Category Archives: Bad Math

Math in the Movies

I’ve been running around this weekend doing important things like appearing in a musical comedy — 22, in which the terrorist threat involves an Infinite Improbability Drive — so thanks for not breaking the Internet whilst I was away! I notice that Mollishka has raised the topic of science and math in the movies, which sounds like a nice way to ease everybody back into the work-week.

Several years ago, I was visiting a friend in a mental institution. (See? Your first story of the week is shaping up just great!) In fact, she was a resident of McLean Hospital, whose wards have housed such notables as John Nash, James Taylor and Sylvia Plath (and oddly enough, I’ve seen the first two of those notables, live). My friends and I had driven out to Belmont to visit our colleague, and while we were chatting in the dining area, another resident of that hall poked into the conversation. He was of average height, but wiry, and spoke as if drawing upon deep reservoirs of energy; he had been placed in McLean by his family, he said, and he let loose a shrill cascade of invective upon the orderlies who eventually took him away.

Before he was hauled back to his room, he got to talking about Darren Aronofsky movies. At the time, those were Pi (1998) and Requiem for a Dream (2000). I’m grateful to him for providing a calibration mark by which I can judge those movies, for in the interval before the orderlies carried him away, he told us that Aronofsky was going to be making more movies, and — children, cover your eyes —
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Rosenhouse on Amanda Shaw

Following up on his previous post, “Is Math a Gift From God?” — calculus students say, “No!” — Jason Rosenhouse has a new essay for your delectation, “Is God Like an Imaginary Number?” Again, the short answer is, “Nope.” The longer answer will take us into the history of mathematics, the role of mysticism in theology and the relationship between science and verbal description.

Rosenhouse sets himself the task of fisking an essay in the religious periodical First Things, by a “Junior Fellow” of that publication named Amanda Shaw. Shaw’s thesis is that the notion of God is akin to that of an imaginary number, and moreover that the same closed-minded orthodoxy which rejected the latter from mathematics for oh so many years is unjustly keeping the former out of science. I find this stance to be, in a word, ironical: if you’re looking for dogmatism and condemnations of the heterodox, your search will be much more rewarding if you look among the people who reject scientific discoveries because they are inconsistent with a Bronze Age folk tale than if you search through science itself!

Still, it’s a fun chance to talk about history and mathematics.

PART A: COMPLEX NUMBERS

As I described earlier, “imaginary” and “complex” numbers arise naturally when you think about the ordinary, humdrum “real numbers” — you know, fractions, decimals and all those guys — as lengths on a number line. In this picture, adding two numbers corresponds to sticking line segments end-to-end, multiplication means stretching or squishing (in general, scaling) line segments, and negation means flipping a segment over to lie on the opposite side of zero. Complex numbers appear when you ask the question, “What operation, when performed twice in succession upon a line segment, is equivalent to a negation?” Answer: rotating by a quarter-turn!

Historically, mathematicians started getting into complex numbers when they tried to find better and better ways to solve real-number equations. Girolamo Cardano (1501–1576), also known as Jerome Cardan, posed the following problem:

If some one says to you, divide 10 into two parts, one of which multiplied into the other shall produce […] 40, it is evident that this case or question is impossible. Nevertheless, we shall solve it in this fashion.

Writing this in more modern algebraic notation, this is like saying [tex] x + y = 10 [/tex] and [tex] xy = 40 [/tex], which we can combine into one equation by solving for [tex] y [/tex], thusly:

[tex] xy = x(10 – x) = 40.[/tex]

In turn, shuffling the symbols around gives

[tex] x^2 – 10x + 40 = 0,[/tex]

which plugging into ye old quadratic formula yields

[tex] x = \frac{10 \pm \sqrt{100 – 160}}{2}, [/tex]

or, boiling it down,

[tex] x = 5 \pm \sqrt{-15}. [/tex]

Totally loony! Taking the square root of a negative number? Forsooth, thy brains are bubbled! Oh, wait, didn’t we just realize that we could maybe handle the square root of a negative number by moving into a two-dimensional plane of numbers? Yes, we did: that’s the prize our talk of flips and rotations won us!
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Forthcoming from Paulos

PZ Myers points to an essay by John Allen Paulos, author of Innumeracy (1989, second edition 2001) among other books. The essay itself doesn’t venture into shockingly new territory — the cranky side of the Web has been spreading pseudomathematical myth-babble for a while now — but Paulos is a clear writer and a good debunker. Most interesting to me was the announcement in the blurb that Paulos has a new book in the works: Irreligion: A Mathematician Explains Why the Arguments for God Just Don’t Add Up, to be published in December. The ad copy says,

Are there any logical reasons to believe in God? Mathematician and bestselling author John Allen Paulos thinks not. In Irreligion he presents the case for his own worldview, organizing his book into twelve chapters that refute the twelve arguments most often put forward for believing in God’s existence. The latter arguments, Paulos relates in his characteristically lighthearted style, “range from what might be called golden oldies to those with a more contemporary beat. On the playlist are the firstcause argument, the argument from design, the ontological argument, arguments from faith and biblical codes, the argument from the anthropic principle, the moral universality argument, and others.” Interspersed among his twelve counterarguments are remarks on a variety of irreligious themes, ranging from the nature of miracles and creationist probability to cognitive illusions and prudential wagers. Special attention is paid to topics, arguments, and questions that spring from his incredulity “not only about religion but also about others’ credulity.” Despite the strong influence of his day job, Paulos says, there isn’t a single mathematical formula in the book.

That Paulos has chosen to write this book and push it towards the nation’s bookshelves is interesting. Back in the original Innumeracy, the book that made him famous, Paulos wrote,
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How Many Books?

God Plays Dice has a good post on that story about people not reading books anymore.

So what do we know about the distribution? One-quarter of people read no books; one-quarter read between one and four; one-eighth read between four and seven; three-eighths read more.

They claim a 3% margin of error, as well, which is standard for polls involving a thousand people (as this one was), but that margin of error only applies to the survey as a whole. The article includes a lot of claims of the form “Xs read more than Ys”, but the number of Xs or Ys that were polled is less than a thousand, so the margin of error is greater.

Furthermore,
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Brief Vacation-but-not-really

Posting will be light this week. With luck, I’ll get a rerun of some old material I wrote at another site up here Wednesday or Thursday. In the meantime, here’s a lengthy discussion at Russell Blackford’s place about emergent properties and consciousness.

When you’re sated on that, check out Mark Liberman’s “Thou shalt not report odds ratios.” See, this is why the denizens of the science-blogging community should include Language Log in their travels: problems like bad math reporting in science journalism affect us all.

QUICK UPDATE: Isabel has more on the bad math in journalism issue at God Plays Dice.

Overbye on Hunting the Higgs

Dennis Overbye has an article in today’s New York Times on the search for the Higgs boson, and naturally, I’ve got complaints about it. It’s a pretty good piece: Overbye can do solid work (he went a little overboard looking for journalistic “balance” in the Bogdanov Affair, but that was a while ago). Still, I wouldn’t be myself if I couldn’t gripe and grouse.

First, I’m definitely not alone in asking people to please stop saying “God particle.” Leon Lederman has a great deal to answer for after coining this term; I’ve never heard or seen physicists use it seriously, and it keeps inviting unwarranted metaphors. (Incidentally, there was once detected an “Oh-My-God Particle,” a cosmic-ray proton of astonishingly high energy; for recent developments in this ultra-high-energy regime, see here. Physicists joke about the term, but they don’t use it.)

Second, this part rubs me the wrong way:
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Inverse Proportionality

I think I first learned the word inverse from Calvin and Hobbes, and in particular the strip in which Calvin ruefully notes, “There’s an inverse relationship between how good something is for you, and how much fun it is” (1 May 1986). The examples leading up to this punchline — he must eat the “slimy asparagus,” he cannot watch the movie Killer Prom Queen — make it clear that “inverse relationship” means that more fun means less good-for-you.

Now, consider what the journalist Johann Hari reports Dinesh D’Souza said on an ocean cruise organized by the National Review.

The nautical counter-revolution has docked in the perfectly-yellow sands of Puerto Vallarta in Mexico, and the Reviewers are clambering overboard into the Latino world they want to wall off behind a thousand-mile fence. They carry notebooks from the scribblings they made during the seminar teaching them “How To Shop in Mexico”. Over breakfast, I forgot myself and said I was considering setting out to find a local street kid who would show me round the barrios — the real Mexico. They gaped. “Do you want to die?” one asked.

The Reviewers confine their Mexican jaunt to covered markets and walled-off private fortresses like the private Nikki Beach. Here, as ever, they want Mexico to be a dispenser of cheap consumer goods and lush sands — not a place populated by (uck) Mexicans. Dinesh D’Souza announced as we entered Mexican seas what he calls “D’Souza’s law of immigration”: “The quality of an immigrant is inversely proportional to the distance travelled to get to the United States.”

Taking D’Souza’s remark at face value, we can translate it into Orwellian sheep-speak as, “Mexicans good, Northern Europeans bad!” Indeed, seeing that D’Souza was born in Bombay, one must consider his “law of immigration” a statement of becoming modesty on his part.

By reasonable extrapolation, people who were born in the United States to families which have lived in the United States for generations are absolutely worthless.

(Tip o’ the G.R.O.S.S. chapeau to PZ Myers.)

Arithmetic and Stoichiometry

Two summers ago, four men tried to “top” the terrorist attacks on the London Tube but failed, for a reason any science educator can appreciate:

The explosives would have caused carnage on the transport network, but the plot mastermind, Ibrahim, miscalculated the ratios of ingredients when making the bombs, Britain’s Daily Mail newspaper has reported.

The court heard Ibrahim personally bought the bleach and chapati flour used to make the devices in his flat, which was also booby-trapped. Police said the highly volatile triacetone triperoxide, known as “mother of Satan” was used as a detonator.

But Ibrahim, who failed maths at school, got his sums wrong when mixing the recipe, making the bombs — which were packed with nails and screws — harmless.

This story doesn’t make explicit that the “bleach” was actually hair bleach, which the plotters tried to distill to get hydrogen peroxide.

The jury was told that, using two saucepans and a frying pan on the flat’s small cooker, the men set up a seven-day rota involving Ibrahim, Asiedu and Omar to oversee the process of boiling it down to the required concentration. Some was re-bottled with labels suggesting it had reached the required strength. Subsequent test firings by forensic scientists suggested that one reason why the bombs had not fully exploded could have been that the chemical had not been sufficiently reduced.

Triacetone triperoxide (TATP) was also used in the 7 July 2005 attack on the London Underground, and many people speculated that the terrorists responsible for the “War on Liquids” were also trying to make TATP in airplane lavatories.

(Via John Armstrong.)

Spam Statistics

Or, “Why oh why don’t people make raw data accessible?”

The Akismet people have made some statistics available on how many spam messages their WordPress plugin has trapped. They use a Flash applet to display their graph, which I hope means that the graph is being updated (instead of merely implying horrible software design). Here’s a screen shot from a moment ago:

This graph shows a few features of interest. First, there’s a big jump — of apparently several hundred thousand — legitimate messages in mid-May. I wonder if this actually represents a new spamming technique. Second, both “ham” and spam show periodicity. Running this time series through a Fourier transform might yield intriguing results.

Sadly, the Akismet folks aren’t providing actual numbers to go along with the pretty pictures, and extracting them from a graph like this doesn’t sound like my idea of a fun Wednesday afternoon.

I’d also be curious to see what the ratio of spams caught to Akismet plugins installed looks like as a function of time.

UPDATE (12 July 2007): The algorithm always finds raw data! The numbers necessary to draw the chart can be retrieved in XML format here, and the snapshots I’ve been playing with are here and here.

Lagrangian Mechanics is Intelligent Design?

Via Kevin Beck I just learned that Sal Cordova, famous (in some circles) for rank dishonesty and general lack of mathematical aptitude, has been claiming that Lagrangian mechanics was inspired by Intelligent Design. For those who are not au courant with physics, Lagrangian mechanics is an alternative take on the classical physics — think billiard balls, pendulums, planets orbiting the Sun — studied by Newton. The enterprise is named for Joseph-Louis Lagrange, who along with Euler and others laid the groundwork. It’s equivalent to Newton’s F = ma approach, but more convenient for some problems, and because it talks about the same physics in a different way, it provides a different and useful starting point for developing new theories. (For example, Barton Zwiebach’s First Course in String Theory generalizes the Lagrangian description of zero-dimensional objects, particles, to invent a theory of one-dimensional objects, strings. This is much easier to do in a Lagrangian rather than a Newtonian formalism.) Phenomena in relativity and quantum field theory are also often studied via a Lagrangian approach.

Many people are familiar with basic characteristics of light. We know, for instance, that light travels in straight lines; when light bounces off a mirror, the angle of incidence equals the angle of reflection; light can be spread out or focused together using lenses; and so forth. When we study optics, we can derive all these disparate facts from a very simple, central premise: when traveling from point A to point B, a light ray takes that path for which the travel time is a minimum. (A more precise statement is that the physical path taken by the light ray is such that a small perturbation to the path does not significantly change the travel time; this is connected to the calculus idea that the slope of a curve at a minimum or maximum is zero. For our purposes, we won’t have to worry about these details.) If there’s nothing in the way to change the light’s propagation speed, or if the material through which the light travels is uniform, then the path of minimum time is a straight line. Requiring that the light go from A down to a mirror and bounce back up to B means — I leave the geometry as an easy exercise to the interested reader — that the angles of incidence and reflection will be equal.

Lagrangian mechanics takes a similar approach, taking the idea of a “minimum principle” and applying it not to light, but to the motion of matter — balls, planets, frightened cats and so forth. Instead of calculating the travel time, as we did with light, we consider the energy of the moving objects; more precisely, our calculations involve the difference between kinetic and potential energies. The “Lagrangian” for classical problems — remember, we can generalize the ideas later — is the difference between the kinetic energy and the potential, and we find the path through space an object will take by adding up, or integrating, the Lagrangian along all possible paths. The physical path, the one the object really follows, is the one whose total Lagrangian, or “action,” is a minimum.

Now, what in blazes does any of this have to do with Intelligent Design?
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Don’t Make Baby Gauss Cry

Cosma Shalizi writes of “Power-Law Distributions in Empirical Data“:

Because this is, of course, what everyone ought to do with a computational paper, we’ve put our code online, so you can check our calculations, or use these methods on your own data, without having to implement them from scratch. I trust that I will no longer have to referee papers where people use GnuPlot to draw lines on log-log graphs, as though that meant something, and that in five to ten years even science journalists and editors of Wired will begin to get the message.

Mark Liberman is not optimistic (we’ve got a long way to go).

Among several important take-home points, I found the following particularly amusing:
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Power-law Distributions in Empirical Data

Throughout many fields of science, one finds quantities which behave (or are claimed to behave) according to a power-law distribution. That is, one quantity of interest, y, scales as another number x raised to some exponent:

[tex] y \propto x^{-\alpha}.[/tex]

Power-law distributions made it big in complex systems when it was discovered (or rather re-discovered) that a simple procedure for growing a network, called “preferential attachment,” yields networks in which the probability of finding a node with exactly k other nodes connected to it falls off as k to some exponent:

[tex]p(k) \propto k^{-\gamma}.[/tex]

The constant γ is typically found to be between 2 and 3. Now, from my parenthetical remarks, the Gentle Reader may have gathered that the story is not quite a simple one. There are, indeed, many complications and subtleties, one of which is an issue which might sound straightforward: how do we know a power-law distribution when we see one? Can we just plot our data on a log-log graph and see if it falls on a straight line? Well, as Eric and I are fond of saying, “You can hide a multitude of sins on a log-log graph.”

Via Dave Bacon comes word of a review article on this very subject. Clauset, Shalizi and Newman offer us “Power-law distributions in empirical data” (7 June 2007), whose abstract reads as follows:
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Oh No, Not Again

Dammit, Warren, we don’t need you spreading pseudoscience along with everybody else. The Heim theory “spacedrive” is a total crock. Remember, this is the nonsense which John Baez called “a run-of-the-mill crackpot theory” and of which Sean Carroll said,

Just so nobody gets too excited — this paper is complete nonsense, not worth spending a minute’s time on. If I find the energy I might post on it, but this is no better than the other hundred crackpot preprints I get in the mail every year.

For details, you can start here and here. It looks like I might have to dig that post out of my “drafts” pile after all. . . .

New Scientist has a lot to answer for.