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 — 7/9/2026
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…
Crooked Timber — 6/28/2026
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…
Crooked Timber — 6/26/2026
For the first time in history, the country in which I live – the Netherlands – has issued a Code Red alert due to the heat. Code red is only issued when the environmental circumstances are such that there is a significant risk of “destabilising of…
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…
Crooked Timber — 6/24/2026
There’s long been a disconnect between concerns about the massive impact of AI data centres on electricity demand and claims by Sam Altman and others that the impact is really modest. Ed Zitron recently posted a summary of OpenAI’s 2025 accounts…
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…
Crooked Timber — 6/24/2026
One Spring Monday in 1852 around thirty gold buyers gathered for the evening at Mrs Black’s Royal Hotel in Bathurst, which was (and is) just on the other side of the Blue Mountains from Sydney. Probably not ordinarily the most collegiate of petty…
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…
Crooked Timber — 6/21/2026
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
Crooked Timber — 6/21/2026
Sorry I’ve been offline for so long. I’m back. For now, anyway. Among other things I’ve been writing a little bit about what it was like for me being a teenager involved in left wing politics at the beginning of the 80’s. This is the first (and far…
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 ……
Crooked Timber — 6/19/2026
I was doing a deep dive into early Canadian history, because reasons, and found a couple of fun stories to share. Because hey — this Sunday is National Indigenous Peoples Day! The Bad Overwinter So a recurring thing in early Canadian history was…
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…
Crooked Timber — 6/17/2026
Sometimes people that know and like each other, and that would never employ snark with each other, can still talk entirely past each other online. Carlo Ludovico Cordasco (Sheffield) wrote a fruitful and prudent sub-stack post (here) on the…
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…
Crooked Timber — 6/15/2026
The SpaceX IPO, valuing a motley collection of dubious business at over a trillion dollars, marks the abandonment of the Efficient (financial) Markets Hypothesis, one of the zombie ideas I criticised in the wake of the Global Financial Crisis. Not…
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…
Crooked Timber — 6/14/2026
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…
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 ……
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…
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…
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…
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…
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…
Good Fibrations — 4/19/2026
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…
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 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 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,…