Category Archives: Mathematics

Salvation

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

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Ordinals

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.

Note. Would it be helpful to write that more verbosely?

  • There is no greatest real number.
  • There is no least real number.
  • The set of real numbers has a countable dense subset, namely the set of rational numbers.
  • Every set of real numbers that has an upper bound has a least upper bound.

  • There is no greatest ordinal number.
  • There is a least ordinal number.
  • Indeed,
    • every nonempty set of ordinal numbers has a least element, and
    • the class of ordinals that are less than a given ordinal is a set.
  • Every set of ordinals 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, occasionally coming to a limit

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On the Idea of History

Note added, March 10–11, 2021. The bulk of this post concerns race in the theory of history, particularly the theory attributed to Johann Gottfried Herder (1744–1803). Not having read Herder for myself, I rely on the accounts of

  • R. G. Collingwood in § 2, “Herder,” of Part III of The Idea of History (1946),
  • Michael Forster in “Johann Gottfried von Herder,” Stanford Encyclopedia of Philosophy (summer 2019).

Somebody like Herder may introduce race as an hypothesis to explain history, but ultimately the hypothesis fails, by denying us the freedom that is essential to history as such. Nonetheless, Forster defends Herder as having

an impartial concern for all human beings … Herder does also insist on respecting, preserving, and advancing national groupings. However, this is entirely unalarming,

because, for one thing, “The ‘nation’ in question is not racial but linguistic and cultural.”

Change Collingwood’s word “race” to “linguistic and cultural grouping” then. I think his conclusion remains sound: “Once Herder’s theory of race is accepted, there is no escaping the Nazi marriage laws.”

More detail is in the post below. I go on to review the philosophy of history that Collingwood presents in the Introduction of The Idea of History. This book provided me with a title for the post.

I wrote a lot in this post, as I often do. Growing self-conscious for being opinionated about the theory of history, I listed the published evidence of my actually being an historian (an historian of ancient Greek mathematics in particular).

I originally wrote that my research had been inspired by a tweet. The author of that tweet also wrote the nice long comment on this post. However, although the tweet can be found on the Internet Archive, the author later deleted his Twitter account, and so the tweet appears on Twitter today as a gap in the thread above my own tweet in response to the other tweet.

That missing tweet referred to another tweet of the author, but the Internet Archive seems not to have saved it. It did save the present post; so if one were curious, one could see the changes that I have made since initial publication, or rather since September 29, 2020, when the Archive took the first snapshot.

The changes are for style and local clarity. Any large-scale changes would need me to recover the spirit that possessed me when I originally wrote.

I return to this post now, simply because a friend mentioned reading Middlemarch, and I remembered quoting George Eliot’s novel in a blog post, and that post turned out to be this one.

Had somebody mentioned reading Herder, I might have recalled writing about him in a blog post; that would be this post too.


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

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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 (as in a particular definition of physicalism) use “iff” needlessly.

  6. A pair of mathematicians who use “iff” needlessly 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

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On Chapman’s Homer’s Iliad, Book XVIII

I analyze Book XVIII of the Iliad into seven scenes.

Branches against sky

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Math, Maugham, and Man

Note added, August 28, 2023. The main purpose of this post of September 1, 2019, seems to have been to assemble some information about the etymologies of “man” and “woman,” because of ongoing controversy about what the words even mean today. I started to take up the controversy itself on December 30 of that year, in “Sex and Gender.” Meanwhile, this post suggests, or points out:

  • a generic “person” may still be male in people’s minds;
  • becoming a woman may be like becoming Jewish;
  • there are no gendered pronouns in Turkish;
  • the series freshman, sophomore, junior, senior is like pinkie, ring finger, middle finger;
  • Greek does not have such an interesting series for the fingers;
  • Greek mathematics includes Thales’s Theorem and Pappus’s Hexagon Theorem.

There does not seem to be any connection between the mathematics and the etymology here, except that I was studying both at the same time. I must have been reading The Razor’s Edge too, where Maugham

  • places himself in a tradition founded by Herodotus;
  • uses “he/him” for for somebody who can be a woman as well as a man.

More themes I took up:

  • what it means to be natural;
  • that I don’t consider myself ADHD;
  • the etymology of “squirrel”;
  • the Etymological Fallacy.

A dog lying in the shade of a beach umbrella looks at us; behind him are a woman and a man sitting facing away from us, towards the sea
Woman, man, and dog
Friday, August 18, 2023
Altınova, Balıkesir, Türkiye

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NL I: “Body and Mind” Again

Index to this series

“We are beginning an inquiry into civilization,” writes Collingwood, “and the revolt against it which is the most conspicuous thing going on at the present time.” The time is the early 1940s.

Human tourists photographing sculptured supine blue ape with chrome testicles outside the Intercontinental Hotel, Prague Continue reading →

Elliptical Affinity

After Descartes gave geometry the power of algebra in 1637, a purely geometrical theorem of Apollonius that is both useful and beautiful was forgotten. This is what I conclude from looking at texts from the seventeenth century on.

Piety

The post below is a way to record a passage in the Euthyphro where Socrates says something true and important about mathematics.

Crude depiction of bug-eyed figure grasping the torso of, and putting into his mouth the arm of, a smaller figure
Goya, [Cronus] Devouring His Son
(see below)

The passage is on a list of Platonic passages that I recently found, having written it in a notebook on May 23, 2018. The other passages are in the Republic; here they are, for the record, with some indication of why they are worth noting (translations are Shorey’s, originally from 1930 and 1935 in the old Loeb edition):

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Logic of Elliptic Curves

In my 1997 doctoral dissertation, the main idea came as I was lying in bed one Sunday morning. Continue reading →