Category Archives: Euclid

Creativity

In the Platonic dialogues, Socrates frequently mentions τέχνη (technê), which is art in the archaic sense: skill or craft. The concern of this post is how one develops a skill, and what it means to have one in the first place.

Books quoted or mentioned in the text, by Midgley, Simone Weil, Thoreau, Amy Mandelker (on Tolstoy), Oliver Byrne (on Euclid), Wittgenstein, Arendt, and Caroline Alexander (on Homer)

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Mathematics and Logic

Large parts of this post are taken up with two subjects:

  1. The notion (due to Collingwood) of criteriological sciences, logic being one of them.

  2. Gödel’s theorems of completeness and incompleteness, as examples of results in the science of logic.

Like the most recent in the current spate of mathematics posts, the present one has arisen from material originally drafted for the first post in this series.

In that post, I defined mathematics as the science whose findings are proved by deduction. This definition does not say what mathematics is about. We can say however what logic is about: it is about mathematics quâ deduction, and more generally about reasoning as such. This makes logic a criteriological science, because logic seeks, examines, clarifies and limits the criteria whereby we can make deductions. As examples of this activity, Gödel’s theorems are, in a crude sense to be refined below, that

  • everything true in all possible mathematical worlds can be deduced;

  • some things true in the world of numbers can never be deduced;

  • the latter theorem is one of those things.

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Multiplicity of Mathematics

I continue with the recent posts about mathematics, which so far have been as follows.

  1. What Mathematics Is”: As distinct from the natural sciences, mathematics is the science whose findings are proved by deduction. I say this myself, and I find it at least implicit in an address by Euphemia Lofton Haynes.
  2. More of What It Is”: Some mathematicians do not distinguish mathematics from physics.
  3. Knottedness”: Topologically speaking, there is a sphere whose outside is not that of a sphere. The example is Alexander’s Horned Sphere, but it cannot be constructed physically.
  4. Why It Works”: Why there can be such a thing as the horned sphere.

When I first drafted the first post above, I said a lot more than I eventually posted. I saved it for later, and later is starting to come now.

Octahedron with edges divided in the Golden Ratio by the vertices of an icosahedron

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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.

Page of Greek text with diagram
From Archimedes, De Planorum Aequilibriis,
in Heiberg’s edition (Leipzig: Teubner, 1881)

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Discrete Logarithms

In the fall of 2017, I created what I propose to consider as being both art and mathematics. Call the art conceptual; the mathematics, expository; here it is, as a booklet of 88 pages, size A5, in pdf format.

More precisely, the work to be considered as both art and mathematics is the middle of the three chapters that make up the booklet. The first chapter is an essay on art, ultimately considering some examples that inspire my own. The last chapter establishes the principle whereby the lists of numbers in Chapter 2 are created.

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Doing and Suffering

To do injustice is worse than to suffer it. Socrates proves this to Polus and Callicles in the dialogue of Plato called the Gorgias.

I wish to review the proofs, because I think they are correct, and their result is worth knowing.

Loeb Plato III cover

Or is the result already clear to everybody?

Whom would you rather be: a Muslim in India, under attack by a Hindu mob, or a member of that mob?

You would rather not be involved; but if you had to choose, which option would be less bad: to be driven to an insane murderous fury, or to be the object of that fury?

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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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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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On Gödel’s Incompleteness Theorem

This is an appreciation of Gödel’s Incompleteness Theorem of 1931. I am provoked by a depreciation of the theorem.

I shall review the mathematics of the theorem, first in outline, later in more detail. The mathematics is difficult. I have trouble reproducing it at will and even just confirming what I have already written about it below (for I am adding these words a year after the original publication of this essay).

The difficulty of Gödel’s mathematics is part of the point of this essay. A person who thinks Gödel’s Theorem is unsurprising is probably a person who does not understand it.

In the “Gödel for Dummies” version of the Theorem, there are mathematical sentences that are both true and unprovable. This requires two points of clarification.

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Victor Vasarely

Tophane-i Amire
Tophane-i Amire, 2017.03.25

Last week I wrote about the Turkish Impressionist Feyhaman Duran, born in 1886. Now my subject is the Hungarian-French Op Artist born twenty years later as Győző Vásárhelyi. His “Rétrospective en Turquie” is at the Tophane-i Amire Culture and Art Center in an Ottoman cannon foundry.

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