Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

430 alasdair urquhartin his mathematical autobiography, Laurent Schwartz, who was eventually tofind the generally accepted interpretation of the delta function as a distribution,remarks:I believe I heard of the Dirac function for the first time in my second year atthe ENS. I remember taking a course, together with my friend Marrot, whichabsolutely disgusted us, but it is true that those formulas were so crazy from themathematical point of view that there was simply no question of accepting them.(Schwartz, 2001, p.218)This attitude is logically impeccable, but it fails to account for the success thatthe physicists attained with their weird ideas. In other words, it is fruitless andunhelpful.There is a second attitude, also of an extreme type, that consists in saying—‘Well,there’s a contradiction, so what? As long as it works, that is all thatmatters’. This is close to the physicists’ own attitudes, since physicists are quitehappy to perform purely formal calculations as long as something resemblingthe right answer comes out in the end. I would also consider the idea ofparaconsistent logic to be somewhat along the same line. Although the idea ofa true contradiction perhaps makes some kind of sense, the idea on the wholedoes not appear to be very fruitful.The third attitude, which I might call the philosophical, or critical attitude,consists in a more nuanced approach to these scandalous invasions of themathematical universe. The idea here is, on the one hand, to maintain theview that the physicists’ calculations are largely correct, but that the objectsthey think they are talking about are in fact of a different nature.Out of this third, or critical, attitude, applied to the case of the Dirac deltafunction, comes the modern and very fruitful theory of distributions of LaurentSchwartz. It’s interesting to see what this move consists in. The idea here is thatthe Dirac delta function is not a function at all, but a distribution, thatistosay,a linear operator on a certain class of functions (the similarity of the strategyfollowed here to that of Rota’s in the case of the umbral calculus is quitestriking). The reason this works is that the delta function, as used by physicists,only appears in a certain restricted type of computation, and it is possibleto redefine this computation using the operators so that all of the physicists’calculations become perfectly rigorous. In this way, the mathematicians canenjoy the best of both worlds.Incidentally, the introduction of infinitesimals makes possible the explicationof the Dirac delta function in a different and perhaps more intuitive way.We simply define a Dirac delta function as a function with a unit integral,all of whose mass is concentrated within an infinitesimal neighbourhood of