John Carlos Baez

I'm a mathematical physicist who likes explaining stuff. Sometimes I work at the Topos Institute. Check out my blog! I'm also a member of the n-Category Café, a group blog on math with an emphasis on category theory. I also have a YouTube channel, full of talks about math, physics and the future.

homepage: http://math.ucr.edu/home/baez/
blog: https://johncarlosbaez.wordpress.com/
math blog: https://golem.ph.utexas.edu/category/
YouTube: https://www.youtube.com/channel/UCy9yxdnj0yfOiMWvMa47PNw

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johncarlosbaez_at_mathstodon.xyz@momostr.pink

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nprofile1qqsf56s63mhapdfa0cxfv64nt0vsg9gjgmcrkxlf37sd958wkj2qxgq82shrm

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Last Notes

2025-05-04 18:25:00

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- yes, I only read the press release. The paper seems find, though about 5 more important math papers appear on the arXiv every day (my judgement).

2025-03-31 23:29:45

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- his thinking that experimentalists are less frustrated than theorists seems like a case of "the grass is greener on the other side". There's a lot more friction in experimental work. So many things break when you're trying to do experiments, or don't work quite as well as you hoped, or cost too much for you to afford.

2025-03-27 02:12:58

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- great, that one could be "better".

2025-03-27 01:18:38

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- you're reminding me of @npub12cu…d38p's work on this, which is explained here:
https://math.ucr.edu/home/baez/visual_insight/20150915%20-%20McGee%20Graph.html
https://media.mathstodon.xyz/media_attachments/files/114/231/758/080/299/155/original/654898ff567a082e.jpg

2025-03-27 01:13:55

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- got it. It matters a bit because there's a thing called a "1-cocycle" which has
D(a c) = D(a) + a D(c)
and then a thing called a "normalize" 1-cocycle which also obeys some other equation. I thought that other equation would be
D(1) = 0
but either it's not, or for some reason all cocycles are automatically normalized here (which sounds fishy).

2025-03-26 21:46:57

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- I'm writing about this now, and I just noticed that
D(a c) = D(a) + a D(c)
implies
D(1) = 0.

2025-03-26 18:23:12

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- another approach would be to avoid linear representations and instead seek subgroups H of the 16-element quasidihedral group G, which give actions of G on the finite sets G/H. To satisfy @npub1whr…qc2t we'd want H to have 2 elements, so G/H would be an 8-element set. There's an obvious 2-element subgroup H ⊆ G consisting of 𝐼 and 𝑠, and this G/H may be Jules' "twisted octagon".
https://media.mathstodon.xyz/media_attachments/files/114/230/119/294/574/977/original/1eeff2efea48b17d.jpg

2025-03-18 19:08:09

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- I'm friends with someone who lived through the Cultural Revolution. She was a lawyer, not a scientist, but maybe I should still talk to her. I kinda doubt Trump will send most scientists to work in the fields to replace the undocumented immigrants he plans to deport, but there could still be useful lessons.

2025-03-17 05:51:48

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- super-cool! I was trying to recognize these equations but I didn't. I wrote a post explaining crossed homomorphisms when I was first trying to understand them:
https://golem.ph.utexas.edu/category/2020/04/crossed_homomorphisms.html

2025-03-10 16:01:49

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You also can't trust AI when it comes to n-categories!
In reality a 1-category is a category, a 0-category is a category, and James Dolan discovered that a (-1)-category is a truth value and a (-2)-category is a true truth value (of which there is only one: "true").
But my friend Allen Knutson asked Gemini and got this garbage. Clearly it's Russian propaganda.
(2/2)
https://media.mathstodon.xyz/media_attachments/files/114/138/963/894/052/197/original/38bf81f0f35f5358.jpg

2025-03-10 15:53:43

A test shows that all the leading AI chatbots have been infected by Russian propaganda created by a Russian organization called the "Pravda network".
According to the American Sunlight Project, a failed attempt to protect democracy in the USA, the Pravda network put 3.6 million articles online in 2024. In a strategy called "LLM grooming", this network seems to be designed to manipulate AI models rather than directly affect people. It's mainly focused on spreading lies connected to the Ukraine war.
https://www.computing.co.uk/news/2025/ai/ai-chatbots-infected-russian-propaganda
(1/2)

2025-03-05 00:26:05

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- what did she say about those?

2025-02-26 02:09:17

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- that's good: so just try to make sure it doesn't speed up work on AI.
You say I have the luxury of retirement, but before I retired I mainly made my living teaching calculus to freshmen.

2025-02-07 20:58:50

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- heh, for Noether that was a mere *lemma*.
Did you know she invented chain complexes and their homology, but never bothered to publish about them?

2025-02-05 01:28:21

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- Thanks! I'm glad I don't have to correct my paper.

2025-02-05 00:54:28

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- by the way, the main person alleging that the zeta function of a scheme of finite type over ℤ is not me but Serre. I got this claim out of his 1963 paper "Zeta and L-functions" - see below. You can't see it below, but that's on the second page of his paper, and on the first he says "all schemes considered below are supposed to be of finite type over ℤ". No extra assumptions are given.
Could he be *implicitly* assuming schemes are not too large? I was wondering about this - I'm glad @npub1d60…mhwm brought it up.
https://media.mathstodon.xyz/media_attachments/files/113/948/541/217/948/445/original/70c228f7e87be8db.jpg

2025-01-24 17:16:28

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- contacting a professor about such a job takes some cheek... or at least mechanical cheek.

2025-01-24 16:44:56

I got an email that says, in part:
"My name is Robert, and I'm a Project Manager at Labelbox / Alignerr, the leading data labeling platform for AI.
Your mathematics background at University of California caught my eye, and wanted to reach out to inform you about a part-time, remote side gig opportunity for math PhD's.
We’re working with companies like Google and Meta to advance their AI models (think ChatGPT) in tasks like complex reasoning in subjects such as mathematics. I believe your expertise could play a significant role in these advancements.
Right now, my team has an urgent need for PhD-level experts in Mathematics (or Physics!). We're offering pay of up to $150 per hour."
I would never do such a thing, but I have a former grad student who is desperate for a job, so I passed it on, warning that it could be a scam and not to give away any bank details that could be abused, etc. They replied:
"I've tried one of these "AI contribution" jobs before and while they aren't outright scams, they might as well be. They give you hours of initial assessments, they take weeks to verify you and then they don't assign you any tasks. At no point do you ever interact with an actual person and I'm pretty sure the assessments that they give you are you just doing free work for them. (The assessments consist of "rating" and editing AI-generated solutions to problems as well as writing solutions to problems from scratch.) And the pay is nowhere near $150 an hour, but more like $15."

2025-01-11 17:43:46

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- I left a comment on your blog. I agree with you!

2024-10-28 22:14:35

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- oh, sorry, I didn't quite understand you before.
https://projects.fivethirtyeight.com/2024-election-forecast/

2024-10-13 14:06:24

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- so like you missed the big fight on my post over whether machine learning stole the physics Nobel?
https://mathstodon.xyz/@johncarlosbaez/113272834785880929

2024-09-28 21:35:49

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- that's what people tell me. I'm sure there must be lots of theories that seek to explain the Titius-Bode rule: astronomers and physicists are not shy. But I guess none of them is widely accepted.

2024-09-22 21:28:23

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- in case it wasn't obvious, @npub12gn…d572 is agreeing with me.
"The gravitational has to be the same everywhere at the surface of the ocean" is another way to say "The force of gravity points perpendicular to the surface of the ocean".

2024-09-14 17:03:46

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- I guess you know about the new "Beat" tour:
https://www.youtube.com/watch?v=QIgxRK914WI

2024-09-14 16:10:09

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- I Zimbra is great, a real blast of energy. With Robert Fripp on guitar, too!
https://www.youtube.com/watch?v=qKFn66dzdz0

2024-09-14 15:54:51

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- the Beatles' "Norwegian Wood" is incredibly catchy, and the first Western pop song to use a sitar.

2024-09-13 01:27:43

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- I forget where, but I claimed they were invented by ancient Vedic mathematicians, hence the Rg Veda.

2024-09-11 14:01:58

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- yes. Al Capone found out the hard way.

2024-09-10 11:49:01

Check out the FBI's files on the mathematician Paul Erdős! You may know him as the author of 500 math papers, endlessly traveling and collaborating. But to the FBI he seemed suspicious, because he came from behind the Iron Curtain, got caught poking around a secret radio tower in 1941, worked with a mathematician from China, and refused to take a loyalty oath.
In 1954 he was barred from re-entering US after going to Amsterdam to address the International Congress of Mathematics. But 5 years later he was granted a visa after Hubert Humphrey intervened on his behalf.
The FBI eventually decided he was ”purely a mathematician with typical atmospheric mind as related to factual things.... a genius type who lives within his own mental scope".
https://www.muckrock.com/news/archives/2015/jul/21/nothing-indicate-nothing-indicate-subject-had-any-/

2024-09-08 19:54:51

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- people need to know a little British English to get that joke. Americans are puzzled to hear British cars wear boots and bonnets.

2024-09-08 19:52:51

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- it'll be fun to think about the 𝑘,ℏ → 0 limit! There do exist 'classical bosons' - identical classical particles such that an n-particle state is unchanged by permutations.

2024-09-08 19:42:15

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- I've neither taken nor taught a typical "modern physics" course - it could be tons of fun to teach, but I was trapped in a math department endlessly teaching calculus (fun the first 50 times). Statistical physics is more mathematically demanding and less sexy than special relativity, but it's a lot more important for understanding the everyday world... and I love it. The Cinderella of the physics family.

2024-09-08 19:34:28

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- Indeed. I'm surprised they don't just put a gun in the trunk so you can shoot your way out.

2024-09-08 19:29:49

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- I haven't seen one in my 2007 Honda. Let me just get in the trunk and see....

2024-09-08 14:38:22

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- modern logician use : a lot in "type theory", and would write 𝑛:ℕ to mean 𝑛 is of type "natural number". Type theory works differently from set theory, so while n:ℕ means something very similar to 𝑛∈ℕ, the former tells us we're not doing set theory.
Thanks to all this, the older usage of : to mean 'such that' is becoming less popular in logic. However, other mathematicians still use it.
There is no one "official standard". Still, there are ways to look like an expert and ways to look strange!
If you read very serious logic books, they will all develop their own "official standard" and stick with it, at least until they introduce abbreviations. A bunch will say that for any formula P and any variable 𝑛 there is a formula ∃𝑛 (P). When P is 𝑛²+1=10 this gives ∃𝑛(𝑛²+1=10). The reason for using the parenthesis is that we need to distinguish between
∃𝑛 (P) ∧ Q
and
∃𝑛 (P ∧ Q)
I wouldn't worry about the difference between ( ) and [ ].

2024-09-08 14:05:50

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- I would write the first one, and even leave out the braces [ ] unless this were part of some larger logical sentence.
The second one looks slightly weird because of the duplicated 𝑛 in
∃𝑛(𝑛∈ℕ)
In the third one I would read the : as 'such that'. So in theory it reads correctly. But no logician would write this, for various reasons.

2024-09-07 21:40:48

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- they're actually not trying to be baroque, they are just mathematicians talking the only way they know how. I know: I'm a mathematician, and I find these entries generally quite clear. The problem is, it's hard to get mathematicians to write in ways that nonmathematicians can understand. At least the first paragraph should be aimed at everyone.

2024-08-31 22:14:43

When Alexander Graham Bell's patent on the telephone gave out in 1894, local phone networks sprang up across the United States – including small 'barbed wire fence networks' set up by farmers.
For more, try this article by Lori Emerson, author of 𝘖𝘵𝘩𝘦𝘳 𝘕𝘦𝘵𝘸𝘰𝘳𝘬𝘴: 𝘈 𝘙𝘢𝘥𝘪𝘤𝘢𝘭 𝘛𝘦𝘤𝘩𝘯𝘰𝘭𝘰𝘨𝘺 𝘚𝘰𝘶𝘳𝘤𝘦𝘣𝘰𝘰𝘬. A quote:
"They would hook up telephones to wire strung from their homes to a nearby fence; at the time, telephones had their own battery which produced a DC current that could carry a voice signal; turning a crank on the phone would generate an AC current to produce a ring at the end of the line. Bob Holmes elaborates on the process: “the barbed wire networks had no central exchange, no operators – and no monthly bill. Instead of ringing through the exchange to a single address, every call made every phone on the system ring. Soon each household had its own personal ringtone… but anyone could pick up…Talk was free, and so people soon began to ‘hang out’ on the phone.” The fence phone lines could also be used to broadcast urgent information to everyone on the line."
Pretty cool. Remind you of anything... like Mastodon?
At first I was hoping the telephone signal was actually transmitted on the barbed wire fence itself. It seemed like such a cool idea: farmers could repurpose miles and miles of metal fencing already present. Then I decided that wouldn't work too well, since the conductivity of the fence is not very high.
But read @npub1w9z…8ple's comment on this post here: it seems farmers actually 𝑑𝑖𝑑 transmit phone signals on the fencing itself! 🎉
https://loriemerson.net/2024/08/31/a-brief-history-of-barbed-wire-fence-telephone-networks/

2024-06-21 08:22:25

Why do we call electrons 'negative' and protons 'positive'? It's an arbitrary convention but it's annoying: it means that when electrons flow through wires, the current is defined to flow in the OTHER DIRECTION.
If you ask people why electrons got called 'negative', you get a bunch of crap answers:
"Electrons are referred to as negative because of their behavior in an electric field. In an electric field, an electron will move from the negative pole to the positive pole, giving it a negative charge by convention."
Yes, but why THAT convention?
"If we re-designate all positive electric charges as negative and vice versa, while keeping their absolute value, the resulting physics would be the same. So exact choice is merely a matter of convention."
Yeah, it's a convention - but why THAT convention?
"In quantum theory of elementary particles (in a sense of irreducible representation of Poincare group with mass m and spin (helicity) s) if some operator Q of internal symmetry commutes (like electric charge charge) with Hamiltonian H...."
Irrelevant crap which never leads up to an answer.
The answer is that Benjamin Franklin chose this convention and nobody knows why.
In Franklin's day there was a 'two-fluid' theory of electricity saying electricity comes in two kinds. Around 1750 he developed a 'one-fluid' theory after showing that a rubbed glass receives an equal but opposite charge as the cloth used to rub the glass. He decided that electrical fluid was going into the glass. So he said the charge of the glass was 'positive' and the cloth was 'negative'.
Unfortunately it turns out that that electrons are going into the cloth.
The whole history is interesting:
https://en.wikipedia.org/wiki/Electric_charge#History
https://media.mathstodon.xyz/media_attachments/files/112/654/420/323/434/029/original/7696ba4c8b5ae3a9.png

2024-06-19 16:26:15

When you have 3 helium atoms, they can stick together and form 'trihelium'. But even more impressively, the different energy states of trihelium are predicted to form an infinite sequence of molecules, each 22.6948 times larger than the previous one!
A while ago I talked about dihelium. This consists of 2 helium atoms held together, not by a chemical bond, but by the weak left-over remnants of the electrical forces holding each atom together. Trihelium works the same way, but it's more stable than dihelium. Indeed if you took one of the larger forms of trihelium and removed any one of the 3 atoms, the other two would fly apart.
This bizarre phenomenon was predicted by the Russian physicist Vitaly N. Efimov in the 1970s. He showed it should happen whenever you have 3 identical bosons attracting each other by a short-range force. As they come closer and closer to having enough energy to no longer be bound together, they form an infinite sequence of bound states, where the radius of the nth state is proportional to the nth power of some number λ. This number λ is close to 22.6948, but Efimov showed that it's actually
exp(π/s)
where s is the unique positive solution of
s cosh(π s/2) = 8 sinh(π s/6)/√3
Alas, I don't understand his calculation.
For a long time Efimov's work was just a theoretical prediction. In 2014 a team observed the first two 'Efimov states' in an ultracold gas of cesium. But trihelium is harder to work with, and it's only with great trouble that people have made it at all. So we have not verified that it has Efimov states... but the math and physics is solid.
(1/2)
https://media.mathstodon.xyz/media_attachments/files/112/644/088/101/918/027/original/ba147bd5de56213a.mp4

2024-06-18 22:04:49

A volunteer search and rescue recently found this object in the Desert National Wildlife Refuge about 16 miles north of Las Vegas. Nobody knows how it got there.
https://archive.is/JKAWe
https://media.mathstodon.xyz/media_attachments/files/112/639/864/856/438/310/original/932be6994a16e7ab.webp

2024-06-16 11:37:52

The last time a big galaxy crashed into ours was between 8 and 11 billion years ago. It added about 50 billion solar masses to our galaxy. And its remains, still visible today, are called the Sausage.
Why the funny name? It's the red thing in the picture! This picture shows orbits of various typical stars in the Milky Way. The z direction is at right angles to the galactic disk. There are 3 quite different kinds of stars:
• The green orbit is a star in the galactic disk, like our Sun.
• The blue orbit is a star in the galactic halo. These stars tend to be very old: over 12 billion years old.
• The red orbit is a star in the Sausage. It moves back and forth in a very narrow orbit!
We learned about the Sausage only after 2014, when an EU-run satellite called Gaia began carefully measuring the velocities of many stars in the Milky Way. Many dwarf galaxies that collided with the Milky Way were already known. But the Sausage was hard to detect until we got good at watching stars move by repeatedly measuring their position.
You see, redshifts are only good for telling if something is moving toward or away from you. Stars in the Sausage are moving back and forth from our point of view. Gaia accurately measured the position of a billion astronomical objects, measuring each 70 times in its five-year life! So it could actually see a bunch of stars slowly moving across the sky.
For more on the Sausage:
https://people.ast.cam.ac.uk/~vasily/gaia_sausage/info.html
https://media.mathstodon.xyz/media_attachments/files/112/625/255/739/022/466/original/9647f90ee430f7d2.png

2024-06-15 09:41:33

Even though helium is the least reactive of the noble gases, you can make a molecule of two helium atoms! Yes, He₂ is a thing! It's called 'dihelium' or the 'helium dimer'.
But the two helium atoms aren't held together by a bond. Instead, they're held together by a much weaker force: the van der Waals force. So this isn't an ordinary molecule. It's huge! The distance between nuclei is 70 times bigger than it is for H₂. And it's very loosely held together, so you'll only see it at extremely low temperatures.
It's called a 'van der Waals molecule'.
But what's the van der Waals force? It's actually a name for a few different forces that exist between electrically neutral objects. For dihelium, the most important one is shown in this gif.
Start with two neutral atoms. Each is a nucleus surrounded by electrons. If they start out spherically symmetric, like helium, and far enough apart that their electron clouds don't overlap, there will be no electrostatic force between them.
But wait - the electrons are randomly moving around! Sometimes they move to make each atom into a 'dipole', with the nucleus on one side, and the electrons over on the other side. If these two dipoles point the same way, this lowers the energy - so it will tend to happen. And now the two atoms attract... since two dipoles pointing the same way attract.
But this effect is small. The gif vastly exaggerates how much the electrons move to one side of each proton. And this van der Waals force dies off like 1/𝑟⁶. So we have two objects held together by a weak force... so they will orbit slowly, and far apart, so that the centrifugal force is equally small!
(1/2)
https://media.mathstodon.xyz/media_attachments/files/112/619/849/994/096/209/original/5a618c066f0f34b2.mp4

2024-06-14 16:50:26

Whenever I see this on the bus it makes me think they're offering some sort of defective wifi access.
https://media.mathstodon.xyz/media_attachments/files/112/615/991/207/242/880/original/cf4f1a70317e2bf2.jpg

2024-06-10 09:54:59

I quit Twitter and moved to Mastodon in October 2022 when Elon Musk took over. Many other academics also came to Mastodon, and I was very happy - see my pinned posts. But most of them later became inactive.
I stick around because I've decided that corporate-owned social media are dangerous - especially now, when authoritarians are spreading their reach. Turns out it's unrealistic to expect that most academics feel this way - yet. They may get the idea when it's too late.
(1/2)
https://arxiv.org/abs/2406.04005

2024-06-09 08:48:29

If you could watch an individual water molecule, about once in 10 hours you'd see it do this!
As it bounces around, every so often it hits another water molecule hard enough enough for one to steal a hydrogen nucleus - that is, a proton - from the other!
The water molecule with the missing proton is called a hydroxide ion, OH⁻. The one with an extra proton is called a hydronium ion, H₃O⁺.
This process is called the 'autoionization' of water. Thanks to this, roughly one in ten million molecules in a glass of water are actually OH⁻ or H₃O⁺, not the H₂O you expect.
And this explains why protons can move through water much more easily than larger ions can. Let's watch how it works.
(1/n)
https://media.mathstodon.xyz/media_attachments/files/112/585/740/156/322/310/original/f4ec78e2d2ff1326.mp4

2024-05-21 13:13:59

When global warming gets bad enough to create millions of refugees, just imagine how nasty the EU and US will become. They may just ship people out to the desert to die.
Oh wait, you don't have to imagine! It's happening already:
"A year-long joint investigation by The Washington Post, Lighthouse Reports and a consortium of international media outlets shows how the European Union and individual European nations are supporting and financing aggressive operations by governments in North Africa to detain tens of thousands of migrants each year and dump them in remote areas, often barren deserts.
European funds have been used to train personnel and buy equipment for units implicated in desert dumps and human rights abuses, records and interviews show. Migrants have been pushed back into the most inhospitable parts of North Africa, exposing them to abandonment with no food or water, kidnapping, extortion, sale as human chattel, torture, sexual violence and, in the worst instances, death."
The full story is here: https://archive.is/1tEgl
(1/2)
https://media.mathstodon.xyz/media_attachments/files/112/479/228/604/482/522/original/17d6aefa15d8fd61.jpg

2024-05-16 06:58:42

Chemistry is like physics where the particles have personalities - and chemists love talking about the really *nasty* ones. It makes for fun reading, like Derek Lowe's column "Things I Won't Work With". For example, bromine compounds:
"Most any working chemist will immediately recognize bromine because we don't commonly encounter too many opaque red liquids with a fog of corrosive orange fumes above them in the container. Which is good."
And that's just plain bromine. Then we get compounds like bromine fluorine dioxide.
"You have now prepared the colorless solid bromine fluorine dioxide. What to do with it? Well, what you don't do is let it warm up too far past +10C, because it's almost certainly going to explode. Keep that phrase in mind, it's going to come in handy in this sort of work. Prof. Seppelt, as the first person with a reliable supply of the pure stuff, set forth to react it with a whole list of things and has produced a whole string of weird compounds with brow-furrowing crystal structures. I don't even know what to call these beasts."
https://www.science.org/content/blog-post/higher-states-bromine

2024-04-14 08:30:40

It looks like they've found protonium in the decay of a heavy particle! 🎉
Protonium is made of a proton and an antiproton orbiting each other. It lasts a very short time before they annihilate each other.
It's a bit like a hydrogen atom where the electron has been replaced with an antiproton! But unlike a hydrogen atom, which is held together by the electric force, protonium is mainly held together by the strong nuclear force.
There are various ways to make protonium. One is to make a bunch of antiprotons and mix them with protons. This was done *accidentally* in 2002 during the first experiment that created antihydrogen. They only realized this upon carefully analyzing the data 4 years later.
This time, people were studying the decay of the J/psi particle. The J/psi is made of a heavy quark and its antiparticle. It's 3.3 times as heavy as a proton, so it's theoretically able to decay into protonium. And careful study showed that yes, it does this sometimes!
The new paper on this has over 550 authors, so I won't list them all. It also has a rather dry title - not "We found protonium!"
• Observation of the anomalous shape of X(1840) in J/ψ→γ3(π+π−), https://arxiv.org/abs/2310.17937
The idea here is that sometimes the J/ψ particle decays into a gamma ray and 3 pion-antipion pairs. When they examined this decay, they found evidence that an intermediate step involved a particle of mass 1880 GeV/c², a bit more than an already known intermediate of mass 1840 GeV/c².
This new particle is a bit lighter than twice the mass of a proton, 938 GeV/c². So, there's a good chance that it's protonium!
https://media.mathstodon.xyz/media_attachments/files/112/268/499/052/720/530/original/ab89c8eac9634b98.jpg