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Finally, re high-school or undergrad “calculus”, I’m tempted to just quote Halmos (one of the foremost expositors of “higher mathematics”—that is, mathematics—of the 20th century) from his classic “How to write mathematics”[1] ... > [T]here are many books that violate the principle of having something to say by trying to say too many things. Teachers of elementary mathematics in the U.S.A. frequently
complain that all calculus books are bad. That is a case in point. Calculus books are bad because there is no such subject as calculus; it is not a subject because it
is many subjects. What we call calculus nowadays in the union of a dab of logic
and set theory, some axiomatic theory of complete ordered fields, analytic geometry and topology, the latter in both the “general” sense (limits and continuous functions) and the algebraic sense (orientation), real-variable theory properly so called (differentiation), the combinatoric symbol manipulation called formal integration, the first steps of low-dimensional measure theory, some differential geometry, the first steps of the classical analysis of the trigonometric, exponential, and logarithmic functions, and, depending on the space available and the personal inclinations of the
author, some cook-book differential equations, elementary mechanics, and a small assortment of applied mathematics. Any one of these is hard to write a good book
on; the mixture is impossible. ... But I should probably explain the general picture somewhat for the reader who’s unable to follow the deluge of terminology here. The problem with calculus, in terms of my parent comment, is that it hasn’t been a singular cluster or clique of ideas in any working mathematician’s association graph since the time of Euler; the list of topics that is presented under that name has never been such—it’s both anachronistically rich in trying to include insights from Weierstrass’s rigour to Tychonoff’s point-set topology and anachronistically poor in avoiding Newton’s motivation from algebraic geometry and differential equations or Euler’s motivation from complex analysis and homotopy theory. So, if we don’t have a justification from either history or state of the art, why do we insist on this low-resolution camrip of Newton and Leibniz’s writings sprinkled with an arbitrary selection of later work? I don’t know, but I suspect that this is simply the best people could fit into the allotted time when they last tried to incorporate actual, live mathematics into the general curriculum at the beginning of the 20th century, after a hundred years of vulgarization erosion and haphazard contradictory pushes for modernization and practicality. Why should you even care how mathematicians think now or thought in the past? You don’t have to, of course, though in that case I would much prefer that you avoid diluting the brand “mathematics” by attaching it to the result[2]. But a course should be about something; either you’re teaching mathematics as a matter of culture and way of thought (you sure as hell don’t have time to teach it as a field of study), or you’re reaching for a particular application (but you better know which one, because an applied course left adrift soon becomes a patchwork zombie). Either way those funny mathematicians in their ivory towers can be of use for you, because they haven’t simply spent all these years playing with impractical abstractions: they broke every subject including old-school calculus into patterns, distilled those patterns into their most elementary possible forms in the form of impractical abstractions, then went back and rearranged their understanding of each subject to make the forms they found more evident, then did it again and again. For decades. Unfortunately for us lovers of simple answers, their conclusion was, “calculus is not a single thing”. Is there a course or two struggling to come out in the general vicinity so labelled? Yes, but while I have some speculations as to what they are, I don’t have a actual plan—that takes years of regular experimentation with and on classes of real students. It probably includes more varied topics compared to the current approach, certainly some formal series and a glimpse beyond dimension one, probably even a bit of linear algebra as a geometric foundation for it all. (Also a pony while we’re at it.) Should we teach them when combinatorics is infinitely more accessible and probability theory is infinitely more practical? I think so, if not for the cultural significance and life wisdom[3], then because any physics worth speaking of is practically inexpressible without it. Sorry, I don’t have an answer for you; if I had, it would’ve already been one of those many, many books. (Took me until the middle of the second of my two answers to realize they both essentially say “Your question is ill-posed” from different points of view; hope they are a bit more helpful than just that.) [1]: https://www.mathematik.uni-marburg.de/~agricola/material/hal... [2]: https://www.maa.org/external_archive/devlin/devlin_03_08.htm... [3]: <https://www.basicbooks.com/titles/jason-wilkes/burn-math-cla...>, <https://www.blackdogandleventhal.com/titles/ben-orlin/change...> |