[wongarsu]

Half of modern mathematics is basically assuming something is true and then trying to work forwards and backwards to show that it is true. But that requires a lot of rigor. If you hand-wave a single step you are just exercising an advanced version of confirmation bias. Working forwards and sanity checking your result is a lot more forgiving.

[adrianN]

Guessing something and then confirming it is completely different from knowing the answer and then working towards it, especially for the kinds of problems that students solve in school. For the kind of proofs that working research mathematicians produce the hard part is not assuming that a Lemma is true, but already coming up with the correct definitions to formulate the Lemma in the first place is a delicate task.