I would like to ask if there is a formal definition of the word "rigorous". I think I have a grasp of the concept of "formal stuffs" but I feel like the word "rigorous" is NOT objective enough.
eg. When reading combinatoric proofs, I always have difficulty in understanding the logic behind them as they often lie on the obscure boundary between logic and common sense.
"Rigorous," in contrast with "formal," is a fuzzy term. A formalist might say that an argument should be thought of as rigorous insofar as it would be straightforward to turn it into a formal (e.g. LEAN-checkable) proof, but broadly speaking an argument is rigorous iff it "meets the standards" of the mathematical community. Those standards might change, of course, and what is considered rigorous one day might not be considered rigorous the next; however, since (for better or worse) most mathematical arguments are not machine-checkable, this subjective distinction is important enough to deserve a name.