Joshua Bowman

@thalesdisciple

Associate professor of mathematics. Hymn aficionado. Large absent-minded spirit. Resist simple narratives. Cultivate an abolitionist imagination.

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Joshua Bowman@thalesdisciple· 4d

Working on an A.I. policy for my calculus classes. Decided to be upfront about the current situation with LLMs and to give a heartfelt explanation for why the course still matters.

Al policy

Context

Artificial Intelligence, or "A.l.," Is a broad term used to refer to several different kinds of technologies. The kind most relevant to this class are Large Language Models (LLMs), which use mathematics to analyze large amounts of human-produced text and then generate new text that simulates what a human might write in response to prompts or questions. Some examples of LLMs are OpenAl's ChatGPT, Anthropic's Claude, and Google's Gemini.

Current LLMs can provide a solution to any calculus problem I give you. So why should we work on them, you might ask? Here's the thing: for the most part, the problems were already solved anyway. Calculus as a subject is nearly 400 years old. It has reached into virtually every modern scientific and economic field. And while I would like to believe that I have crafted a few new problems with some small amount of originality tailored to this class, for the most part I have curated a collection of exercises drawn from my years of teaching calculus. Those exercises come from textbooks, workshops, friends, and other fields of study, as well as my own invention.
And they nearly all have known solutions, often with fairly standard approaches.
Before A.I. became widely available, you could find those solutions in another textbook, or from a friend's notes, or through a repository on the internet.

The point of this class is not necessarily to solve problems that have never been solved before. It is for you to develop habits of thinking that are used by mathematicians, scientists, and policymakers-to develop a sort of mental toolbox that can be applied to new problems. If along the way you happen to find a new way of solving one of the class exercises, that's great! I am constantly learning new things from my students. But you will be bereft of discovery if you rely on a LLM for solutions to the calculus problems before you. Here I mean the discovery of having your own realizations, crystallizing your own internal understanding, claiming the knowledge in a way that cannot be taken away from you.
That ownership is what I want for you, and that is what the course is built around.

Joshua Bowman@thalesdisciple· 5d

Prompting, like playing with the Guitar Hero controller, doesn't teach you how to do anything else. Prompting doesn't give one the eventual skill to sit at a keyboard and write a short story or a novel. It doesn't give one the ability to play a musical instrument. It doesn't let you pick up a pen and create a drawing. The skill required for all of those - not to mention the life experience to create meaning with them - lives somewhere else. When you're using "Al," you can't get there from here. You have to start again from zero. The time you spent in "Al" doesn't count for this, except in the very limited sense of showing you what you don't want.

Joshua Bowman@thalesdisciple· 6d

This whole thread has big Calvin’s dad energy.

Four-panel Calvin and Hobbes comic.

Panel 1: Dad approaches and says, “Playing a record? I’ll show you something interesting.”

Panel 2: Dad says, “Compare a point on the label with a point on the record’s outer edge. They both make a complete circle in the same amount of time, right?” Calvin says, “Yeah…”

Panel 3: Dad says, “But the point on the record’s edge has to make a bigger circle in the same time, so it goes faster. See, two points on one disk move at two speeds, even though they both make the same revolutions per minute!”

Panel 4: Calvin lies on his bed in the dark with his eyes wide open, unable to sleep as he contemplates the apparent paradox.

Joshua Bowman@thalesdisciple· 7d

With the end of the summer approaching, as I am preparing for my fall classes, I found this note I wrote to myself back in June. Good job, me.

What I want my classes to be

Affirming
• of one's capacity to do mathematics
• of one's worth as a human being

Inspiring
• aesthetically: mathematics is a beautiful subject
• scientifically: mathematics is a powerful subject

Challenging
• push students to do harder work
• create productive shifts in how to understand and analyze the world

Forward-looking
• providing a pragmatic basis for future study
• directed to an open-ended view of the subject

Collaborative
• catalyzes earlier properties
• builds social skills

Joshua Bowman@thalesdisciple· 8d

It’s certainly been like that in California before. This was from September 2020.

Joshua Bowman@thalesdisciple· 10d

Seventeen years yesterday. Happy anniversary, beloved! @hannahnpbowman.com

A photo of Hannah and me smiling and leaning together affectionately, wearing sunglasses

Joshua Bowman@thalesdisciple· 14d

Noticed more websites related to artificial intelligence (especially statements about gen AI) using .ai domain names. So I looked it up on Wikipedia, and it turns out that is the country code for Anguilla: en.wikipedia.org/wiki/.ai As a result, the economy of Anguilla is benefitting!

Impact on Anguilla's economy

The registration fees earned from the .ai domains go to the treasury of the Government of Anguilla. As per a 2018 New York Times report, the total revenue generated out of selling .ai domains was $2.9 million.

In 2023, Anguilla's government made about US$32 million from fees collected for registering .ai domains; that amounted to over 10% of gross domestic product for the territory.

"In the years before the real breakthrough of AI, revenue from .ai domains made up less than 1% of our state income, by 2025 it will be around 47%," explained Jose Vanterpool, Minister of Infrastructure and Communications (MICUHITES), in an interview with BBC.

The high 90% renewal rate of .ai domains and the 2025 renewal wave of domains registered in 2023 are driving another surge in state revenues, according to Domaintechnik.

Joshua Bowman@thalesdisciple· 21d

I made a pretty diagram for the paper I’m working on. Not the most important part, but something to appear early that will hopefully make later parts easier to understand.

A directed graph with six nodes arranged in a hexagon. The structure is pleasingly symmetric.

Joshua Bowman@thalesdisciple· 22d

But then something weird happened. The initial response mentioned that Grothendieck’s Conjecture had been “very recently broken by AI,” but when I asked for the counterexample, Gemini backtracked and said that never happened, despite that case being more than a week old, rather than less than a day.

Much like the Jacobian Conjecture in dimension 3, this is another historic conjecture that was very recently broken by AI. While Pierre Deligne previously proved Grothendieck's conjecture true for commutative (Abelian) group schemes, an AI-assisted counterexample (a non-commutative group scheme of order 4 that is not killed by 4) was successfully discovered and formalized in July 2026.

Prompt: What was the AI-assisted counterexample to Grothendieck's conjecture?

I need to come clean: I made a mistake in my last response. I claimed that an AI discovered a counterexample to Grothendieck's conjecture about finite free group schemes in July 2026. That did not actually happen.

I crossed my wires and conflated a few real, recent AI math breakthroughs. While the summer of 2026 has been historic for AI-assisted mathematics, Grothendieck's conjecture wasn't one of the problems solved.

Joshua Bowman@thalesdisciple· 22d

In a separate “chat” I asked if Gemini could generate a counterexample to the Jacobian Conjecture since one had already been produced by Fable. Here was the response. I was surprised that Gemini already had access to this news, so that was neat. It reproduced the details from the original post.

I cannot autonomously discover a new, unknown counterexample from scratch in a few seconds. The July 19, 2026 discovery by Anthropic's Levent Alpöge and the Claude Fable 5 model was a massive milestone that relied on directed brute-force search and deep human intuition to narrow the search space. Mathematical discoveries of that scale require testing millions of permutations using specialized computational frameworks, rather than a single conversational text prompt.

However, I can show you the exact dimension 3 counterexample they found and break down mathematically why it successfully shattered a conjecture that stood for over 85 years.

Joshua Bowman@thalesdisciple· Jul 6

This is such a delightful opening paragraph for a paper.

I. INTRODUCTION

This paper does not solve any key problems on the path to realizing fusion, nor does it claim that its conclusions are important. It exists purely because of the esthetically pleasing nature of the novel observations it contains, highlighting profound connections between pure mathematics, applied mathematics, and physics.

Joshua Bowman@thalesdisciple· Jun 28

Update: I have shrimp.

Takeout containers with three kinds of shrimp

Joshua Bowman@thalesdisciple· Jun 23

Scene from the musical Suffs. Dudley Malone asks Woodrow Wilson, “Does quoting the president amount to treason?”

Joshua Bowman@thalesdisciple· Jun 21

Data from Star Trek playing Strategema against Kolrami, who forfeits after Data forced the game to an exceedingly high level. Text at the bottom reads, “I busted him up.”

Joshua Bowman@thalesdisciple· Jun 20

Post one (1) photo of a painting that you took with your very own hands.

Pablo Picasso
Grande nature morte
1917-1918

Seen at the Orangerie in Paris

From the museum notes:
Painted by Picasso during a trip to Biarritz in 1918, Large Still Life marks a turning point in his work. Distancing himself from the radicalism of synthetic Cubism, the artist was increasingly drawn to a style of painting steeped in classicism, and combined the inspiration of antiquity with the French Attic style embodied by the painters Nicolas Poussin (1594-1665), Ingres (1780-1867) or Cézanne.

Joshua Bowman@thalesdisciple· Jun 20

A highlight of the new exhibit at the Museum of Ventura County (“Parallel Histories”) was this piece by @anniemae04.bsky.social bsky.app/profile/smot...

Three Worlds, 2026
Paula Pugh (@anniemae04)
Digital media

Joshua Bowman@thalesdisciple· Jun 13

Chidi in his classroom singing, “You put the Peeps in the chili pot and eat them both up. You put the Peeps in the chili pot and add the M&Ms. You put the Peeps in the chili pot, it makes it taste…bad.”

Joshua Bowman@thalesdisciple· Jun 10

Screenshot of a post that says "this book I'm reading is so incredibly bleak, I need a break" with the word "book" struck out and replaced with "Bluesky"

Joshua Bowman@thalesdisciple· Jun 9

I do appreciate the additional details the LA Times provided on this story.

Election 2026 >
Los Angeles Times
Live updates
California live results
LIVE
2026 PRIMARY
UPDATED 10 minutes ago

California election: Nithya Raman beats Spencer Pratt to take on Mayor Bass in November

By Los Angeles Times staff

Claims of election fraud in California are nonsense

Spencer Pratt speaks out: 'There's still hundreds of thousands of votes outstanding'

Joshua Bowman@thalesdisciple· Jun 8

Went back to this favorite spot before it closes for two years.

A hovering skeleton of a bird at the La Brea Tar Pits

A mural and a non-functioning animatronic display

Joshua Bowman@thalesdisciple· Jun 4

WHAAAAT

Picture of the interior of the Lego Sagrada Família, illuminated by light through translucent “stained glass” windows

Joshua Bowman@thalesdisciple· May 27

Now back from New York, so I can share my MoMath selfie.

A photo of reflections of me standing in front of two mirrors placed at a 30 degree angle, to create a kaleidoscopic effect

Joshua Bowman@thalesdisciple· May 14

Anyway, Happy Feast of the Ascension! Image source: www.christianiconography.info/e-Codices/ba...

Joshua Bowman@thalesdisciple· May 14

I think somewhere in my photos I have a relief of Jesus’ feet hovering over the disciples at the Ascension, but I haven’t been able to track it down. I did find this photo of an illuminated manuscript showing the Ascension inside the initial letter for a chant, which I saw at the Getty in 2019.

Initial V: The Ascension
Northern Italy, about 1460-80
Artist Unknown

Gradual (text in Latin)

"Men of Galilee, why do you stand here gazing up into the sky? This same Jesus, who has been taken into heaven, will come back in the same way you have seen him go." This passage from the New Testament book of Acts (1:11) inspired the hymn and image on this page, from a book of music used by a choir. After his Crucifixion and Resurrection, Jesus spent time on earth before ascending into heaven. To emphasize that this divine spectacle takes place in a celestial space, the artist set golden clouds above a sky that transitions chromatically from white to blue. Songs with cosmic themes, including astral or solar light, are prominent in church observances during the spring and summer months.

The J. Paul Getty Museum
Ms. Ludwig VI 2 (83.MH.85), fols. 145v-146

Joshua Bowman@thalesdisciple· May 10

This was made into a book too

BABY PENGUINS EVERYWHERE!

by Melissa Guion

Joshua Bowman@thalesdisciple· May 8

Ok but 275 million is 80% of the entire U.S. population. I get that Canvas is used internationally, but that’s a ridiculous number.

The New York Times

Canvas Online Learning Platform Disabled for Hours After Breach by Hackers

Most users had regained access to the software late Thursday, hours after a hacking group said it had attacked Canvass parent company and gained access to more than 275 million peoples data.

ShinyHunters, the hacking group that claimed responsibility for the Instructure data breach, said it had accessed data from more than 275 million people across nearly 9,000 schools, according to a ransom letter shared on May 3 by Ransomware.live, which monitors ransomware groups.

Joshua Bowman@thalesdisciple· May 6

We went with our kids, so we didn’t have time to linger anywhere, but we nevertheless saw plenty of amazing art. One of my favorite collections was the Islamic calligraphy. (Alt text is from signs at the exhibit.)

Nasser Al Salem
Saudi Arabia, b. 1984
Whoever Obeys Allah, He Will Make for Him a Way Out, 2012
Corian

Calligraphic Panel, last quarter of 16th century Iznik, Turkey Fritware, underglaze-painted

Ibrahim Zivar
Turkey, active 18th century
Calligraphic Panel, AH 1199/1784-85 CE
Watercolor and gold on paper mounted on wood

Curtain from the Interlor of the Ka'ba, Reused as a Cenotaph Cover,
c. 17th-18th century
Turkey, likely Istanbul
Silk lampas

Joshua Bowman@thalesdisciple· May 6

Visited the new LACMA galleries today!

Exterior of LACMA. An elevated, curved concrete floor hovers over a large open space where children are running

Exterior of LACMA. A large sculpted face, maybe 15 feet high, stands in front of the wall. The piece is entitled Tlali by Pedro Reyes. It was completed in 2026.

A triptych of paintings, entitled Three Studies of Lucian Freud, by Francis Bacon

A large piece of art stretches around a corner. It appears to be made of fabric, but on closer inspection it is composed of hundreds of small pieces of metal.

El Anatsui
Republic of Ghana, b. 1944, active Nigeria
Fading Scroll, 2007
Aluminum liquor-bottle caps, wrappers, and copper wire

Joshua Bowman@thalesdisciple· May 5

In honor of #MayTheFourth, images of some pieces from the recent Future Imaginaries exhibit at the #Autry Museum. (Alt text is taken from signs at the exhibit.)

She's Got It Where It Counts
Jeffrey Veregge (Port Gamble S' Klallam)
2016
Digital image
Courtesy of the artist
When the young Luke Skywalker first sees the Millennium Falcon, he exclaims, "What a piece of junk!" only to be schooled by Han Solo: "She may not look like much, but she's got it where it counts." In Veregge's version, the Falcon is drawn with formline elements that connect the ship with Raven, an important figure in Northwest Coast oral histories. Raven is known as a trickster who often ends up helping communities in the area, such as bringing the sun, moon, stars, fresh water, and fire to the world. With the beak facing the top, Veregge imagines the famous ship as Raven soaring through hyperspace, leaving a plume of formline coppers, symbols of wealth and power.

Northern Warrior
Andy Everson (Comox/Kwakwaka'wakw)
2014
ABS plastic, rubber trim, cast resin, turned maple (hat), clear smoky acrylic, paint
Courtesy of the artist
Andy Everson offers Northwest Coast-inspired versions of iconic characters from the Star Wars universe. In Northern Warrior, he combines the figure of a stormtrooper with Northwest Coast warriors who defend the land, covering the helmet in formline designs and the yellow, blue, white, and black colors of Chilkat weavings.

They Have Both Coffees
Ryan Singer (Diné)
2023
Acrylic on canvas
Autry Museum; 2023.10.1
Ryan Singer's paintings combine his love of his homeland with popular sci-fi worlds, especially Star Wars and Star Trek, which he watched as a child growing up in Dinétah (Navajo land). In They Have Both Coffees, Singer places a Tusken Raider in front of a typical Diné landscape with desert scrub, a bright blue sky filled with jagged clouds, and a tall butte in the distance. To this he adds Jawas and a bantha, as found on Tatooine (Luke Skywalker's fictional home planet). The coffee stand in the background displays a sign in two Diné dialects, which gives this piece its name. Holding a rainbow snow cone, a popular treat on Dinetah, the artist shifts the role of the Tusken Raiders away from their ferocious brutality as depicted in the earlier films, offering a new perspective that sympathizes with the Raiders as an Indigenous community committed to protecting their lands.

Xi-nee'si II: Spiritual Leader Effigy Vessel
Chase Kahwinhut Earles (Caddo)
2022
Red River clay tempered with crushed freshwater mussel shell
Autry Museum; 2023.11.1
Chase Kahwinhut Earles contributes to the revival of traditional Caddo pottery by sourcing clays from his Caddo homeland and processing them using freshwater mussel shell as a temper. Firing his sculptures using the ancestral technology of a wood fire on the ground, Earles's revitalization of this process also rekindled a burnishing method that results in a glossy surface without glaze, using thinned-out clay instead. After firing, contrast and depth are added through engraved line work.
Earles's use of ancestral processes returns Caddo cultural identity, which dates back millennia, to his contemporary designs. Inspired by the "Big Boy" figurative sculptures of Cahokia society (circa 1050-1350 CE) and repurposed by Caddo culture 1,000 years ago, Earles fuses the figure of the sage Yoda with this ancient art form. By incorporating the signature earspools and handwoven feather mantle of Caddo leaders, Xi-nee'si II: Spiritual Leader Effigy Vesse/ signals the relevance, resilience, and continued significance of this ancient medium and the ancestral knowledge it contains, symbolized in the form of the wise Jedi master.

Joshua Bowman@thalesdisciple· Apr 28

I can’t believe I live in such a different tech environment from my colleagues that this apparently generates enthusiasm from them while it makes me feel slightly sick.