Not in pure mathematics but in applied mathematics we have the case Ludwig Boltzman, the Austrian physicist (also the founder of the Austrian Mathematical Society) whose greatest achievement was in the development of statistical mechanics. He spent a life time trying to defend the now famous equation

$$ S = k\log W $$

Boltzmann's mentor and colleague Josef Loschmidt criticized Boltzmann's demonstration of entropy increase on the grounds that dynamical laws are reversible. If all the particles could be turned around exactly (or if time could be reversed), Boltzmann's work indicated the entropy should decrease, violating the second law. Eventually he committed suicide out of depression.

Today the above equation is one of the most important and fundamental equations of science.