Category Archives: Mathematics

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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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 →

Knottedness

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.

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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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What Mathematics Is

Mathematics “has no generally accepted definition,” according to Wikipedia on September 15, 2020, with two references. On September 14, 2023, the assertion is, “There is no general consensus among mathematicians about a common definition for their academic discipline”; this time, there are no references.

I suggest that what really 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 says also (as of either date given above),

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.

A 7×7 grid of squares, divided into four 3×4 rectangles arranged symmetrically about one square; the rectangles are divided in two by diagonals, which themselves describe a square
The right triangle whose legs are 3 and 4 has hypotenuse 5, because the square on it is
(4 − 3)2 + 2 ⋅ (4 ⋅ 3),
which is indeed 25 or 52. This is also
42 + 32.

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LaTeX to HTML

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.

A stack of books of and about mathematics: The Princeton Companion to Mathematics at the bottom, volume 2 of Heath’s edition of The Elements of Euclid at the top

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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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An Exercise in Analytic Geometry

This past spring (of 2020), 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.

Diagram from textbook page shows, centered at the origin of coordinates, a circle and an ellipse whose four points of intersection are traversed by two lines in red through the origin
From Weeks & Adkins, Second Course in Algebra, p. 395

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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.”

Summary added August 23, 2020: Mathematics may bring out such emotions as poetry does; but in the ideal, a work of mathematics is correct or not, in a sense that everybody will agree on. Here I review work of

  1. Lisa Morrow, writing in Meanjin as an immigrant to Istanbul, like me.
  2. Wendell Berry, in “The Peace of Wild Things,” which things “do not tax their lives with forethought / of grief,” and include the stars.
  3. Randall Jarrell, in The Animal Family.
  4. Mary Midgley, in Evolution as a Religion, on how we see animals.
  5. James Beall, astronomer, poet of the stars, tutor at my college.
  6. Edith Södergran, in “God,” as translated by Nicholas Lawrence in Cordite.
  7. Lukas Moodysson, in Fucking Åmål, where Agnes’s father notices that his daughter is reading Edith Södergran.
  8. Thomas J.J. Altizer, in The Gospel of Christian Atheism, a book that I kept from my father’s collection.
  9. Özge Samancı, in Dare to Disappoint, where the character to be disappointed is the father of the artist, and where Özlem (the artist’s friend and mine) praises the poetry of mathematics.
  10. Fiona Hile, writing, quâ editor of an issue of Cordite featuring poetry of mathematics, about the set theory of Maryanthe Malliaris and Saharon Shelah.
  11. Anupama Pilbrow, a poet writing in Meanjin about studying mathematics.
  12. Robert Pirsig, about students who ask their teacher, “Is this what you want?”
  13. R. G. Collingwood, who in Speculum Mentis analyzes Art, Religion, Science, History, and Philosophy as modes of existence.
  14. Michael Oakeshott, supposedly influenced by Collingwood, but also considered a forefather of “postmodern conservatism,” and analyzing existence into different modes from Collingwood’s, the latter according to the article in the Stanford Encyclopedia of Philosophy by Terry Nardin, who reports, “to insist on the primacy of any single mode is not only boorish but barbaric.”
  15. Allan Bloom, who suggests, in The Closing of the American Mind, that for Ronald Reagan, for the Soviet Union to be “the evil empire” and to “have different values” from the United States is the same thing.
  16. Galen Strawson, who seems to belie the possibility of different modes of being by saying, “we know exactly what consciousness is,” and also, “The nature of physical stuff is mysterious except insofar as consciousness is itself a form of physical stuff,” when (according to me) consciousness is simply not physical, not in the sense of being studied by physics.

A Twitter friend living here in Istanbul announced (on June 16) her pleasure in having a memoir published in Meanjin.

Meanjin cover, Winter 2020: a bird crushed by a stone heart

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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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