Claude-Louis Navier was an engineering professor in Paris in the 1820s, where he was known for his deep analytical knowledge of bridge-building, but also blamed for relying too much on theory when a bridge he designed cracked and had to be dismantled.
asserted
he → know → theory
George Gabriel Stokes was the child of a church rector from the small Irish village of Skreen, whose talent for doing sums eventually led him, in 1849, to become the Lucasian professor of mathematics at Cambridge University, a position once held by Isaac Newton.
asserted
talent → do → Newton
Born 34 years apart in different countries, the two never met.
asserted
two → bear → countries
Yet, their names are forever entwined thanks to a set of equations they separately formulated, spawning a notoriously pertinacious math problem.
asserted
they → entwine → problem
This week, the Navier-Stokes existence and smoothness problem, as it is known, made headlines after it was declared cracked (like a puzzle, not a bridge) by artificial intelligence.
asserted
it → know → intelligence
If verified, the breakthrough is both spectacular and controversial.
asserted
breakthrough → verify → ?
Mathematicians witnessing first-hand the transformation of their profession by AI are finding it hard to keep up with the growing capabilities of the technology.
asserted
it → witness → technology
For some, this week’s news is another sign that, instead of competing with AI, their skills may be better put toward managing its risks.
uncertain
skills → compete → risks
“I think a lot of us are constantly recalibrating, almost minute by minute, given the pace we’re going at here,” said Deanna Needell, a Vancover-based professor of mathematics at UCLA and the University of British Columbia.
asserted
Needell → think → Columbia
The Navier-Stokes problem concerns the motion of fluids in three dimensions, and whether or not there are circumstances where the equations used to describe them can fail.
asserted
equations → concern → them
Last Tuesday, OpenAI, the tech company behind ChatGPT, said that its AI agents, after 88 hours of computation, had found a situation where this occurs.
asserted
this → say → situation
This immediately captured media attention, in part because Navier-Stokes is one of seven “millennium problems” that were listed in 2000 as outstanding challenges by the Colorado-based Clay Institute of Mathematics.
asserted
that → capture → Mathematics
Each problem is attached to a US$1-million prize for anyone who can provide a verified solution.
asserted
who → attach → solution
Prior to this week, only one had been solved, in 2003.
asserted
one → solve → 2003
No sooner had OpenAI made its announcement than Tristan Buckmaster, a mathematician at New York University, revealed that he had been working with a collaborator, Levent Alpöge, at Anthropic, an OpenAI rival, and they were closing in on their own AI-assisted solution to the problem.
asserted
they → make → problem
The situation is complicated by the fact that Dr. Buckmaster used OpenAI’s tools for some his research, though OpenAI has said it’s not possible that its model somehow found and used this work to gain the lead in the discovery.
asserted
model → complicate → discovery
Then there’s the question of whether such advancements should be made public in the traditional way, through publication in a peer-reviewed journal.
asserted
advancements → ’ → journal
It’s too soon to know how credit and prize money (if any) will be apportioned, or how history will view the matter.
asserted
history → ’ → matter
What is not in doubt is that AI has ascended to the highest levels of mathematics with remarkable speed – and likely changed the field forever.
asserted
AI → ascend → field
The prospect that it could soon climb higher than the brightest human minds is both fascinating and scary.
uncertain
it → climb → minds
“I expect most of our research projects ongoing in the world right now to be solved fairly quickly,” said Jacob Tsimerman, a University of Toronto professor.
asserted
Tsimerman → expect → Toronto
In July, he became the first Canada-based winner of the prestigious Fields Medal, known unofficially as Nobel Prize of mathematics.
asserted
he → become → mathematics
He added that machines may also “autonomously go on and do math that we haven’t even thought of yet.
uncertain
we → add → that
It’s fair to say that many people think of advanced mathematics as the hardest thing the human brain can do.
asserted
brain → ’ → thing
At the professional level, it requires raw intellectual power and years of training to make a meaningful contribution.
asserted
it → require → contribution
One might therefore conclude that mathematics would be among the domains least susceptible to competition from AI.
uncertain
mathematics → conclude → AI
Sure, ChatGPT can write a greeting card poem, but solve a millennium problem?
asserted
ChatGPT → write → problem
In fact, Dr. Tsimerman said, what makes math hard for humans is also what makes it amenable to exploration by AI.
asserted
it → say → AI
Theorems can be proven true or false, with no grey area in between.
asserted
Theorems → prove → between
And, most importantly, from the days of Pythagoras to the present, math is something we perform in an idealized, abstract environment that AI is well-equipped to operate in.
asserted
AI → perform → environment
“Math is a closed loop.
asserted
Math → close → ?
You can do it just by thinking,” Dr. Tsimerman said.
asserted
Tsimerman → do → it
It was Dr. Tsimerman who instigated the second big math story of the week.
asserted
who → instigate → week
A few months ago, after winning the Fields Medal, he revealed he would be working with OpenAI on safety in the use of artificial intelligence – a topic that has become his primary focus.
asserted
that → win → intelligence
On Tuesday, Dr. Tsimerman said he was launching a new centre dubbed the Mathematical AI Safety Institute in California’s Bay Area, where OpenAI is based.
asserted
OpenAI → say → Area
“Specifically, we’re going to be developing theories with the intention of building safe superintelligence,” Dr. Tsimerman said.
asserted
Tsimerman → go → superintelligence
The point, he said, is not simply to program AI to avoid bad behaviour, but to make it think in ways that align with human well-being, and follow instructions “not just to the letter, but also in spirit.
asserted
that → say → spirit
Until now, Dr. Needell said, AI safety issues have often been regarded as a “masking” problem, which programmers try to address by building guardrails to prevent AI from revealing dangerous knowledge it may have acquired.
uncertain
it → say → knowledge
Such a stopgap is problematic, however, because it means the knowledge is still in the machine and discoverable in principle.
asserted
knowledge → mean → principle
What may be more effective is adjusting the algorithms so they are unable learn certain things in the first place.
uncertain
they → adjust → place
…and 12 more, not listed.
The Aporia
