One of the best books I ever picked up second hand  begins with the very old problem of the vibrating string, as studied by Euler
and contemporaries. In the author's historical opinion, the young Lagrange
took on board both (i) Euler's preference for introducing the concept of non-differentiable
functions and (ii) Euler's confusion over, and lack of enthusiasm for, expressing general solutions in terms of differentiable
periodic functions. In other words, Lagrange was happy to work with generalised functions, but unhappy about Euler's Leibnizian
infinitesimals. This prompted Lagrange to reinterpret calculus using Taylor series, in his own words "independent of all metaphysics"
. Nobody can know exactly what he meant by this, but perhaps it is a reference to the philosophy
These events must strike a chord with anyone educated in physics in the 20th century, heavily influenced by the thinking of Lagrange. Of course, modern physicists also have Fourier
analysis, but this was a later development. In fact, Fourier's first paper on the heat equation
(1807) was never published because of Lagrange's objections to it.
 I. Grattan-Guinness, The development of the foundations of mathematical analysis from Euler to Riemann
, MIT Press (1970)