For a more current example of a mathematical technique that preceded formalization by a considerable amount, consider renormalization. Particularly renormalization over a calculation that takes place over Feynman diagrams.<p>For decades physicists were happily using this to predict experiment, while mathematicians were tearing their hair out trying to make some formal sense of this, even if only in a limited context. I'd have to do some poking around to find out whether mathematicians are happy about it yet, even though the idea is older than I am.