soft question, but is there a difference between rigorous and formal, and if so, what is it? I have seen those terms used interchangeably, and I use them interchangeably myself, but is there a distinction between the two?

user107952
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My soft answer to a soft question.

Formal is relatively clear. A proof is formal if it is a step by step transformation from axioms and hypotheses to conclusion using precisely specified transformation rules. So essentially machine checkable.

My personal take on rigor is that a proof is rigorous if it is sufficiently detailed so that a professional mathematician sufficiently familiar with the area can check its correctness.
No handwaving.

In a sense, rigor varies with time and place. Euclid and Euler were rigorous enough when they wrote, although their arguments don't meet modern standards. So "sufficient unto the day is the rigor thereof".

For students, I ask that their proofs convince me that they have convinced themselves for good reasons. Students who try to write very formal proofs often fail at it.

Ethan Bolker
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    Soft comment: sometimes the word "formal" is used for manipulations that are not rigorous. The Feynman path integral or solving an ODE by separation of variables might be examples. To give tribute to rigor: The former leads to results that can be tested by experiment. The results of the latter can be checked if they solve the ODE. – Kurt G. Aug 24 '23 at 06:40