Category Archives: Mathematics


Index to this series

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, Weil, Thoreau, Tolstoy, Byrne, Wittgenstein, Arendt, and Alexander

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On Playing Zpordle

After Wordle appeared, a number of variants came out. One of the least popular may be Zpordle, or ℤp-ordle. I imagine it could be more popular, if people knew it did not require advanced mathematics. It still involves numbers, to which some people declare an allergy. Nonetheless, I think Zpordle can be explained in elementary-school terms, and that is what I shall try to do here.

Zpordle screenshot
Screenshot of the Zpordle game won on May 27, 2022

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Gödel, Grammar, and Mathematics


This attempt at exposition of Gödel’s Incompleteness Theorem was inspired or provoked by somebody else’s attempt at the same thing, in a blog post that a friend directed me to (by means of a Twitter message on November 17, 2020). I wanted in response to set the theorem in the context of mathematics rather than computer science.

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The Society of Mathematics 2

This post is a response by Alexandre Borovik to my previous post. The following words then are Sasha’s:

Dear David,

I joined the AMR. In my view, its areas of activities are sufficiently clearly defined [on its homepage]:

The AMR has several initiatives under development. These include:

  • AMR colloquia, lectures and workshops, exploring new ways to present research
  • Updates and reviews of new research
  • Reviews of classic influential papers
  • Discussions of open problems
  • Video expositions of mathematical research
  • AMR journals and publications leveraging new technological opportunities
  • Interviews of mathematicians
  • Developing new ideas for the flourishing of the international mathematical research community

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The Society of Mathematics

Mannequin in front of summation formula

This post concerns the Association for Mathematical Research, or AMR. A number of people are upset by its existence. I am not exactly one of them, but am suspicious, mainly because I do not know why a new organization would be needed, when we already have

The Twitter account of the AMR is dated to April, 2021. The website of the AMR supplies a list of founding members, but no account of when, how, or why they became founders. The site has a brief mission statement:


Are those other organizations not doing a good job? Continue reading

Feminist Epistemology

To this post, I am adding this introduction in July 2021. I have returned to some of the ideas of the post, and I see that I left them in a jumble. They may still be that, but I am trying to straighten up a bit.

Beyond this introduction, the post has three parts. Part III takes up more than half of the whole post and consists of my notes on

  1. Elizabeth Anderson, “Feminist Epistemology and Philosophy of Science,” Stanford Encyclopedia of Philosophy, February 13, 2020. 61 pages.

In Anderson’s article I see – as I note below – ideas that are familiar, thanks to my previous reading of philosophers such as Robin George Collingwood, Mary Midgley, and Robert Pirsig. Henry David Thoreau may not exactly be one of those philosophers, but he is somehow why I came to write this post in the first place.

Here is a table of contents for the whole post:

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Pacifism is properly pacificism, the making of peace: not a belief or an attitude, but a practice. Mathematics then is pacifist, because learning it means learning that you cannot fight your way to the truth. Might does not make right. If others are going to agree with you, they will have to do it freely. Moreover, you cannot rest until they do agree with you, if you’ve got a piece of mathematics that you think is right; for you could be wrong, if others don’t agree.

The book *Dorothy Healey Remembers,* with photo of subject

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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 actually be physically constructed.

  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.

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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