In 2013, a new user named Cleo stormed a digital sports forum with unverified answers. Today, she is an urban legend. But who was she? Editor’s choice for 2023.
By Tolika Boss, Allison Barschall, and Karen Leung
Mathematics
Boss: Hello, math geniuses! I’m Tolika Boss, the Editor-in-Chief of Digital at Scientific American. If you’re missing that famous Martin Gardner corner of Scientific American from the 1950s – fear not, we have some great mathematical content in the new year.
But for now, I’d like to leave you with one of my favorite stories from last year, presented and edited by the talented Allison Barschall. It tracks the story of Cleo, a mysterious user in an online sports forum known for unleashing a series of quick and unverified answers in the forum without showcasing any of her work. Between 2013 and 2015 – the user named Cleo did this about 37 times, thrilling all those with degrees in science, technology, engineering, and mathematics. But who was Cleo?
Anthony Bonato: It’s a legendary story in the world of mathematics. There’s a kind of romance to it, in a way.
Barschall: I’m Allison Barschall, and you’re listening to “Science in a Flash.” Today we have an episode about a mysterious figure in the world of online mathematics. They disappeared years ago but continue to stir controversy and speculation.
Barschall: We all love a good puzzle. Some people enjoy crosswords. Some play Sudoku. Others are still playing “Wordle.”
But Ron Gordon, a patent agent and former physicist in Massachusetts, is doing complex mathematical calculations. In 2013, when our story takes place, he spent enough time on an online math forum called “Math Stack Exchange” that it could have been a full-time job.
Gordon: I was working my full-time job, and then I was on Stack Exchange. Additionally, I had a family too. I enjoyed it so much that I didn’t even track how many hours I was devoting to it.
Barschall: Mathematics Stack Exchange is like Yahoo Answers, if the people on Yahoo Answers had advanced degrees in science, technology, engineering, and mathematics.
Now, Ron solved 2954 mathematical problems over the course of a decade on Stack Exchange, but he’s most famous for his answer to one particular integral. On November 11, 2013, a user on Stack Exchange asked a question:
“I need help with this integral: the integral from negative one to one of one over x times the square root of one plus x over one minus x times the natural logarithm of 2x squared plus 2x plus one, all divided by 2x squared minus 2x plus one, dx.”
Jay Cummings: Well, this is a crazy integral. There are many parts to it; any change in any of these parts makes the answer completely different.
Barschall: This is Jay Cummings. He’s an associate professor of mathematics at California State University, Sacramento. I asked him to help me figure out what I was looking at.
As much as I’ve been haunted by solving integrals since my calculus days, the idea of integration is actually quite simple. Imagine a line on a graph. Now imagine taking a colored pencil and shading the area under that line, down to the bottom edge of the graph.
What we’re trying to find is the area of that shaded color. For a straight line, this is very straightforward – it’s basic geometry. But the more complex and wavy and strange your line becomes, the harder it is to know the area under it. And the integral in the post on November 11, 2013 – was hard. The line on the graph looks like the spine of a long-necked dinosaur.
Try
The original poster used few computer programs, but none of them managed to provide what is called “the closed form” of the answer – and this is the exact and concise solution. Five minutes after it was posted, someone commented:
“Do you have any reason to believe there is a closed form for this ghastly thing?”
Gordon: That was a really good question… because it would save a lot of time if someone said, “This thing is impossible. Forget it. There’s no way.”
Parshall: Then, after four and a half hours from the original post, there was an answer:
“I = 4 pi times the inverse tangent of the square root of the golden ratio.”
The answer came from a user named Cleo. She was a new account with only one prior answer. Cleo provided no comments, evidence, or explanation – just one link above the code for the golden ratio, which takes you to a definition of the golden ratio.
Cummings: It’s a really silly answer. It makes you feel like you’re dealing with a supercomputer, or a theorem-proving program that hasn’t been released yet. Did ChatGPT start solving integrals in 2012?
Parshall: The Stack Exchange community, which always showcased its work, exploded into discussions in the comments section. Here’s one of them:
“I stick to saying Hamming: ‘The purpose of computing is insight, not numbers. Unless the result itself is particularly enlightening, I do not agree that it is an answer.’”
Parshall: This last comment came from Ron Gordon, a patent agent and former physicist, who didn’t see much value in Cleo’s simple answer.
Gordon: I think the value of a site like Stack Exchange lies in the knowledge you can impart to people. And I believe that the abstract answer to the question, on its own, isn’t worth much.
But it definitely influenced my determination to find a definitive solution. I spent most of the weekend writing and analyzing it. It took me about half a legal pad to accomplish that.
Parshall: It turned out Cleo was right. Ron posted the full answer, which immediately began to collect positive votes from community members. Many were astonished by the techniques he used to solve the problem. It was eventually posted on subreddit r/Math under the title “Integration Master.”
Gordon: It’s crazy. This is something I did ten years ago. I think I have better answers in the Stack Exchange world than that answer, believe it or not. But yes, Cleo too, I think she strikes a nerve, of course.
Parshall: Cleo’s quick answer unleashed madness on the Stack Exchange math forum. Between 2013 and 2015, she did it 37 more times, often appearing with unreasonable speed to solve extremely complex integral problems with complete answers. She didn’t even leave a trace of her work. Then she would disappear again into thin air.
Anthony Bonato: Experts are really divided about Cleo. She’s definitely someone who masters real integration techniques… like she mentioned some bizarre functions that I haven’t heard of.
Parshall: This is Anthony Bonato. He’s a mathematics professor at Toronto Metropolitan University.
Bonato: Some people suggested that Cleo might be Stephen Hawking – or has been Stephen Hawking – or Maryam Mirzakhani, a Fields Medalist.
Parshall: Food for thought, I guess.
Cummings: Or is this Terence Tao, you know, just relaxing in the evening?
Parshall: For the record, Terence Tao, who is sometimes described as one of the greatest living mathematicians, confirmed via email that he is not Cleo.
Cummings: Or is this Ramanujan…? Is Cleo another math genius from southern India doing this in her spare time?
Parshall: That genius he’s talking about is Srinivasa Ramanujan, one of the most enigmatic figures in the history of mathematics. You may have heard of him – he was portrayed by Dev Patel in a biopic released in 2016 titled “The Man Who Knew Infinity.”
Parshall:
Born in Tamil Nadu in 1887, he often comes up when talking about Cleo.
Cummings: He had an intuitive sense of mathematics that was … truly inspiring … he had no advanced math education. Yet somehow, he came up with these stunning theories.
Parshall: It seems they struck the same nerves over 100 years later. Because he didn’t include proofs. And that was Ramanujan’s gift and curse. I mean, he was incredibly talented, but he wasn’t placed in an educational box that said, “This is how you prove things; this is the path you follow to do mathematics.”
Gordon: I think a lot of people who hated being told “Show your work, show your work, show your work,” here’s someone who flaunts not showing his work, and people encourage it.
Parshall: But for Ron and many of the members of Math Stack Exchange, all the fun in their shared hobby is in showing their work. It’s not a dry explanation – it’s an adventure. Take Ron’s answer on the famous 2013 integral.
Gordon: By the time I got to where I wanted, I had eighth-grade fun in the denominator
Source: https://www.scientificamerican.com/podcast/episode/podcasts-of-the-year-cleo-the-mysterious-math-menace/#transcripts-body
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