Mathematical ideas that took long to define rigorously It often happens in mathematics that the answer to a problem is &quot;known&quot; long before anybody knows how to prove it.  (Some examples of contemporary interest are among the Millennium Prize problems: E.g. Yang-Mills existence is widely believed to be true based on ideas from physics, and the Riemann hypothesis is widely believed to be true because it would be an awful shame if it wasn't.  Another good example is Schramm–Loewner evolution, where again the answer was anticipated by ideas from physics.)
More rare are the instances where an abstract mathematical &quot;idea&quot; floats around for many years before even a rigorous definition or interpretation can be developed to describe the idea.  An example of this is umbral calculus, where a mysterious technique for proving properties of certain sequences existed for over a century before anybody understood why the technique worked, in a rigorous way.
I find these instances of mathematical ideas without rigorous interpretation fascinating, because they seem to often lead to the development of radically new branches of mathematics$^1$.  What are further examples of this type?
I am mainly interested in historical examples, but contemporary ones (i.e. ideas which have yet to be rigorously formulated) are also welcome.


Footnote: I have some specific examples in mind that I will share as an answer, if nobody else does.
The notion of probability has been in use since the middle ages or maybe before. But it took quite a while to formalize the probability theory and giving it a rigorous basis in the midst of 20th century. According to wikipedia:


  There have been at least two successful attempts to formalize probability, namely the Kolmogorov formulation and the Cox formulation. In Kolmogorov's formulation, sets are interpreted as events and probability itself as a measure on a class of sets. In Cox's theorem, probability is taken as a primitive (that is, not further analyzed) and the emphasis is on constructing a consistent assignment of probability values to propositions. In both cases, the laws of probability are the same, except for technical details.
  
  There are other methods for quantifying uncertainty, such as the Dempster–Shafer theory or possibility theory, but those are essentially different and not compatible with the laws of probability as usually understood.
The Egyptians and the Babylonians knew a lot of mathematical facts (e.g. the Pythagorean theorem, the quadratic formula, the volumes of prisms and pyramids, etc.) which were later proved by the Greeks.

Also, Pascal and Fermat used mathematical induction and the well ordering of the naturals (in the form of infinite descent) 250 years before Peano formalized the axioms defining the natural numbers.