Some old traditions in mathematics and in mathematical education (Q1962978)

The author takes four cases. 1) Many applications of the calculus assume either left- or right-differentiability of a function, according as the interpretation of rate of change refers to the past or to the future respectively. 2) Similarly, temporal causality requires that velocity be conceived as a function of a function of time, the latter fulfilling certain conditions to ensure temporal priority. 3) Methods of optimization are compromised in generality of application by side or other conditions; but some methods manage to sidestep such restrictions. 4) `Integral calculus can be taught before differential calculus' (p. 16), and indeed there is a scrappy history of such procedures. It was John Bernoulli who imposed the priority of the differential, \textit{not} that the derivative was developed first (p. 16); in Leibniz `\(d\)' and `\(\int\)' are together, and on a par.