Posts Tagged ‘Mathematics’
“Beauty is the first test: there is no permanent place in this world for ugly mathematics”*…
Is mathematical beauty real? Or is it just a subjective, human ‘wow’ that is becoming redundant in an AI age? Rita Ahmadi explores…
It is a hot July day in London and I take the bus to Bloomsbury. I often come here for the British Library, the British Museum or the London Review Bookshop. More than a location, Bloomsbury feels like stepping into a work of art – maybe one of Virginia Woolf’s stories, or Duncan Grant’s paintings.
This time, I am here for mathematics: the Hardy Lecture at the London Mathematical Society (LMS), named after G H Hardy, a professor of mathematics at the University of Cambridge, a member of the Bloomsbury Group, and a president of the LMS. You may know him from the film The Man Who Knew Infinity (2015), in which he’s played by Jeremy Irons.
The 2025 lecture is by Emily Riehl of Johns Hopkins University in Baltimore, who is talking about a complex mathematical ‘language’ known as infinity category theory: could we teach it to computers so that they could understand it? If successful, computer programs could verify proofs and construct complex structures in this area.
A few seats to my left, I recognise Kevin Buzzard, wearing the multi-coloured, patterned trousers he’s known for among mathematicians. Based at Imperial College London, Buzzard is working on a computer proof assistant called Lean. His interest is personal: after long disputes with a colleague over a flawed proof, he lost trust, as he often puts it, in ‘human mathematicians’. His mission now is to convince all mathematicians to write their proofs in Lean. In the Q&A after one of his talks, he said of the debate between truth and beauty in mathematics: ‘I reject beauty, I want rigour’ – though his vibrant sense of fashion suggests otherwise.
Interest in an AI-driven approach to mathematics has been exponential, and many mathematicians have left traditional academic research to explore its potential. Recently, one group of distinguished mathematicians designed 10 active, research-level questions for AI to tackle. At the time of writing, various AI companies and researchers had claimed to find solutions, which were under evaluation by the community.
Sitting in the room in Bloomsbury, I stared at the Hardy plaque and wondered: would Hardy find proofs generated by AI beautiful? I wasn’t sure. He believed there should be a strong aesthetic judgment in mathematics, drawing parallels with poetry, and argued that beauty is the first test of good mathematics. He went as far as to say that there is no permanent place in the world for ugly mathematics.
If asked, many mathematicians today still talk about the aesthetic appeal of one approach over another.
Yet we live in a different century to Hardy and his Bloomsbury peers, with different technologies and techniques, so perhaps we need a clearer definition of what mathematical beauty actually is. Over the history of mathematics, we can find examples where both rigour and the pursuit of beauty have shaped mathematics itself. So, if we’re completely replacing this with a computer-assisted quest for truth and rigour, we ought to know what we’d be abandoning, if anything. Is mathematical beauty like the beauty in literature and art – or is it something else?…
[Ahmadi explores the idea of “beauty,” generally and in mathematics; traces the rise of AI as a tool, and concludes…]
… my own definition of beauty in mathematics would be as follows:
“Asimplemathematical structure that surprises even the most experienced mathematicians and transfers a sense of vitality.”
But is an AI-assisted proof simple or surprising? How do we define vitality in a machine? On these questions, the jury is out. Myself, I am torn. Maybe models just need more training to match our creativity. But I also wonder whether our limbic system is required. Can we write proofs without emotional kicks? I am also unsure if perfectly efficient brains can come up with novel revolutionary ideas.
Ultimately, this debate is about more than aesthetics; it is closely tied to the development of AI-assisted mathematics. If AI models can produce novel mathematical structures, how should we direct them? Is it a search for beautiful or truthful structures? A question that possibly guides the years to come.
Some mathematicians say they prefer the ‘truth’ and only the ‘truth’. However, my recent discussions with mathematicians showed me that most immediately recognise, enjoy, and even wholeheartedly smile at a beautiful piece of maths. In fact, they spend their whole lives in search of one…
Fascinating: “The eye of the mathematician,” from @ritaahmadi.bsky.social in @aeon.co.
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As we embrace elegance, we might send garcefully-calculated birthday greetings to Eduard Heine; he was born on this date in 1821. A mathematician, he is best remembered for his introduction of the concept of uniform continuity, for the Mehler–Heine formula, and for the Heine–Cantor theorem… all of them, quite beautiful.
“I am never forget the day I first meet the great Lobachevsky. / In one word he told me secret of success in mathematics: / Plagiarize!”*…
In an 1874 paper, Georg Cantor proved that there are different sizes of infinity and changed math forever. But as Joseph Howlett reports, a trove of newly unearthed letters shows that it was also an act of plagiarism…
When Demian Goos followed Karin Richter into her office on March 12 of last year, the first thing he noticed was the bust. It sat atop a tall pedestal in the corner of the room, depicting a bald, elderly gentleman with a stoic countenance. Goos saw no trace of the anxious, lonely man who had obsessed him for over a year.
Instead, this was Georg Cantor as history saw him. An intellectual giant: steadfast, strong-willed, determined to bring about a mathematical revolution over the clamorous objections of his peers.
It was here, at the University of Halle in Germany, that Cantor launched his revolution 150 years ago. Here, in 1874, he published one of the most important papers in math’s 4,000-year history. That paper crystallized a concept that had long been viewed as a mathematical malignancy to be shunned at all costs: infinity. It forced mathematicians to question some of their longest-held assumptions, rocking mathematics to its very foundations. And it gave rise to a new field of study that would eventually bring about a rewriting of the entire subject.
Now Goos, a 35-year-old mathematician and journalist, had come to Halle — a five-hour train ride from his home in Mainz — to look at some letters from Cantor’s estate. He’d seen a scan of one and was pretty sure he knew what the others would say. But he wanted to see them in person.
Richter — who, like Cantor, had spent her entire career here, first as a research mathematician and then, after retiring, as a lecturer on the history of mathematics — gestured for Goos to sit. She lifted a thin blue binder from the scattered piles of books and papers on her desk. Inside were dozens of plastic sheet protectors, each one containing an old, handwritten letter.
Goos began flipping through, contemplating the letters with the relish of an archaeologist entering a long-lost tomb. Then he reached a particular page and froze. He struggled to catch his breath.
It wasn’t the handwriting. At this point in his research on Cantor, he’d become accustomed to the strange, nearly indecipherable Gothic script known as kurrentschrift, which Germans used until around 1900.
It wasn’t the signature. He knew that the German mathematician Richard Dedekind had been a key player in Cantor’s quest to understand infinity and solidify math’s foundations, and that the two had exchanged many letters.
It was the date: November 30, 1873.
He’d never seen this letter before. No one had. It was believed to be lost, destroyed in the tumult of World War II or perhaps by Cantor himself.
This was the letter that had the power to rewrite Cantor’s legacy. The letter that proved once and for all that Cantor’s famous 1874 paper, the one that would go on to reshape all of mathematics, had been an act of plagiarism…
The extraordinary story of unearthing this extraordinary story: “The Man Who Stole Infinity,” from @quantamagazine.bsky.social.
See also: “How Can Infinity Come in Many Sizes?“
* Tom Lehrer (not just a glorious songwriter, but also a gifted mathematician), “Lobachevsky” (referring to the mathematician Nikolai Ivanovich Lobachevsky— “not intended as a slur on [Lobachevsky’s] character [but chosen]”solely for prosodic reasons”)
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As we confer credit where credit is due, we might spare a thought for Charles-Jean Étienne Gustave Nicolas, baron de la Vallée Poussin; he died on this date in 1962. A Belgian mathematician, he is best known for proving the prime number theorem (which formalized the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs). So great was the contribution that the King of Belgium ennobled him with the title of baron.
“Never tell me the odds!”*…
How likely is it that one will be born on a Leap Day? That one will find a pearl in an oyster? That one will solve Wordle on the first guess? That one will die on a tornado? That two people will share the same fingerprint?
The good folks at R74n (@r74n.com) have these probabilities– and so many more: “What Are The Odds?”
(Image above– and tutorial on the odds ratio: source)
* Han Solo (Harrison Ford) in Star Wars: Episode V– The Empire Strikes Back
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As we place our bets, we might spare a thought for Harvey Kurtzman; he died on this date in 1993. A cartoonist and editor, he is best know for writing and editing the parodic comic book Mad from 1952 until 1956. Kurtzman scripted every story in the first twenty-three issues. (The New York Times‘ obituary for Kurtzman in 1993, alluding to the role of publisher William Gaines, said Kurtzman had “helped found Mad Magazine.” This prompted an angry response to the newspaper from Art Spiegelman, who complained that awarding Kurtzman partial credit for starting Mad was “like saying Michelangelo helped paint the Sistine Chapel just because some Pope owned the ceiling.”)
Kurtzman, who mentored many younger cartoonists (including Terry Gilliam and Robert Crumb), is considered, with cartoonists like Will Eisner, Jack Kirby, and Carl Barks, one of the defining creators of the Golden Age of American comic books. The prestigious Harvey Awards (for achievement in comic books) are named in his honor.
“Juggling is sometimes called the art of controlling patterns, controlling patterns in time and space”*…
A skill for our times…
The Library of Juggling is an attempt to list all of the popular (and perhaps not so popular) juggling tricks in one organized place. Despite the growing popularity of juggling, few websites are dedicated to collecting and archiving the various patterns that are being performed. Most jugglers are familiar with iconic tricks such as the Cascade and Shower, but what about Romeo’s Revenge or the 531 Mills Mess? The goal of this website is to guarantee that the tricks currently circulating around the internet and at juggling conventions are found, animated, and catalogued for the world to see. It is a daunting task, but for the sake of jugglers everywhere it must be done.
For every trick found in the Library, there will be an animated representation of the pattern created via JugglingLab, in addition to general information about the trick (siteswap, difficulty level, prerequisite tricks, etc.). If I am able to run the pattern, then I will provide a text-based tutorial for the trick with the help of animations. I will also include links to other tutorials for the trick that can be found online, ranging from YouTube videos to private sites like this one. If I am unable to provide my own tutorial, there will still be a short description of the trick in addition to outside tutorials and demonstrations…
… if you have come to the Library looking to find out how to start juggling, than it would be best to begin with the Three Ball Cascade pattern. If you are a juggler who is already familiar with the basics, then the various tricks included in the Library can be accessed via the navigation tree on the left, or you can click here to view all of the tricks by difficulty…
Enjoy “The Library of Juggling.”
And see also: “The Museum of Juggling History,” the resources at the International Jugglers’ Association, and “The world cannot be governed without juggling.”
* mathematician (and juggler) Ronald Graham
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As we toss ’em up, we might send carefully-calculated birthday greetings to G. H. Hardy; he was born on this date in 1877. A mathematician who made fundamental contributions to number theory and mathematical analysis, Hardy juggled other interests as well– for example his Hardy–Weinberg principle (“allele and genotype frequencies in a population will remain constant from generation to generation in the absence of other evolutionary influences”) is now a basic principle of population genetics.
In Hardy’s own estimation, his greatest contribution was something else altogether: from 1917, Hardy was the mentor of the Indian mathematician Srinivasa Ramanujan, a relationship that has become celebrated. Hardy almost immediately recognised Ramanujan’s extraordinary (albeit untutored brilliance), and the two became close collaborators. When asked by a young Paul Erdős what his greatest contribution to mathematics was, Hardy unhesitatingly replied that it was the discovery of Ramanujan, remarking that on a scale of mathematical ability, his own ability would be 25, Littlewood would be 30, Hilbert would be 80, and Ramanujan would be 100.
“They are strange times, times of beginnings and endings. Dangerous and powerful. And we feel it even if we don’t know what it is.”*…
Back in 2019, (R)D considered a piece from the remarkable Freeman Dyson on what the biotech revolution could mean (itself further to thoughts in an earlier piece of his). Those thoughts popped back into my mind when I read Quentin Hardy‘s recent recounting of his lunch with a friend…
We’re at an outdoor table in Mission Bay, the wet tech hotspot of San Francisco, home to Biopharma, Biotech, and Techbio research labs, known and emerging, plus big hospitals and research outfits.
Across from my salad and his sandwich, Ashlee sweeps his arm in a big arc across Long Bridge Street, towards all the residential and mixed-use buildings.
“There’s dozens of tiny labs up there,” he says, “somebody’s got a mouse, they’re doing something – growing organs, playing with neurons, injecting them with a virus to change their genetics. All kinds of weird shit. It’s wild, man.”
“All kinds of weird shit” and Ashlee have been intimates for years. They have been good to each other. We met around the time computers started moving from the closet to the cloud, and we both wrote about dirt-cheap satellites, and how cell phone guts were ending up in strange places, changing our world with cheap drones and voluminous data. Back when everything really started changing.
I went to Google to write about how those really big data sets and massive amounts of cloud computation were enabling Artificial Intelligence. Ashlee wrote the first biography of Elon Musk, which took him into Musk’s interests in non-governmental rocketry and neural implants. For many years Bloomberg paid him to do a show called Hello World, where he covered Doomsday preppers, fake meat, Nigerian hackers, and all kinds of strange things. All the creative journalists were jealous of him, not least because they couldn’t touch his talent for finding and admiring this abundance of exotic invention.
He now has his own show, Core Memory, which has unsurpassed reporting on all sorts of cutting-edge robotics, life reprogrammers, amateur space stations, body hackers, and new materials manufacturers. Highly recommended.
Back to his current interests. “I know this guy who’s harvesting rat neurons,” he says, “he talks about using them to power data centers.”…
… We start talking about biohacking and self-medication, all the people shooting up peptides, and the places around town where the kids are mixing their AI with their biohacking, and all the quasi-legal stuff people are doing, growing new human and animal parts.
On one level, they’re just following the “lots of data, lots of compute” model, only into the infinitely more complex wet world. Just as enough people posted tagged photos online to enable Fei Fei Li to make and exploit ImageNet, a major milestone in the creation of image-recognition AI, so these new hackers hope to tag, track, remix and scan enough biological data to remake biological understanding. And capability.
I’ve got my kale and he’s got his meat, partly liberated from the bread. Some of the fun in hanging out with Ashlee is the way we can free-associate over years of covering this kind of stuff, knowing that some things blow up and some things don’t work out, good ideas go down while the mad and the lucky are proclaimed geniuses. In other words, we get to bullshit about the weird shit.
“Maybe it’s going to turn into some kind of ghost gun thing, where people take drugs and perform genetic procedures that are legal on their own and turn it into some kind of illegal treatment,” I say. “You’ll go on a luxury cruise into international waters to get your genetic makeup altered, or blend two different animals into a third. Like ‘The Floating Offshore Platform of Dr. Moreau,’” after the H.G. Wells’ story about a mad scientist making human-animal hybrids…
… But we’re also talking about Biology, that most intimate and complex of sciences, being colonized by a trend we’ve seen elsewhere in tech for years: Prices fall far enough to change the rules of access, newcomers hack the system in defiance of the old standards and business models. Oceans of new data turn up, changing the entire process of understanding.
We’ve seen it happen in enough places to know the pattern. Open source Linux, cheap and attractive enough for all kinds of people to improve it for free, wiped out the old computer server industry. WiFi was open source too, so the price was right and interest surged.
The tech doesn’t have to be open source, or free, either. Economic cycles play a part. When the Internet bubble burst, space companies like Iridium and Globalstar, Rotary Rocket and Kistler, crashed. Lots of cheap talent and parts hit the market, which enabled Elon to do Space X. I once did a story about how entertainment in Africa changed after the price of satellite dishes fell below $200, and the tech moved from expatriate compounds to local bars.
The cost of biological experimentation is on a far crazier decline, giving Ashlee a lot of material. Twenty-three years after the first human genome was sequenced at a cost of $2.7 billion, a “complete genetic engineering home lab,” with a refurbished DNA sequencing machine and a “Bioengineering 101 Course” can be yours for $2500. Neurotechnology tools are available for sale or rent, so you can try neural implants at home. China is spinning up dozens of brain-computer interface startups.
“They’ve got a city in China that’s just doing brain technology stuff,” says Ashlee. When I lived in Asia 30 years ago, cities in China were famous for specializing in things like athletic socks and bras, wiping out the competition worldwide by cranking out more stuff more cheaply than anyone else. Now that the abundance of data and the cheapness of commute have kicked off the AI revolution, they have turned to brain tech. I pick at my kale.
Of course, just because the prices are a fraction of what they used to be, and these new hackers are descending on San Francisco, Cambridge, Miami, and who knows where else, it doesn’t mean breakthroughs are at hand. Biology is a lot more complex than electronics – a lot. Perhaps even more important, the new AI technology that people hope will enable all kinds of bio breakthroughs requires enormous amounts of data. The data set has to be huge, it has to be gathered in a single place the AI can access, and perhaps most critically of all, it has to be standardized to the highest quality…
… The biohackers face a big quality issue too. The Nobel Prize-winning protein information made use of some of the cleanest data possible, and Waymo came out of Alphabet’s cutting-edge sensor- and data-analysis labs. The guy in some converted Apartment 3G doing the thing with the iguana liver, the woman in the co-working space with the rat pituitary, they’re probably not going to bring the same magic.
“Yeah, but they’re not the only ones doing this,” says Ashlee. “I just had on Jennifer Doudna.” Doudna, who won a Nobel prize for her work on gene editing, now runs the Innovative Genomics Institute, a place rigorously pursuing this knowledge following traditional standards. She makes a couple of excellent points in Ashlee’s interview. She thinks a lot of the gunslinger biohackers will find biology much more complex and problematic than they think. At the same time, she expects a lot of the regulatory hurdles to new ways of doing things will become familiar over time, lowering the steps and costs of bringing out new drugs and treatments.
These lower costs will make more things possible, and attract more innovation. This will drive crazy a health and insurance industry built around high costs. If history is any guide, the incumbents won’t surrender their high-cost businesses without a fight. That may be one reason why Doudna thinks that big genetic alterations, will show up in agriculture first…
… Which, apparently, at this point isn’t weird enough. “I’ve got to catch up with this university researcher I met at a party,” he says, pushing away his plate. “She’s working on transplanting the personality of one animal, like a dolphin, to another, like a cow.”
“You mean, like you get a cow that wants to body surf in the wake of a tourist boat?”
He nods. “I know. Weird shit, right?”
I barely know what to do with this one, but I’m still in my “Dr. Moreau” zone.
“So maybe someday, instead of capital punishment, a convicted murder will receive the personality of a Labrador Retriever?”“Could be,” he says. “Who knows what people do with this stuff.”
“Has there ever been a time when people were creating a future this weird, when people were going to live in ways they couldn’t even recognize?”
“I dunno,” he says. “Explorer times?”
“I mean yeah, maybe for the Aztecs at first, when they saw the Conquistadors on their horses and thought it was some new kind of hybrid god/animal. But pretty soon the Spanish guys got off their horses and just started messing up the city and killing people. Pretty much like the Aztecs had been doing for a couple of generations. Business as usual.”
“I feel you,” he says. “Hey, I got to go. There’s some guys in Argentina who have this satellite and space tug that went off course. It’s like 50 million kilometers from Earth, but they think they can bring it back.” Weird stuff…
Biohacking in SF, where Dr. Moreau’s a piker, & humanoid robots are a happy delusion. Eminently worth reading in full: “Kale Salad with Ash.”
For more on the dizzying pace of experimentation (this time, in AI), pair with “Agent Claw.”
[Image above: source]
* “At such times the universe gets a little closer to us. They are strange times, times of beginnings and endings. Dangerous and powerful. And we feel it even if we don’t know what it is. These times are not necessarily good, and not necessarily bad. In fact, what they are depends on what we are.” – Terry Pratchett, I Shall Wear Midnight
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As we FAFO, we might recall that it was on this date in 1897 that the Indiana State House of Representatives passed Bill No.246 which gave pi the exact value of 3.2– a nice, round– and wrong– number.
Hoosier Dr. Edwin J. Goodwin, M.D, a mathematics enthusiast, satisfied himself that he’d succeeded in “squaring the circle.” Hoping to share with his home state the fame that would surely be forthcoming, Dr. Goodwin drafted legislation that would make Indiana the first to declare the value of pi as law, and convinced Representative Taylor I. Record, a farmer and lumber merchant, to introduce it. As an incentive, Dr. Goodwin, who planned to copyright his “discovery,” offered in the bill to make it available to Indiana textbooks at no cost.
It seems likely that few members of the House understood the bill (many said so during the debate), crammed as it was with 19th century mathematical jargon. Indeed, as Peter Beckmann wrote in his History of Pi, the bill contained “hair-raising statements which not only contradict elementary geometry, but also appear to contradict each other.” (Full text of the bill here.) Still, it sailed through the House.
As it happened, Professor Clarence Abiathar Waldo, the head of the Purdue University Mathematics Department and author of a book titled Manual of Descriptive Geometry, was in the Statehouse lobbying for the University’s budget appropriation as the final debate and vote were underway. He was astonished to find the General Assembly debating mathematical legislation. Naturally, he listened in… and he was horrified.
On February 11 the legislation was introduced in the Senate and referred to the Committee on Temperance, which reported the bill favorably the next day, and sent it to the Senate floor for debate.
But Professor Waldo had “coached” (as he later put it) a number of key Senators on the bill, so this time its reception was different. According to an Indianapolis News report of February 13,
…the bill was brought up and made fun of. The Senators made bad puns about it, ridiculed it and laughed over it. The fun lasted half an hour. Senator Hubbell said that it was not meet for the Senate, which was costing the State $250 a day, to waste its time in such frivolity. He said that in reading the leading newspapers of Chicago and the East, he found that the Indiana State Legislature had laid itself open to ridicule by the action already taken on the bill. He thought consideration of such a propostion was not dignified or worthy of the Senate. He moved the indefinite postponement of the bill, and the motion carried.
As one watches state governments around the U.S. enacting similarly nonsensical, unscientific legislation (e.g., here… perhaps legislators went to school on this), one might be forgiven for wondering “Where’s Waldo?”
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