Take the tour ×  Recently, I realized that there are some famous
problems in mathematics whose solution depended heavily on the
right formulation of an intuitive concept.  For example, there
was no precise definition of an algebraic integer before
Dedekind. As Milne says in his book on algebraic number theory,
Euler's proof of Fermat's Last theorem for the exponent $3$ does
only become correct when you replace $\Bbb Z[\sqrt{-3}]$ by
$\Bbb Z[\frac{1+\sqrt{-3}}{2}]$.  Do you know any other examples
of theories where the correct formulation of a concept was an
important step in the evolution of that theory?  (As usual, one
example per answer.)  share|improve this question community
wiki? –  Jason DeVito Nov 25 '12 at 20:55 add comment  2 Answers
up vote 1 down vote accepted  How about analysis? The idea of
continuity wasn't well defined at the foundation of analysis,
and this leads to many interesting discrepancies stemming from
different definitions. The most famous of which is probably
"Darboux continuity" vs