Blog - Logic Matters — 5/2/2026
I am still working again through the third edition of the category theory book, correcting typos in the first printed version (fortunately, relatively few are possibly misleading), correcting thinkos (cheeringly not many), rewording a few…
The n-Category Café — 5/2/2026
In the last episode of my column in Notices of the American Mathematical Society, we looked at a particle moving in an attractive central force whose strength is proportional to the inverse cube of the distance from the origin….
The Aperiodical — 5/1/2026
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of April 2026, is now online courtesy of Karrie Liu. The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical….
Computational Complexity — 4/29/2026
My favorite quote about networking came from Jim Kurose. The Internet works so well because it doesn’t have to. The IP and lower layers of the internet stack make no promises of delivery. Complete failure fulfills the protocol. This allows for…
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…
The Aperiodical — 4/29/2026
Double Maths First Thing no longer runs marathons. Hello! My name is Colin and I am a mathematician on a mission to spread the joy and delight of doing maths, figuring things out and generally taking pleasure in doing clever things. Like everyone…
The Aperiodical — 4/28/2026
There’s a story about a child mathematician talking to an older mathematician and saying “I think the biggest number is a TRILLION.” The grown-up says “OK, but what about a trillion and one?” The child mathematician looks crestfallen, but only for…
Crooked Timber — 4/28/2026
That’s the actual name of the paper. Isn’t that great? Here’s a prologue: a post I wrote a while back about the Portuguese Man-o’-War. (It’s kind of long — I was new to CT back then, and still figuring stuff out). To summarize: the Portuguese…
Computational Complexity — 4/26/2026
A few months ago in Oxford, Bernard Sufrin, an emeritus fellow, said he’s looking to hire a student to implement LEAP (Logic Engine for Argument by Pointing), a way to teach logic by proving basic logic theorems via pointing and clicking. Rahul…
Crooked Timber — 4/26/2026
Crooked Timber — 4/24/2026
“I admire the many federal prosecutors across the country who have chosen to resign rather than carry out illegal or immoral orders. To my knowledge, no department head, dean, or other administrator at Texas A&M has taken any meaningful action to…
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…
Computational Complexity — 4/23/2026
Michael Rabin passed away on April 14, 2026 at the age of 94. (Scott Aaronson has also blogged about his passing, see here.) I had many points to make about him; however, the first one got so long that I will just do that one for today’s blog…
The Aperiodical — 4/22/2026
Double Maths First Thing: Northerly 3-5, moderate, fair, good, some kraken later. Hello! My name is Colin and I am a mathematician on a mission to spread the joy and delight of doing maths, solving puzzles, and making the world a better place. I’m…
Crooked Timber — 4/20/2026
I said a while back that nobody’s going to Mars any time soon. Which is true. But that doesn’t mean Mars isn’t interesting! Mars is very interesting. So today’s paper is about Mars. Okay, it’s about a moon of Mars. TLDR: one of Mars’ moons may…
Crooked Timber — 4/19/2026
Blog - Logic Matters — 4/19/2026
Another six large-carrier bags of books have gone to Oxfam. And my tiny redecorated study is at last put together again, and — a miracle! — with no books in piles on the floor. Rather late in the day, it is the nicest-looking room I have ever had….
Crooked Timber — 4/19/2026
Back in the 1980s, I was (among other things) a writer and singer of satirical folk songs. Going to the National Folk Festival in Canberra at Easter, I caught up with old friends and was reminded that I had produced a book of my songs. Returning…
Crooked Timber — 4/17/2026
What does it mean to be an academic in different parts of the world? What comes along as the same job description – a bundle of teaching, research, and impact tasks – varies enormously from place to place. Not only the financial conditions of…
Computational Complexity — 4/16/2026
At Oxford I focused my research and discussions on how we can use the tools of computational complexity to help us understand the power and limitations of machine learning. Last week I posted my paper How Does Machine Learning Manage Complexity?, a…
Crooked Timber — 4/15/2026
Do you know Baba Yetu? Take three minutes and listen to this performance of Baba Yetu. (Our ancient blogging platform doesn’t like embedded video, so you’ll have to click through to YouTube. Go ahead and click, nothing bad will happen.) Some…
The Aperiodical — 4/15/2026
Double Maths First Thing is playing 1D chess. Hello! My name is Colin and I am a mathematician on a mission to spread the delight and joy of doing maths, and trying really hard to communicate that it’s not just about arithmetic. The kids have…
Computational Complexity — 4/14/2026
Guest post from Peter Brass, Former NSF Theory director (though not affiliated with the NSF now) on the White House NSF budget for FY 2027.———————————————Dear ColleaguesA week ago the White House released the NSF…
Crooked Timber — 4/14/2026
Before I depart this world, I would like to visit St. Giles’ Cathedral in Edinburgh, Scotland, and see the Jenny Geddes memorial. I’m told it’s open to the public. Why? What’s interesting about a stool? Well, it’s probably impossible to point to a…
Crooked Timber — 4/14/2026
Among other things, the unlamented former autocrat Viktor Orban was one of the leading proponents of pro-natalist policies, and more open than most about the racist underpinnings of his view. However, like others who have tried to raise birth…
Computational Complexity — 4/14/2026
I posted about using the Banach-Tarski Paradox(BT) to explain the miracle of Loaves and Fishes (LF) here.Darling says that whenever I fool my readers or my students then I have to tell them later, so I’ll tell you now: The story about me meeting…
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…
The Aperiodical — 4/13/2026
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of March 2026, is now online at Tom Rocks Maths. The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See…
Crooked Timber — 4/13/2026
The news from Hungary’s election is so good that I need to write about it, even if not all the implications are clear yet, and even in a disorganised and way, repeating lots of what others are saying. Although the polls predicted Orban’s defeat,…
Crooked Timber — 4/12/2026
The Aperiodical — 4/12/2026
Here’s a quick round-up of maths internet news this month! Progress has been made on the Lonely Runner problem, which concerns when runners of different speeds going round a track will meet up, and has connections to higher-dimensional geometry. A…
Blog - Logic Matters — 4/10/2026
With my mind partly on revising the Study Guide, I have been browsing through three relatively recently published logic books. None of them impress as candidates for self-study. I will be brief (unfairly? life is short …). Robert André’s Set…
Crooked Timber — 4/10/2026
The New South Wales gold rush began more than 400 million years ago. It was an age of fire, that ended with ice. Australia was part of the super-continent Gondwana, which was not yet south. By continent standards it was moving fast. By the end of…
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…
Crooked Timber — 4/9/2026
A journalist from the Wall Street Journal wrote to me a week ago to ask what the numbers that I use in the opening pages of my book Limitarianism would look like today. In particular, she asked whether I could calculate for her the “lifetime…
The Aperiodical — 4/8/2026
Double Maths First Thing could be Rotterdam or anywhere Hello! My name is Colin and I am a mathematician on a mission to spread the joy and delight in doing maths. I’m on a bit of a buzz at the moment – I went to my local open mic on Friday and had…
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,…
Crooked Timber — 4/7/2026
(A piece I wrote for the Guardian) A couple of weeks ago, just before my 70th birthday, I completed the Mooloolaba standard distance triathlon (1,500m swim, 40km cycle, 10km run). There was nothing exceptional about my performance, placing 1,509…
Blog - Logic Matters — 4/6/2026
Alex Oliver and Tim Smiley’s magnum opus Plural Logic runs to some 340 pages before the end matter in the second edition. The final chapters are heavy going, involving detailed developments of some cumbersome (indeed, uninviting) formal theories. I…
Computational Complexity — 4/5/2026
Here are the solutions to the problems I posted last week. Problem 1 A language (L) is commutative if for all (u), (v) in (L), (uv = vu). Show that (L) is commutative if and only if (L) is a subset of (w^*) for some string…
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…
Blog - Logic Matters — 4/2/2026
Well, that all took rather longer than expected … A lot of repairing and filling and preparation (the joys of a late Victorian house). But at last the miniature room that counts as my study is redecorated. A carpet remains to be laid in ten days,…
Computational Complexity — 4/1/2026
I blogged about Pope Leo XIV here. Pope Leo XIV has an undergraduate degree in mathematics. He saw my post and asked for my help with his latest encyclical. LEO: Let’s have lunch together at Popeyes.BILL: Why Popeyes?LEO: The name is Pope-yes so I…
The Aperiodical — 4/1/2026
Double Maths First Thing insists it’s March 32nd Hello! My name is Colin and I am a mathematician on a mission to spread the joy and delight in mathematical thinking. After the clock change and a trip to that London – we chanced on a Rob Eastaway…
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
Computational Complexity — 3/29/2026
Occasionally I run into what I consider fun problems in complexity, that require just a little bit of out of the box thinking. They require some background in theory, but nothing too deep. Some of these problems have been mentioned before on my…
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…
The n-Category Café — 3/27/2026
Slides for a talk on features of the octonionic projective plane.
Blog - Logic Matters — 3/25/2026
Some of my very favourite paintings are Annunciations … the stunning Leonardo and Botticelli in the Uffizi, the lovely small painting from a predella by Domenico Veneziano close to home in the Fitzwilliam, the wonderful Filippo Lippi in the…
Computational Complexity — 3/25/2026
High table dinner at MagdalenMy time in Oxford has come to an end and I head back to Chicago this week. I was a visiting Fellow at Magdalen (pronounced “maudlin”) College for the Hilary Term.There’s a six week break between the eight-week Hilary…
The Aperiodical — 3/25/2026
Double Maths First Thing saw two shooting stars last night Hello! My name is Colin and I am a mathematician on a mission to spread the joy of thinking mathematically, solving problems and generally doing creative things with one’s brain. Such as,…
Computational Complexity — 3/22/2026
If I get an email offering me a $1000 for I DON”T KNOW SINCE I ignore it and don’t even bother looking for other signs it is a scam. If I get an email offering me $100 I may look more carefully and often they are legit (most common is to give a…
The n-Category Café — 3/20/2026
A large language model has very little self-awareness. But it is easy to give it some rudimentary but useful forms of self-awareness using the “plumbing” language.
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
Computational Complexity — 3/18/2026
Gilles Brassard and Charlie BennettCharlie Bennett and Gilles Brassard will receive the 2025 ACM Turing Award for their work on the foundations of quantum information science, the first Turing award for quantum. Read all about it in The New York…
Blog - Logic Matters — 3/16/2026
In the last couple of years, Dirk Hoffmann has published English translations of two of his books which originally appeared in German. In 2024, he gave us Gödel’s Incompleteness Theorems (Springer), which I will discuss in a future post. And then…
Computational Complexity — 3/16/2026
(If you live in Montgomery County Maryland OR if you care about Education, you MUST read this guest blog by Daniel Gottesman on Scott Aaronson’s blog HERE.) (This post is a sequel to a prior post on this topic that was here. However, this post is…
The n-Category Café — 3/12/2026
A category-theoretic approach to “agent frameworks”: that is, frameworks for coordinating “agents” that are large language models.
Computational Complexity — 3/10/2026
Turing Award winner and former Oxford professor Tony Hoare passed away last Thursday at the age of 92. Hoare is famous for quicksort, ALGOL, Hoare logic and so much more. Jim Miles gives his personal reflections.Jill Hoare, Tony Hoare, Jim Miles….
Computational Complexity — 3/8/2026
In my graduate Ramsey Theory class I taught Kruskal’s tree theorem (KTT) which was proven by Joe Kruskal in his PhD thesis in 1960. (Should that be in a graduate Ramsey Theory class? There are not enough people teaching such a course to get a…
Blog - Logic Matters — 3/4/2026
Some good things come out of America — Dvořák’s String Quintet Op. 97, for example, written while he was staying with a Czech-speaking community in Iowa for the summer of 1893. And there was a quite joyous performance a few days ago at Wigmore Hall…
Computational Complexity — 3/4/2026
In discussions of AI and Mathematics, the discussion often goes to mathematical proofs, such as the the First Proof challenge. So let’s look at the role of proofs in mathematics.Without a proof, you don’t even know whether a theorem is true or…
Computational Complexity — 3/2/2026
Goodhart’s law: When a measure becomes a target, it stops being a measure. I was watching the show Masterminds where Ken Jennings is one of the Masterminds. Here is what happened: Brook Burns (the host): The only vice president in the 20th century…
Blog - Logic Matters — 2/28/2026
As I said some weeks ago, I am slowly revising my Introduction to Gödel’s Theorems; I am still only about a third of the way through. I haven’t yet spotted any real horrors, but I’ve found some ways of re-arranging the material for the better, and…
Computational Complexity — 2/25/2026
Last week I had the pleasure of meeting Alex Bellos in Oxford. Among other things Bellos writes the Guardian Monday puzzle column. He gave me a copy of his latest book, Puzzle Me Twice, where the obvious answer is not correct. I got more right than…
Computational Complexity — 2/25/2026
A commenter on this post asked for me (or anyone) to solve the problem without AI:A,B,C,D,E are digits (the poster said A could be 0 but I took A to be nonzero) such thatABCDE + BCDE + CDE + DE + E = 20320.(CLARIFICATION ADDED LATER: We allow two…
The n-Category Café — 2/22/2026
Making precise the idea that equivalent structures are indistinguishable.
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…
Blog - Logic Matters — 2/18/2026
It is time at last to redecorate my small but now rather depressingly shabby study. Already books from the top shelves are piled in a hallway, and the rest are covered in plastic sheeting as I start tackling the late Victorian ceiling (which isn’t…
Computational Complexity — 2/18/2026
Computer Science Professor Joseph Halpern passed away on Friday after a long battle with cancer. He was a leader in the mathematical reasoning about knowledge. His paper with Yoram Moses, Knowledge and Common Knowledge in a Distributed Environment,…
Computational Complexity — 2/15/2026
I sometimes assign open problems as extra credit problems. Some thoughts:1) Do you tell the students the problems are open?YES- it would be unfair for a student to work on something they almost surely won’t get.NO- Some Open Problems are open…
DEONTOLOGISTICS — 2/14/2026
A couple updates for readers. There will be an impromptu book launch for The Revenge of Reason at Newcastle University on the 25th of February, from 5-7pm in HDB.1.02 in the Henry Daysh Building. I’ll be having a conversation about the book with…
Computational Complexity — 2/12/2026
A reader worried about the future.I am writing this email as a young aspiring researcher/scientist. We live in a period of uncertainty and I have a lot of doubts about the decisions I should make. I’ve always been interested in mathematics and…
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 →
Computational Complexity — 2/9/2026
(I thought I had already posted this but the blogger system we use says I didn’t. Apologies if I did. Most likely is that I posted something similar. When you blog for X years you forget what you’ve already blogged on.) Historians who study ancient…
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…
Computational Complexity — 2/4/2026
Wandering around maze known as the Computer Science building at Oxford I found the computer science library. Rarely these days do you see a library (and a librarian) devoted to computer science. The librarian found their copy of The Golden Ticket…
Computational Complexity — 2/2/2026
Lance and I had a blog-debate about What to do about students using ChatGPT to do their Homework.Some commenters pointed out that we’ve been here before. I will now list past technologies that looked like they were a problem for student assignments…
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
The n-Category Café — 1/26/2026
David Jaz Myers has some thoughts about Riemann’s functional equation for the zeta function.
The n-Category Café — 1/6/2026
Dynkin diagrams have always fascinated me. They are magically potent language — you can do so much with them!…
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 approaches.
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.
The n-Category Café — 12/21/2025
There are two ways to stick SU(2) × SU(3) in Spin(10). One is good for physics; the other, alas, is easily obtained using the octonions.
The n-Category Café — 12/4/2025
An introduction to the bioctonionic plane and the mathematics needed to understand it.
The n-Category Café — 12/3/2025
A complex-analytic perspective on the indefinite integral of 1/x.
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?
The n-Category Café — 11/30/2025
Tymoczko gave a good talk on the math of music theory.
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…