June 2009

What's the Real Story?

STOCKHOLM (AFP) - A 16-year-old Iraqi immigrant living in Sweden has cracked a maths puzzle that has stumped experts for more than 300 years, Swedish media reported on Thursday.

In just four months, Mohamed Altoumaimi has found a formula to explain and simplify the so-called Bernoulli numbers, a sequence of calculations named after the 17th century Swiss mathematician Jacob Bernoulli, the Dagens Nyheter daily said.

That was the opening of a news story that landed in my email inbox on 28 May, courtesy of a listserve subscribed to by a (non-mathematical) friend of mine. The story didn't make a lot of sense, even when I read the entire article. The Bernoulli numbers are not calculations but, as the name indicates, numbers. To be sure, they are important, particularly in number theory, so any discovery involving them is definitely newsworthy, at least in the mathematical world. But, I was not aware of any 300-year-old problem about the Bernoulli numbers that had resisted all attempts at a solution. Nor did I know how exactly a new result might "explain and simplify" them.

So I dug around on the Web for more details. There were a lot of news stories about the topic, but they all said more or less the same as the article I had already seen. Eventually, however, I found a Swedish news Website with an English-language story that was close to the source (Uppsala University).

"Swedish teen tackles centuries-old numbers challenge" was the headline. The story began, "A 16-year-old Iraqi immigrant in central Sweden has single-handedly figured out a formula with Bernoulli numbers that is normally reserved for much more seasoned mathematicians, earning him praise from professors at prestigious Uppsala University." Ah. Much more believable.

The reporter went on to explain that Altoumaimi, the young high school pupil, had developed some equations involving the Bernoulli numbers. When his school math teachers were unable to tell him whether what he had done was correct, the student contacted a professor at Uppsala University, who, after examining his work, declared that it was indeed correct. Not new, however. As the story continued,

"While it's not the first time that someone has shown such Bernoulli number relationships, it's highly unusual for a first year high school student to make his way through the complicated calculations, according to Uppsala University senior maths lecturer Lars-ke Lindahl."

Now we have the real story. Not an original result at all. For sure it was an impressive demonstration of mathematical ability from a young high school pupil, whom the mathematical world may well hear a lot more of in future years, and definitely newsworthy. But most of the story that circulated around the press was entirely fabricated. That's right, literally made up. Sure, it made for a more exciting story, but it wasn't true! Now, I don't know about you, but much as though I love to see math in the news, I also like it to be fairly close to the truth.

Notice that I did not say it should be totally accurate. That is simply too much to expect, given the circumstances under which newspapers, in particular, are produced. The journalist is generally not an expert in the domain, has limited time, and has to get the story out regardless of whether she or he has managed to get hold of a helpful expert to check the facts and give a nice quote. Moreover, the research is almost invariably carried out over the phone, which is not the most ideal way to convey mathematics.

With a magazine it's possible to take a bit more time, but other factors come into play that can lead to errors - even if the writer is a professional mathematician. This has happened to me on a number of occasions. It starts well enough. The editor contacts me asking for a story on a particular topic, say a major new discovery. I have a week or two to produce it and send it off, after which ... well, for a long time nothing. During this period, which can stretch over several weeks, the editor is busy soliciting other articles, and looking to produce a balanced magazine.

Then, out of the blue, I get an email or a phone call requesting various changes to my article, perhaps additional material for a sidebar, or a suggestion of illustrations, maybe a caption for a particular illustration. And this time the deadline is short - possibly only one or two days. To make it worse, this task usually lands on my desk when I am already overloaded with other, equally pressing duties. On top of which, I have not given the article a single look since I originally sent it off, and have forgotten the reasons why I wrote the piece the way I did.

Occasionally, my response leads to a rapid flurry of exchanges involving several of the magazine staff. Passages are added, deleted, moved around, lengthened, shortened, and otherwise amended. Sometimes a suggested alteration will come in over the phone, with an on-the-spot response required. Everything moves so fast, the ground is highly fertile for errors to creep in. And they usually do.

On one occasion some years ago, I was writing an article for Discover magazine about a surprising new theorem that marked significant progress toward a possible proof of the Twin Primes Conjecture (that there are infinitely many pairs of prime numbers separated by exactly 2). Somehow, the version that ended up in print, with my name as the author, claimed that the new theorem actually proved the Twin Primes Conjecture. (On that occasion, I finally realized that the error was the result of an entire line of text being dropped when the article was formatted for the actual printed page. I had been sent a PDF file of the final page just prior to publication, so I had probably "seen" the erroneous final version. But I had not spotted the mistake. I read what I thought it said.)

On another occasion, I was writing a piece about the large number of false claims about the Golden Ratio, also for Discover, when a last minute request came in to provide a sidebar deriving the ratio from its definition in terms of subdividing a line segment. With no free time, I hastily did some copying and pasting from a set of old lecture notes from an undergraduate course given many years earlier. "Yes, that's the one," I thought, seeing the short algebraic derivation of a number that began 1.61803.

What I had forgotten was that in the course in question, I had provided the students with examples of proofs, some of which were correct, others containing one or more errors. Their task had been to identify the correct arguments and indicate the errors in the false ones. Okay, you can guess the rest. Of course, I was not aware of the error until the day after publication, when my email inbox filled to overflowing with emails that typically began, "Professor Devlin, I am surprised that someone with your experience could make such an elementary mistake." And of course, it did have to occur in an article devoted to pointing out false claims about the Golden Ratio!

Having written many articles for newspapers and magazines over the years, I'm used to this happening, and the truth is I'm actually surprised I don't make more such blunders. Faced with a very tight time deadline, it's all too easy to read what you think the words say, rather than what they actually say.

With those kinds of experiences behind me, I am therefore very forgiving of errors made in mathematics stories by writers who know far less about the subject than I do. On the other hand, there is a big difference between getting something wrong and simply making stuff up, as in the news story about the Swedish high school student. (The real story here was not the solution by a young schoolboy of a problem that had defied the experts for 300 years; it was how a young schoolboy had shown precocious mathematical talent. Altogether very different.)

At the end of the day, however, I don't think it matters very much if a story does have errors, and even if, within reason (which the Swedish news story is not, in my view), it contains some creative embellishment. (This comment will infuriate many of my colleagues. I know that because I've said it before. But I'll sail on regardless.) I learned that lesson from my editor when I first started to write occasional articles for The Guardian newspaper in the UK back in the early 1980s.

"A tiny fraction of your readers will understand what the story is about," he said. "They will spot any errors, and be able to correct them. Some of them will then have the added pleasure of writing to you to point out the mistake." (He was right on that score!)

"But this is a national newspaper, not a mathematics journal, and people who know some mathematics are not your main audience. The reader you should focus on is the person who knows nothing about maths. They won't understand any of the details, but if you write the story in an engaging fashion, they will read to the end and be left with several impressions: (i) that there has been a new development in the subject, (ii) what kind of person or persons made the advance, (iii) what it is about in general terms, (iv) perhaps s sense of why some people chose to do mathematics for a career, and (v) how it may affect their lives." (News stories about mathematics almost always include a remark about possible applications precisely to provide this link to the readers' lives.)

To my mind, taken together, (i) through (v) constitute a home run in terms of reminding people that mathematics is a living, growing subject that is important to society and should continue to be supported. And that's my main goal in writing for a newspaper.

Oh, and by the way, I made up those quotes. My editor did say something like that, and I took it to heart. But I did not take notes and I forget his actual words. I just thought it would add a human element to my story to format it as a quote. (I even wrote "maths" instead of "math" since, as a New Zealander, he would have used the British form.) So there!

And if you got to this point in my column, I kept your attention right to the end, so it all worked, right? Okay, I realize that, as a reader of MAA Online, you may well have been reading simply to catch me out!

Now, I wonder what error I have made in this article.

Devlin's Angle is updated at the beginning of each month. Devlin's most recent book for a general reader is The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern, published by Basic Books.
Mathematician Keith Devlin (email: devlin@stanford.edu) is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guy on NPR's Weekend Edition.