At the library the other day I requested an old book from off-site storage. When I called for it at the circulation desk, I felt as I would when buying some lurid tabloid at the drugstore checkout counter. The book was Erreurs de Mathématiciens des origenes à nos jours, a tell-all tattle-tale account that promised to reveal shamefaced failures of judgment and lapses of logic from throughout the long history of mathematics. When I got the book home, though, it was a sad disappointment. I was expecting to read about errors with grave consquences—false proofs and misconceptions that sent mathematics down the wrong path for decades or centuries. Instead I learned that Leonhard Euler incorrectly indentified 1,000,009 as a prime, and J. J. Sylvester once got some signs wrong in a paper. What a scandal! There are a few famous goofs among the listings (Pierre de Fermat’s conjecture that 2^2^n + 1 is always prime), but other equally celebrated mistakes go unreported (for example, the flawed proofs of the four-color theorem by Alfred Kempe and Peter Guthrie Tait). Some of the listings are simply mystifying. St. Augustine is cited for his assertion that the creation of the world took six days because six is a perfect number; I can’t see a problem with Augustine’s arithmetic here (6 is equal to 1 + 2 + 3 and 1 × 2 × 3, isn’t it?), so I suppose the flaw must lie in the theology, which is beyond my competence.

Erreurs de Mathématiciens was compiled by Maurice Lecat, who had it privately printed in 1935. Lecat was a Belgian chemist, mathematician and know-it-all; it seems that nothing was beyond his competence. In another of his books I found a long and diverse list of works du même autour. Within mathematics he wrote on determinants and the calculus of variations; in chemistry his specialty was azeotropic distillation (which sounds interesting, though I know zip about it); he also ventured into cosmology and the theory of relativity and published a monograph on intelligence among social insects. It appears he wrote at least seven books about his countryman Maurice Maeterlinck. Maybe some of these works are of lasting value.

But when it comes to errors of mathematicians, surely we can do better than Lecat’s nitpicking. Anyone have a favorite mathematical faux pas?

Potentates and plutocrats get their names plastered all over the landscape, but as far as I know you can’t yet buy the naming rights to a theorem in mathematics. You’ve got to actually do the math.

I’ve just (belatedly) discovered a thoughtful catalog and discussion of some mathematical eponyms. Dave Rusin of Northern Illlinois University has sifted through the Mathematical Subject Classification—a list of some 5,000 topics—and extracted all names of persons mentioned there. Rusin’s analysis was first posted to the Usenet newsgroup sci.math in 1998; the latest revision (updated in 2005) is available at Rusin’s web site.

Working with the 1991 version of the MSC, Rusin identified 357 terms that seem to refer to persons, and he put considerable effort into identifying which persons. It’s not always obvious. There’s only one Euclid in this crowd, but Hopf algebras and Weiner-Hopf operators are named for different Hopfs. And it’s always a nightmare sorting out the Bernouilli brothers—and cousins and uncles.

Many prominent mathematicians have their names attached to multiple concepts and so are mentioned more than once in the MSC ontology. Rusin’s top-ten list of mathematicians with the most MSC citations reveals some surprises. Riemann and Hilbert make the cut, but not Euler or Gauss, Archimedes or Newton. The No. 1 spot, with 54 citations, is occupied by Sophus Lie, a Norwegian best known for his contributions to group theory and algebra. Niels Henrik Abel, another Norwegian algebraist and group theorist, is at No. 6. And at No. 8 we find Evariste Galois, who, though not Norwegian, worked in some of the same areas. Perhaps there is some career advice in this cluster for those seeking mathematical immortality?

It so happens that I stumbled upon Rusin’s list while googling about for information on Ernst Jacobsthal, whom I have twice (link 1 and link 2) mentioned here recently. Jacobsthal’s name is indeed present in the MSC (Subject 11L10: “Jacobsthal and Brewer sums; other complete character sums”), and I think I know how he got there. When he was driven out of Germany by Nazi persecution, where did he take refuge? In Norway of course! But he made the strategic mistake of studying number theory rather than algebra or group theory and thus has only one entry in the MSC.

Christopher Joyce of National Public Radio is about to launch a series of reports on the “Industriosphere,” in which I am taking part. As in my book Infrastructure: A Field Guide to the Industrial Landscape, the subject matter is all the manmade bits of the environment we live in. (What is that stuff? What does it do? Why is it there?) Two weeks ago I spent a couple of days exploring and recording in and around Washington, DC, with Joyce and his NPR crew. The first report is scheduled for broadcast tomorrow afternoon (November 26) on Weekend All Things Considered. Check your local NPR station for broadcast times, or listen on the Internet. After the broadcast, some additional material (including photographs) will be posted on the NPR web site. And there is already an Industriosphere Flickr group.

In the photo above Christopher Joyce (foreground), producer Tina Tennessen (right) and audio engineer Drew Reynolds are “getting ambi” (recording ambient sounds) near Thomas Circle in Washington, where we all spent some time playing in traffic.

Update 2006-12-03: A brief followup interview, discussing the wooden rooftop water tanks of New York City, was broadcast on Saturday 2 December 2006. The interview is online here.

Driving over the river and through the woods yesterday, I was running low on fuel. My car has two kinds of instruments to tell me that I’ll soon be standing by the side of the road feeling foolish. A conventional gas gauge shows the fraction of a tankful remaining, presumably based on readings from some sort of float mechanism inside the tank. The second instrument measures the rate of fuel flow to the engine, showing the result on a digital display that can be set to any of three modes, labeled “consumption,” “average consumption” and “range.” The two “consumption” modes are calibrated in miles per gallon; the “range” mode gives an estimate of the distance remaining until the tank is empty, in miles. When you’re nervous about whether or not you can make it to the next gas station, the range is clearly of interest.

But keeping an eye on the range estimate is also somewhat disconcerting. If the meter says you can last another 23 miles, and then you drive a mile, it seems reasonable to expect that the meter will report a remaining range of 22 miles. In fact it may well say 19 miles, or 23, or even 26. It’s particularly bizarre to see the range increase as you continue driving.

What’s going on here? It’s not hard to guess. The estimated range is simply the number of gallons remaining in the tank multiplied by the fuel-use rate in miles per gallon. Both measurements doubtless have some noise in them, but variations in the fuel flow rate are the major cause of fluctuations in the range estimate. For my car, the instantaneous fuel economy dips down below 10 miles per gallon under hard acceleration, and it appears to go well above 100 miles per gallon when coasting downhill. (The meter tops out at 99.9 mpg.) These variations could alter the range estimate by a factor of 10 or more. Strictly speaking, the fluctuating estimates are not wrong—they indicate the actual range if you were to continue driving exactly as you were at the moment of measurement—but some averaging or filtering would seem sensible.

In fact, I think the range readings I see on my dashboard instrument are smoothed to some extent. The number is updated at intervals of about 30 seconds, and it may reflect an average calculated over a somewhat longer period. The question I want to ask is this: What is the optimum averaging interval—optimal in the sense that it minimizes some measure of error in the estimates? I doubt there can be any definitive answer without making some assumptions about the nature of the fluctuations, but I have a heuristic proposal that seems pretty good to me.

Here’s how I was thinking about the problem during my Thanksgiving pilgrimmage. Suppose you keep driving until the tank runs dry, all the while recording your distance and rate of fuel consumption, moment by moment. Retrospectively, then, it’s easy to determine the number that the range meter should have been displaying at any point during the trip: You just measure the distance backward from the point where the engine died, and by definition that’s the range remaining. But note that for every point along the route, this “retrodicted” range should be equal to the number of gallons left in the tank at that point multiplied by the fuel-use rate (in miles per gallon) averaged over the remainder of the distance. This fact suggests a perfect estimation strategy: You should always average the fuel-use rate over the remaining range. Unfortunately, the remaining range is exactly what you’re trying to calculate, so this algorithm is not very practical. But perhaps we can approximate it.

To reiterate: If the remaining range is n miles, then the ideal is to estimate this range by averaging the fuel consumption over the next n miles. We can’t quite do that, for two reasons. First, we don’t know what the average consumption will be in the n miles to come; it will depend on terrain, speed, traffic conditions, and many other imponderables. Second, we don’t even known what n is; that’s what we’re trying to estimate. But don’t despair. To cope with the first problem, we can choose some other interval of n miles as a surrogate for the n miles just ahead; the obvious choice is the n miles just behind us. As for the unknown quantity n, we calculate it iteratively. Make some initial guess r₀ about the fuel consumption rate, perhaps using the long-term average since the car was manufactured. Multiply r₀ by the gallons in the tank to get a first estimate n₀ of the remaining range. Then take the average fuel consumption over the preceding n₀ miles and again multiply by the number of gallons to get a better range estimate, n₁. Only a few repetitions of this process ought to be needed to converge on a pretty good estimate of n. That estimate, of course, is what the dashboard meter will report.

With this scheme, as the estimated range gets smaller, it will also get more volatile, because the consumption rate will be averaged over a smaller interval. I argue that this tendency to wider fluctuations is not a failure of the algorithm. In the last few miles before the tank runs dry, the range really does depend sensitively on whether you’re descending a hill on the open highway or stopping and starting at a series of city traffic lights.

Something tells me I’m not the first person to think about this problem or the first to propose this solution. The same issues arise in lots of other contexts, such as predicting how long the battery will last in a laptop computer. If anyone has a plan that can beat mine, I’d be pleased to hear about it.

By the way, I arrived on time for Thanksgiving dinner, with a few drops left in the tank.

In an item published here last May I stumbled across the sequence

1 3 5 11 21 43 85 171 341 683 1365 2731 5461 10923 21845 43691
87381 174763 349525 699051

which I dubbed “the oddest numbers” but which Neil J. A. Sloane’s Encyclopedia of Integer Sequences calls the Jacobsthal sequence. I asked: Who is or was Jacobsthal? Keith Matthews, creator and maintainer of the Number Theory Web, has stepped forward with an answer. Ernst Jacobsthal (1882-1965) was a German-born mathematician who fled to Norway when the Nazis came to power and eventually became a Norwegian citizen and professor at Trondheim. The story is told in an obituary that Matthews has made available on the Number Theory Web both in the original Norwegian and in an English translation by Jan Kristian Haugland. Matthews also mentions a brief biography in German. And the Mathematics Genealogy Project offers the further information that Jacobsthal was a student of Georg Frobenius and Issai Schur.

A bit more poking around reveals that Jacobsthal’s doctoral dissertation is available online, published by the Humboldt-Universität zu Berlin. In this work Jacobsthal gives a proof that every prime of the form 4n+1 can be written as a sum of two squares.

I’m very grateful to Matthews for answering my idle question about Jacobsthal—but there’s no end to questions. What I’d like to know now is when and where and in what context Jacobsthal discussed his sequence. The numbers are most readily defined by the recurrence J_n = J_n-1 + 2J_n-2, with J₀ = J₁ = 1. How did this idea come up in his work? Several journal articles and web sites mention the Jacobsthal numbers, but I’ve yet to find a reference to any specific publication.

Whether or not anyone else is reading the stuff I’m posting here, the spammers have discovered me. And I’m afraid that “comment spam” is not as amusing as the e-mail variety. I have therefore installed one of those annoying sentry programs that will ask for the answer to a trivial question before allowing you to post a comment. I’m choosing questions that I’m sure every reader of this blog will be able to answer instantly — things like, “Who published the first proof of the impossibility of trisecting an angle?” and “True or false: The present king of France is bald?” Please give it a try. If you have any trouble — and you’re not a spambot — e-mail me at brian@bit-player.org.

Update 2006-12-14: A few days ago I thought I had solved the comment-spam problem by other means, and so I turned off the interrogation. But the spammers are back, and so I have put the sentry on duty again. This time the questions all take the same form: a sequence of positive integers. You’ll be asked to name the next element of the sequence. None of them are tricky (no lists of subway stops), or even interesting. If the spambots learn to solve the problems in this set, well, good for them; at that point we can think about escalating to the next level.

Three postings here (1, 2, 3) have discussed what happens when you form all pairwise sums and differences from a finite set of integers. The number of differences almost always exceeds the number of sums—a fact that lends special interest to the occasional exceptional sets, with More Sums Than Differences (MSTD).

A question left up in the air when I last wrote on this subject was the size of the smallest MSTD set. The smallest known set was {0, 2, 3, 4, 7, 11, 12, 14}, which has 26 sums but only 25 differences. This set has eight elements. Could there be an MSTD set with seven or fewer elements? The question has now been answered in the negative by Peter V. Hegarty of the Chalmers University of Technology and Göteborg University. He proves there is no smaller set, and furthermore that {0, 2, 3, 4, 7, 11, 12, 14} is the only MSTD set of size eight. (Apart from other eight-element sets generated from {0, 2, 3, 4, 7, 11, 12, 14} by affine transformations.)

Read all about it at the arXiv.

I want to thank those of my friends who sent me links to Jenn Shreve’s amusing collection of bloggers’ “haven’t posted in awhile” excuses and apologies. What has kept me away all this time is the search for an adequate excuse.

Thanks, too, to those of my friends who didn’t send me the links.