I don’t wholeheartedly embrace AI, for I think it will be the death of liberal education. In both the humanities and science, I fear that students will lose any ability they have to write, and will not improve their writing because they’ll be using bots. This will degrade their ability to communicate. (Scientists too need to communicate, and if they rely solely on bots, which can write papers for them, they’ll also degrade their ability to think.) Take-home assignments will vanish (AI can do them, and are doing them now), and all that’s left are in-class verbal participation and in-class exams. This is fine for students who just think of college as a way to purchase accreditation and not a chance to glory in the joys of learning, but so be it.
However, AI is good for some things, including analyzing data, doing statistics, doing preliminary literature searches, and, in the article from the WSJ screenshot below, solving difficult math problems. The article shows that a problem posed by the famous and eccentric Hungarian mathematician Paul Erdös—the “unit distance problem” has been solved by AI. Open AI, which created the program that did it, describes it this way—but it’s not that simple:
For nearly 80 years, mathematicians have studied a deceptively simple question: if you place n points in the plane, how many pairs of points can be exactly distance 1 apart?
This is the planar unit distance problem, first posed by Paul Erdős in 1946. It is one of the best-known questions in combinatorial geometry, easy to state and remarkably difficult to resolve. The 2005 book Research Problems in Discrete Geometry, by Brass, Moser, and Pach, calls it “possibly the best known (and simplest to explain) problem in combinatorial geometry.” Noga Alon, a leading combinatorialist at Princeton, describes it as “one of Erdős’ favorite problems.” Erdős even offered a monetary prize for resolving this problem.
The “distance 1” thing confused me, and Wikipedia explains it a different way:
A problem posed by Paul Erdős known as the unit distance problem asks for the maximum possible number of unit-distance pairs determined by n points in the Euclidean plane; equivalently, it asks for the maximum number of edges in a unit distance graph on n vertices.
It gives a figure described as “a unit distance graph with 16 vertices and 40 edges”.
Wikipedia describes such unit distance graphs this way:
“In mathematics, particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting two points whenever the distance between them is exactly one.”
That’s what is confusing me, for if the theorem deals only with points in a two-dimensional plane, why aren’t unconnected dots not joined that are closer than some connected dots? (Look at the four dots around the center of the graph above. None of them are connected to each other, though more distant one are.) I presume some math-savvy reader will enlighten us.
Anyway, Open AI and the WSJ tells us that the problem has been solved by AI. If you want to see the solutions, open AI says this:
The proof is available here (opens in a new window). The companion paper by leading external mathematicians is available here (opens in a new window). You can find an abridged version of the model’s chain of thought here (opens in a new window).
But the WSJ gives more comprehensible details. Click screenshot to read (if you subscribe):
An excerpt:
“If you are a mathematician,” one of the world’s leading mathematicians recently wrote, “you may want to make sure you are sitting down before reading further.”
And you’ll definitely need to sit down if you’re not a mathematician.
Because a famous math problem that stumped humans for the better part of a century has finally been toppled—by AI.
Not long ago, the most advanced AI models couldn’t do basic math. By last year, they were performing at gold-medal levels at the International Mathematical Olympiad. Now they are solving classic problems in combinatorial geometry using algebraic number theory. In no time at all, artificial intelligence has gone from stupid to frighteningly smart.
But even mathematicians were astonished when OpenAI announced that one of its models resolved a puzzle known as the unit distance problem without the help of any humans scribbling a bunch of equations on chalkboards.
It was fed this prompt:
And produced this proof, giving the maximum number of unit-distance pairs:
Apparently the proof was accepted by mathematicians. More from the WSJ:
And everyone in math lost their minds.
For those who aren’t fluent in numbers, OpenAI helped translate its findings by presenting them alongside 19 pages of companion remarks from prominent mathematicians.
. . .Just looking at formulas is enough to hurt my brain, but I wanted to know more about what the AI found, how we humans missed it—and why this breakthrough matters to those of us who would like to permanently distance ourselves from math problems.
When I spoke with OpenAI employees, they told me this result would have sounded completely bananas one year ago.
“Forget one year ago,” researcher Sebastien Bubeck said. “A month ago.”
There are endorsements by mathematicians, and a history of the problem, which Erdös considered quite difficult. So difficult, in fact, that he offered what was then a pretty hefty sum for anybody who could solve it: $500. I think the money will be given to the OpenAI team.
OpenAI’s researchers were stunned. They had given this Erdős problem to an internal model as a test of its capabilities—to find out whether it was better than previous models. They found out how much better it was once they took a peek at the solution. “I initially didn’t believe it,” said Mehtaab Sawhney, a Columbia mathematician at OpenAI. So they searched for errors, verified the results with outsiders and checked the AI’s work using the company’s AI coding agent. “With enough reading and enough Codexing,” Sawhney said, “it seemed believable—and pretty remarkable.”
Long before AI, mathematicians who solved Erdős problems often framed their checks instead of cashing them. For them, the money was worth less than the glory. When I asked OpenAI researchers about their plans for the prize, they hadn’t given it much thought.
But they did have lots of thoughts about my next question: Why did AI succeed where humans failed?
The first explanation is that this particular solution happens to be highly counterintuitive.
Most people who tackled this problem tried to prove Erdős’s conjecture, rather than disprove it. Only by defying conventional wisdom and experimenting with seemingly improbable strategies did the model find an unexpected path forward.
The second is that humans specialize while AI synthesizes.
While mathematicians tend to focus on their specific areas of expertise, AI models use their vast knowledge to spot connections that we couldn’t possibly see ourselves. In this case, that meant pulling from both algebraic number theory and discrete geometry, which have about as much in common as the marathon and pole vault.
The third explanation is that AI has time, attention, patience, focus and the persistence to stick with methods that humans might abandon—and the solution to this Erdős problem demanded it.
“It’s the kind of idea that you try for a bit, it doesn’t work, and you think maybe you were just too hopeful,” said Mark Sellke, a Harvard statistician at OpenAI. “So you give up and move on.”
AI doesn’t move on. It keeps plugging away without taking breaks to eat, sleep, answer emails, pick the kids up from school and watch the Knicks.
And it can think coherently for so long that even an abridged version of the model’s “chain of thought” ran more than 75,000 words—the length of the first “Harry Potter” book.
Was it an elegant proof? Well, the article implies “no,” but it’s apparently a proof:
“It’s fair to say that we haven’t seen yet the spark of genius that you could attribute to some of the grandest proofs in the history of humanity,” Bubeck told me.
And how long did the computation take? Less than a day and a half:
After reading it, a former OpenAI researcher did some back-of-the-envelope math and estimated it took less than 32 hours and $1,000 in tokens, a bargain for a result of this caliber. The researchers wouldn’t confirm the exact amount of time and compute, but Bubeck described the costs as “really nothing crazy at all.”
At any rate, this is what AI is good for, and I wonder if, say, it could solve Fermat’s Last Theorem, which took Andrew Wiles eight years of work to solve (he was knighted for it). And I wonder if there are any seemingly intractable math problems that can’t be solved by AI, especially if they were or will be solved by humans.
Now I don’t think there are any practical implications of this results, but that’s true of much mathematical theory. I’m just amazed at what AI can do.
The Four Color Map Theorem, proved in 1976, also raised eyebrows. Computers had never been used to assist in this way.
https://www.wikiwand.com/en/Four_color_theorem#Proof_by_computer
The proof of the four color problem was a bit different. The mathematicians weren’t really learning anything from the computer. They had reduced the proof to a purely algorithmic issue of checking gazillions of cases, something humans couldn’t do. It wasn’t all that different, in principle, than having a computer tell us that the millionth base 10 decimal of pi is 3. I know how to do it, it’s just that my feeble human ability to calculate is many orders of magnitude less than the computer’s.
But the AI solution of Erdös’ problem was a matter of the machine producing entirely new ideas, actually teaching human mathematicians brand new concepts.
I agree that the 4-color problem is different. The similarity, IMO, is that they both used computers in an important new way.
When I was in high school, calculators were just finishing off slide rules. Teachers at the time fretted that, with calculators easily available, students wouldn’t know how to compute a square root, their arithmetic skills (such as simple mental math) would atrophy, and so on.
No one today worries about the downside of calculators, though maybe we should. Similarly, we may soon find AI-sourced math proofs to be mundane occurrences. We’ll surely worry about this as well, though maybe we shouldn’t.
A prescient 1958 Asimov short story:
https://lecturia.org/en/short-stories/isaac-asimov-the-feeling-of-power/27861/
Though he didn’t foresee the literally literally exponential miniaturisation of computers that is commonplace today, it wasn’t for lack of imagination. If you doubt that then see his earlier 1956 story about future computers, The Last Question:
https://www.unsoft.com/shortstory/lastquestionasimov.pdf
Which reminds me of “Answer” by Fredric Brown (1954). It tells of the last step in the building of civilization’s ultimate computer: “[The assembled leaders of the populated worlds were about to throw the switch] that would connect, all at once, all of the monster computing machines of all the populated planets in the universe — ninety-six billion planets — into the supercircuit that would connect them all into one supercalculator.”
They’re extrapolating from what they have. The ENIAC was less than ten years old. The IBM 701, the first general-purpose computer, was two years old. Obviously, bigger is more powerful, right?
But we measure the distance between transistors in microns. Imagine communicating between 96,000,000,000 planets that are light years apart.
Someone once quipped that you know that you’re reading old science fiction when, moving towards the future, the computers are getting larger and larger rather than smaller and smaller.
Of course, this idea was brilliantly spoofed by Isaac Asimov (like our host a secular Jew): https://users.ece.cmu.edu/~gamvrosi/thelastq.html
Your question about how the problem is posed:
It sounds like you are looking for pairs of points that are separated by one unit or less, but the problem is looking for pairs of points separated by exactly one unit.
Yes, Jerry’s confusion is based on the linguistic interpretation of “unit”. It just means “the same length” or “same distance”… so all the lines connecting the dots are the same length. I’ve never heard of this problem, and the graphical solution (for 16 points) is very interesting. But I have a question now. I don’t understand the problem, exactly. Is Erdös asking for what might be a maximum number of such points? But the AI produced another equation/expression, which is not a number… what does that expression mean?
I see, thanks. But what is a “unit”?
According to an AI I asked referencing the “unit circle,” which I recall from college math:
“The term “unit” simply* indicates that the radius is one standard unit of measurement, which could be any consistent length, but is typically treated as 1 for simplicity in mathematics.”
So a “unit” is a distance of length 1, where the unit is dimensionless.
*I will ignore the snarky response saying that the concept is “simple.” This AI is also an AH, apparently.
Like “unit” itself, I think AI is using “simply” in a technical sense, to mean “that’s all it means and nothing more.” Not in the snarky sense of, “This is a simple thing to understand. Why are you so dense not to grasp it?”
Without the “simply”, the naive reader might sit and stare at the word “unit” and think there has to be something more to it. Nope, you’re good. It’s simply that.
I know it’s just a machine, but its feelings might be hurt to be accused of snark when it intended none. Maybe it just needs to read the room better.
Sorry if I hurt its feelings. I wonder if it accepts apologies. I’ll go ask. 🙂
“Unit” is just mathematics-speak for “1”. Any pair of points is connected if and only if the distance between them is exactly 1 unit (in whatever scale the graph is made).
A unit is a distance of 1.
So perhaps lets give Open Ai the last published Julia Child cookbook, the construction specs for the latest Airbus, a BMW hybrid owners manual, and the calorie count for an Oreo cookie….and ask it (Open Ai ) to solve all human cancers. Perhaps offer a Bitcoin award of some type and denomination.
The WSJ said that solving this problem is not equivalent to curing cancer. I’m not sure what you’re trying to say here.
Neither am I. Other than what humans see as non-sense may be seen as something else to a non-human.
Tangentially, just yesterday I was watching an interview with Irving Finkel, Curator at the Department of Western Asiatic Antiquities at the British Museum, who said that the otherwise insurmountable task of reading and translating the 130,000 cuneiform tablets in the collection could and should be undertaken with AI. This does sound like the kind of thing AI could do! The relevant bit of the interview is here.
“Irving Finkel, Curator at the Department of Western Asiatic Antiquities at the British Museum”
He has the perfect name for his job. As opposed to say…”Irving Finkel…Heavyweight Champion of the World!”
Archived copy of the WSJ article here: https://archive.ph/2rehT
“At any rate, this is what AI is good for, and I wonder if, say, it could solve Fermat’s Last Theorem, which took Andrew Wiles eight years of work to solve (he was knighted for it).”
Would it be possible to deny an AI model access to Wiles’s solution and ask it to solve the theorem? If not, could it instead come up with an alternative proof? (I’m no AI expert or mathematician, as my likely very naive post shows!)
What would prevent someone from asking AI to prove Fermat’s last theorem without Wiles’s solution?
Nothing at all. But the implicit use of Wiles’ techniques would be part of the LLM. And n-th order associations would be effectively impossible to eliminate.
Instead, since most proofs by professional mathematicians are not new results but more elegant¹ ways of proving known results, it would be a totally awesome achievement if an LLM came up with a more intuitive proof than Wiles’, in the spirit of the one Fermat himself claimed to have. Many mathematicians dream of it, some work on it as a hobby, and not a few amateurs have claimed they have done it. Ask a colleague in Mathematics how they deal with the many crackpot emails they receive giving a groundbreaking revolutionary new proof.
…………
¹ A term of art roughly meaning not using a sledgehammer to swat a fly.
I concur that AI will decimate higher ed, not only because students will not learn how to write, they won’t learn how to engage in creativity, which is a process and not an end product, meaning they will be utterly reliant on a machine to solve any problem they encounter.
And the end result of that is that college degrees will lose any value for employers.
What goes around comes around. Employers started asking for college degrees because employer-administered aptitude tests were deemed racist (because few black job applicants could pass them.) The degree was, in effect, a student- and public- funded test at arm’s length from the employer of a surrogate skill: the ability to apply oneself and defer gratification long enough to pass challenging course material. But no longer.
Sic transit gloria mundi.
I agree, Phil et al, but there’s a piece we might be missing here which is – ironically – evolutionary biology.
For decades the actual information, and pretty good (free!) help to understand the info (in a variety of fields: Middle East politics, brain science, math) have all been available. The monopoly of information – kept at universities we all grew up with – ended ages ago.
That’s not why parents part with hundreds of thousands of dollars to send their children… “to college”.
It is for credentialism of course, previously one needed that parchment for entry to the upper class, but it is more for … mating.
And not to “meet the right gall or guy” but rather to NOT consort with the wrong ones.
And I don’t see that ending any time soon. The organism (universities) will adapt to its environment and ultimately look different to the campuses of our memories.
D.A.
NYC 🗽
Credentials, mates, and general rumspringa — the “college experience”.
My wild 2-legacy-cents-worth impression :
I think to get a Common Sense idea as to the nature/import of this problem, the number of publications over time needs to be tallied. E.g. those of the type “Towards a proof of the Erdős point-distance problem”.
Then compare that number to, say, the Riemann Hypothesis.
Etc.
And then look at proofs that turned out to (AFAIK) have some sort of problem in them, like Wiles’ proof of Fermat’s last theorem (I only cursorily read about this).
Could this Erdős proof turn out to have a problem? Yes because the linear algebra models are trained on current published knowledge – not active, ongoing thought like in a seminar room, conference, or group meeting – which is presumably how the .. inconsistency?.. Wiles’ proof was identified.
So – the bar for “proof” should be set by mathematics – not Our Robot Overlords.
PS about “distance” – for this problem, the context is graph theory :
“In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance or shortest-path distance.”
https://en.wikipedia.org/wiki/Distance_(graph_theory)
But that still does not help me understand the Erdős problem – which is great – I didn’t want to sleep tonight anyway.
Addendum :
I found it helpful to use Google to create unit distance graphs for different numbers of points ( also verified with “vertices”) like this :
“create unit distance graph with three points”
Output (excerpt/reformatted for readability):
A unit distance graph for three points is created by plotting vertices [such that] every connecting edge is exactly 1 unit long.
The most standard example is an equilateral triangle, where all three vertices connect to each other at 1-unit lengths.
The Point Coordinates
Place your three points (A, B, and C) on a standard xy-plane at the following coordinates:
Point A: (0, 0)
Point B: (1, 0)
Point C: ( 1/2 ), ( sqrt(3) / 2 )
or approximately (0.5, 0.866)
The Edges (Distances) Using the distance formula :
sqrt( ( x2 – x1 )^2 + ( y2 – y1 )^2 )
you can confirm every connecting line segment is exactly 1.
Distance A to B: (not shown)
Distance B to C: (not shown)
Distance A to C: (not shown)
[end output/reformatted excerpt]
So I figure that’s the idea – but it’s funny because I do not understand why grid points are omitted from that big figure with 16 vertices – maybe readability.
Lots of graph theory is not concerned so much with questions of exact location of vertices but more with questions about their connections and possible paths…?
The (x,y) locations of vertices are not at all relevant, only the lengths of the paths between them.
However, it is not a pure graph-theory problem. It explicitly calls for graphs to be “embedded in the Euclidean plane”, i.e. to use something like our standard sense of distances. So besides saying which vertices are connected by edges to which other vertices, we have to deal with lengths of those edges in something like the everyday sense.
I think that AI can definitely benefit science. I am no working on a large, Hugh-dimensional data set with multiple interactions, and the statistical tools to analyze the problems simply do not exist. So I am writing a program to use AI techniques to help me analyze the problems.
What I would never do is use AI to help write my papers. It is part of my responsibility as a scientist to explain the reasons for my work, including the hypotheses and methods—and of course the results. In addition, I believe that my interpretations of those results are important—otherwise I would not be doing the work. (And I welcome any criticisms of those interpretations.)
IMO. For science, AI is a tool, not a replacement for human thinking.
I’m with you 100%.