I did not mean my archive pieces this week to all be patching up stuff omitted from my Leibniz essay. But that essay touched a bunch of mathematical points, some of which I already had essays about. One of them was published in the 2018 A-to-Z. We really, really want to use the idea of an infinitesimally tiny change to understand how calculus works. But we can’t do that with logical rigor. (Unless we redefine what “real numbers” are, and some mathematicians go in for that.) Ah, but: what if we could get all the things these infinitesimals give us without having to do anything weird? And that is the modern idea of the limit, which we sorted out about 150 years ago and are pretty satisfied with.
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