Euler's work on divergent series was guided by computational procedures, rather than any definition of the "value" of such a series. E.g., he was happy to have half a dozen procedures that indicated that $0!-1!+2!-3!+\dots$ equals 0.59637..., even though he had no definition of what an expression like $0!-1!+2!-3!+\dots$ means.

This style of mathematics is not very common nowadays; some purists would say you're not even doing math if you can't give a definition of the objects you're studying. But I don't think many people would disparage the work of Renaissance algebraists who studied complex numbers without benefit of the modern concept of complex numbers as ordered pairs.

What are some good modern examples of mathematical research in which one is guided by computational procedures that seem to give mutually consistent results, rather than by definitions of what the procedures are purportedly computing? (I suspect that quantities that physicists computed in the late 20th century using ad hoc regularization techniques would be one class of examples.)

Feel free to make this a Community Wiki (I'm still not sure when this is appropriate).