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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.)

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community wiki? –  Jason DeVito Nov 25 '12 at 20:55
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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. "regular continuity" of functions. (By regular continuity I mean continuity in the sense that it is defined nowadays.)

A function $f: \mathbb{R} \rightarrow \mathbb{R} $ is called Darboux continuous if for any $[x,y]$ , for each $z$ in the interval bounded by $f(x)$ and $f(y)$ there exists $\theta \in [x,y] $such that $f(\theta) = z$. i.e. the function obeys the "intermediate value property". It then follows from the intermediate value theorem that any continuous function is Darboux continuous. The theorem of Darboux states that that if $f$ is differentiable with derivative $f'$ then $f'$ is Darboux continuous. Darboux himself also gave examples of continuous functions (in the usual sense) with discontinuos derivative, hence the derivatives are Darboux continuous but not continuous, so the two definitions are not the same.

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Conways base 13 function (en.wikipedia.org/wiki/Conway_base_13_function) is a particlarly nice example. –  Dominik Nov 25 '12 at 21:51
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Maybe the parallel postulate / different geometries fall under this category: Until one thought about the problem differently, i.e. that the axioms of geometry are not evident truths about THE geometry, the research on the parallel postulate was fruitless. Only after taking a different view on what axioms and gemoetries are, one could come up with variants of the postulate (and thus show that it is independant of the other axioms of geometry).

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