Well, Feynman formulates this as "along every possible path from the emitter to the detector, calculate a complex number amplitude" then "Add up the amplitudes for all the paths" and then "take the absolute magnitude of the result"

The problem for a classically trained mathematician is that "every possible path from the emitter to the detector" is too big of a class of things. It's not clear that the "set of all functions that take on the value f(0)=(0,0) and f(t) = (1,d) where d is the y location of the detector and 1 is the x location of the detector" is a well defined set, or even if it is, that it is physically relevant. There are for example functions representing a photon traveling at a billion times the speed of light out to Jupiter and then swinging around and coming back in time for tea at the detector...

The early days of doing these calculations people used simply a grid of points and photons traveled from the origin to the first set of grid points, and then from wherever they landed on that grid to a point at the next set of grid points, and soforth. At no time could the photon fly off to Jupiter between infinitesimal timepoints.

The important point is that the technique gave PHYSICALLY CORRECT frequencies of observations. So to the extent that "along every possible path from the emitter to the detector" is not a mathematically well posed idea, it's probably because the well posed idea is actually something more like "along each of a nonstandard number of paths constructed in such and such a way" is the correct posing of the problem. Thinking in terms of nonstandard construction is usually a much better way to develop a scientific model. If there is no mechanism whereby we could construct a path involving some strange Cantor set... it's the case that the strange Cantor set no matter how mathematically pure it is, is not relevant to the modeling problem.

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