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

I shall explain; but first I want to look at such words as promote the book, even on Strogatz’s own website:

A brilliant and endlessly appealing explanation of calculus—how it works and why it makes our lives immeasurably better.

Without calculus, we wouldn’t have cell phones, TV, GPS, or ultrasound. We wouldn’t have unraveled DNA or discovered Neptune or figured out how to put 5,000 songs in your pocket.

How much would I mind not having any of those things? A year ago, I did have an MRI; this revealed the cause of a problem, which could therefore be solved (by surgery). I don’t care about the planet called Neptune, and I don’t carry 5,000 songs in my pocket. I have a lot of songs in my head, put there mainly by radio broadcasts of recordings when I was younger. The songs were put inside me, only in a manner of speaking; I remember them because I listened to them. I might be better off, had all of my musical impressions come from live performances; I don’t dwell on this though.

Something that would make my life immeasurably better would be an end to certain things:

  1. the flouting or abuse of law for the sake of imprisoning one’s supposed enemies or just enriching oneself and one’s supposed friends;

  2. war in Syria and elsewhere;

  3. habitat destruction, for farms in Brazil, oil pipelines in Canada, gold mines in Turkey;

  4. production of nuclear waste;

  5. burning of fossil fuels.

Concerning the pollution and destruction involved in spreading and maintaining a certain way of life, I need not be told that I would like the alternatives even less. The point is that there are problems in the world whose solutions are not obvious. If you think they are obvious, then you still have the problem of getting others to agree with you. If how to solve this problem were obvious, you would just do it, thereby solving the former problems as well.

The last two or three problems on my list may have such solutions as are described as technological. Thus they would involve calculus. I don’t think the first two problems have technological solutions. Technology may increase material wealth, but this does not automatically make us more liberal and peaceful.

I wrote that, then received email notification of the latest issue of the Nonzero Newsletter. I looked at the previous issue in my last post, “Evolution of Reality.” This week, Robert Wright makes a remarkable proposal based on one by Bertrand Russell. In what Wright says below, I suppose the phrase “an AI” is a slang expression for a computer program that has the kind of unpredictability referred to as “artificial intelligence”:

Is it too far-fetched to think that someday an AI could adjudicate international disputes? Fifteen years ago I might have said it was. But each year Google Assistant seems to do a better job of understanding my questions and answering them. And each year more and more highly skilled American workers see computers as a threat: paralegals, radiologists, sports writers … Can we be so sure that judges will be forever immune? Is it crazy to imagine a day when an AI can render a judgment about which side in a conflict started the trouble by violating international law?

It’s not crazy to think a computer program could make such a judgment. What’s crazy is to think the judgment would automatically be respected. Wright imagines that a computer algorithm could eliminate biased judgments. I think it could not; but here let me just defer to Cathy O’Neil’s Weapons of Math Destruction: How big data increases inequality and threatens democracy (2016). I would only add that, even if a program elimited bias to your satisfaction, you would still have to convince everybody else that it did.

book cover: O'Neil, Weapons of Math Destruction

That’s a problem; and apparently it is a problem, just to get people to see that it is a problem.

This is where mathematics come in—and Jane Austen.

I have seen it asserted, and I agree, that the world can be saved by Jane Austen, because she teaches us that other persons have an inner life. Calculus can teach us something similar, just by being mathematics.

Mathematics is both personal and universal:

  • personal, for giving us the right to decide for ourselves what is true;

  • universal, for giving us the obligation

    • to share what he have learned,

    • to seek the corroboration of others, and

    • to reconsider our work if we find disagreement.

You cannot browbeat people into accepting your theorem; much less can you imprison them. If your theorem is correct, you have to find the right words to explain this; if not, you have no alternative but to go back to the drawing board. This is a message that I should like to see conveyed by a popular book (not to mention the textbooks used to train teachers).

I have been alluding to the remarkable instrument of mathematical proof. This is how we confirm, or perhaps refute, our hypotheses and suspicions about what it true. I have tried to distinguish mathematical proof from physical proof in two posts:

Steven Strogatz may not be particularly interested in the distinction between mathematical and physical proof; for he is an applied mathematician. He says what this means to him in an online excerpt from his book:

As should be obvious by now, I’ll be giving an applied mathematician’s take on the story and significance of calculus. A historian of mathematics would tell it differently. So would a pure mathematician. What fascinates me as an applied mathematician is the push and pull between the real world around us and the ideal world in our heads. Phenomena out there guide the mathematical questions we ask; conversely, the math we imagine sometimes foreshadows what actually happens out there in reality. When it does, the effect is uncanny.

I suppose the “uncanny effect” is what Wigner alludes to in “Unreasonable Effectiveness of Mathematics in the Natural Sciences,” a 1960 article that I examined in “Effectiveness.”

It is good to note, as Strogatz does, that there are different ways of going about one’s business. To insist that there was only one true path would be to deny the unity of the universe, as I have argued in yet another post.

I do have my doubts about Strogatz’s understanding of the real and the ideal. He continues:

To be an applied mathematician is to be outward-looking and intellectually promiscuous. To those in my field, math is not a pristine, hermetically sealed world of theorems and proofs echoing back on themselves. We embrace all kinds of subjects: philosophy, politics, science, history, medicine, all of it. That’s the story I want to tell—the world according to calculus.

Thinking that the adjective “hermetic” alluded to the monastic hermit, I was going to suggest that mathematicians are not really cloistered. Some of us may come close to living like Carthusian monks, but I think our purpose is rather different.

It turns out that the word “hermetic” is an irregular formation alluding to Hermes Trismegistus quâ originator of alchemy. I don’t know whether Strogatz intends such an allusion. In any case, I say again that what we do in mathematics, even “pure” mathematics, cannot remain sealed off, but must be shared.

As one of the greatest mathematicians ever to walk the Earth, though he did so more than two thousand years ago, Archimedes shared what he found. Enough copyists, particularly monks, thought his work worth preserving that we still have a good deal of it. Strogatz sees the origins of calculus in Archimedes, but I think he also has a misunderstanding. In a passage I noted in Pandora bookshop, on page 33, Strogatz writes,

It may seem strange to modern minds that pi doesn’t appear in Archimedes’s formula for the area of a circle, A = rC / 2, and that he never wrote down an equation like C = πd to relate the circumference of a circle to its diameter. He avoided doing all that because pi was not a number to him. It was simply a ratio of two lengths, a proportion between a circle’s circumference and its diameter. It was a magnitude, not a number.

Page 33 of Strogatz

I have a three-fold response:

  1. If Archimedes avoided writing down modern equations, this suggests, probably wrongly, that he somehow was aware of our modern understanding, but didn’t like it.

  2. Ratios, proportions, and magnitudes are not the same kind of thing. Two magnitudes may have a ratio; four magnitudes, a proportion.

  3. A number is a magnitude, or at any rate a multitude of magnitudes.

Thus I question Strogatz’s assertions.

I hope that my students will learn to make their own assertions

  • with confidence, because they have confirmed the assertions to their own satisfaction; but

  • with humility, because they know the truth is independent of any imagined power to impose it.

Here I am talking about my students’ mathematical assertions. Historical assertions are different: your audience may not be able to confirm them out of their own mental resources.

You don’t get mathematical history right, just by being a mathematician, even an applied mathematician. Strogatz has his history wrong, in a way that an editor or fact-checker should have caught, if indeed the error is in his book; I find it in a transcription of an interview, where Strogatz says,

geometry, before Archimedes, could not handle smoothly curved shapes—like circles and spheres—as far as finding—as far as measuring them. Finding their area, or the circumference, or their volume.

You won’t find the formula, say, the formula pi r squared. That’s not in Euclid’s geometry. That had to wait for Archimedes and his incredibly ingenious use of infinity to find that formula.

On the contrary, Euclid “handled” the area of a circle in the second proposition of the twelfth book of the Elements, in what is the earliest instance of calculus that I know. The proposition is that circles are to one another as the squares on the circles’ diameters. In other words, there is a proportion, whereby the ratio of one circle to another is the same as the ratio of the squares on their respective diameters. The latter ratio is easily the same as the ratio of the squares on the radii. We may say then today that the circle varies as the square on its radius. If we name the “constant of proportionality” here as π, we arrive at the modern equation, A = πr². In a post mainly about Thales, I called this equation a modern summary of the most difficult theorem in Euclid.

Two ratios can be the same as one another, but they are never then described as being equal to one another, as far as I can tell. Equality is a relation that two distinct magnitudes can have. The magnitudes of Euclid and Archimedes are lengths, areas, and volumes. Magnitudes thus have a dimension, in the modern sense, and hence we may refer to a mass or a velocity or anything else that we can measure as a magnitude. Two magnitudes of the same dimension have a ratio, but then this ratio is dimensionless.

A number is a multitude of units. As I read Euclid, a unit is a magnitude, and so a number is a magnitude, albeit considered as a multitude. Another scholar has told me that a number is not a magnitude, but I have not received a clear reason. Euclid does have two theories of proportion: one for arbitrary magnitudes, another for numbers. The latter theory is obscure for modern readers; I have spelled out my understanding of it in “The Geometry of Numbers in Euclid.”

In the preface to A Comprehensive Introduction to Differential Geometry, Volume One (2d ed. 1979), Michael Spivak writes,

in order for an introduction to differential geometry to expose the geometric aspect of the subject, an historical approach is necessary,

but also,

Of course, I do not think that one should follow all the intricacies of the historical process, with its inevitable duplications and false leads. What is intended, rather, is a presentation of the subject along the lines which its development might have followed … When modern terminology finally is introduced, it should be as an outgrowth of this (mythical) historical development.

It is fine and perhaps even essential to teach mathematics according to a mythology; but then the point of doing history would be to question the myths.

I just chanced on Strogatz’s mythologizing about Greek mathematics in his book. I had picked up the book to see what the author said about Abraham Robinson’s so-called non-standard analysis. He said nothing, as far as I could tell, though I relied mainly on the index. Strogatz may not be too excited that Robinson has given us a rigorous justification of the “infinitesimal” magnitudes used by Archimedes and Newton; as an applied mathematician, Strogatz may not have felt the need for this justification in the first place.

In the post on Thales already pointed to, I suggest it is not good criticism to complain about what an author leaves out. I cannot complain that Strogatz does not mention non-standard analysis. As he himself says, a pure mathematicians would have a different telling of the story of calculus. That happens to be the telling I would rather hear. If Feynman is right that God talks the language of calculus, I want to know more precisely whether it is the language of infinitesimals or of epsilons and deltas. But this is a subject that I took up in “Limits,” almost seven years ago.

4 Trackbacks

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