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.)



Procedure-based (as opposed to definition-based) concepts
Having mentioned Euler and twentieth century physicists in the question, Pierre Cartier's '[Mathemagics? (A Tribute to L. Euler and R. Feynman)](http://www.kurims.kyoto-u.ac.jp/EMIS/journals/SLC/wpapers/s44cartier1.pdf)'
should provide good cases for you.



> 
> My thesis is: there is another way of doing mathematics, equally
> successful, and the two methods should supplement each other and not
> fight.
> 
> 
> This other way bears various names: symbolic method, operational
> calculus, operator theory . . . Euler was the first to use such
> methods in his extensive study of infinite series, convergent as well
> as divergent. The calculus of differences was developed by G. Boole
> around 1860 in a symbolic way, then Heaviside created his own symbolic
> calculus to deal with systems of differential equations in electric
> circuitry. But the modern master was R. Feynman who used his diagrams,
> his disentangling of operators, his path integrals . . . The method
> consists in stretching the formulas to their extreme consequences,
> resorting to some internal feeling of coherence and harmony.
> 
> 
> 


Presumably you already know of John Baez [The Mysteries of Counting: Euler Characteristic versus Homotopy Cardinality](https://math.ucr.edu/home/baez/counting/)


