OpenAI says it cracked 90-year-old maths problem in 88 hours

OpenAI, the pioneering artificial intelligence research laboratory behind the groundbreaking ChatGPT, has announced a potentially monumental achievement: the successful resolution of a long-standing, highly complex mathematical problem that has eluded human mathematicians for nearly a century. In a startling display of AI’s burgeoning capabilities, the company claims its advanced AI model, empowered by a distributed network of approximately 10,000 autonomous AI agents, or "bots," managed to crack the notoriously difficult Navier-Stokes equations in a mere 88 hours. This rapid advancement underscores the accelerating pace of AI development and its potential to tackle challenges previously considered the exclusive domain of human intellect.

The Navier-Stokes equations, a cornerstone of fluid dynamics, describe the motion of viscous fluid substances, from the swirling patterns of atmospheric currents to the intricate flow of blood through our veins. For 90 years, however, fundamental aspects of these equations, particularly concerning their existence and smoothness in three dimensions, have lacked rigorous mathematical proof. This void has represented a significant hurdle in understanding phenomena like turbulence, a chaotic and unpredictable aspect of fluid motion that remains one of physics’ most profound mysteries. OpenAI has hailed its purported solution as a "milestone," offering compelling evidence of the rapidly evolving prowess of AI tools.

However, the scientific community, accustomed to rigorous peer review and independent verification, awaits further scrutiny. OpenAI’s claimed solution has yet to be independently verified or formally accepted by the prestigious Clay Mathematics Institute, the organization behind the renowned Millennium Prize. This prize, offering a substantial reward of $1 million for the solution of seven specific, highly challenging mathematical problems, includes the Navier-Stokes existence and smoothness problem. OpenAI has explicitly stated that its intention is not to claim this prize, but rather to showcase the "substantial progress of our AI models."

The genesis of this breakthrough can be traced back to late August, when OpenAI began training a novel AI model. This new iteration quickly demonstrated an exceptional aptitude for mathematical reasoning, a characteristic that sets it apart from previous models. AI models, in essence, are sophisticated computer programs trained on vast datasets, enabling them to recognize intricate patterns and make predictions within complex information. While this advanced model remains an internal tool, deemed "significantly more capable" than OpenAI’s most recently released public models, its researchers saw an opportunity to apply it to some of the most celebrated unsolved problems in mathematics.

The company’s decision to tackle the Navier-Stokes problem was spurred by a confluence of events. On September 1st, OpenAI researchers reportedly "heard rumors that two Millennium Prize problems had been resolved." This news prompted them to deploy their newly trained AI model, orchestrating a vast swarm of approximately 10,000 AI bots, to rigorously pursue solutions to some of the remaining unsolved problems. The collective effort paid off remarkably swiftly. By September 5th, a mere 88 hours after the task was initiated, OpenAI reported that its AI agents had indeed devised a solution to the Navier-Stokes existence and smoothness problem.

The sheer scale of computational effort involved in this endeavor is staggering. Despite the seemingly swift resolution, the AI bots engaged in an exchange of nearly 3 million messages and processed an immense 130 billion output tokens. These tokens represent the individual units of text and code that an AI model generates in its responses. Extrapolating from OpenAI’s own pricing for its most advanced models, this computational undertaking would have incurred an estimated cost of around $10 million.

The solution that OpenAI claims to have achieved addresses two of the four critical statements required for a complete proof of the Navier-Stokes existence and smoothness problem as stipulated by the Millennium Prize. While the company has publicly stated its disinterest in claiming the prize, its announcement has inevitably stirred significant debate and controversy within the mathematical community.

Adding a layer of complexity to the narrative, a mathematics professor at New York University, Tristan Buckmaster, has come forward with a statement alleging a more intricate timeline and a potential overlap in research efforts. Buckmaster, along with Levent Alpöge, a mathematician at OpenAI rival Anthropic, had also been actively pursuing solutions to the Navier-Stokes problem, utilizing OpenAI’s Codex tool in their work. According to Buckmaster’s account, he discovered on September 3rd that "information about our progress had been passed to OpenAI." His statement, released on the same day as OpenAI’s publication, contends that OpenAI did not commence its work on the Navier-Stokes equations until after receiving intelligence about his and Alpöge’s research. Buckmaster has provided email correspondence as evidence, raising questions about the timing and methodologies employed by OpenAI.

While Buckmaster admits he has not yet had the opportunity to review OpenAI’s complete proof, he felt compelled to publicly share his account. He stated his motivation was to counteract what he perceived as a potentially misleading sequence of announcements. "The alternative is to let a sequence of announcements say something I know to be false," he explained.

In response to these claims, OpenAI extended congratulations to Buckmaster and Alpöge for their "concurrent work," describing it as "remarkable." The company maintains its position that it had not accessed "any of their work through any means until they released it publicly" and asserts that no user data was compromised during its investigation of the Navier-Stokes problem. Nevertheless, OpenAI acknowledged a slight possibility: "While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models." The company further clarified that despite any potential indirect influence, "our proofs differ significantly and even the precise results proved are different."

The implications of OpenAI’s announcement extend far beyond the realm of pure mathematics. It signals a profound shift in the potential capabilities of artificial intelligence, suggesting that AI can now not only process vast amounts of data and identify patterns but also engage in complex, abstract reasoning and contribute to the advancement of fundamental scientific understanding. The development raises crucial questions about the future of scientific discovery, the role of human researchers, and the ethical considerations surrounding AI’s increasing influence in intellectual pursuits. As the scientific community delves deeper into OpenAI’s purported proof, the world watches with anticipation, eager to understand the full impact of this extraordinary technological leap. The resolution of a 90-year-old mathematical enigma in a matter of hours by an AI underscores the transformative power of artificial intelligence and hints at a future where the boundaries of human knowledge may be significantly expanded by our intelligent creations. The ensuing verification process will be critical in determining the true extent of this achievement and its ramifications for the future of both mathematics and artificial intelligence.