Having mentioned Euler and twentieth century physicists in the question, Pierre Cartier's '[Mathemagics? (A Tribute to L. Euler and R. Feynman)][1]'
 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][2]


  [1]: http://www.kurims.kyoto-u.ac.jp/EMIS/journals/SLC/wpapers/s44cartier1.pdf
  [2]: http://math.ucr.edu/home/baez/counting/