Category Archives: Bad Math

Today in Incoherent Numerology

Or, “Oh, Wikipedia, How I Love Thee. Let me count the ways: one, two, phi…”

From Wikipedia’s page on Duchamp’s Nude Descending a Staircase, No. 2 (today’s version):

It has been noted disquisitively [link] that the number 1001 of Duchamp’s entry at the 1912 Indépendants catalogue also happens to represent an integer based number of the Golden ratio base, related to the golden section, something of much interest to the Duchamps and others of the Puteaux Group. Representing integers as golden ratio base numbers, one obtains the final result 1000.1001φ. This, of course, was by chance—and it is not known whether Duchamp was familiar enough with the mathematics of the golden ratio to have made such a connection—as it was by chance too the relation to Arabic Manuscript of The Thousand and One Nights dating back to the 1300s.

Euhhhhh, non.

As best I can tell, all this is saying is that the catalogue number of Duchamp’s painting contains only 0s and 1s.
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One and One and One Make Three

Every once in a while, a bit of esoteric mathematics drifts into more popular view and leaves poor souls like me wondering, “Why?”

Why is this piece of gee-whizzery being waved about, when the popularized “explanation” of it is so warped as to be misleading? Is the goal of “popularizing mathematics” just to inflate the reader’s ego—the intended result being, “Look what I understand!,” or, worse, “Look at what those [snort] professional mathematicians are saying, and how obviously wrong it is.”

Today’s instalment (noticed by my friend Dr. SkySkull): the glib assertion going around that

$$ 1 + 2 + 3 + 4 + 5 + \cdots = -\frac{1}{12}. $$

Sigh.

It’s like an Upbuzzdomeworthy headline: These scientists added together all the counting numbers. You’ll never guess what happened next!

“This crazy calculation is actually used in physics,” we are solemnly assured.

Sigh.

The physics side of the story is, roughly, “Sometimes you’re doing a calculation and it looks like you’ll have to add up $$1+2+3+4+\cdots$$  and so on forever. Then you look more carefully and realize that you shouldn’t—something you neglected matters. It turns out that you can swap in $$-1/12$$ for the corrected calculation and get a good first stab at the answer. More specifically, swapping in $$-1/12$$ tells you the part of the answer which doesn’t depend on the particular details of the extra effect you originally neglected.”

For an example of this being done, see David Tong’s notes on quantum field theory, chapter 2, page 27. For the story as explained by a mathematician, see Terry Tao’s “The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation.” As that title might hint, these do presume a certain level of background knowledge, but that’s kind of the point. This is an instance where the result itself requires at least moderate expertise to understand, unlike, say, the four-colour theorem, where the premise and the result are pretty easy to set out, and it’s the stuff in between which is much harder to follow.

ADDENDUM (19 January 2014): I’ve heard the argument in favour of this gee-whizzery that it “gets people excited about mathematics.” So what? A large number of people are misinformed; a tiny fraction of that population goes on to learn more and realize that they were, essentially, lied to. Getting people interested in mathematics is a laudable goal, but you need to pick your teaser-trailer examples more carefully.

And I see Terry Tao has weighed in himself with a clear note and some charming terminology.

Reflections

Prompted by this review of Colin McGinn’s Basic Structures of Reality (2011), I read a chapter, courtesy the uni library. It was endumbening. To the extent that he ever has a point, he says in many words what others have said more clearly in few. He confuses the pedagogy of a particular introductory book with the mature understanding of a subject, displays total ignorance of deeper treatments of his chosen topic, blunders into fallacies, and generally leaves one with the impression that he has never done a calculation in all the time he spent “studying physics”. Truly an amazing achievement.

A few years ago, I might have blogged my way through the whole darn book. I must be getting old (“REALLY? NO WAY!” declares my weak knee). But is it a healthy and mature sense of priorities, or a senescent academic crustiness? Have I become one of those people, concerned with my vita to the exclusion of all else? Dark thoughts for this cold autumn evening, dark as our current season of superhero movies—Fimbulwinter 3: Flame of Despair….

Precalculus -> Statistics

Now that 2.2 metric Ages of Internet Time have passed since Andrew Hacker’s ill-advised “math is hard!!” ramble, I figure it’s a good day to propose my own way of improving high-school mathematics education. Be advised: this is a suggestion about the curriculum, not about how to train teachers, buy books and all that un-TED-friendly stuff which reformers happily gloss over. And I’ll be talking about changes late in the game, which won’t address problems at the “why can’t Johnny add?” level.

When I was in high school—at a pretty well-supported public school, out in the ‘burbs at the comparatively unimpoverished end of town—I took a “precalculus” class my eleventh-grade year. Most of the advanced-track students I knew did the same thing. (If you’d gotten yourself on the even-more-advanced track back in eigth grade, you took precalculus in tenth.) This was supposed to prepare us for taking the AP Calculus class our senior year, which would allow us to get college credit. Instead, it was a thoroughgoing waste of time. The content was a repeat of Algebra II/Trigonometry, which we’d taken the year before, with two exceptions thrown in. The first, probability, was a topic our teacher didn’t know how to teach. In fact, she admitted as much: “I don’t know how to teach probability, so you’re all going to read the book today.” The second, limits, served no purpose. I’ll explain why in a moment.

I suggest the following: scrap “precalculus” and replace it with a year-long statistics course. This plan has several advantages:
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“Is Algebra Necessary?” Are You High?

“This room smells of mathematics!
Go out and fetch a disinfectant spray!”

A.H. Trelawney Ross, Alan Turing’s form master

It’s been a while since I’ve felt riled enough to blog. But now, the spirit moves within me once more.

First, I encourage you to read Andrew Hacker’s op-ed in The New York Times,Is Algebra Necessary?” Then, sample a few reactions:
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Signature in the Cell (Repost)

For your convenience:

The following is a list of debunkings of Stephen C. Meyer’s Signature in the Cell, arranged more or less in chronological order. I have not included every blog post I’ve seen on the topic; as I did for Behe’s The Edge of Evolution, I’ve focused on the most substantive remarks, rather than keeping track of every time somebody just quoted somebody else. (I’ve also probably overlooked, forgotten, mistakenly thought I’d already included or never been made aware of some worthwhile essays.) In some cases, additional relevant posts can be found by following links within the essays I have listed.
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How Not to be a Network-Theory n00b

Copied from my old ScienceBlogs site to test out the mathcache JavaScript tool.

Ah, complex networks: manufacturing centre for the textbook cardboard of tomorrow!

When you work in the corner of science where I do, you hear a lot of “sales talk” — claims that, thanks to the innovative research of so-and-so, the paradigms are shifting under the feet of the orthodox. It’s sort of a genre convention. To stay sane, it helps to have an antidote at hand (“The paradigm works fast, Dr. Jones!”).

For example, everybody loves “scale-free networks”: collections of nodes and links in which the probability that a node has $k$ connections falls off as a power-law function of $k$. In the jargon, the “degree” of a node is the number of links it has, so a “scale-free” network has a power-law degree distribution.
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The Necessity of Mathematics

Today, everything from international finance to teenage sexuality flows on a global computer network which depends upon semiconductor technology which, in turn, could not have been developed without knowledge of the quantum principles of solid-state physics. Today, we are damaging our environment in ways which require all our fortitude and ingenuity just to comprehend, let alone resolve. More and more people are becoming convinced that our civilization requires wisdom in order to survive, the sort of wisdom which can only come from scientific literacy; thus, an increasing number of observers are trying to figure out why science has been taught so poorly and how to fix that state of affairs. Charles Simonyi draws a distinction between those who merely “popularize” a science and those who promote the public understanding of it. We might more generously speak of bad popularizers and good ones, but the distinction between superficiality and depth is a real one, and we would do well to consider what criteria separate the two.

Opinions on how to communicate science are as diverse as the communicators. In this Network age, anyone with a Web browser and a little free time can join the conversation and become part of the problem — or part of the solution, if you take an optimistic view of these newfangled media. Certain themes recur, and tend to drive people into one or another loose camp of like-minded fellows: what do you do when scientific discoveries clash with someone’s religious beliefs? Why do news stories sensationalize or distort scientific findings, and what can we do about it? What can we do when the truth, as best we can discern it, is simply not politic?

Rather than trying to find a new and juicy angle on these oft-repeated questions, this essay will attempt to explore another direction, one which I believe has received insufficient attention. We might grandiosely call this a foray into the philosophy of science popularization. The topic I wish to explore is the role mathematics plays in understanding and doing science, and how we disable ourselves if our “explanations” of science do not include mathematics. The fact that too many people don’t know statistics has already been mourned, but the problem runs deeper than that. To make my point clear, I’d like to focus on a specific example, one drawn from classical physics. Once we’ve explored the idea in question, extensions to other fields of inquiry will be easier to make. To make life as easy as possible, we’re going to step back a few centuries and look at a development which occurred when the modern approach to natural science was in its infancy.

Our thesis will be the following: that if one does not understand or refuses to deal with mathematics, one has fatally impaired one’s ability to follow the physics, because not only are the ideas of the physics expressed in mathematical form, but also the relationships among those ideas are established with mathematical reasoning.

This is a strong assertion, and a rather pessimistic one, so we turn to a concrete example to investigate what it means. Our example comes from the study of planetary motion and begins with Kepler’s Three Laws.

KEPLER’S THREE LAWS

Johannes Kepler (1571–1630) discovered three rules which described the motions of the planets. He distilled them from the years’ worth of data collected by his contemporary, the Danish astronomer Tycho Brahe (1546–1601). The story of their professional relationship is one of clashing personalities, set against a backdrop of aristocracy, ruin and war. From that drama, we boil away the biography and extract some items of geometry:
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Statistics: Tool of an Elitist Bastard

I was prepared to enjoy Susan Jacoby‘s Op-Ed in the New York Times, “Best Is the New Worst” (30 May 2008). Yes, the title “Best Is The New Worst” is a snowclone, the sort of trite phrasal template with plug-and-play slots which is the refuge of lazy writing. In fact, “X is the new Y” is a classic of the genre. (Is this headline choice an example of such, or a clever meta-reference intended to mock declining standards? Discuss.) However, this bit was more troubling, and made me look askance:

Another peculiar new use of “elitist” (often coupled with “Luddite”) is its application to any caveats about the Internet as a source of knowledge. After listening to one of my lectures, a college student told me that it was elitist to express alarm that one in four Americans, according to the National Constitution Center, cannot name any First Amendment rights or that 62 percent cannot name the three branches of government. “You don’t need to have that in your head,” the student said, “because you can just look it up on the Web.”

Ahem. Laziness in a student may well be a cause for alarm, but uncritically accepting the spin which innumerate and agenda-driven reportage has placed upon a survey is, I dare to suggest, even worse. I’m going to go out on a limb here and say that these numbers, intimidating though they may be, are in fact completely worthless. They don’t measure what they’re claimed to measure. The reason goes back to the First Amendment itself:

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances.

This breaks down to six different rights. The first half of the religion bit — the Establishment Clause — covers a different territory than the second half, the Free Exercise Clause. It’s possible to have the right to follow your personal religion even in a country which has an established church (hint, hint: the United Kingdom has Muslims). Speech and the press get one “freedom” each, but the people have a “right” to assemble and another “right” to petition.

The National Constitution Center website features a 1997 survey which speaks of “four rights” in the First Amendment, and lists them as “speech,” “religion,” “press” and “assembly.” Now, I’d argue that this obscures the constitutional status of religion, reflecting or enforcing an oversimplified view of the matter, but more importantly, the right to petition vanished down a memory hole.

If it were universally acknowledged that, say, petitioning and assembly were two faces of the same freedom, then I wouldn’t mind so much. However, a 2006 survey by the McCormick Tribune Freedom Museum asked respondents to name their First Amendment rights, and their “correct” answers were the freedoms of religion, speech, press, petition and assembly. When two different love-the-Constitution groups can’t even be consistent on how many freedoms exist, can the results be considered reliable? What, exactly, are we citizens supposed to know, and will we lose credit for knowing the wrong “right” answer? Suppose that the NCC called you one September morning in 1997, and you said that the First Amendment covered speech, religion, press and peaceable assembly. You would be a sterling citizen in 1997, but nine years later, the Freedom Museum would stamp you defective.

Both surveys share another flaw. As was pointed out back in 2006,
several “freedoms” which are likely to be uppermost in an American’s mind are found in other, later amendments: bearing arms, avoiding self-incrimination, avoiding unreasonable search and seizure, not being enslaved (took a while for that one) and so forth. What happens if you remember a whole list of these rights, but can’t recall which amendment they go under?
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Mathification

Megan Garber writes the following in the Columbia Journalism Review‘s daily blog:

We currently find ourselves in, to put it mildly, a lull in the 2008 campaign’s primary season. The delegate tallies are in limbo. Parsing them seems to require a postgraduate degree in calculus.

I call mathification! The analogue of linguification, this term refers to statements which, as Isabel Lugo puts it, “clearly intend to get across a true point about the real world by making a false point about mathematics.”
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Shorter Sal Cordova, Redux

BPSDBAn interesting development has unfolded in the math-blogging world. Sal Cordova, famous for calling Charles Darwin a puppy-killer, has attempted to show that he, Cordova, is not a stupefied ignoramus on the subject of quantum mechanics. Naturally, such ignorance would not be a crime, except that Cordova is hell-bent on using quantum physics to prop up his “Advanced Creation Science.” See here, here, here, here and here if you’ve been suffering a lack of reading material. If, on the other hand, you’re a busy citizen of the high-speed modern world, let us summarize:

If I post a comment in which I fail to address the criticisms leveled at me on a long-dead blog discussion thread and, two days later, crow about it before a sycophantic audience while intentionally mangling my critic’s name, not only will I demonstrate my intellectual superiority over the filthy Darwinists, but also, Jesus will bring me 72 virgins in Heaven.

Oh, by the way, an integral sign is not the same thing as an upper-case S.

Your Funny for Today

This one comes from Ellen Wulfhorst on the Reuters wire:

Unlike traditional, mainstream media, blogs often adopt a specific point of view. Critics complain they can contain unchecked facts, are poorly edited and use unreliable sources.

I sense a great disturbance in the Schwartz, as if a million monitors were just sprayed with soda. (Well, no, I don’t have that many readers, but Dave Neiwert, Athenae and Coturnix have already picked up on it.) And here’s another puzzling statement from the same piece, describing the poll which found that “a majority of Americans do not read political blogs.”

The poll was conducted online from January 15 to January 22 among 2,302 adults. Harris said it does not calculate or provide a margin of error because it finds such figures can be misleading.

Is anyone else concerned by the sampling bias which this procedure could entail? If you ask people online what websites they read, you’re going to get (at best) a measure of what people who spend time online read, not what Americans in general are reading. Sure, that’ll probably increase the percentage of blog readers, skewing the poll towards “new media,” but it’s still bias. (They claim to have used “propensity score weighting” to “adjust” for this.)

On their website, Harris Interactive lays out their rationale for not reporting margins of error. Basically, they assert that people are too poorly educated to know what “margin of error” means: people don’t know that the phrase refers only to sampling error, not to other possible sources of obfuscation (which are harder to get a quantitative handle on). Therefore, people will assume that polls are more accurate than they really are; to avoid this problem, and to save the wear-and-tear on a newscaster’s mouth which the tediously long phrase “margin of sampling error” would produce, Harris will not admit fallibility at all.

Gee, how nice of them to make that decision for us, so that even the people who know statistics can’t get the figures. I just loooove suffering for the sins of innumerate America.

UPDATE: I also get a kick out of this:

Just one in ten (19%) Echo Boomers (those aged 18-31) regularly read a political blog

One in ten. Nineteen percent. Oops.

Yet Another Relativity Denier

BPSDBExercise: find the mistake in this attempt to challenge Einstein. Hint: if an observer in one Lorentz frame measures the position of a particle to be changing as [tex]x = ct[/tex], then that particle is traveling at the speed of light, and all observers in other Lorentz frames will agree.

Bonus point: explain the difference between the speed of light in a vacuum and the speed of light as measured when light is passing through matter.

(Thanks to the reader who noticed the “relativity challenge” Google ad in my sidebar. You know, it’s not quite cricket for me to plead that the Gentle Reader click on those links, but I can’t help it if other people appreciate the irony of pseudoscience making micropayments to science.)

Shorter Uncommon Descent

BPSDBTyler DiPietro finds a fresh example of steaming creationist nonsense at the weblog of Dembski and sycophants, Uncommon Descent. I summarize for the busy reader:

Because running an image of a face through a software filter makes it look less like a face, Darwin was a fuddy-duddy and materialism is on its last leg.

The cream on the cocoa is, however, this bit, from a commenter:

I’m not sure about the CSI [complex specified information] decreasing, however. Anyone able to get values of the CSI for each picture?

Gee, given that “complex specified information” has never been consistently defined, I wonder how one could compute it.

An Item of Prime Importance

I was busy with something or other, so I didn’t get to see the event Dennis Overbye describes in the New York Times, where the director and the star of the new film Jumper chatted with MIT professors Ed Farhi and Max Tegmark before a live audience in lecture hall 26-100. Not having been there, I don’t have very much to say, but I do feel the need to quote one paragraph of Overbye’s article and add just a tiny bit of emphasis:

The real lure, [Farhi] said, is not transportation, but secure communication. If anybody eavesdrops on the teleportation signal, the whole thing doesn’t work, Dr. Farhi said. Another use is in quantum computing, which would exploit the ability of quantum bits of information to have different values, both one and zero, at the same time to perform certain calculations, like factoring large prime numbers, much faster than ordinary computers.

Ahem.

In any other circumstance, I’d probably pontificate on how exponential parallelism is not the source of quantum computing’s calculation-fu, but I think we have a few other points to address first. . . .

Attentive readers will recall that Dr. Farhi was the guy who signed my paperwork when I was an undergraduate. For an amusing story from those days, see my post of last November, “Pay No Attention to the Man.”

(Tip o’ the fedora to Dave Bacon.)