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…
Computational Complexity — 5/27/2026
The technical paper for the Erdős Unit Distance Problem lists only “OpenAI” as an author. When Bill posted on Sunday about the Erdős distance problems, he mentioned the names of OpenAI researchers who prompted and checked the proof. Sebastien…
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…
The Aperiodical — 5/26/2026
We were sent a free copy of this book by the publisher, and invited irregular contributor Elinor Flavell to read and review it. Been feeling finite recently? Bounded by life? Those larger numbers feeling out of reach? Then you need “A Guide to…
Computational Complexity — 5/25/2026
In a 1946 paper in the American Mathematical Monthly, Paul Erdős posed the Erdős Distinct Distance Problem and the Erdős Unit Distance Problem.——————————————————————–THE ERDŐS DISTINCT DISTANCE PROBLEMA…
Crooked Timber — 5/24/2026
Blog - Logic Matters — 5/23/2026
The Logic Matters blog started on 9 March 2006. Bother! — I missed its twentieth birthday. Raise a belated glass, as that was a milestone of sorts. I do plan to continue with the blog and website. But I am going to experiment (initially in a local…
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
Crooked Timber — 5/21/2026
One my betes noires has been in the news lately. Jonathan Haidt has been annoying me since at least 2012, when I was critical of his bothsidesism on the culture wars. At the time, he was a concern troll, posing as a liberal worried about other…
Computational Complexity — 5/20/2026
Let (f) be a function mapping binary strings of length (m) to strings of length (n) with (n>m). Since there are more strings of length (n) than (m), (f) is not onto. Can you find a string not in the range? This is known as the range…
The Aperiodical — 5/20/2026
Double Maths First Thing has been blown away by a puzzle Hello! My name is Colin and I am a mathematician on a mission to spread the joy and delight in mathematical thinking, in figuring things out, and in doing things that make you feel clever….
Blog - Logic Matters — 5/18/2026
The revised/corrected third edition of Introducing Category Theory is now available as a paperback. To keep the cost minimal, this is again via print-on-demand from Amazon US, Amazon UK, and of course other Amazons too, with ISBN 1068346728. There…
Crooked Timber — 5/18/2026
Mostly I leave Sunday photography to our colleague, the estimable Chris Bertram. Still, this Sunday I was walking the dog in the hills above my town. (“My town” being a modest community of a couple of thousand people in the rolling countryside of…
Blog - Logic Matters — 5/18/2026
There have been so many moving obituaries and fond tributes. Here, to mark her passing, is Felicity Lott at the start of her stellar career, singing Pamina at Glyndebourne, almost fifty years ago. How joyous. There was, by the way, a wonderful…
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,…
Computational Complexity — 5/17/2026
1) Congratulations to Scott Aaronson for winning the first Trevisan Award.The Trevisan Award is in memory of Luca Trevisan and recognizes expository work in Theoretical Computer Science. It is given out by the ACM. The ACM announcement of Scott’s…
Blog - Logic Matters — 5/17/2026
What have I learnt over recent weeks, preparing a corrected version of Introducing Category Theory, invoking — at various stages — my friends Claude, Gemini and ChatGPT as proof-reading assistants? Not very helpfully, the take-home message is that…
Crooked Timber — 5/17/2026
Blog - Logic Matters — 5/14/2026
The end is in sight. I have just uploaded another PDF of Introducing Category Theory (now identified as Version 3.4 on the verso of the title page — you might need to clear your brower’s cache to force a download of the latest, greatest, version)….
Computational Complexity — 5/14/2026
For those very long-time readers this blog extensively covered prediction markets from 2006 to 2008. In a prediction market, you have a future event, such as the winner of an election, and a market that pays off one dollar if that event happens and…
The Aperiodical — 5/13/2026
Double Maths First Thing is petitioning the weather to be nicer Hello! My name is Colin and I am a mathematician on a mission to spread joy and delight in all things mathematical. I had a nice “oho!” moment in class with a student today looking at…
Crooked Timber — 5/12/2026
Academics, especially in the humanities, produce texts, and they teach students to produce text. This is a standard assumption, often taken for granted, and maybe not too surprising in times in which productivity is a supreme social norm. Think of…
Computational Complexity — 5/11/2026
In David Marcus’s guest post on good coding style (see here) he reviewed a book from 1986 called “Professional Pascal.”I wondered if it was still in print and could be bought:1) I went to Amazon and searched all products for Professional Pascal….
Crooked Timber — 5/11/2026
Last week Australia’s central bank (Reserve Bank of Australia, RBA) raised interest rates. Again. Political economists have been talking for decades about the RBA’s tendency to redistribute wealth from the bottom upwards. But now it seems most…
Crooked Timber — 5/10/2026
Blog - Logic Matters — 5/9/2026
A very interesting new blog post by Tim Gowers on his recent experiences with ChatGPT 5.5 Pro. A headline: “I would judge the level of the result that ChatGPT found in under two hours to be that of a perfectly reasonable chapter in a combinatorics…
Blog - Logic Matters — 5/7/2026
The Victorian floor of my tiny study slopes and dips quite badly and it would have been expensive to engineer a flat, level, surface. So, to finish the redecoration, it has had to be new carpet. No majolica tiles for me. Not like this one (now in…
Computational Complexity — 5/6/2026
As the blog of record in computational complexity, we like to bring attention to those in the community who have left us. When we learn of someone in our field who has died, Bill and I will talk to each other and decide whether we should do a…
The Aperiodical — 5/6/2026
Double Maths First Thing has Youri Tielemans in swaps Hello! My name is Colin and I am a mathematician on a mission to spread the joy and delight of figuring stuff out. I’ve been thinking a lot about MathsJam and Parkrun and similar activities,…
Computational Complexity — 5/4/2026
Michael Rabin passed away on April 14,2026. I blogged about him here. My post listed results of his that proved upper and lower bounds on problems. My point was that he proved upper and lower bounds for MANY different levels- from decidable to…
Crooked Timber — 5/3/2026
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…
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…
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…
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…
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,…
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…
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…
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.
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…
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…
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…
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…
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…
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…
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
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…