Quote from Wigner’s <a href=https://math.dartmouth.edu/~matc/MathDrama/reading/Wigner.html>The Unreasonable Effectiveness of Mathematics in the Natural Sciences</a>:
<ul>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.</ul>
Doyle has Sherlock Holmes say:
<ul>When you have eliminated the impossible, whatever remains, however improbable, must be the truth.</ul>
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 <a href=../MathPhys/Formal/believe.html>Banach-Tarski Paradox</a>.
It also yields the greatest and many of the most useful mathematical achievements.