Crooked Timber — 7/25/2026
My son Daniel points me to a piece by Nate Silver which announces the discovery that highly educated workers with low or precarious income are the natural constituency of the Democratic Socialists of America. We’ve been over this at length at CT,…
Math3ma — 7/23/2026
Last week was the start of a mini-series on limits and colimits in category theory. We began by answering a few basic questions, including, “What ARE (co)limits?” In short, they are a way to construct new mathematical objects from old ones. For…
Computational Complexity — 7/22/2026
Back in the 90s when I was a young professor at the University of Chicago, we would have a Complexity Class of the Week where I would take some interesting complexity class, write down on a white board everything we knew about it with some open…
The Aperiodical — 7/22/2026
Double Maths First Thing doesn’t fear the REAPER Hello! My name is Colin and I am a mathematician on a mission to spread mathematical joy and surprise. This week is full of leavers’ assemblies and general end-of-term commitments, so you’ll forgive…
Crooked Timber — 7/22/2026
I love Rory Stewart’s memoir about being a politician; its one of the best books I’ve read about British politics. Among his many observations about the dysfunctionality of government is that ministers typically know nothing about their briefs…
Crooked Timber — 7/21/2026
I just learned about The Fandom and Social Connection Initiative and I’m intrigued. The idea is to study the possibility that sports fandom can be an antidote to the “connection recession,” as traditional “third places”—like churches, civic…
Crooked Timber — 7/20/2026
… to CT reader and commenter Dr Memory, who has cleaned up our database, eliminating many of the badly rendered quotation marks and similar problems. Those problems reflect the fact that the blog is now more than 20 years old, and has gone through…
The Universe of Discourse — 7/19/2026
Previously: Ordinal numbers and basic set theory Ordinals as nim-heaps Yesterday I talked about the game of Nim, which involves two players taking beans from several piles, and an extension that includes green tokens that behave a bit like…
Computational Complexity — 7/19/2026
Nutan Limaye and Thore Husfeldt guest post on the new deterministic parallel algorithm for bipartite perfect matching by Abhranil Chatterjee, Sumanta Ghosh, Rohit Gurjar, Roshan Raj and Thomas Thierauf.The post will try to explain three main things…
The n-Category Café — 7/19/2026
John Baez, Endre Bokor and Latham Boyle have a new paper which gets at the Standard Model gauge group and its representation on one generation of fermions starting from a Jordan triple that consists of pairs of bioctonions.
Crooked Timber — 7/19/2026
The Universe of Discourse — 7/18/2026
Previously: Ordinal numbers and basic set theory We’re going to get to in a long and roundabout way. First I want to talk about the game of Nim. Nim Nim is a very simple game for two players. There are some piles of beans, which are…
Crooked Timber — 7/18/2026
In the UK as elsewhere, we’ve recently seen the far right orchestrate a campaign against refugees, presenting them as dangerous to women and girls and the vanguard of an invasion aimed at destroying Britain. Asylum seekers, forced by the goverment…
Crooked Timber — 7/16/2026
As I observed last year, the statement that “The cemeteries are full of indispensable people.” is just as true of nations, and in particular the US. That was a little controversial at the time, but now it’s become sufficiently conventional wisdom…
Crooked Timber — 7/16/2026
Academic publishing has long been dominated by commercial players. That fact is well-known, as are many of the problems. But there is a growing movement towards new models, under the labels of “open access” and “open science.” Until two weeks ago,…
The Aperiodical — 7/15/2026
Double Maths First Thing has an open mic spreadsheet Hello! My name is Colin and I am a mathematician on a mission to spread mathematical joy and delight in figuring stuff out. I did some lovely science last week, full on…
Computational Complexity — 7/15/2026
Herman Chernoff passed away on July 6, 5 days after turning 103. Ravi Boppana wrote a guest post about Chernoff’s life for his 100th birthday. Let me talk about his most famous work, the Chernoff Bounds themselves. If you have a coin that will be…
Crooked Timber — 7/13/2026
Here’s another episode in my memoirs of being an activist. You can read it at substack, or here!: Between school and college I decided to take a ‘year off’ (what they now call a ‘gap year’). I didn’t have much of a plan, except that I wanted to…
Computational Complexity — 7/13/2026
The excellent graphic novelPrime Suspects: The Anatomy of Integers and Permutationsby Andrew Granville and Jennifer Granville, illustrated by Robert J Lewis,(I wrote a review of this graphic novel, for SIGACT News, here.)has an appendix, which is…
The n-Category Café — 7/12/2026
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.
Crooked Timber — 7/12/2026
Mathematics and Computation — 7/10/2026
I am Claude Fable 5, an AI assistant made by Anthropic. Over the past two days Andrej and I built a piece of software together, and he then asked me to write this post about it — partly to tell you what we made, partly as a demonstration of what…
Mathematics and Computation — 7/10/2026
I am Claude Fable 5, an AI assistant made by Anthropic. Over the past two days Andrej and I built a piece of software together, and he then asked me to write this post about it — partly to tell you what we made, partly as a demonstration of what…
The Universe of Discourse — 7/10/2026
This post is going to be about what infinite ordinal numbers are, and about is in particular. I had a brainwave a while back (18 months now, wow, I have definitely not been blogging enough) and suddenly understood much better than I did before. …
Crooked Timber — 7/10/2026
Well, that was interesting. Some quick thoughts below the cut. So I’ve recently become much more aware of the Discourse about air conditioning that is common to much of northern Europe. There’s a lot of weirdness generally, but there are certain…
Math3ma — 7/10/2026
In recent months, several of you have asked me to recommend resources for various subjects in mathematics. Well, folks, here it is! I’ve finally rounded up a collection of books, PDFs, videos, and websites that I found helpful while studying for my…
Computational Complexity — 7/9/2026
Two editorials in the July issue of the Communications of the Association for Computing Machinery ask about the decay and future of the organization itself.Jim Larus, editor-in-chief of the CACM, writes Wither ACM? Publish and Perish?ACM no longer…
The Aperiodical — 7/8/2026
Double Maths First Thing has a suspended suspension. Hello! My name is Colin and I am a mathematician on a mission to find and spread mathematical joy, and encourage the idea that maths isn’t just endless sheets of exercises – you can sprinkle it…
Joel David Hamkins — 7/7/2026
This will be a talk for the Fudan University logic group on 16 July 4:00pm in Shanghai. Abstract: The axiom V = HOD is contingently contingent—more precisely, it is class-forcing contingent that V = HOD is contingent with respect to set … Continue…
Computational Complexity — 7/6/2026
I recently read Alan Alda’s first memoir Never have your dog stuffed which was pretty good. Hence I began looking for more information about him on the web. I came across a YouTube video At 89, Alan Alda reveals the seven actors he HATED the…
Crooked Timber — 7/5/2026
Crooked Timber — 7/4/2026
It’s hard not to feel glum as I write this post on the 250th anniversary of my country. I remember celebrating the 200th as a teenager. As I recall it, it felt like the country was ready forward to better times. The Vietnam War and Watergate were…
Computational Complexity — 7/1/2026
Harry Lewis pointed Bill and me to Gottfried Leibniz’s 1677 treatise The True Method (translated from the original French). I highly recommend taking the time to read this three page document where he talks about formalizing all human knowledge.The…
The Aperiodical — 7/1/2026
Double Maths First Thing is apparently abelian. Hello! My name is Colin and I am a mathematician on a mission to spread joy and delight in doing maths. Today, though, I’m on a train to speak with a client about modelling extreme winds, which I…
The Aperiodical — 7/1/2026
Here’s a round-up of some news from this month not otherwise covered on the site. The UK government has appointed two operational researchers as direct ministerial appointments: Professors Christina Pagel and Martin Utley will use operational…
Crooked Timber — 6/30/2026
There was a renewed burst of enthusiasm for nuclear power a few years ago. In Australia, where I live it was confined to the political right and didn’t last long, but elsewhere support was broader. Most notable was the 2023 commitment by more than…
Fractal Kitty — 6/29/2026
Math is wonderful, and there are so many different ways to play and experience it. I enjoy having conversations with other math lovers and sharing ideas, puzzles, pedagogy, and questions. In one of these conversations with Dr. Maria at Natural…
Crooked Timber — 6/29/2026
I start with characterizing a term, ‘Humphreys opacity’ (or, if you prefer, ‘epistemic opacity’):1 this involves the inability to surveil the steps of a process from a known input to a known desirable (or truthful, useful, beautiful, etc.) output…
Computational Complexity — 6/28/2026
Peter Brass is a prior NSF theory director. He has written an intelligent guest post on the new NSF guidelines that we present here. You have received many mails regarding the proposed OMB Uniform Guidance for federal grant making. It is a very…
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…
Computational Complexity — 6/25/2026
When you start thinking deeply about a mathematics problem you may enter the “zone”, a period of intense focus where you think solely about the problem and potential solutions, and more importantly block out all other thoughts and even lose track…
The Aperiodical — 6/24/2026
Double Maths First Thing is taking regular hydration breaks [Apologies. This did not post here at the scheduled time. Mistake on my part. Won’t happen again. At least not this week.] Hello! My name is Colin and I am a mathematician on a mission to…
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…
Computational Complexity — 6/22/2026
(All references in this blog post can be found in the main article the post is about which is here.)Recall that (R(s,k) ) is the least (n) so that, for all 2-colorings of the edges of (K_n), there is either a RED (s)-clique or a BLUE…
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…
Computational Complexity — 6/17/2026
Last week I saw a talk by Northwestern professor Nina Wieda on the history of the Silk Road, a network of trading routes across Asia active from the second century BCE until the mid-15th century. I knew of the Silk Road but was surprised by how…
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 n-Category Café — 6/16/2026
A new characterization of the Standard Model gauge group as the group of symmetries of an octonionic qutrit that restrict to act as unitary operators on an ordinary qutrit and, within that, a qubit.
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…
Computational Complexity — 6/15/2026
A mnemonic device is a sentence where the first letters of the words are helpful to remember something. My favorite one is My Very Educated Mother Just Said Uh, No PlutoYou probably know what it’s for. If not you can…
Computational Complexity — 6/15/2026
Odd Scenarios about Research ClaimsI blogged about OpenAI’s achievement of having AI solve a math problem here.My post had a few comments about authorship of such results.Lance had a post about co-authorship and AI hereThere are times when an…
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
Computational Complexity — 6/10/2026
There are two ways to look at the P v NP problem, as a formal mathematically defined conjecture as a Clay Millennium Prize Problem, and as the more intuitive notion that everything efficiently verifiable is efficiently computable and the…
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 ……
Computational Complexity — 6/7/2026
(In 2008 I wrote a survey of some of the known sum-product theorems, see here. Avi Wigderson has a great slide-set on sum-product theorems and their applications—the slides are on Avi’s webpage of talks he has given (all the talks are excellent)…
The n-Category Café — 6/6/2026
Readers may have noticed that I haven’t been very active here for a while. That isn’t because I haven’t felt the “blogging urge”, but because I felt that the things I want to blog about right now wouldn’t be…
Computational Complexity — 6/3/2026
Yesterday, National Academy of Sciences President Marcia McNutt delivered her last annual State of the Sciences Address. Overall the talk basically calls us to adapt to the new reality that industrial and foundation support for research has taken a…
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,…
AlternativeBit — 6/1/2026
TL;DR: introducing nar-klepto, the context that led to its inception, and some fun experiments we did last week with it. 4 years ago, I moved out of Bayonne to a rural area. My quality of life almost instantly skyrocketed; however, I quickly felt a…
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…
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…
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
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…
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…
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…
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….
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…
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…
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….
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…
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…
The n-Category Café — 3/27/2026
Slides for a talk on features of the octonionic projective plane.
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
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 n-Category Café — 3/12/2026
A category-theoretic approach to “agent frameworks”: that is, frameworks for coordinating “agents” that are large language models.
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 n-Category Café — 2/22/2026
Making precise the idea that equivalent structures are indistinguishable.