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