“Thus, a schism appeared between the theory and the practice of the calculus as the level of rigour in the calculus was raised: the foundationalists had one set of rules, the practitioners another. The situation has persisted to this day, with quite the unfortunate and unnecessary confusions for students. It is common experience for them to learn in calculus lectures that infinitesimally small differentials do not exist, but to use them constantly in the mathematical physics lectures. While Eulerian calculus is not rigorous, it should be taught for what it is: a powerful tool for the analysis of physical and geometric phenomena, which has left its considerable mark on the conceptions, terminology, and notations of later presentations of the subject. As things are, the treatment in textbooks is unsatisfactory. Some basically follow Cauchy’s practice of notating the derivative by \(f^\prime\) and defining the differential by some equivalent of this, while others notate the derivative by the single symbol \(\dfdx{y}{x}\) and omit differentials altogether; and neither treatment warns the reader of the existence of the other. Further, both treatments give a prime place to limits without explaining why the standard of rigour and generality obtainable from this very difficult concept is desirable in the first place, or what kinds of less rigorous approaches are being superseded.” (From Calculus to Set Theory 1630–1910, p. 116)
“This modern limit–theory cannot be recommended to beginners of the calculus. The physical notion of ’velocity’ and the ’slope of a curve’ must be retained as great aids to the young student.” (from Grafting the Theory of Limits on the Calculus of Leibniz, The American Mathematical Monthly, Vol. 30 #5)
“To think that there is one calculus for the pure mathematician and another for the physicist, the engineer, the geometer, or the cultured layman, is to fail to appreciate that that which is most central in the calculus is its quantitative character, through which it measures and estimates the things of the world of our senses. And instruction in the calculus that does not point out — not merely at the beginning or at the end, but all through the course — this close contact with nature, has not done its duty by the student.” (from The Calculus in our Colleges and Technical Schools)
