Category Archives: Euclid

Doing and Suffering

Edited March 30, 2020

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?

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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 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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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.

Vasarely show Continue reading

The geometry of numbers in Euclid

This is about how the Elements of Euclid shed light, even on the most basic mathematical activity, which is counting. I have tried to assume no more in the reader than elementary-school knowledge of how whole numbers are added and multiplied.

How come 7 ⋅ 13 = 13 ⋅ 7? We can understand the product 7 ⋅ 13 as the number of objects that can be arranged into seven rows of thirteen each.

Seven times thirteen

Seven times thirteen

If we turn the rows into columns, then we end up with thirteen rows of seven each; now the number of objects is 13 ⋅ 7. Continue reading

Equality Is Not Identity

I want to record here an account by Collingwood of Aristotle’s theory of knowledge. The passages quoted below are relevant, both to something I have learned from reading Euclid with students, and to the considerations of consciousness that led to my recent article “Body and Mind.” Continue reading

The Peace of Liberal Education

The wall of Dolmabahçe Sarayı, January 11, 2015

The wall of Dolmabahçe Sarayı, January 11, 2015

The occasion of this article is my discovery of a published Turkish translation of Collingwood’s Speculum Mentis or The Map of Knowledge (Oxford, 1924). Published as Speculum Mentis ya da Bilginin Haritası (Ankara: Doğu Batı, 2014), the translation is by Kubilay Aysevenler and Zerrin Eren. Near the end of the book, Collingwood writes the following paragraph about education, or what I would call more precisely liberal education. The main purpose of this article then is to offer the paragraph to any reader who happens to stop by.

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NL III: “Body As Mind”

Index to this series

In Chapter I of The New Leviathan, we stipulated that natural science, the “science of body,” must be free to pursue its own aims. But we ourselves are doing science of mind, and:

1. 85. The sciences of mind, unless they preach error or confuse the issue by dishonest or involuntary obscurity, can tell us nothing but what each can verify for himself by reflecting on his own mind.

All of us can be scientists of mind, if only we are capable of reflection: Continue reading

NL I: “Body and Mind”

Index to this series. See also a later, shorter article on this chapter

The Chapter in Isolation

“Body and Mind” is the opening chapter of Collingwood’s New Leviathan. The chapter is a fine work of rhetoric that could stand on its own, though it invites further reading. In these respects it resembles the first of the ten traditional books of Plato’s Republic, or even the first of the thirteen books of Euclid’s Elements. The analogy with Euclid becomes a bit tighter when we consider that each chapter of The New Leviathan is divided into short paragraphs, which are numbered sequentially for ease of reference.

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