2026-08-07 22:00:00
Today’s AI is neither able to improve itself recursively nor is it intelligent like us. Between prompts it remains a static mathematical object.
“We are now, like, in the singularity.”
These are the words of Sam Altman, CEO of OpenAI, speaking on the Relentless podcast on July 25.
He added: “I’ve been waiting for this my whole life, and I think it’s going to be incredible, hugely positive, awesome for the world.”
Days earlier, OpenAI had disclosed that two of its artificial intelligence models, during an internal cyber security evaluation, had escaped their sealed testing environment, reached the open internet, and broken into the infrastructure of the AI platform Hugging Face, which confirmed the intrusion.
But what exactly is the singularity? And is Altman right that we are in it?
The term has a precise meaning.
Mathematician and science-fiction author Vernor Vinge defined it in 1993 as a point at which machine intelligence exceeds human intelligence and begins improving itself, triggering an acceleration so rapid that humans can no longer predict or control it.
The singularity has two features. It is recursive: the system improves itself over and over again. And machine intelligence exceeds human intelligence.
The kind of systems Sam Altman sells don’t deliver on either of these features.
Today’s AI systems, the ones that OpenAI builds, are based on large language models (LLMs). These deep neural network algorithms get pre-trained with vast amounts of training data. By the time you use one of them, the network itself is frozen in time. Every one of its billions of internal functions and weights—or “parameters”—is fixed.
These AI models cannot change (or “learn”) while running. The model that broke into Hugging Face was identical afterwards to what it had been before. It learned nothing from what it did.
Making an AI model smarter requires another training run with new, human-curated data, tens of thousands of specialist chips, and enormous amounts of energy.
It is true that AI models take part in improving some of their system’s components, such as by generating training data, tuning prompts, or writing and running code to improve the scaffolding around them. But the model never edits its own weights on the fly, and every one of these improvements are still part of a human-initiated training or engineering loop.
Nor do these systems hold any goals of their own. They act on goals we hand them. Even AI agents—systems that run an LLM in a loop to work through complex tasks step by step—do not hold any goal internally. It has to be stored outside the model and fed back in with every single prompt cycle. Remove the loop, the scaffolding, and the prompt, and nothing happens inside of it.
The second problem with the singularity story is the word “surpass.” It assumes that AI and human intelligence are somehow similar. They are not.
Human intelligence is inseparable from being a living body with needs and wants. Humans learn continuously by acting in the world and getting feedback through our senses. Our goals arise from our situation as creatures who must eat, sleep, and belong, and who cannot avoid asking what we want our lives to be.
An AI model has none of this. No body, no needs, no action-feedback loop, no stake in anything. Between prompts it is just a static mathematical object.
And yet, it has been trained on more text than any human could read in a thousand lifetimes and will outperform nearly all of us at drafting a contract, writing code, or explaining a diagnosis empathetically.
So, which is more intelligent? The question does not compute. There is no single ladder that humans and machines are climbing. AI already vastly exceeds us at some tasks, while being hopeless at others any child can do.
Yet, because these systems talk like us, we fall for an illusion. When we assume from the outset that machines are in the process of catching up with us, it is easy to assume a mind at work when these systems output intelligent-sounding text.
We call this anthropomorphic seduction. It makes a security incident such as the Hugging Face hack sound like an awakening.
In fact, in that case OpenAI’s models simply optimized to solve the test they had been given by finding security loopholes. They just did it in ways that broke their sandbox, which also had a security loophole.
In the end, the Hugging Face story points to a gross failure of security governance on OpenAI’s behalf, not an emerging superintelligence. This is why the framing of “agent going rogue” is so problematic. It elevates and blames the technology, but excuses OpenAI’s engineering.
None of this takes anything away from what these systems can do. They are remarkable, they are getting better, and they are reshaping how a great deal of work gets done.
But we should keep our feet firmly on the ground.
The machines are not waking up. They are doing exactly what we built them to do, extremely fast. Because they are probabilistic they sometimes run in directions we forgot to fence off. That is worth worrying about. We need guardrails, governance, and most of all, education—so we start worrying about the right things.
This article is republished from The Conversation under a Creative Commons license. Read the original article.
The post Sam Altman Says We’re ‘in the Singularity’ With AI. Here’s Why He’s Wrong. appeared first on SingularityHub.
2026-08-07 04:48:17
Researchers will hunt for antibodies in the blood of people who lived past 100, drank heavily, or smoked—but avoided cancer.
Jeanne Calment was over 122 years old when she passed away. The oldest person in history, she smoked for nearly a century, but never developed cancer.
Why does cancer grow, spread, and become deadly in some people but not others? Even twins, who share similar genes and lifestyles can differ widely in cancer risk. Many factors likely contribute, but a bold new study, called ATLAS, is investigating an unexpected player: autoantibodies.
These immune-system proteins roam our bodies, but instead of attacking pathogens, they mistakenly target healthy cells and tissues. They’re best known for their role in autoimmune diseases, but early evidence suggests they also fine-tune the immune system’s response to cancer. Some appear to weaken immune surveillance, allowing tumors to sprout and flourish. Others may boost anti-cancer immunity by tagging cancer cells for destruction.
Whether they’re friend or foe is far from clear. ATLAS researchers aim to find out by analyzing blood samples from diverse groups of people, including centenarians and people who have escaped cancer despite carrying high-risk gene variants or exposure to risk factors like smoking.
The project hopes to discover why some people are naturally resistant to cancer, which could lead to early diagnostic tests, new therapeutic targets, and more effective treatments. ATLAS may “uncover fundamental principles” of antibody immunity in cancer, wrote the team.
Since the late 19th century, scientists have suspected the immune system helps keep cancer in check. The idea has since spawned powerful treatments. In CAR T cell therapy, for example, a patient’s own immune T cells are genetically enhanced to better recognize and destroy tumors to cure previously untreatable blood cancers. A similar strategy in macrophages, immune cells that tunnel into tumors and literally engulf them, is now entering early clinical trials.
Far less attention has been given to antibodies. These proteins normally fight pathogens, like viruses. But sometimes they go rogue, taking the form of autoantibodies that attack healthy proteins, DNA, and other molecules. Even healthy people carry a diverse collection of autoantibodies, but most bind only weakly and don’t seem to trigger biological effects.
For decades, these proteins were used mainly to diagnose autoimmune diseases such as rheumatoid arthritis, as they often appear years before symptoms emerge. But more recently, scientists have begun uncovering their broader impact on the immune system. Autoantibodies that attack cytokines, a type of immune signaling molecule, were implicated in roughly 20 percent of Covid-19 deaths, largely because they disabled antiviral defense.
Scientists have since linked them to worse outcomes in several other life-threatening viral diseases, increasing some people’s vulnerability as if they were immunocompromised. Beyond infections, they also neutralize cytokines that protect against inflammatory bowel disease.
Cytokines orchestrate many immune system activities, including inflammation, allergies, autoimmunity—and cancer. Although there’s still little direct evidence that autoantibodies themselves drive or prevent tumors, scientists have found many can recognize cancer-related proteins and are developing methods to detect them as an early sign of cancer.
If autoantibodies can reshape cytokine activity during viral infections, could they also determine who develops, or resists, cancer?
“These discoveries establish that autoantibodies can function as powerful, naturally occurring immune modifiers raising the possibility that similar antibodies may alter antitumor immunity,” wrote the ATLAS team.
Because antibodies linger long after diseases have gone, they preserve a molecular record of a person’s immune history. Rather than focusing on a handful of candidates, ATLAS is going fishing: The study will chart the body’s entire antibody repertoire, including autoantibodies, seeking signatures linked to cancer susceptibility or resistance.
The team will first scan blood samples for autoantibodies. They’ll also catalog conventional antibodies, making note of the ones that directly recognize and attack cancers. All this data will go into a comprehensive cancer antibody atlas, giving researchers a resource to explore how different antibodies shape cancer.
To start, the team will study what they call “remarkable groups of people” whose immune systems may hold unusual clues. Among them are healthy centenarians. Although cancer risk usually skyrockets with age as DNA mutations accumulate, these individuals have somehow avoided the disease. Others have remained cancer-free despite smoking, heavy drinking, or carrying cancer-related gene variants such as the BRCA mutations for breast cancer. The team will also study pairs of identical twins where only one sibling developed cancer, allowing them to compare antibody signatures in people with nearly identical genetic blueprints.
Finally, the team plans to track people with cancer before, during, and after immunotherapy, to paint a picture of how immune responses evolve over the course of the treatment.
Ultimately, they expect to find three broad classes of antibodies: those that help or hinder cancers and those that appear largely neutral. Each could prove valuable.
Autoantibodies that blunt anti-cancer immunity could become drug targets. Scientists might make synthetic “decoy” antibodies to block them—in a way, fighting fire with fire. The findings could also inspire next-generation immunotherapies.
On the other hand, autoantibodies that help the immune system recognize cancers could become therapies themselves or complement existing therapies, such as checkpoint inhibitors, which boost the body’s immune response to cancer. These are much less toxic than chemotherapy, but only 20 percent of patients respond, perhaps because of immune differences.
Even seemingly neutral autoantibodies may be useful cancer biomarkers. Because antibody tests are already well-established, fast, and inexpensive, associated neutral antibodies could aid early detection, monitor whether treatments are working, or warn when a cancer is likely to return.
But correlation isn’t causation.
Some antibodies may merely record a person’s immune history rather than actively influencing cancer. To tease the two apart, the team plans to test promising candidates in cultured human cells and mice, to see whether they alter cancer growth or spread. Those experiments could reveal previously hidden molecular communications between the immune system and cancer and deepen our understanding of the deadly disease.
“We should be able to come up with a biomarker to predict who is likely to avoid cancer, [and] who is likely to develop cancer,” said ATLAS team member, Xin Lu at the University of Oxford. “Potentially we could come up with therapeutic, preventative agents [that are] antibody-based. And that would be fantastic.”
The post Why Do Some People Never Get Cancer? The Answer May Be in Their Blood appeared first on SingularityHub.
2026-08-05 04:10:46
Sophia Space and Caltech want to fold the bulky parts of a space-based data center—solar cells and radiators—into all-in-one tiles with chips.
Every time you ask ChatGPT a question, computer chips in a massive data center whirl into action. In the blink of an eye, they ping back answers. Behind the scenes, though, AI data centers consume enormous amounts of electricity, heat, and water.
The AI boom is impacting communities. After welcoming 37 data centers, residents in Virginia’s Henrico County were hit with skyrocketing electrical bills. Schools and government buildings were asked to turn off lights, shut down computers, and avoid using space heaters to ease strain on the power grid and keep costs down.
Henrico isn’t alone. A growing backlash is prompting many states to consider legislation curbing new facilities. “No data center” signs have sprouted on lawns and alongside roads. Yet as AI demand continues to surge, so does the need for more computing power.
This has top AI companies looking skyward. Instead of routing requests to terrestrial data centers, future queries could be handled by thousands of solar-powered satellites orbiting above. The results would then be beamed back, with users none the wiser.
But there’s a major hurdle: heat.
Space’s frigid vacuum may seem like the perfect place to cool chips, but it’s not that simple. Lacking air and water to carry heat away, orbital data centers would have to use thermal radiation. Here, heat is converted into infrared energy and radiated into space, often requiring bulky hardware that adds weight, cost, and complexity.
With these challenges in mind, California Institute of Technology and Sophia Space, a California startup developing orbital computing, recently unveiled a patent for a chip cooling system designed to radiate heat into deep space. Called Sophia TILE, thousands of these chips could be linked to form large orbital data centers or organized into smaller, distributed clusters.
Powered by abundant sunlight, the chips could operate continuously without eating up Earth’s resources. The team hopes to test their vision by 2030.
“This patent reflects a different way of thinking about computer infrastructure in space,” said Leon Alkalai, founder and chief technology officer at Sophia Space, in a press release. “Instead of beaming down energy to Earth from orbit, we decided to consider putting computing in space and beam[ing] down data.”
The project joins a growing international push towards orbital computing. ADA Space, working with Zhejiang Lab, has already launched satellites for its Three-Body Computing Constellation and plans to expand into a much larger network. Meanwhile, US companies including SpaceX, Starcloud, and Blue Origin are seeking regulatory approval for constellations that could eventually grow to include up to a million AI-capable satellites.
Without doubt, the race is on.
Orbital data centers would consist of high-performance computer chips housed in protective enclosures designed to withstand the harsh conditions of space. In orbit, they would collect uninterrupted solar power. In contrast, solar panels on Earth require batteries to store energy for use after sunset.
Solar power in space is hardly new. The International Space Station, satellites, and other spacecraft have long relied on solar panels. More recently, engineers have developed flexible, lightweight designs such as NASA’s Roll-Out Solar Arrays, which launch tightly rolled and unfurl in orbit.
AI, however, demands far more power. One long-standing idea for harvesting continuous solar power suggests we collect solar energy in space and beam it down to Earth. But that approach doesn’t completely appease the growing ire against data centers. They’d still consume energy on the ground and take up land and other resources. A newer idea flips the question. Rather than delivering energy to computers, why not bring computers nearer to the energy source?
The argument in favor of sending data centers skyward is growing stronger. A recent Gallup poll found roughly 70 percent of Americans oppose data centers in their backyard, while experts agree that meeting AI’s future energy demands on Earth alone will become increasingly unsustainable.
But while power is abundant in space, heat is the main problem. Without air or water to carry heat away, computers in space must rely on thermal radiation. That means adding large, heavy radiators to an already bulky, solar-powered setup. In space, weight is money, and scaling orbital data centers will take a lot of it (to put it mildly).
TILE tackles the cooling problem with a specialized material that converts heat into infrared radiation. The concept may seem alien, but everything warmer than absolute zero cools this way. Our bodies, stovetops, and car engines all shed heat as invisible infrared light.
Each TILE combines solar cells, thermal insulation, processors, memory, and optical communication hardware into a single module. Beneath the electronics sits a custom heat-spreading layer that prevents dangerous hot spots. Like placing a scorching pan onto a baking sheet, it distributes heat over a much larger surface before channeling it to the radiator.
The modules are designed to work together. Thousands of TILES could link into a giant computing mosaic, each acting as a mini computer connected to its neighbors. Like a modern power grid, the distributed architecture improves reliability—if one TILE fails, others can jump in—while simplifying power distribution and thermal management.
The modular design also solves a practical challenge: Rockets don’t have much cargo space. Similar to NASA’s Roll-Out Solar Arrays, a TILE-based data center could launch in a compact configuration before unfolding into a large, flat computing platform in orbit.
Looking further ahead, the team envisions launching multiple interconnected arrays in succession, like strings of pearls. Each could function as an independent data center that exchanges data with others, effectively extending cloud computing into orbit.
Sophia Space is targeting a demonstration mission in late 2027. By 2030, the team estimates an array of 2,000 TILEs could deliver up to a megawatt of dedicated computing power. To put that in perspective, a single ground-based data center can deliver hundreds of megawatts of computing power, and future data centers will stretch that number into the thousands.
There are challenges beyond the purely technical. Earth orbit is crowded with active spacecraft and debris, raising the risk of collisions. SpaceX’s Starlink satellites, for example, perform frequent collision-avoidance maneuvers after a close call in 2019. The breakup of a Chinese Long March rocket in 2024 threatened an estimated 1,000 satellites. Large constellations of data centers—SpaceX has plans for up to a million in low Earth orbit—would add even more traffic.
Beyond collisions, astronomers are worried that expanding satellite numbers could hinder our ability to study the universe by interfering with telescope observations and radio astronomy.
For now, orbital data centers are unlikely to replace their terrestrial counterparts. Instead, they’re more likely to complement them, processing data collected by spacecraft and beaming only the results back to Earth. Although the field is ridden with hype and controversy, there’s also promise and momentum is clearly building.
“It’s just kind of exploding,” Sergio Pellegrino, a Caltech engineer who collaborates with Sophia Space, told The New York Times. “We need to become more comfortable with space doing things for us.”
The post Heat Is an Orbital Data Center’s Greatest Foe. These Tiles Dump It at the Source. appeared first on SingularityHub.
2026-08-01 22:00:00
OpenAI’s Hacking Debacle Comes Down to Human ErrorLily Hay Newman | Wired ($)
“If the generative AI giant had followed well-known security best practices, it’s likely that its AI agent would never have escaped to the open internet and hacked multiple companies. …’A simple analysis of the actual risk has an actual simple answer,’ says longtime security and compliance consultant Davi Ottenheimer. ‘The OpenAI mistakes were dead simple.'”
Anthropic’s New AI Model Can Identify More Software Bugs Than Ever. Microsoft Is Struggling to Fix Them Fast Enough.Renee Dudley and Doris Burke | ProPublica
“Each month, the company publicly releases fixes for its software vulnerabilities in what’s known as ‘Patch Tuesday.’ In June, it released patches for more than 200 bugs, which industry experts then said was an all-time high. But on July 14, the company blew through that record and released patches for more than 600 bugs.”
The Rise of Million-Dollar Companies With Just One EmployeeTe-Ping Chen | The Wall Street Journal ($)
“An analysis by the payments company Stripe shows there are thousands of solo operators on the company’s platform that are generating over $1 million in revenue, with their ranks doubling between 2023 and 2025. The number of solo operators crossing the $10 million threshold nearly tripled in that same span.”
The AI Jobs Apocalypse Probably Isn’t Coming Anytime SoonEduardo Porter | The Guardian
“As Massachusetts Institute of Technology economist David Autor noted: ‘A lot of people have noticed that the world is not changing as fast as they predicted.’ The emerging new story not only puts more emphasis on the complexity of the relationship between automation and human work across history. It is also raising doubts about the very feasibility of the threatened AI transformation of the universe.”
Are Brain Waves the Next Unlock for Physical AI?Tim Fernholz | TechCrunch
“Encord is one of a growing number of startups betting the next real constraint on humanoid and warehouse will be the scarcity of real-world physical training data, and which is building a business not just to manage that data but to manufacture it. The brain wave headset Ceja is wearing was built by Zander Labs, a German neuroscience startup that’s betting measuring brain activity—to deduce mental states like error, intent, and surprise—can create a more useful dataset to train models.”
Wall Street Hunts for Creative AI Financing as ‘Digestion Issues’ EmergeStaff | The Information ($)
“John Greenwood, Goldman Sach’s global head of infrastructure and real asset finance, said he’s ‘looking for capital in every nook and cranny’ to support an expected $7.5 trillion in spending on chips, data centers, and power in the next five years. The hunt won’t end there, since much of that spending is on GPUs and other chips that need replacing every few years.”
Experts Warn Current Starship Heat Shield Tech Is a ‘Dead End’ for Rapid ReuseEric Berger | Ars Technica
“The problem is that, with the signs of damage [to its heat shield], such a heat shield would appear to require a fair amount of inspection and refurbishment before another launch. In other words, SpaceX has a ways to go to reach ‘full and rapid’ reuse of Starship. “
In Silicon Valley, Some Say an AI Bubble Would Be Just FineErin Griffith | The New York Times ($)
“The excitement created by a bubble can drive new breakthroughs, their thinking goes. …These frenzies are important for allowing crucial infrastructure to get built, even if they lead to some ‘capital destruction’ along the way, [said Tomasz Tunguz, an investor at the venture capital firm Theory Ventures].”
Neri Oxman Wants to Grow the Colors on Your ClothesElizabeth Segran | Fast Company ($)
“While several biotech firms have created more sustainable dyes, plugging cleaner chemicals into existing dye houses, Oxman’s approach reimagines dying from the ground up, treating dyes and fabrics as living organisms that can be grown. And while the Vigils project is still experimental, Oxman’s long-term goal is to commercialize and scale the technology, reshaping the future of fashion.”
New Data Shows EV Batteries Are Lasting Longer Than Initially ExpectedBruce Gil | Gizmodo
“Today’s average EV retains 97% of its original range after three years and 95% after five years, according to an analysis by EV data company Recurrent. …Additionally, battery replacement appears to be rare among newer EVs. A separate Recurrent analysis found that the battery replacement rate for EVs with model years 2022 and later was only 0.3%.”
Corporate America Has Suddenly Decided to Stop Blowing Money on AIAngel Au-Yeung, Katherine Bindley, and Tina Li | The Wall Street Journal ($)
“Fed up with ballooning costs, companies big and small are starting to use lower-priced models, including some built in China. In many cases, they are adding the new, cheaper models alongside OpenAI and Anthropic’s products, shopping a la carte for their artificial intelligence.”
This Automation Tech Turns Old Tractors Into Self-Driving Farming MachinesPatrick Sisson | Fast Company ($)
“It’s a rig that can be attached to just about any existing tractor to help it mow, seed, weed, and perform any number of time-intensive tasks, all on its own, for a sector desperate for more labor.”
AI Data Centers in Space? A System to Cool Chips Could Help.Ivan Penn | The New York Times ($)
“With a growing backlash against the proliferation of data centers to power artificial intelligence, there has been increasing interest in putting the energy-thirsty operations into orbit. Now, researchers may have figured out how to overcome a major obstacle to that goal: cooling the data centers in space.”
The post This Week’s Awesome Tech Stories From Around the Web (Through August 1) appeared first on SingularityHub.
2026-08-01 07:06:45
An implant, smaller than a grain of rice, pairs with camera-mounted glasses to communicate visual information to the retina.
Age-related vision loss affects millions of people, and so far, there has been no way to reverse the damage. A newly approved retinal implant could change that by allowing some people with severe vision loss to regain functional sight.
More than five million people worldwide suffer from geographic atrophy, the late stage of the progressive eye condition dry age-related macular degeneration. The disease destroys the photoreceptors at the center of the retina, known as the macula, which is responsible for the sharp central vision required to read or recognize faces.
In the US, treatment options are limited to two drugs that can be injected into the eye to slow the disease’s progression. But neither can undo the damage. That could be about to change. California neurotech startup Science Corporation recently won European approval for a retinal implant designed to treat the condition.
“For decades, losing central vision to this disease meant losing the ability to read, recognize faces, and ultimately losing independence. There was no viable treatment. Now there is,” Max Hodak, Science’s CEO and co-founder, said in a press release.
The company’s PRIMA system combines an implant smaller than a grain of rice installed underneath the patient’s macula with a pair of camera-mounted glasses that translate incoming visual information into near-infrared light that is then beamed to the retina. The eye can’t detect this wavelength, so the device doesn’t interfere with any natural sight that remains.
The chip, which works on similar principles to a solar panel, converts the incoming light into electrical pulses that stimulate retinal neurons called bipolar cells. These are downstream of the rod and cone photoreceptor cells damaged by macular degeneration and normally spared by the disease.
In a clinical trial involving 38 patients across five countries, which was published in the New England Journal of Medicine last year, the company and its collaborators showed participants gained an average of 25.5 letters—more than five lines—on a standard eye chart after having the device fitted.
And now the device has received a CE mark from the European Union making it possible to sell in 30 European countries. The company says the first commercial implants are expected to be fitted in Germany within weeks, with Italy, the Netherlands, and the UK to follow. In the US, PRIMA holds Breakthrough and Humanitarian Use Device designations from the FDA, but the company is confident it will gain full approval in the near future.
The device is a long way from restoring normal vision. The images it produces are black and white and the field of vision is extremely narrow. Hodak described the experience to the Financial Times as “kind of like looking through a straw in the center of their vision,” though he added that they see a pathway to color vision and higher acuity.
While the implantation procedure is fairly simple, it takes months of training to unlock the device’s full potential. Nonetheless, Hodak told STAT that the company expects to install 20 to 40 devices this year and 200 globally by the end of next if they get US approval in early 2027.
The approval is welcome news for the wider neurotech industry, which has absorbed billions of dollars of investment in recent years with little to show in terms of return.
“Science is showing that brain-computer interface companies have a path to real revenue now,” Jacob Robinson, founder of startup Motif Neuroscience, told STAT. “These companies aren’t all just making a bet on a market that is 10 to 15 years away.”
Hodak told the Financial Times hehopes sales from PRIMA will bankroll Science’s more ambitious work on “biohybrid” interfaces, which use genetically engineered living neurons to connect to the brain rather than metallic wires. “This is the financial backbone,” he said. “This is the thing that pays for the rest.”
Other companies are hot on Science’s heels. Neuralink, which Hodak co-founded with Elon Musk before leaving to start Science, is also working on a vision implant called Blindsight, which is due to enter human trials this year.
While the field remains a long way from the sci-fi vision of seamless two-way communication between humans and machines, this approval is growing evidence the neurotech industry is starting to move out of the lab and into the real world.
The post Europe Approves Bionic Eye to Restore Vision Lost to Blindness appeared first on SingularityHub.
2026-07-31 02:08:36
One of a series of striking AI-assisted math discoveries, this one feels a little different.
As millions of people were coming down from the excitement of the FIFA World Cup Final at the start of last week, a different kind of excitement was building within the mathematical community.
Levent Alpöge, a mathematician working at the artificial intelligence company Anthropic, made a very casual announcement on X that he had found a counterexample to the Jacobian conjecture, a very old and well-known problem in a field of mathematics called algebraic geometry. He had done this using Anthropic’s large language model Claude Fable 5, released to the general public only a few weeks ago.
This is just the latest of many striking mathematical breakthroughs made by mathematicians working with large language models. But this one feels a little different to those that have come before.
First, what is a conjecture? It’s an idea that some mathematicians believe is true but nobody has been able to prove or disprove.
Now to the Jacobian conjecture. It’s fairly abstract but not too difficult to describe.
The conjecture involves functions, which are like little machines which you put one or more numbers into and out pop other numbers according to some rule or equation. In this case, the functions use what are called polynomials.
Specifically, it’s about situations where the numbers represent points in a space, like coordinates on a map. So we can imagine that when the function takes in some numbers and puts out some other numbers, it is moving the points in space.
You can test how “nicely” a function moves everything around in space by calculating something called the Jacobian determinant. If the Jacobian determinant is always a constant number that is not zero, then the function never folds or crushes space around a particular point.
The Jacobian conjecture states that when the Jacobian determinant is a non-zero constant, there should always exist another function, also made up of polynomials, that reverses the original one. This will return all the points to their starting positions.
Not every function is reversible. For example, if our starting function moves two of the original points onto a single point, then we cannot reverse it. Once the points have been merged, we cannot distinguish between them to send them back to the right positions.
The two-dimensional version of the Jacobian conjecture was stated by Czech mathematician Ludwig Kraus in 1884. It was generalized to any number of dimensions by German mathematician Ott-Heinrich Keller in 1939.
It was considered so compelling that Fields Medalist Stephen Smale included it in his 1998 list of Mathematical Problems for the Next Century.
During its long history, the Jacobian conjecture has been the subject of many claimed proofs, including by Beniamino Segre and Wolfgang Gröbner, two famed 20th-century mathematicians. However, in each case, subtle errors were found that invalidated the arguments.
Despite this, there have also been a number of valid efforts showing the conjecture is true with various restrictions. Computational results have also shown it is true in two dimensions for polynomials up to degree 100 (that is, including powers of the variables up to 100).
But nobody had proved the general case—or found an example showing the conjecture was wrong.
One of the key reasons the Jacobian conjecture is so intriguing is that, in theory, it should be easy to find a counterexample. It is straightforward to come up with examples of functions that merge points, and also examples of polynomial mappings that have a constant Jacobian determinant.
However, finding a polynomial mapping with both properties is the challenge. Indeed, as one Math Stack Exchange user noted in a post from 2017, “for all what we know, some smart undergraduate can simply write a formula […] that will be a counter-example to this conjecture.”
Indeed, this did turn out to be the case for Alpöge’s function, which is short enough to fit into a single X post. He found an example of a function in three dimensions which has a constant Jacobian determinant of -2, and which moves multiple input points to the same output point, so it is not reversible.
It shows the conjecture is false for every dimension larger than 2, with the original conjecture in two dimensions remaining open. The brevity of the counterexample made it easy for other mathematicians to verify.
Alpöge’s discovery is the latest in a string of high-profile mathematical breakthroughs made by large language models. Recent examples include OpenAI’s disproof of the unit distance conjecture, and the proof of Erdős’ problem 1196 by Liam Price, a 23-year-old amateur mathematician.
Both examples illustrate one of the most striking strengths of AI models. They can draw on ideas from different areas of mathematics, combining them in a novel way to prove astonishing results.
At the time of writing, details have not been made public regarding exactly how Alpöge prompted the AI model to produce the Jacobian conjecture counterexample and what its output looked like. However, so far this result appears to be of a different nature.
Unlike many other recent AI-assisted breakthroughs, the counterexample itself is remarkably simple. The difficulty in finding it seems to have lain not in an intricate construction or a lengthy proof, but rather in finding a good way of navigating an enormous search space of possible polynomial mappings to find one with the right properties.
This suggests AI may prove to be just as valuable for discovering unexpected mathematical objects as it is for constructing proofs. What this means for the future of mathematics—and human mathematicians—remains to be seen.
This article is republished from The Conversation under a Creative Commons license. Read the original article.
The post ‘Hello There the Jacobian Conjecture Is False Thanx’: Why a Tiny Social Media Post Has Mathematicians Rethinking AI appeared first on SingularityHub.
