The book of numbers
The recent Nautilus article by Siobhan Roberts about the mathematician Neil Sloane, titled “How to Build a Search Engine for Mathematics,” is the most interesting thing I’ve read online in months. I stumbled across it around six this morning, at a point when I was thinking about little more than my first cup of coffee, and when I was done, I felt energized, awake, and excited about the future. At first glance, its subject might not seem especially promising: Sloane’s baby, The On-Line Encyclopedia of Integer Sequences, sounds about as engaging as the classic bestseller A Million Random Digits with 100,000 Normal Deviates. But the more you think about Sloane and his life’s work, the more it starts to seem like what the Internet was meant to do all along. It’s a machine for generating connections between disciplines, a shortcut that turns good hunches into something more, and a means of quickly surveying an otherwise unnavigable universe of information. In short, it does for numbers, or anything that can be expressed as a sequence of integers, what Google Books theoretically should do for words. The result is a research tool that led Rutgers University professor Doron Zeilberger to call Sloane “the world’s most influential mathematician,” although, if anything, this understates the possible scope of his accomplishments. And even if you’re already familiar with OEIS, the article is well worth reading anyway, if only for how beautifully Roberts lays out its implications.
The appeal of Sloane’s encyclopedia can best be understood by going back to its origins, when its creator was a graduate student at Cornell. While writing his doctoral dissertation on a problem in artificial intelligence, he calculated an integer sequence—0, 1, 8, 78, 944, and so on—that described the firing of neurons in a neural network. As Roberts writes:
The sequence looked promising, though Sloane couldn’t figure out the pattern or formula that would give him the next and all further terms, and by extension the sequence’s rate of growth. He searched out the sequence at the library to see if it was published in a math book on combinatorics or the like, and found nothing. Along the way, however, he came upon other sequences of interest, and stashed them away for further investigation. He eventually computed the formula using a tool from 1937, Pólya’s enumeration theorem.
But the roundabout process had been frustrating. The task should not have been so difficult. He should have been able to simply look up his sequence in a comprehensive reference guide for all extant integer sequences. Since no such thing existed, he decided to build it himself. “I started collecting sequences,” he said. “I went through all the books in the Cornell library…And articles and journals and any other source I could find.”
Reading this, I was inevitably reminded of the experience of writing my own senior thesis, in the days before universal book search was available, and the kind of random scavenging through the stacks that was required back then to track down references and make connections. Sloane’s impulse to collect such sequences initially took the form of a set of punchcards, followed years later by A Handbook of Integer Sequences, published by his employers at Bell Labs. Finally, about twenty years ago, he put it online. Before long, the database began to prove its value, as when it revealed that a sequence related to the problem of placing cell towers matched one from an unrelated subject in number theory. It’s the closest thing we have to a search engine for math, as long as you can express whatever you’re doing in terms of a sequence of numbers:
Ultimately, it all comes back to counting things, and counting is a universally handy tool. Which in turn makes the encyclopedia handy, too. “Suppose you are working on a problem in one domain, say, electronics, and while solving a problem you encounter a sequence of integers,” said Manish Gupta, a coding theorist by training who runs a lab at the Dhirubhai Ambani Institute of Information and Communication Technology. “Now you can use the encyclopedia and search if this is well known. Many times it happens that this sequence may have appeared in a totally unrelated area with another problem. Since numbers are the computational output of nature, to me, these connections are quite natural.”
As Roberts concludes: “The encyclopedia’s impact on scientific research broadly speaking can be measured by its citations in journals, which currently Sloane has tallied to more than 4,500, ranging through biology, botany, zoology, chemistry, thermodynamics, optics, quantum physics, astrophysics, geology, cybernetics, engineering, epidemiology, and anthropology. It is a numerical database of the human canon.” And although the humanities go mostly unrepresented in that list, that’s probably because the translation of such concepts into numbers isn’t always intuitive. But researchers in other areas can at least appreciate its usefulness by analogy. When I think of how I use Google as a creative tool, it’s less to find specific information than to unearth connections—as when I spent a month looking up pairs of concepts like “Dadaism” and “Vehmgericht” to populate the conspiracy theory in The Icon Thief—or to verify a hunch I’ve already had. (As E.L. Doctorow once put it: “[Research] involved finding a responsible source for the lie I was about to create, and discovering that it was not a lie, which is to say someone else had thought of it first.”) Sloane’s encyclopedia essentially allows mathematicians and scientists to do the same, once they’ve converted their ideas into a searchable sequence, which can be a useful exercise in itself. And even if you aren’t in one of those fields, a few minutes browsing in OEIS is enough to remind you of how large the world is, how patterns can emerge in unexpected places, and how the first step to insight is making sure that those connections are accessible.