Category Archives: Calculus

More of What It Is

I say that mathematics is the deductive science; and yet there would seem to be mathematicians who disagree. I take up two cases here.

From Archimedes, De Planorum Aequilibriis,
in Heiberg’s edition (Leipzig: Teubner, 1881)

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Because Herman Wouk was going to put physicists in a novel, Richard Feynman advised him to learn calculus: “It’s the language God talks.” I think I know what Feynman meant. Calculus is the means by which we express the laws of the physical universe. This is the universe that, according to the mythology, God brought into existence with such commands as, “Let there be light.” Calculus has allowed us to refine those words of creation from the Biblical account. Credited as a discover of calculus, as well as of physical laws, Isaac Newton was given an epitaph (ultimately not used) by Alexander Pope:

Nature and Nature’s laws lay hid in night:
God said, Let Newton be! and all was light.

I don’t know, but maybe Steven Strogatz quotes Pope’s words in his 2019 book, Infinite Powers: How Calculus Reveals the Secrets of the Universe. This is where I found out about Wouk’s visit with Feynman. I saw the book recently (Saturday, February 22, 2020) in Pandora Kitabevi here in Istanbul. I looked in the book for a certain topic that was of interest to me, but did not find it; then I found a serious misunderstanding.

book cover: Steven Strogatz, Infinite Powers Continue reading

Learning mathematics

This is mostly reminiscences about high school. I also give some opinions about how mathematics ought to be learned. This article originally formed one piece with my last article, “Limits.”

I learned calculus, and the epsilon-delta definition of limit, in Washington D.C., in the last two years of high school, in a course taught by a peculiar fellow named Donald J. Brown. The first of these two years was officially called Precalculus Honors, but some time in that year, we started in on calculus proper. Continue reading


This is about limits in mathematics: both the technical notion that arises in calculus, and the barriers to comprehension that one might reach in one’s own studies. I am going to say a few technical things about the technical notion, but there is no reason why this should be a barrier to your reading: you can just skip the paragraphs that have special symbols in them.

Looking up something else in the online magazine called Slate, I noted a reprint of an article called “What It Feels Like to Be Bad at Math” from a blog called Math With Bad Drawings by Ben Orlin. Now teaching high-school mathematics, Mr Orlin recalls his difficulties in an undergraduate topology course. His memories help him understand the difficulties of his own students. When students do not study, why is this? It is because studying makes them conscious of how much they do not understand. They feel stupid, and they do not like this feeling. Continue reading