Quote from Wignerâ€™s The Unreasonable Effectiveness of Mathematics in the Natural Sciences:
The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible. That his recklessness does not lead him into a morass of contradictions is a miracle in itself: certainly it is hard to believe that our reasoning power was brought, by Darwin's process of natural selection, to the perfection which it seems to possess.

Doyle has Sherlock Holmes say:
When you have eliminated the impossible, whatever remains, however improbable, must be the truth.

This is actually not a very good precept in real life, but in mathematics it shines.
It takes you places that disconcert some mathematicians, such as the Banach-Tarski Paradox.
It also yields the greatest and many of the most useful mathematical achievements.