Category Archives: Mathematics

Why It Works

The last post, “Knottedness,” constructed Alexander’s Horned Sphere and proved, or sketched the proof, that

  • the horned sphere itself is topologically a sphere, and in particular is simply connected, meaning

    • it’s path-connected: there’s a path from every point to every other point;

    • loops contract to points—are null-homotopic;

  • the space outside of the horned sphere is not simply connected.

This is paradoxical. You would think that if any loop sitting on the horned sphere can be drawn to a point, and any loop outside the horned sphere can be made to sit on the sphere and then drawn to a point, then we ought to be able to get the loop really close to the horned sphere, and let it contract it to a point, just the way it could, if it were actually on the horned sphere.

You would think that, but you would be wrong. Continue reading


If you roll out a lump of clay into a snake, then tie a string loosely around it, can you contort the ends of the snake, without actually pressing them together, so that you cannot get the string off?

You can stretch the clay into a Medusa’s head of snakes, and tangle them as you like, again without letting them touch. If you are allowed to rest the string on the surface of the clay, then you can get it off: you just slide it around and over what was an end of the original snake.

Continue reading

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.

From Archimedes, De Planorum Aequilibriis,
in Heiberg’s edition (Leipzig: Teubner, 1881)

Continue reading

What Mathematics Is

Mathematics “has no generally accepted definition,” according to Wikipedia today. Two references are given for the assertion. I suggest that what has no generally accepted definition is the subject of mathematics: the object of study, what mathematics is about. Mathematics itself can be defined by its method. As Wikipedia currently says also,

it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions.

I would put it more simply. Mathematics is the science whose findings are proved by deduction.

Continue reading


This is a little about mathematics, and a little about writing for the web, but mostly about the nuts and bolts of putting mathematics on the web. I want to record how, mainly with the pandoc program, I have converted some mathematics from a LaTeX file into html. Like “Computer Recovery” then, this post is a laboratory notebook.

The mathematics is a proof of Dirichlet’s 1837 theorem on primes in arithmetic progressions. This is the theorem that, if to some number you keep adding a number that is prime to it, there will be no end to the primes that you encounter in this way.

Continue reading

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.

Continue reading

An Exercise in Analytic Geometry

This past spring, when my university in Istanbul was closed (like all others in Turkey) against the spread of the novel coronavirus, I created for my students an exercise, to serve at least as a distraction for those who could find distraction in learning.

From Weeks & Adkins, Second Course in Algebra, p. 395

The exercise uses no more mathematical tools than may be found in an algebra course in high school; yet it serves the purposes of university mathematics, as I understand them.

Continue reading

Poetry and Mathematics

This reviews some reading and thinking of recent weeks, pertaining more or less to the title subjects, of which it may be worth noting that

  • poetry is from ποιέω “make”;

  • mathematics is from μανθάνω “learn.”

Continue reading

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?

Continue reading


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