The Universe of Discourse — 6/27/2026
A while back I related how I had been mocked by an English person for using the word “burglarize”. I ended by saying: Okay, whatever. Brits have been mocking the American language for centuries now. Let them go ahead. We all know who won that…
The Universe of Discourse — 6/27/2026
This is my great-grandfather, born Dominusz Andor in Szeged, Hungary in 1886. In the picture he is in Brooklyn, New York, probably sometime in the early 1950’s. By 1911 Andor had moved from Hungary to Vienna and had changed the spelling of his…
The Universe of Discourse — 6/23/2026
Making the rounds last week was this magnificent article on the complications of Arabic typesetting, An interactive introduction to the terrific experience of rendering Arabic typography and its technical debt. The author, Saleh, promises: The…
The Aperiodical — 6/22/2026
In this series of posts, we’ll be featuring mathematical video and streaming channels from all over the internet, by speaking to the creators of the channel and asking them about what they do. We spoke to Stanley, who runs the StanDoesMath…
Fractal Kitty — 6/21/2026
IntroductionIn this inquiry, we build a sequence from a single 2. The first rule of this sequence is that it has to describe itself.Starting with TwoHere is a 2. 2It says, “There are two here.” The first number is a 2, so the next
Joel David Hamkins — 6/20/2026
This is a talk for the Workshop on Mereology at Shandong University in Jinan, China, a part of the week-long conference Week of Fusion Philosophy, 22-26 June 2026. The mereology talks are on 22 June 2026. Title: Set-theoretic mereology as ……
The Universe of Discourse — 6/18/2026
While cleaning out my office today, I found this, which I wrote in 1992: In the middle 1970’s, the IBM corporation did (and perhaps still does) most of their in-house programming in a computer language called FORTRAN. They had a pretty good…
The Universe of Discourse — 6/17/2026
(Very much previously: Egyptian Fractions) Back in March, I had been reading On the Egyptian method of decomposing into unit fractions by Abdulrahman A. Abdulaziz, and I reported that: There is some indication that Ahmes preferred fractions…
The Aperiodical — 6/17/2026
Double Maths First Thing is 3D-printing Cristiano Ronaldo Hello! My name is Colin and I am a mathematician on a mission to spread the joy and delight that comes from engaging with maths. I understand there’s a soccerball elimination in progress at…
The Aperiodical — 6/15/2026
The UK Government have announced the new set of King’s Birthday Honours. Here’s our selection of particularly mathematical entries for this year. If you spot any more, let us know in the comments and we’ll add to the list. Get the full UK list from…
The Universe of Discourse — 6/13/2026
(Previously) In 2003 I visited Barcelona and spent all day wandering around the mighty Basilica de la Sagrada Família, the architectural masterpiece of Antoni Gaudí. It had been under construction since 1882, and at the time only four of its 18…
The Aperiodical — 6/12/2026
We were kindly sent a copy of Claudia de Rham’s new book ‘The Beauty of Falling’, and asked irregular contributor Elinor Flavell to write this review. Claudia de Rham’s “The Beauty of Falling” is not just a book about gravity: it is a book about…
The Universe of Discourse — 6/12/2026
The ancient Egyptians had a terrible notation for fractions. They had notations for for each , for , but everything else was written as a sum of these, with repeats forbidden, so that for example had to be written as . (Wikipedia) In an older…
Fractal Kitty — 6/11/2026
Let me tell you about a language. But if you wish to go play instead go here. It’s going to get a little abstract below.There is a Canvas to Compose UponThe canvas is a square with the largest circle that has a radius of ρ.There
The Aperiodical — 6/10/2026
Double Maths First Thing is NOT procrastinating in the group chat. Why would you even suggest that? Hello! My name is Colin and I am a mathematician on a mission to spread the joy and delight of doing maths and, from time to time, the absurdities…
Joel David Hamkins — 6/8/2026
This will be a series of graduate lectures at Peking University, two lectures per week beginning mid-June and proceeding into July. Topics. We shall aim to cover the central results in the modal logic of forcing, including an exploration of ……
The Aperiodical — 6/4/2026
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of May 2026, is now online courtesy of Sophia Wood at Fractal Kitty. The Carnival rounds up maths blog posts from all over the internet, including some from our…
Fractal Kitty — 6/3/2026
Welcome to the 252nd Carnival of Mathematics! This post brings together submissions and other posts from the mathy web. Thanks all for participating.Let’s start with the number: 252Divisors: 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84,…
The Aperiodical — 6/3/2026
Here’s a short round-up of maths news stories from the last two months that we didn’t otherwise cover on the site. Thomas Dieterrich, a representative of the arXiv, has clarified the site’s AI policy – in a Twitter thread (non-Twitter mirror link)…
The Aperiodical — 6/3/2026
Double Maths First Thing doesn’t know where to start Hello! My name is Colin and I am a mathematician on a mission to spread the joy and delight that comes from working things through and wrapping them up with a neat mathematical bow. Before…
The Aperiodical — 5/28/2026
This is a guest post from museum educator and mathematician Tom Briggs, about his session at last August’s Talking Maths in Public conference. 2025’s Talking Maths in Public (TMiP) conference took place on the Warwick University campus, but the…
The Aperiodical — 5/27/2026
Double Maths First Thing adds a Holy C Hello! My name is Colin and I am a mathematician on a mission to spread the joy and delight of figuring things out. Here in Weymouth, we’re sweltering but lucky 12yo is about to jet off to Barcelona for a week…
Math ∩ Programming — 5/22/2026
Here are some more whimsical OEIS sequences I came across. XKCD 2016 joked that “OEIS keeps rejecting my submissions,” including one that gives “Integers in increasing order of width when printed in Helvetica.” Well, two days after that comic was…
Fractal Kitty — 5/21/2026
unedited human writing before bedOriginIn the beginning there was a point. …And the beginning was but a period in which time was noted by a wisp of this existence
Fractal Kitty — 5/17/2026
Thanks to Sam Graf for introducing me to this and suggesting some toys. IntroductionMultiplication tables can be fun. Line up your numbers, multiply, and find patterns. Like with 5x5, we can fill it out and highlight symmetry, divisibility,…
Math ∩ Programming — 4/29/2026
Table of Contents In this tutorial series, I will introduce the CKKS homomorphic encryption scheme from the ground up, in rather intricate detail. Each article in this series corresponds to a pull request on a GitHub repository. The code for this…
Joel David Hamkins — 4/23/2026
This will be a talk for the Philosophy Department Colloquium at Ohio University in Athens, OH on April 30th, 2026. I am very grateful for the invitation. A potentialist perspective on ultrafinitism, Ohio University Abstract. Ultrafinitism is the…
Fractal Kitty — 4/23/2026
IntroductionPentominoes are shapes made from 5 squares joined edge-to-edge. There are 12 of them:Next, let’s define what an enclosed area is with these shapes. The pentominoes must create a fence where they touch edge-to-edge with no overlaps. Note…
Good Fibrations — 4/19/2026
Mathematics and Computation — 4/13/2026
After spending many irritating hours with ChatGPT and Copilot, I finally tried out Claude. I told it to update photos of mathematicians from a derelict Perl script to a shiny new Python script with JSON, face recognition and modern CSS. It worked…
Mathematics and Computation — 4/13/2026
After spending many irritating hours with ChatGPT and Copilot, I finally tried out Claude. I told it to update photos of mathematicians from a derelict Perl script to a shiny new Python script with JSON, face recognition and modern CSS. It worked…
Math ∩ Programming — 4/13/2026
I went hunting for references to the OEIS in open source code, and found some weird ones. There are not one, but two live-coding music frameworks that use OEIS sequences as a source for “anything that can be sequenced” in music. I’m guessing that’s…
Math ∩ Programming — 4/9/2026
A051070 is a sequence about OEIS sequences. a(n) is the n-th term in sequence A_n (or -1 if A_n doesn’t have enough terms). So the first term in A051070 is 1 because A000001 is the number of groups of order n, and that sequence has 1 as its entry…
Math ∩ Programming — 4/7/2026
Problem: Determine if a 32-bit number is prime (deterministically) Solution: (in C++) // Bases to test. Using the first 4 prime bases makes the test deterministic // for all 32-bit integers. See https://oeis.org/A014233. int64_t bases[] = {2, 3, 5,…
Joel David Hamkins — 4/4/2026
This will be a talk for the Logic Seminar at the University of Notre Dame, 14 April 2026, 2pm, Room 125 Hayes-Healey. Abstract After establishing several general features of the hierarchy of consistency strength, we shall consider the possible…
Math ∩ Programming — 4/1/2026
It’s the 5th annual April Cools! Here are my previous April Cools articles This year it’s a book review of Ben Recht’s book, The Irrational Decision: How We Gave Computers the Power to Choose For us, released Mar 10, 2026. The publishing industry…
Fractal Kitty — 4/1/2026
IntroductionLet’s start with E. Its opposite is O. So if we flip E, we get O. Let’s make a pattern. EE OE O O EE O O E O E E OHow is this pattern constructed? What comes next? Write
Joel David Hamkins — 3/28/2026
I am very pleased to announce that The Book of Infinity is now available for pre-order. Check it out at your favorite booksellers. From the preface: Come, let us explore infinity! We shall visit all my favorite paradoxes and conundrums. The ancient…
Fractal Kitty — 3/20/2026
When I create art, I do so for many reasons. Some of these are:to engage in an expression of beingto explore a concept or experiment with an ideato grow as a person through creativity and struggleto immerse myself in a spiritual actto have a coping
The Universe of Discourse — 3/17/2026
A couple of years back I was discussing the Rhind Mathematical Papyrus (RMP). It includes a table expressing as a sum \(\frac1{a_1}+\frac1{a_2}+\dots+\frac1{a_k}\) fractions with numerator 1 (“unit fractions”). I said: Getting the table of…
The Universe of Discourse — 3/9/2026
A couple of days ago I recounted a common complaint: I keep seeing programmers say how angry it makes them that people are willing to write detailed CLAUDE.md and PROJECT.md files for Claude to use, but they weren’t willing to write them…
The Universe of Discourse — 3/8/2026
A number of years ago I wondered how many movies I had seen. The only way I could think of finding out was just to make a list. This I did as best I could. (It turned out to be around 700.) I found, though, that I could not include all the James…
The Universe of Discourse — 3/5/2026
Our company is going to a convention later this month, and they will have a booth with big TV screens showing statistics that update in real time. My job is to write the backend server that delivers the statistics. I read over the documents that…
The Universe of Discourse — 3/3/2026
Bo Diddley’s cover of “Sixteen Tons” sounds very much like one of my favorites, “Can’t Judge A Book By Its Cover”. It’s interesting to compare. Thinking on that it suddenly occured to me that his name might have been a play on “diddley bow”,…
Abuse of Notation — 2/22/2026
In the 19th century, Copernicus, Newton, Galilei et al pushed a revolutionary new idea that reshaped the way we think… but no, it’s not talking about cosmology, but about theology. This idea, (which was also the real reason they were in so much…
Joel David Hamkins — 2/12/2026
I am honored to be invited to give the Owen G. Owens Memorial Lecture at Wayne State University on 16 April 2026, joining a distinguished list of luminaries giving previous Owens lectures, including Gregory Margulis, John Milnor, Mikhael Gromov,…
Joel David Hamkins — 2/11/2026
I have been asked by the ASL to fill in as a last-minute substitute speaker for the ASL session at the upcoming 2026 APA Central Division Meeting in Chicago, February 18-21, 2026, due to a late cancellation of one of … Continue reading →
Fractal Kitty — 2/6/2026
Whether you call this triangle Pascal’s triangle, Binomial Expansion Coefficients, Yang Hui’s triangle, or any other name, it is beautiful.Finding patterns in this triangle is fun - from counting numbers, to looking at parity (even/odd-ness), to…
Joel David Hamkins — 2/4/2026
This will be a talk at the CUNY Logic Workshop on 13 March 2026, held at the CUNY Graduate Center. Abstract. I shall introduce the elementary theory of surreal arithmetic (SA), a first-order theory that is true in the surreal field when equipped…
Proses.ID — 1/31/2026
I miss writing. That’s a strange thing to say because I’ve been employed as a full-time writer for the past 14 months. And I have…
Proses.ID — 1/31/2026
I’ve been down a rabbit hole for the past few months, obsessed with a single question: “What makes a writing human?” It started, ironically, because…
Fractal Kitty — 1/29/2026
Have you ever been to a quilt store and bought fabric without a plan? You just saw the pretty colors and patterns and went for it? Well, I did - with a jelly roll of white, beige, grays, and black with mathy patterns (Note: A jelly roll is a roll of
Fractal Kitty — 1/1/2026
Happy New Year! It’s time for Genuary 2026! I am not sure how many prompts I will do (or combine), but I hope to share my code and progress here. I hope to get at least 5-10 done this year with a mix of different languages and
Fractal Kitty — 12/31/2025
IntroductionIn this inquiry, we explore chords, which are lines drawn across circles, using different rules to create various patterns, curves, and shapes. This inquiry will be different from those in the Inquiries Series in that it will be more…
Abuse of Notation — 12/24/2025
Everything we do to secure ourselves, every decision we make out of fear of the future, ends up destroying us, ends up making our future a little more bleak — the closer we are to the public ideal of “success”, the farther we go from our own…
Abuse of Notation — 12/24/2025
When the tower crumbles, some will laugh some will cry, some will fall from the top, some will be buried below. When the tower crumbles, better not be around better go all the way down, so you can run away.
Fractal Kitty — 12/2/2025
December Adventure was started by Eli_oat at Oatmeal. I love seeing what others do this month - here is a log of logs.This December, I plan to make a generative quilt, play with origami, doodle some mossy mandalas, set up next year’s journal, and…
Abuse of Notation — 12/1/2025
I should stop doing category theory. What’s the point?
Math ∩ Programming — 11/17/2025
In an earlier article, I covered the basic technique for performing matrix-vector multiplication in fully homomorphic encryption (FHE), known as the Halevi-Shoup diagonal method. This article covers a more recent method for matrix-matrix…
Joel David Hamkins — 11/12/2025
This will be a talk for the Logic Seminar at the University of Notre Dame, Tuesday 18 November 20215 2pm 125 Hayes-Healy Building. Abstract. I shall introduce what I call the first-order elementary theory of surreal arithmetic, a theory that is…
Fractal Kitty — 11/8/2025
A coded poem and trackThe Track:Initiated on a piano, and realized on an OP-1 Field. The Visual:Human coded in p5js with p5sound - enjoy in fullscreen. Click here for the visual with music - (CW: Strobing effects).The Coded Poem (that makes the…
Fractal Kitty — 11/6/2025
IntroductionIn this inquiry, nodes are connected one at a time. How many lines can you draw before a triangle emerges?Starting with FourLet’s start with four nodes - draw them on a sheet of paper. How many lines (called edges) can you draw before a…
Joel David Hamkins — 11/4/2025
I am honored to be giving the 2025-26 Charles R. DePrima Memorial Lecture for the Mathematics Department of the California Institute of Technology. This lecture series aims to bring mathematical researchers to Caltech to give talks for a primarily…
Math ∩ Programming — 10/19/2025
Polyhedral optimization is a tool used in compilers for optimizing loop nests. While the major compilers that use this implement polyhedral optimizations from scratch,1 there is a generally-applicable open source C library called the Integer Set…
Abuse of Notation — 10/16/2025
In my last post about generality, I tried to show how our ambition to discover ideas that are all-encompassing and eternal makes our worldview crumble, leaving us unable to think clearly even about simple issues with obvious solutions. Today, I…
Joel David Hamkins — 9/10/2025
Recently I had the pleasure to give a talk at the Conference on the occasion of Jörg Brendle’s 60th birthday at Kobe University in Japan, and I was invited to make remarks at the conference banquet given in his honor. … Continue reading →
Joel David Hamkins — 8/20/2025
This will be a talk for the Conference on the occasion of Jörg Brendle’s 60th birthday at Kobe University in Kobe, Japan, 2-5 September 2025. Many years ago, I was a JSPS Fellow at Kobe University, at the same time … Continue reading →
Joel David Hamkins — 8/18/2025
This will be a talk I shall give for the History and Philosophy of Science (HPS) Colloquium at the University of Notre Dame, 17 October 2025, 12:30-1:30 pm, 201 O’Shaughnessy Hall. Did Turing ever halt? Abstract. Alan Turing’s 1936 paper … Continue…
Math ∩ Programming — 7/25/2025
On Monday, July 14th 2025, I hosted a mini-workshop on homomorphic encryption at Google’s Portland, Oregon office. Though Portland is a small city, it’s becoming a hub for homomorphic encryption. Intel and Google both have a presence here, as well…
Math ∩ Programming — 7/18/2025
I work on homomorphic encryption (HE or FHE for “fully” homomorphic encryption) and I have written a lot about it on this blog (see the relevant tag). This article is a collection of short answers to questions I see on various threads and news…
Joel David Hamkins — 7/16/2025
This will be a talk for the Fudan Logic Seminar at Fudan University, to be followed immediately by two talks for the Fudan Logic student seminar, forming a mini-conference for the logic group on 23 July 2025. Abstract. I shall … Continue reading →
Joel David Hamkins — 7/14/2025
This will be a talk for the Seminar on Frontier Issues in Logic and Philosophy The First Forum on Logic and Philosophy 逻辑与哲学前沿问题研究”学术研讨会暨第一届逻辑与哲学论坛 Changchun, China, 18-20 July 2025 Pointwise definable end-extensions of models of arithmetic and set…
Abuse of Notation — 5/27/2025
title: A monad layout: microblog category: microblog tags: programming haskell — A monad is when you know how to convert $M (M a)$ to $M a$, but not $M a$ to $a$.
Abuse of Notation — 5/27/2025
The level of progress in programming language design: Shortly after the first ever programming language was created, it’s author said that the language’s whole paradigm is flawed and we should do functional programming instead*. That was 46 years…
Abuse of Notation — 5/27/2025
I want to forget Gaza. I want to forget the pain. I want to forget all dumb jokes, old song lyrics. I want to forget that most things ever happened, as remembering makes everything tedious. Forces you to become a bureaucrat of your memories, to…
Abuse of Notation — 5/27/2025
Every time you eat mushrooms, I have to try them too. Cause, if the mushrooms turn out to be poisonous and you die of slow and painful dead, I will die as well. I once dreamed that I was dying, You were beside me, I turned to you and said, “I don’t…
Abuse of Notation — 5/27/2025
The road I walk is new. It’s made not of stone, but dirt. There are no road signs yet. And no bridges to cover the rivers. The road I walk is new, so people think that it is hard. “How do swim?” “Don’t you get dirty?” “How do you cope with all that…
Good Fibrations — 5/23/2025
I’ve felt for a long time that automorphisms of curves should control or at least exert serious force on the slopes on their Jacobians. Symmetry forces height, as I’ve written about previously in Models of Formal Groups Laws of Every Height, and…
Math ∩ Programming — 5/13/2025
I was inspired to browse some of Edsger Dijkstra’s essays today, and came across his speech, “Under the spell of Leibniz’s Dream”. It’s the sort of personal history I love to read, which gives one person’s sense of the world over a period of…
Math ∩ Programming — 5/12/2025
Editor’s note: This essay was originally published on Medium on 2016-03-05. I have made minor edits in this republishing and added a few small retrospective notes. 2010–2011 (Year 0) I had just switched my major at Cal Poly State University from…
Proses.ID — 4/18/2025
lately I’ve been watching a lot of videos on YouTube about China. it’s partly due to all the chaos around Trump’s tariff, but mostly these…
Math ∩ Programming — 4/15/2025
Last month I gave a talk on the HEIR compiler project at the FHE.org conference in Sofia, Bulgaria. The video is on YouTube now, and the slides are public. I plan to write more about HEIR in the coming months, because it’s been an exciting and…
Math ∩ Programming — 4/1/2025
It’s April Cools! Last year I wrote about parenting, in 2023 about friendship bracelets. and in 2022 about cocktails. This year it’s a bit of a meandering stroll through some ideas around mutual aid and self-reliance. Maternity wards If you walk…
Proses.ID — 3/17/2025
What is reality? It might seem like a straightforward question, but Ursula Franklin, a pioneering thinker in technology and society, saw it as layered and…
Math ∩ Programming — 2/23/2025
My four-year-old son has declared 36 to be the best number. His reason: 36 is the only number (he knows of) that is both a square and a staircase number AND an up-and-down-staircase number. “Staircase numbers” are what he calls triangular numbers…
Good Fibrations — 2/14/2025
Here’s something enticing and strange: there are two “cut and paste” invariants of the same group which are equal to dual zeta values! the euler characteristic ( \chi(SL_2(\mathbb{Z})) = \zeta(-1), ) and the tamagawa measure of (…
Math ∩ Programming — 2/10/2025
A colleague of mine recently lent a hand implementing a polynomial approximation routine I could port to our compiler, though it wasn’t the method I was expecting. As I had written about previously, I was studying the Remez algorithm and…
Math ∩ Programming — 2/7/2025
Back in 2020, when I worked in the supply chain side of Google, I had a fun and impactful side project related to human-level explanations of linear programs. A linear program is a mathematical model that defines some number of variables, linear…
Good Fibrations — 1/28/2025
Good Fibrations — 1/17/2025
Cech Covers (click the link to listen to us). I wrote this song with my beloved old room mate Christian Gorski in my last year of grad school while I was wrapping up my thesis. For weeks, I was doing nothing but computing etale sheaf cohomologies…
Good Fibrations — 1/17/2025
This drawing is an old drawing I made when I was preparing for my qualifying exam in my second year of grad school at Northwestern. It is the crystalline period map. The tower to the left is the “Lubin-Tate” tower, the deeper it goes the more level…
Good Fibrations — 1/16/2025
I have been working recently to counter the writers block that has formed insidiously from an unhealthy creeping perfectionism. In order to do this, I will post some old art and music which at the time I felt was “not good enough to share” or…
Math ∩ Programming — 1/8/2025
I’ll be at the Joint Mathematics Meeting in Seattle (starting tomorrow). If you see me there, say hi! I will have a very light schedule, plenty of time for coffee chats. I’ll be attending many of the crypto sessions for the homomorphic encryption…
Math ∩ Programming — 1/4/2025
The Hyperfixed Podcast had a lovely episode recently about tape measures. It started from “why does my tape measure seem to always be off a little bit” and went all the way to the inherent limitations of physical measurement at small scales. In…
Math ∩ Programming — 1/3/2025
In this living document, I will document reactions to uses of homomorphic encryption by members of the public. By “member of the public,” I mean people who may be technical, but are not directly involved in the development or deployment of…
Math ∩ Programming — 11/15/2024
In my little corner of the FHE world, things have been steadily heating up. For those who don’t know, my main work project right now is HEIR (Homomorphic Encryption Intermediate Representation), a compiler toolchain for fully homomorphic encryption…
Math ∩ Programming — 11/15/2024
Editor’s note: This essay was originally published in 2019. I have made minor edits in this republishing. There was a MathOverflow thread about mathematically interesting games for 5–6 year olds. A lot of the discussion revolved around how young…
Math ∩ Programming — 11/1/2024
Welcome to the 233rd Carnival of Mathematics! Who can forget 233, the 6th Fibonacci prime? Hey, not all numbers are interesting. Don’t ask me about the smallest positive uninteresting number. You can’t make it interesting with your feeble mind…
Math ∩ Programming — 10/31/2024
This article will explain how the blog is organized at a technical level, and show how I implemented various IndieWeb features. Table of Contents: Motivation Structure and Deployment Static search index Running scripts via GitHub Actions Social…
Proses.ID — 10/16/2024
I’ve been wanting to reflect of the past 2 years and 2 months I spent not working, but it never felt urgent or important (to…