Author Archives: David Pierce

Mathematician & logician; amateur of philosophy; relation of journalists; alumnus of St John’s College (USA); living in Ankara & Istanbul since 2000


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 I would 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 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. 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 written, 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 thus 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.

Evolution of Reality

I enjoy and recommend Robert Wright’s Nonzero Newsletter, which presents thought on both American politics and thought itself.

Tiny green plants on red tile roof, cloudy day

In a 2017 post of this blog, I quoted Wright’s 1988 article in The Atlantic Monthly about Edward Fredkin. Somewhat differently from Fredkin, I spelled out my title, “What Philosophy Is,” without actually being a professional philosopher. I touched on a theme that I shall take up now: that thinkers today could benefit from knowing the thought of R. G. Collingwood.

In Wright’s newsletter this week, the thought that is thought about is more precisely thought about “reality.” The person thinking about it (besides Wright and interested readers) is billed as a “cognitive scientist.”

Stone house lit by sun, bare trees in front

I would say I too am a cognitive scientist, just for being a logician. Every science aims to produce cognition that is satisfactory on its own terms; thus a science of cognition will be neither purely descriptive, nor normative in the sense of imposing standards from outside. A science of cognition as such will be criteriological, in the sense of Collingwood, which I have discussed in “A New Kind of Science.” Logic is the traditional name for the study of theoretical cognition; the study of practical cognition is ethics.

So I say, as somebody who began pondering Collingwood’s voluminous œuvre more than thirty years ago, albeit with little attention to the more recent thinkers now called analytic philosophers. As far as I can tell, a proper reading of Collingwood could straighten them out. They may respond that I lack their training and knowledge; but how will they know this matters, if they have not read Collingwood for themselves?

Two figures assembled from gnarled wood

This week, as far as I understand from Wright’s interview with him, cognitive scientist Donald Hoffman accepts a syllogism that I formulate as follows.

  1. (See below.)

  2. What evolution favors is not our perception of reality as such (called “reality as it is,” “objective reality,” “actual reality”), but whatever traits we may have that

    • are determined by genes,

    • promote the propagation of those very genes.

  3. Therefore, as evolved beings, we cannot expect to grasp reality as such.

This makes a certain sense, but prompts the question: What did we think reality was in the first place, before our study of evolution straightened us out? Beyond this, there is the missing major premiss: Evolution itself is real. How do we know?

Here is some of what Hoffman thinks, in his own words from Wright’s interview:

I’m happy to contemplate the idea that the notion of causality itself is not a fiction, but that the specific causality that we all know and love—namely that [of] a physical object like a billiard ball hitting another billiard ball and making it move—that that’s genuine causality, that I think we will have to give up.

So when the white ball hits the eight ball into the corner pocket, we can say that the cue ball caused the eight ball to move, and for practical purposes, that’s fine. It’s a useful fiction. But strictly speaking, it’s a fiction.

Of course it’s a fiction that the cue ball causes the eight ball to move. It is we who cause the latter to move, by means of the former. We are the agent, and a cause needs an agent. In speech and thought, we may find it convenient to transfer our agency to the cue ball (though here I speak theoretically, not actually having spent much time with billiards). I imagine anybody would agree that this transference is, in Hoffman’s term, a fiction.

Archway of brick and stone

So I imagine; but then I see in the world a lot of (what I think is) confusion about causation. I have tried to address this in my post “On Causation,” which is based on an article by Collingwood that ended up in his Essay on Metaphysics (1940) as an example of how to do metaphysics.

The post mentions Collingwood’s argument that our notion of causation in the natural world is a remnant of Neoplatonism. Our notion of evolution might be explained in similar terms. In any case, it is a fiction that evolution “favors” anything. According to Hoffman again,

the assumption in the field has been that the perceptual strategies that will actually be favored by that kind of natural selection are perceptual strategies that see reality as it is. Not exhaustively—very, very few people would claim that we see all of reality as it is—but that those aspects of the world that we do see, we do see accurately; and we see the ones that we need to survive and reproduce.

Hoffman takes issue with the particular assumption discussed. I take issue with the assumption that any kind of “perceptual strategy” can be “favored” by an abstraction called “natural selection.” This assumption is anthropomorphism.

Passage downhill between green roof and hammam dome, tower in distance

Perhaps it is inevitable. As Collingwood says,

We cannot help thinking anthropomorphically; but we are provided with a remedy: our own laughter at the ridiculous figure we cut, incorrigibly anthropomorphic thinkers inhabiting a world where anthropomorphic thinking is a misfit.

That’s paragraph 14. 61 of the New Leviathan (1942), alluded to in my post about the “Reason” chapter, but actually quoted in the post about Chapter XVIII, “Theoretical Reason.” Collingwood’s theme is that the science we do—our study of the world—will reflect how we think about one another and ourselves. He concludes Chapter XVIII with paragraph 18. 92:

It is in the world of history, not in the world of Nature, that man finds the central problems he has to solve. For twentieth-century thought the problems of history are the central problems: those of Nature, however interesting they may be, are only peripheral.

Collingwood may be justified here, writing as he is in response to a war that is said to have arisen from problems not resolved by an earlier war. No historical problems are ever permanently resolved, and it is dangerous to think they are, as he writes in Chapter XXVI, “Democracy and Aristocracy.” Victory over the Nazis did not eradicate fascism, as unfortunately we are seeing now. I think this is a reason why part of the Nonzero Newsletter is called “Mindful Resistance.” It is a reason why I have found the New Leviathan worth studying. The last book that Collingwood saw to press is itself an instance of mindful resistance.

The photographs above are from the Nesin Mathematics Village, during the two winter weeks of January 27 and February 3, 2020, when I taught courses on the ordinal numbers. I took the photos with my feature phone, mentioned in “Computer Recovery” as an alternative means of web access to the laptop I have used to compose the present post. I took the photos on February 7 and 9; I included three similar photos from the former, cloudy, day in a tweet. I also have a dedicated camera, which would have taken better photos, had I brought it with me.

Sex and Gender

A certain thesis is reasonable to me, and yet it would seem to anger persons whom I wish to respect. I am trying to understand why it does.

The hypothesis of the homunculus in the sperm
by Nicolaas Hartsoeker, 1695

Perhaps the manner of expression of the thesis is the problem. Thus one person tweets:

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On the Odyssey, Book II

Having been put to bed by Eurycleia at the end of Book I of the Odyssey, Telemachus gets up in the morning and has the people summoned to council, at the beginning of Book II.

Three books with beads

There is no mention of a breakfast. Perhaps none is eaten. On the other hand, Telemachus probably relieves his bladder at least, and there is no mention of that either.

Telemachus straps on a ξίφος, but arrives at the assembly with a χάλκεον ἔγχος in hand. Wilson calls it a sword in either case; for Fitzgerald and Lattimore, the first weapon is a sword, but the second a spear and a bronze spear, respectively. Cunliffe’s lexicon supports the men; however, for Liddell and Scott, an ἔγχος can also be a sword, at least in Sophocles. For Beekes, ξίφος is Pre-Greek, and ἔγχος may be so. Continue reading


This is about the ordinal numbers, which (except for the finite ones) are less well known than the real numbers, although theoretically simpler.

The numbers of either kind compose a linear order: they can be arranged in a line, from less to greater. The orders have similarities and differences:

  • Of real numbers,

    • there is no greatest,

    • there is no least,

    • there is a countable dense set (namely the rational numbers),

    • every nonempty set with an upper bound has a least upper bound.

  • Of ordinal numbers,

    • there is no greatest,

    • every nonempty set has a least element,

    • those less than a given one compose a set,

    • every set has a least upper bound.

One can conclude in particular that the ordinals as a whole do not compose a set; they are a proper class. This is the Burali-Forti Paradox.

Diagram of reals as a solid line without endpoints; the ordinals as a sequence of dots, periodically coming to a limit Continue reading

On the Odyssey, Book I

  • In reading his rendition of the Iliad, having enjoyed hearing Chapman speak out loud and bold;

  • having enjoyed writing here about each book, particularly the last ten books in ten days on an Aegean beach in September of this year (2019);

  • having taken the name of this blog from the first line of the Odyssey;

  • having obtained, from Homer Books here in Istanbul, Emily Wilson’s recent translation (New York: Norton, 2018);

  • Book on table, Wilson's Odyssey Continue reading

Computer Recovery

I record here how I fixed my computer, because

  • I am pleased to have been able to do it, and

  • I may have to do it again.

Briefly, when Windows on my laptop failed, I installed Ubuntu, but this failed. Somebody else installed Ubuntu again, and this worked for a while before failing. I managed to fix that problem for myself; but later an upgrade failed. Now I have fixed that.

Computer on table by window at dawn

This post is some kind of laboratory notebook. Continue reading

On the Idea of History

Our environment may influence our feelings, but what we make of those feelings is up to us. Thus we are free; we are not constrained by some fixed “human nature”—or if we are, who is to say what its limits are?

Rembrandt van Rijn (and Workshop?), Dutch, 1606-1669,
The Apostle Paul, c. 1657, oil on canvas,
Widener Collection, National Gallery of Art

Insofar as we humans have come to recognize our freedom, we have done so after thinking that what we did depended on our class—our kind, our sort, even our “race.” We might distinguish three stages of thought about ourselves.

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Anthropology of Mathematics

This essay was long when originally published; now, on November 30, 2019, I have made it longer, in an attempt to clarify some points.

The essay begins with two brief quotations, from Collingwood and Pirsig respectively, about what it takes to know people. The Pirsig quote is from Lila, which is somewhat interesting as a novel, but naive about metaphysics; it might have benefited from an understanding of Collingwood’s Essay on Metaphysics. A recent article by Ray Monk in Prospect seems to justify my interest in Collingwood; eventually I have a look at the article. Ideas that come up along the way include the following.

  1. For C. S. Lewis, the reality of moral truth shows there is something beyond the scope of natural science.

  2. I say the same for mathematical truth.

  3. Truths we learn as children are open to question. In their educational childhoods, mathematicians have often learned wrongly the techniques of induction and recursion.

  4. The philosophical thesis of physicalism is of doubtful value.

  5. Mathematicians and philosophers who ape them use “iff” needlessly.

  6. One pair who do this seem also to misunderstand induction and recursion.

  7. Their work is nonetheless admirable, like the famous expression of universal equality by the slave-driving Thomas Jefferson.

  8. Mathematical truth is discovered and confirmed by thought.

  9. Truth is a product of every kind of science; it is not an object of natural science.

  10. The distinction between thinking and feeling is a theme of Collingwood.

  11. In particular, thought is self-critical: it judges whether itself is going well.

  12. Students of mathematics must learn their right to judge what is correct, along with their responsibility to reach agreement with others about what is correct. I say this.

  13. Students of English must learn not only to judge their own work, but even that they can judge it. Pirsig says this.

  14. For Monk, Collingwood’s demise has meant Ryle’s rise: unfortunately so since, for one thing, Ryle has no interest in the past.

  15. In a metaphor developed by Matthew Arnold, Collingwood and Pirsig are two of my touchstones.

  16. Thoreau is another. He affects indifference to the past, but his real views are more subtle.

  17. According to Monk, Collingwood could have been a professional violinist; Ryle had “no ear for tunes.”

  18. For Collingwood, Victoria’s memorial to Albert was hideous; for Pirsig, Victorian America was the same.

  19. Again according to Monk, some persons might mistake Collingwood for Wittgenstein.

  20. My method of gathering together ideas, as outlined above, resembles Pirsig’s method, described in Lila, of collecting ideas on index cards.

  21. Our problems are not vague, but precise.

When Donald Trump won the 2016 U.S. Presidential election, which opinion polls had said he would lose, I wrote a post here called “How To Learn about People.” I thought for example that just calling people up and asking whom they would vote for was not a great way to learn about them, even if all you wanted to know was whom they would vote for. Why should people tell you the truth?

Saturn eclipse mosaic from Cassini

With other questions about people, even just understanding what it means to be the truth is a challenge. If you wanted to understand people whose occupation (like mine) was mathematics, you would need to learn what it meant to prove a theorem, that is, prove it true. Mere observation would not be enough; and on this point I cite two authors whom I often take up in this blog.

  • In the words of R. G. Collingwood in Religion and Philosophy (1916, page 42), quoted in An Autobiography (1940, page 93) as well as in the earlier post here, “The mind, regarded in this external way, really ceases to be a mind at all.”

  • In the words of English teacher and anthropologist Verne Dusenberry, quoted by Robert Pirsig in Lila (1991, page 35), “The trouble with the objective approach is that you don’t learn much that way.”

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On Translation

Achilles is found singing to a lyre, in a passage of Book IX of the Iliad. Homer sets the scene in five dactylic hexameters; George Chapman translates them into four couplets of fourteeners.

I wrote a post about each book of the Iliad, in Chapman’s version of 1611. As I said at the end, I look forward to reading Emily Wilson’s version. Meanwhile, here I examine the vignette of the lyre in several existing English translations, as well as in the original.

Three books mentioned in the text Continue reading