Charles Seife’s ‘Proofiness,’ reviewed by John Allen Paulos

PROOFINESS

The Dark Arts of Mathematical Deception

By Charles Seife. Viking. 295 pp. $25.95

The title of Charles Seife’s new book, “Proofiness” is a takeoff on Stephen Colbert’s notion of truthiness, the property of statements that have the ring of truth to them but upon a little reflection are seen to be bogus. Likewise, proofiness refers to numbers and statistical arguments that seem convincing but are really somewhere between unwarranted and ludicrous. Seife begins by pointing out that numbers in the news do not inhabit some ideal Platonic realm but result from very fallible measurements that are often based on vague definitions or faulty assumptions.

He tells the story of the museum guard who claimed the dinosaur on exhibit was 65,000,038 years old. When pressed about the precision of the number, the guard says that a scientist told him the dinosaur was 65 million years old when he was hired 38 years before. Seife calls this error “disestimation,” the mathematical sin of underestimating the uncertainties associated with most numbers.

He gives cute names to other types of proofiness as well: Potemkin numbers, those without even a tenuous connection to reality, such as Joe McCarthy’s charge of the State Department harboring 205 communists; cherrypicking, examining only data favorable to one’s position; randumbness, the tendency to see patterns and order in random data; causuistry, ascribing causes to accidental associations; risk mismanagement, misjudging or even lying about risk, often so that the people assuming it are not the ones reaping its rewards; and so on.

These mathematical solecisms are illustrated with many topical examples, ranging from the non-perils of NutraSweet and the inequities of the death penalty to the numerical skullduggery behind the U.S. abandonment of the nuclear test ban treaty with the Soviet Union in 1982. That last case involved a nonsensical formula that led to a spurious allegation of Soviet cheating. The second half of the book is devoted to somewhat fuller discussions of issues such as the U.S. census and its problems; gerrymandering, which seems to be tolerated if it is politically, rather than racially, motivated; the mortgage crisis, including a particularly clear and brief discussion of it via an analogy involving fire insurance, whereby fire insurance policies are imagined to be bundled and sold the way mortgages were; and close elections, particularly the 2008 Franken-Coleman Senate race in Minnesota and the 2000 Bush-Gore contest.

With regard to the latter, Seife urges acceptance of the uncomfortable fact that some races are essentially ties, that the official and meaningless “exact totals” are sometimes overwhelmed by systemic errors in counting, the vagueness of the election laws and the unclear criteria, definitions and protocols governing them. As I wrote in 2000, trying to determine whether Gore or Bush received more votes in Florida was a bit like trying to measure bacteria with a yardstick. The official 537-vote difference between them recalls the absurdity of the 65,000,038-year-old dinosaur.

Throughout the book, Seife’s practiced journalist’s eye results in trenchant nonmathematical observations. He notes, for example, that in a very close election the candidate in the lead almost always takes the position that “rules are rules” and the candidate who is behind almost always argues to “count every vote.”

Seife condemns the 2000 Supreme Court decision forbidding the use of statistical sampling to ascertain the number of households missed by census workers. Likening the decision to poll taxes, literacy tests and voter ID laws, he notes that its effect is an apportionment of House seats that disenfranchises millions of largely minority voters. The decision seems not only unfair but also inconsistent. For example, if an answer on a census form looks wrong — say someone claims 1,000,000 children — census workers are allowed to replace this number with one reflecting what statistically similar households have reported. This imputation and others in common use also violate the prohibition of statistical techniques, but the court has accepted them.

Polls, of course, play an important role in a journalist’s craft(iness), and Seife writes that they are one of the primary sources of proofiness in modern life. Real, unpredictable events provide fodder for journalists and commentators, but polls and pseudoevents, which he defines as synthetic, planned and occurring at convenient times and locations, are almost as good for this purpose. Moreover, polls are subject not just to the statistical margin of error but, more crucially, to systemic shortcomings and tendentious special pleading as well.

Disposing of arithmetical mistakes, misjudgments and misunderstandings is, like trash removal, a never-ending job. Seife performs it cogently and entertainingly without resorting to arcane mathematics. The effort is important because the cumulative effect of proofiness is, as he sagely concludes, “toxic to a democracy.”

John Allen Paulos is a professor of mathematics at Temple University and the author of “Innumeracy,” “A Mathematician Reads the Newspaper” and, most recently, “Irreligion.”

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Full article: http://www.washingtonpost.com/wp-dyn/content/article/2010/10/08/AR2010100802980.html

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Step 1: Post Elusive Proof. Step 2: Watch Fireworks.

The potential of Internet-based collaboration was vividly demonstrated this month when complexity theorists used blogs and wikis to pounce on a claimed proof for one of the most profound and difficult problems facing mathematicians and computer scientists.

Vinay Deolalikar, a mathematician and electrical engineer at Hewlett-Packard, posted a proposed proof of what is known as the “P versus NP” problem on a Web site, and quietly notified a number of the key researchers in a field of study that focuses on problems that are solvable only with the application of immense amounts of computing power.

The researcher asserted that he had demonstrated that P (the set of problems that can be easily solved) does not equal NP (those problems that are difficult to solve, but easy to verify once a solution is found). As with earlier grand math challenges — for example, Fermat’s last theorem — there is a lot at stake, not the least of which is a $1 million prize.

In 2000 the Clay Mathematics Institute picked seven of the greatest unsolved problems in the field, named them “Millennium Problems” and offered $1 million for the solution of each of them. P versus NP is one of those problems. (In March, the first prize was awarded to a reclusive Russian mathematician, Grigory Perelman, for the solution to the century-old Poincaré conjecture. A few months later he refused the prize.)

P versus NP has enormous practical and economic importance, because modern cryptography is based on the assumption, which is workable so far, that P does not equal NP. In other words, there are problems that are impossible for computers to solve, but for which the solutions are easily recognizable. If these problems were shown to be solvable, that could undermine modern cryptography, which could paralyze electronic commerce and digital privacy because transactions would no longer be secure.

In a note sent to a small group of researchers on Aug. 6, Dr. Deolalikar wrote: “The proof required the piecing together of principles from multiple areas within mathematics. The major effort in constructing this proof was uncovering a chain of conceptual links between various fields and viewing them through a common lens.”

An outsider to the insular field, Dr. Deolalikar set off shock waves because his work appeared to be a concerted and substantial challenge to a problem that has attracted intense scrutiny since it was first posed in 1971 by Stephen Cook, a mathematician and computer scientist who teaches at the University of Toronto.

“The reason there was such excitement is there have been many alleged proofs,” said Moshe Vardi, a professor of computer science at Rice University and the editor in chief of The Communications of the Association for Computing Machinery. “This looks like a serious paper. In particular what he has done is bring forward a new idea that is worth exploring.”

In this case, however, the significant breakthrough may not be in the science, but rather in the way science is practiced. By the middle of last week, although Dr. Deolalikar had not backed away from his claim, a consensus had emerged among complexity theorists that the proposed proof had several significant shortcomings.

“At this point the consensus is that there are large holes in the alleged proof — in fact, large enough that people do not consider the alleged proof to be a proof,” Dr. Vardi said. “I think Deolalikar got his 15 minutes of fame, but at this point the excitement has subsided and the skepticism is turning into negative conviction.”

What was highly significant, however, was the pace of discussion and analysis, carried out in real time on blogs and a wiki that had been quickly set up for the purpose of collectively analyzing the paper. This kind of collaboration has emerged only in recent years in the math and computer science communities. In the past, intense discussions like the one that surrounded the proof of the Poincaré conjecture were carried about via private e-mail and distribution lists as well as in the pages of traditional paper-based science journals.

Several of the researchers said that until now such proofs had been hashed out in colloquiums that required participants to be physically present at an appointed time. Now, with the emergence of Web-connected software programs it is possible for such collaborative undertakings to harness the brainpower of the world’s best thinkers on a continuous basis.

In his recently published book “Cognitive Surplus: Creativity and Generosity in a Connected Age” (Penguin Press), Clay Shirky, a professor of interactive telecommunications at New York University, argues that the emergence of these new collaborative tools is paving the way for a second scientific revolution in the same way the printing press created a demarcation between the age of alchemy and the age of chemistry.

“The difference between the alchemists and the chemists was that the printing press was used to coordinate peer review,” he said. “The printing press didn’t cause the scientific revolution, but it wouldn’t have been possible without it.”

Now, he says, the new tools are likely to set off a similar transformation.

“It’s not just, ‘Hey, everybody, look at this,’ ” he said, “but rather a new set of norms is emerging about what it means to do mathematics, assuming coordinated participation.”

The computer science community has long been an innovator in the design of science-collaboration tools. Indeed, the ARPAnet, the forerunner of the Internet, was initially created in 1969 to make one of the first computerized collaboration tools, Douglas Engelbart’s oNLine System, or NLS, available from remote locations. During the 1980s physicists at the physics research center CERN near Geneva created the World Wide Web to facilitate the sharing of scientific research.

In 2009, a Cambridge mathematician, Timothy Gowers, created the Polymath Project, a blog and wiki-oriented collaboration tool that used the comments section of a blog to pursue mathematics collaboratively. Related efforts like the Web site Mathoverflow help attack unsolved mathematical problems by using new Internet tools to help stimulate collaboration.

In the case of the P versus NP paper, most of the action has taken place in several blogs maintained by researchers in the field, like a computer scientist, Richard Lipton, at Georgia Tech and a theoretical physicist, Dave Bacon, at the University of Washington, as well as in a wiki by a quantum theoretician, Michael Nielsen.

Passions have run high. A computer scientist at the Massachusetts Institute of Technology, Scott Aaronson, literally bet his house last week — $200,000 — that the Deolalikar paper would be proved incorrect: “If Vinay Deolalikar is awarded the $1,000,000 Clay Millennium Prize for his proof of P-NP, then I, Scott Aaronson, will personally supplement his prize by the amount of $200,000.”

Despite his skepticism, he acknowledged that this was, to date, one of the most impressive attempts to settle the question.

“So far this is not your typical P versus NP crank solution, which I hear about once a week,” he said.

John Markoff, New York Times

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Full article and photo: http://www.nytimes.com/2010/08/17/science/17proof.html

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The Hilbert Hotel

In late February I received an e-mail message from a reader named Kim Forbes.  Her six-year-old son Ben had asked her a math question she couldn’t answer, and she was hoping I could help:

Today is the 100th day of school. He was very excited and told me everything he knows about the number 100, including that 100 was an even number. He then told me that 101 was an odd number and 1 million was an even number, etc.  He then paused and asked: “Is infinity even or odd?”

I explained that infinity is neither even nor odd.  It’s not a number in the usual sense, and it doesn’t obey the rules of arithmetic.  All sorts of contradictions would follow if it did.  For instance, “if infinity were odd, 2 times infinity would be even.  But both are infinity!  So the whole idea of odd and even does not make sense for infinity.”

Kim replied:

Thank you.  Ben was satisfied with that answer and kind of likes the idea that infinity is big enough to be both odd and even.

Although something got garbled in translation (infinity is neither odd nor even, not both), Ben’s rendering hints at a larger truth.  Infinity can be mind-boggling.

Some of its strangest aspects first came to light in the late 1800s, with Georg Cantor’s groundbreaking work on “set theory.”  Cantor was particularly interested in infinite sets of numbers and points, like the set {1, 2, 3, 4,…} of “natural numbers” and the set of points on a line.  He defined a rigorous way to compare different infinite sets and discovered, shockingly, that some infinities are bigger than others.

At the time, Cantor’s theory provoked not just resistance, but outrage.  Henri Poincaré, one of the leading mathematicians of the day, called it a “disease.”  But another giant of the era, David Hilbert, saw it as a lasting contribution and later proclaimed, “No one shall expel us from the Paradise that Cantor has created.”

My goal here is to give you a glimpse of this paradise.  But rather than working directly with sets of numbers or points, let me follow an approach introduced by Hilbert himself.  He vividly conveyed the strangeness and wonder of Cantor’s theory by telling a parable about a grand hotel, now known as the Hilbert Hotel.

It’s always booked solid, yet there’s always a vacancy.

For the Hilbert Hotel doesn’t merely have hundreds of rooms — it has an infinite number of them.  Whenever a new guest arrives, the manager shifts the occupant of room 1 to room 2, room 2 to room 3, and so on.  That frees up room 1 for the newcomer, and accommodates everyone else as well (though inconveniencing them by the move).

Now suppose infinitely many new guests arrive, sweaty and short-tempered.  No problem.  The unflappable manager moves the occupant of room 1 to room 2, room 2 to room 4, room 3 to room 6, and so on.  This doubling trick opens up all the odd-numbered rooms — infinitely many of them — for the new guests.

Later that night, an endless convoy of buses rumbles up to reception.  There are infinitely many buses, and worse still, each one is loaded with an infinity of crabby people demanding that the hotel live up to its motto, “There’s always room at the Hilbert Hotel.”

The manager has faced this challenge before and takes it in stride.

First he does the doubling trick.  That reassigns the current guests to the even-numbered rooms and clears out all the odd-numbered ones — a good start, because he now has an infinite number of rooms available.

But is that enough?  Are there really enough odd-numbered rooms to accommodate the teeming horde of new guests?  It seems unlikely, since there are something like “infinity squared” people clamoring for these rooms.  (Why infinity squared?  Because there were an infinite number of people on each of an infinite number of buses, and that amounts to infinity times infinity, whatever that means.)

This is where the logic of infinity gets very weird.

To understand how the manager is going to solve his latest problem, it helps to visualize all the people he has to serve.

Of course, we can’t show literally all of them here, since the diagram would need to be infinite in both directions.  But a finite version of the picture is adequate.  The point is that any specific bus passenger (your Aunt Inez, say, on vacation from Louisville) is sure to appear on the diagram somewhere, as long as we include enough rows and columns.  In that sense, everybody on every bus is accounted for.  You name the passenger, and he or she is certain to be depicted at some finite number of steps east and south of the diagram’s corner.

The manager’s challenge is to find a way to work through this picture systematically.  He needs to devise a scheme for assigning rooms so that everybody gets one eventually, after only a finite number of other people have been served.

Sadly, the previous manager didn’t understand this and mayhem ensued. When a similar convoy showed up on his watch, he became so flustered trying to process all the people on bus 1 that he never got around to any other bus, leaving all those neglected passengers screaming and furious.  Visualized on the diagram below, this myopic strategy would correspond to a path marching eastward along row 1, never to return.

The new manager, however, has everything under control.  Instead of tending to just one bus, he zigs and zags through the diagram, fanning out from the corner as shown below.

He starts with passenger 1 on bus 1 and gives her the first empty room.  The second and third empty rooms go to passenger 2 on bus 1, followed by passenger 1 on bus 2, both of whom are depicted on the second diagonal from the corner of the diagram.  After serving them, the manager proceeds to the third diagonal and hands out a set of room keys to passenger 1 on bus 3, passenger 2 on bus 2, and passenger 3 on bus 1.

I hope the manager’s procedure — progressing from one diagonal to another — is clear from the picture above, and that you’re convinced that any particular person will be reached in a finite number of steps.

So, as advertised, there’s always room at the Hilbert Hotel.

The argument I’ve just presented is a famous one in the theory of infinite sets.  Cantor used it to prove that there are exactly as many positive fractions (ratios p/q of positive whole numbers p and q) as there are natural numbers (1, 2, 3, 4, …).  That’s a much stronger statement than saying both sets are infinite.  It says they are infinite to precisely the same extent, in the sense that a “one-to-one correspondence” can be established between them.

You could think of this correspondence as a buddy system in which each natural number is paired with some positive fraction, and vice versa.  The existence of such a buddy system seems utterly at odds with common sense — it’s the sort of sophistry that made Poincaré recoil.  For it implies we could make an exhaustive list of all positive fractions, even though there’s no smallest one!

And yet there is such a list.  We’ve already found it.  The fraction p/q corresponds to passenger p on bus q, and the argument above shows that each of these fractions can be paired off with a certain natural number 1, 2, 3,…, given by the passenger’s room number at the Hilbert Hotel.

The coup de grace is Cantor’s proof that some infinite sets are bigger than this.  Specifically, the set of real numbers between 0 and 1 is “uncountable” — it can’t be put in one-to-one correspondence with the natural numbers.  For the hospitality industry, this means that if all these real numbers show up at the reception desk and bang on the bell, there won’t be enough rooms for all of them, even at the Hilbert Hotel.

The proof is by contradiction.  Suppose each real number could be given its own room.  Then the roster of occupants, identified by their decimal expansions and listed by room number, would look something like this:

Room 1:          .6708112345…

Room 2:          .1918676053…

Room 3:          .4372854675…

Room 4:          .2845635480…

Remember, this is supposed to be a complete list.  Every real number between 0 and 1 is supposed to appear somewhere, at some finite place on the roster.

Cantor showed that a lot of numbers are missing from any such list; that’s the contradiction.  For instance, to construct one that appears nowhere on the list shown above, go down the diagonal and build a new number from the boldface digits:

Room 1:          .6708112345…

Room 2:          .1918676053…

Room 3:          .4372854675…

Room 4:          .2845635480…

The decimal so generated is .6975…

But we’re not done yet.  The next step is to take this decimal and change all its digits, replacing each of them with any other digit between 1 and 8.  For example, we could change the 6 to a 3, the 9 to a 2, the 7 to a 5, and so on.

This new decimal .325… is the killer.  It’s certainly not in Room 1, since it has a different first digit from the number there.  It’s also not in Room 2, since its second digit disagrees.  In general, it differs from the nth number in the nth decimal place.  So it doesn’t appear anywhere on the list!

The conclusion is that the Hilbert Hotel can’t accommodate all the real numbers.  There are simply too many of them, an infinity beyond infinity.

And with that humbling thought, we come to the end of this series, which began 14 weeks ago with a scene in another imaginary hotel.   A Sesame Street character named Humphrey, working the lunch shift at The Furry Arms, took an order from a roomful of hungry penguins — “fish, fish, fish, fish, fish, fish” — and soon learned about the power of numbers.

It’s been a long journey from fish to infinity.  I hope you’ve enjoyed it as much as I have. The next trip leaves in 2012, when these columns and many more will appear as a book.  Thanks for joining me, especially to those readers who shared their comments, questions and insights on this blog.

NOTES

1. For more about Cantor, including the mathematical, philosophical and theological controversies surrounding his work, see:
   J.W. Dauben, “Georg Cantor” (Princeton University Press, 1990).
2. The classic biography of Hilbert is a moving and non-technical account of his life, his work and his times:
   C. Reid, “Hilbert” (Springer, 1996).
   His contributions to mathematics are too numerous to list here, but perhaps his greatest is his collection of 23 problems — all of which were unsolved when he proposed them — that he thought would shape the course of mathematics in the 20th century.    For the ongoing story and significance of these Hilbert Problems and the people who solved some of them, see:
   B.H. Yandell, “The Honors Class” (A K Peters, 2002).
   Several of the problems still remain open.
3. Hilbert’s parable of the infinite hotel is mentioned in George Gamow’s evergreen masterpiece:
   G. Gamow, “One Two Three … Infinity” (Dover, 1988), p. 17.
   Gamow also does a good job of explaining countable and uncountable sets and related ideas about infinity.
4. The comedic and dramatic possibilities of the Hilbert Hotel have often been explored by writers of mathematical fiction.  For example, see:
   S. Lem, “The extraordinary hotel or the thousand and first journey of Ion the Quiet,” reprinted in “Imaginary Numbers: An Anthology of Marvelous Mathematical Stories, Diversions, Poems and Musings,” edited by W. Frucht (Wiley, 1999);
   I. Stewart, “Professor Stewart’s Cabinet of Mathematical Curiosities” (Basic Books, 2009).
   A children’s book on the same theme is:
   I. Ekeland, “The Cat in Numberland” (Cricket Books, 2006).
5. If you haven’t read it yet, I recommend last year’s surprise best-seller “Logicomix,” a brilliantly creative graphic novel about set theory, logic, infinity, madness and the quest for mathematical truth:
   A. Doxiadis and C.H. Papadimitriou, “Logicomix” (Bloomsbury, 2009).
   It stars Bertrand Russell, but Cantor, Hilbert, Poincaré and many others make memorable appearances.
6. For a more mathematical but still very readable discussion of infinity, as well as many other ideas discussed throughout this series, see:
   J. Stillwell, “Yearning for the Impossible” (A K Peters, 2006).
7. Readers wishing to go deeper into infinity might enjoy Terry Tao’s blog post about “self-defeating objects.”  In a very accessible way, he presents and elucidates a lot of fundamental arguments about infinity that arise in set theory, philosophy, physics, computer science, game theory and logic.
8. A tiny finesse occurred in the argument for the uncountability of the real numbers when I required that the diagonal digits were to be replaced by digits between 1 and 8.  This wasn’t essential.  But I wanted to avoid using 0 and 9 to sidestep any fussiness caused by the fact that some real numbers have two decimal representations.  For example, .200000…. equals .199999…  Thus, if we hadn’t excluded the use of 0’s and 9’s as replacement digits, it’s conceivable the diagonal argument could have inadvertently produced a number already on the list (and that would have ruined the proof).  By forbidding the use of 0 and 9 we didn’t have to worry about this annoyance.

Thanks to Margaret Nelson, for her artistry and sense of humor; Paul Ginsparg and Jon Kleinberg, for their comments and suggestions on this piece; and my wife, Carole Schiffman, for her lightheartedness and for giving new meaning to the concept of “infinite support.”

Steven Strogatz, New York Times

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Full article and photos: http://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert-hotel/

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Group Think

My wife and I have different sleeping styles — and our mattress shows it.  She hoards the pillows, thrashes around all night long, and barely dents the mattress, while I lie on my back, mummy-like, molding a cavernous depression into my side of the bed.

Bed manufacturers recommend flipping your mattress periodically, probably with people like me in mind.  But what’s the best system?  How exactly are you supposed to flip it to get the most even wear out of it?

Brian Hayes explores this problem in the title essay of his recent book, “Group Theory in the Bedroom.”  Double entendres aside, the “group” in question here is a collection of mathematical actions — all the possible ways you could flip, rotate or overturn the mattress so that it still fits neatly on the bed frame.

By looking into mattress math in some detail, I hope to give you a feeling for group theory more generally.  It’s one of the most versatile parts of mathematics. It underlies everything from the choreography of contra dancing and the fundamental laws of particle physics, to the mosaics of the Alhambra and their chaotic counterparts like this image.

As these examples suggest, group theory bridges the arts and sciences.   It addresses something the two cultures share — an abiding fascination with symmetry.  Yet because it encompasses such a wide range of phenomena, group theory is necessarily abstract.  It distills symmetry to its essence.

Normally we think of symmetry as a property of a shape.  But group theorists focus more on what you can do to a shape — specifically, all the ways you can change it while keeping something else about it the same.  More precisely, they look for all the transformations that leave a shape unchanged, given certain constraints.  These transformations are called the “symmetries” of the shape.  Taken together they form a “group,” a collection of transformations whose relationships define the shape’s most basic architecture.

In the case of a mattress, the transformations alter its orientation in space (that’s what changes) while maintaining its rigidity (that’s the constraint).  And after the smoke clears, the mattress has to fit snugly on the rectangular bed frame (that’s what stays the same). With these rules in place, let’s see what transformations qualify for membership in this exclusive little group.  It turns out there are only four of them.

The first is the “do-nothing” transformation, a lazy but popular choice that leaves the mattress untouched.  It certainly satisfies all the rules, but it’s not much help in prolonging the life of your mattress. Still, it’s very important to include in the group.   It plays the same role for group theory that zero does for addition of numbers, or that 1 does for multiplication.  Mathematicians call it the “identity element,” so I’ll denote it by the symbol I.

Next come the three genuine ways to flip a mattress.  To distinguish among them, it helps to label the corners of the mattress by numbering them like so:

The first kind of flip is depicted near the beginning of this post.  The handsome gentleman in striped pajamas is trying to turn the mattress from side to side by rotating it 180 degrees around its long axis, in a move I’ll call H, for “horizontal flip.”

A more reckless way of overturning the mattress is a “vertical flip” V.  This maneuver swaps its head and foot.  You stand the mattress upright, the long way, so that it almost reaches the ceiling, and then topple it end over end.  The net effect, besides the enormous thud, is to rotate the mattress 180 degrees about the axis shown below.

The final possibility is to spin the mattress half a turn while keeping it flat on the bed.

Unlike the H and V flips, this “rotation” R keeps the top surface on top.  That difference shows up when we look at a top view of the mattress — now imagined to be translucent — and inspect the numbers at the corners after each of the possible transformations.

The horizontal flip turns the numerals into their mirror images.  It also permutes them so that 1 and 2 trade places, as do 3 and 4.

The vertical flip permutes the numbers in a different way and stands them on their heads, besides mirroring them.

The rotation, however, doesn’t generate any mirror images.  It merely turns the numbers upside down, this time exchanging 1 for 4 and 2 for 3.

These details are not the main point.  What matters is how the transformations relate to one another.  Their patterns of interaction encode the symmetry of the mattress.

To reveal those patterns with a minimum of effort, it helps to draw the following diagram.  (Images like this abound in a terrific new book called “Visual Group Theory,” by Nathan Carter.  It’s one of the best introductions to group theory — or to any branch of higher math — I’ve ever read.)

The four possible “states” of the mattress are shown at the corners of the diagram.  The upper left state is the starting point.  The colored arrows indicate the moves that take the mattress from one state to another.

For example, the green arrow pointing from the upper left to the lower right depicts the action of the rotation R.  The same green line also has an arrowhead on the other end, because if you do R twice, it’s tantamount to doing nothing.

That shouldn’t come as a surprise.  It just means that turning the mattress head to foot and then doing that again returns the mattress to its original state.  We can summarize this property with the equation RR = I, where RR means do R twice, and I is the do-nothing identity element.  For that matter, the horizontal and vertical flip transformations also undo themselves: HH = I and VV = I.

The diagram embodies a wealth of other information.  For instance, it shows that the death-defying vertical flip V is equivalent to HR, a horizontal flip followed by a rotation — a much safer path to the same result.   To check this, begin at the starting state in the upper left.  Head due east along H to the next state, and from there go diagonally southwest along R.  Because you arrive at the same state as if you’d simply followed V to begin with, the diagram demonstrates that HR = V.

Notice, too, that the order of those actions is irrelevant: HR = RH, since both roads lead to V.  This indifference to order is true for any other pair of actions.  You should think of this as a generalization of the commutative law for addition of ordinary numbers, x and y, according to which x + y = y + x.  But beware: the mattress group is special.  Many other groups violate the commutative “law.”  Those fortunate enough to obey it are particularly clean and simple.

Now for the payoff.  The diagram shows how to get the most even wear out of a mattress.  Any strategy that samples all four states periodically will work. For example, alternating R and H is convenient — and since it bypasses V, it’s not too strenuous.  To help you remember it, some manufacturers suggest the mnemonic “spin in the spring, flip in the fall.”

The mattress group also pops up in some unexpected places, from the symmetry of water molecules to the logic of a pair of electrical switches.  That’s one of the charms of group theory.  It exposes the hidden unity of things that would otherwise seem unrelated … like this anecdote about how the physicist Richard Feynman got a draft deferment.

The army psychiatrist questioning him asked Feynman to put out his hands so he could examine them.  Feynman stuck them out, one palm up, the other down.  “No, the other way,” said the psychiatrist. So Feynman reversed both hands, leaving one palm down and the other up.

Feynman wasn’t merely playing mind games; he was indulging in a little group-theoretic humor.  If we consider all the possible ways he could have held out his hands, along with the various transitions among them, the arrows form the same pattern as the mattress group!

But if all this makes mattresses seem way too complicated, maybe the real lesson here is one you already knew — if something’s bothering you, just sleep on it.

NOTES

1. Two recent books inspired this piece:
   N. Carter, “Visual Group Theory” (Mathematical Association of America, 2009).
   B. Hayes, “Group Theory in the Bedroom, And Other Mathematical Diversions” (Hill and Wang, 2008).
   Carter introduces the basics of group theory gently and pictorially.  He also touches on its connections to Rubik’s cube, contra dancing and square dancing, crystals, chemistry, art and architecture.
   An earlier version of Hayes’s mattress-flipping article appeared in American Scientist in the issue of September/October 2005.
2. The mattress group is technically known as the “Klein four-group.”  It’s one of the simplest in a gigantic zoo of possibilities.  Mathematicians have been analyzing groups and classifying their structure for about 200 years.  Among the earliest pioneers were two brilliant men who died tragically young: Évariste Galois, killed in a duel at age 20, and Niels Henrik Abel, dead from tuberculosis at age 26.  The questions that concerned them were purely mathematical, having to do with the finding the roots of polynomials and proving the unsolvability of the quintic equation in terms of simple formulas involving radicals.  For more about their stories, see:
   M. Livio, “The Equation That Couldn’t Be Solved” (Simon and Schuster, 2005).
   A. Alexander, “Duel at Dawn” (Harvard University Press, 2010).
   And for an engaging account of the quest to classify all “finite simple groups,” see:
   M. du Sautoy, “Symmetry” (Harper, 2008).
3. A word about some potentially confusing notation used above: in equations like HR = V,  the H was  written on the left to indicate that it’s the transformation being performed first.  Carter uses this notation for functional composition in his book, but the reader should be aware that many mathematicians use the opposite convention, placing the H on the right.
4. Readers interested in seeing a definition of what a “group” is should consult any of the authoritative online references or standard textbooks on the subject.  The treatment I’ve given here emphasizes symmetry groups rather than groups in the most general sense.
5. Michael Field and Martin Golubitsky have studied the interplay between group theory and nonlinear dynamics.  In the course of their investigations, they’ve generated stunning computer graphics of symmetric chaos.  For the art, science and mathematics of this topic, see:
   M. Field and M. Golubitsky, “Symmetry in Chaos,” 2nd edition (Society for Industrial and Applied Mathematics, 2009).
6. For the anecdote about Feynman and the psychiatrist, see:
   R. P. Feynman, “ ‘Surely You’re Joking, Mr. Feynman!’ ” (Norton, 1985), p. 158.
   J. Gleick, “Genius” (Random House, 1993), p. 223.

Thanks to Mike Field and Marty Golubitsky for sharing their images of symmetric chaos; Margaret Nelson for preparing the illustrations; and Paul Ginsparg, Jon Kleinberg, Tim Novikoff, Diana Riesman and Carole Schiffman for their comments and suggestions.

Steven Strogatz, New York Times

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Full article and photos: http://opinionator.blogs.nytimes.com/2010/05/02/group-think/

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Chances Are

Have you ever had that anxiety dream where you suddenly realize you have to take the final exam in some course you’ve never attended?  For professors, it works the other way around — you dream you’re giving a lecture for a class you know nothing about.

That’s what it’s like for me whenever I teach probability theory.  It was never part of my own education, so having to lecture about it now is scary and fun, in an amusement park, thrill-house sort of way.

Perhaps the most pulse-quickening topic of all is “conditional probability” — the probability that some event A happens, given (or “conditional” upon) the occurrence of some other event B.  It’s a slippery concept, easily conflated with the probability of B given A.  They’re not the same, but you have to concentrate to see why.  For example, consider the following word problem.

Before going on vacation for a week, you ask your spacey friend to water your ailing plant.  Without water, the plant has a 90 percent chance of dying.  Even with proper watering, it has a 20 percent chance of dying.  And the probability that your friend will forget to water it is 30 percent.  (a) What’s the chance that your plant will survive the week?  (b) If it’s dead when you return, what’s the chance that your friend forgot to water it?  (c) If your friend forgot to water it, what’s the chance it’ll be dead when you return?

Although they sound alike, (b) and (c) are not the same.  In fact, the problem tells us that the answer to (c) is 90 percent.  But how do you combine all the probabilities to get the answer to (b)?  Or (a)?

Naturally, the first few semesters I taught this topic, I stuck to the book, inching along, playing it safe.  But gradually I began to notice something.  A few of my students would avoid using “Bayes’s theorem,” the labyrinthine formula I was teaching them.  Instead they would solve the problems by a much easier method.

What these resourceful students kept discovering, year after year, was a better way to think about conditional probability.  Their way comports with human intuition instead of confounding it.  The trick is to think in terms of “natural frequencies” — simple counts of events — rather than the more abstract notions of percentages, odds, or probabilities.  As soon as you make this mental shift, the fog lifts.

This is the central lesson of “Calculated Risks,” a fascinating book by Gerd Gigerenzer, a cognitive psychologist at the Max Planck Institute for Human Development in Berlin.  In a series of studies about medical and legal issues ranging from AIDS counseling to the interpretation of DNA fingerprinting, Gigerenzer explores how people miscalculate risk and uncertainty. But rather than scold or bemoan human frailty, he tells us how to do better — how to avoid “clouded thinking” by recasting conditional probability problems in terms of natural frequencies, much as my students did.

In one study, Gigerenzer and his colleagues asked doctors in Germany and the United States to estimate the probability that a woman with a positive mammogram actually has breast cancer, even though she’s in a low-risk group: 40 to 50 years old, with no symptoms or family history of breast cancer.  To make the question specific, the doctors were told to assume the following statistics — couched in terms of percentages and probabilities — about the prevalence of breast cancer among women in this cohort, and also about the mammogram’s sensitivity and rate of false positives:

The probability that one of these women has breast cancer is 0.8 percent.  If a woman has breast cancer, the probability is 90 percent that she will have a positive mammogram.  If a woman does not have breast cancer, the probability is 7 percent that she will still have a positive mammogram.  Imagine a woman who has a positive mammogram.  What is the probability that she actually has breast cancer?

Gigerenzer describes the reaction of the first doctor he tested, a department chief at a university teaching hospital with more than 30 years of professional experience:

“[He] was visibly nervous while trying to figure out what he would tell the woman.  After mulling the numbers over, he finally estimated the woman’s probability of having breast cancer, given that she has a positive mammogram, to be 90 percent.  Nervously, he added, ‘Oh, what nonsense.  I can’t do this.  You should test my daughter; she is studying medicine.’  He knew that his estimate was wrong, but he did not know how to reason better.  Despite the fact that he had spent 10 minutes wringing his mind for an answer, he could not figure out how to draw a sound inference from the probabilities.”

When Gigerenzer asked 24 other German doctors the same question, their estimates whipsawed from 1 percent to 90 percent.   Eight of them thought the chances were 10 percent or less, 8 more said 90 percent, and the remaining 8 guessed somewhere between 50 and 80 percent.  Imagine how upsetting it would be as a patient to hear such divergent opinions.

As for the American doctors, 95 out of 100 estimated the woman’s probability of having breast cancer to be somewhere around 75 percent.

The right answer is 9 percent.

How can it be so low?  Gigerenzer’s point is that the analysis becomes almost transparent if we translate the original information from percentages and probabilities into natural frequencies:

Eight out of every 1,000 women have breast cancer.  Of these 8 women with breast cancer, 7 will have a positive mammogram.  Of the remaining 992 women who don’t have breast cancer, some 70 will still have a positive mammogram.  Imagine a sample of women who have positive mammograms in screening.  How many of these women actually have breast cancer?

Since a total of 7 + 70 = 77 women have positive mammograms, and only 7 of them truly have breast cancer, the probability of having breast cancer given a positive mammogram is 7 out of 77, which is 1 in 11, or about 9 percent.

Notice two simplifications in the calculation above.  First, we rounded off decimals to whole numbers.  That happened in a few places, like when we said, “Of these 8 women with breast cancer, 7 will have a positive mammogram.”  Really we should have said 90 percent of 8 women, or 7.2 women, will have a positive mammogram.  So we sacrificed a little precision for a lot of clarity.

Second, we assumed that everything happens exactly as frequently as its probability suggests. For instance, since the probability of breast cancer is 0.8 percent, exactly 8 women out of 1,000 in our hypothetical sample were assumed to have it.  In reality, this wouldn’t necessarily be true.  Things don’t have to follow their probabilities; a coin flipped 1,000 times doesn’t always come up heads 500 times.  But pretending that it does gives the right answer in problems like this.

Admittedly the logic is a little shaky — that’s why the textbooks look down their noses at this approach, compared to the more rigorous but hard-to-use Bayes’s theorem — but the gains in clarity are justification enough.  When Gigerenzer tested another set of 24 doctors, this time using natural frequencies, nearly all of them got the correct answer, or close to it.

Although reformulating the data in terms of natural frequencies is a huge help, conditional probability problems can still be perplexing for other reasons.  It’s easy to ask the wrong question, or to calculate a probability that’s correct but misleading.

Both the prosecution and the defense were guilty of this in the O.J. Simpson trial of 1994-95.  Each of them asked the jury to consider the wrong conditional probability.

The prosecution spent the first 10 days of the trial introducing evidence that O.J. had a history of violence toward his ex-wife, Nicole.  He had allegedly battered her, thrown her against walls and groped her in public, telling onlookers, “This belongs to me.”  But what did any of this have to do with a murder trial?  The prosecution’s argument was that a pattern of spousal abuse reflected a motive to kill.  As one of the prosecutors put it, “A slap is a prelude to homicide.”

Alan Dershowitz countered for the defense, arguing that if even the allegations of domestic violence were true, they were irrelevant and should therefore be inadmissible.  He later wrote, “We knew we could prove, if we had to, that an infinitesimal percentage — certainly fewer than 1 of 2,500 — of men who slap or beat their domestic partners go on to murder them.”

In effect, both sides were asking the jury to consider the probability that a man murdered his ex-wife, given that he previously battered her.  But as the statistician I. J. Good pointed out, that’s not the right number to look at.

The real question is: What’s the probability that a man murdered his ex-wife, given that he previously battered her and she was murdered by someone?  That conditional probability turns out to be very far from 1 in 2,500.

To see why, imagine a sample of 100,000 battered women.  Granting Dershowitz’s number of 1 in 2,500, we expect about 40 of these women to be murdered by their abusers in a given year (since 100,000 divided by 2,500 equals 40). We can estimate that an additional 5 of these battered women, on average, will be killed by someone else, because the murder rate for all women in the United States at the time of the trial was about 1 in 20,000 per year. So out of the 40 + 5 = 45 murder victims altogether, 40 of them were killed by their batterer.  In other words, the batterer was the murderer about 90 percent of the time.

Don’t confuse this number with the probability that O.J. did it.  That probability would depend on a lot of other evidence, pro and con, such as the defense’s claim that the police framed him, or the prosecution’s claim that the killer and O.J. shared the same style of shoes, gloves and DNA.

The probability that any of this changed your mind about the verdict?  Zero.

NOTES:

1. For a good textbook treatment of conditional probability and Bayes’s theorem, see:
   S.M. Ross, “Introduction to Probability and Statistics for Engineers and Scientists,” 4th edition (Academic Press, 2009).
2. The answer to part (a) of the “ailing plant” problem is 59 percent.  The answer to part (b) is 27/41, or approximately 65.85 percent.  To derive these results, imagine 100 ailing plants and figure out (on average) how many of them get watered or not, and then how many of those go on to die or not, based on the information given. This question appears, though with slightly different numbers and wording, as problem 29 on p. 84 of Ross’s text.
3. The study of how doctors interpret mammogram results is described in:
   G. Gigerenzer, “Calculated Risks” (Simon and Schuster, 2002), chapter 4.  For more on the O.J. Simpson case and a discussion of wife battering in a larger context, see chapter 8.
4. For many entertaining anecdotes and insights about conditional probability and its real-world applications, as well as how it’s misperceived, see:
   J.A. Paulos, “Innumeracy” (Vintage, 1990);
   L. Mlodinow, “The Drunkard’s Walk” (Vintage, 2009).
5. The quotes pertaining to the O.J. Simpson trial, and Alan Dershowitz’s estimate of the rate at which battered women are murdered by their partners, appeared in:
   A. Dershowitz, “Reasonable Doubts” (Touchstone, 1997), pp. 101-104.
6. Probability theory was first correctly applied to the Simpson trial by the late I.J. Good, in:
   I.J. Good, “When batterer turns murderer,” Nature, Vol. 375 (1995), p. 541.
   I.J. Good, “When batterer becomes murderer,” Nature, Vol. 381 (1996), p. 481.
   Good phrased his analysis in terms of odds ratios and Bayes’s theorem, rather than the more intuitive “natural frequency” approach presented here and in Gigerenzer’s book.
   Good had an interesting career.  In addition to his many contributions to probability theory and Bayesian statistics, he helped break the Nazi Enigma code during World War II, and introduced the futuristic concept now known as the “technological singularity.”
7. Here is how Dershowitz seems to have calculated that fewer than 1 in 2,500 batterers go on to murder their partners, per year.  On page 101 of his book “Reasonable Doubts,” he cites an estimate that in 1992, somewhere between 2.5 and 4 million women in the United States were battered by their husbands, boyfriends, and ex-boyfriends.  In that same year, according to the FBI Uniform Crime Reports, 913 women were murdered by their husbands, and 519 were killed by their boyfriends or ex-boyfriends.  Dividing the total of 1,432 homicides by 2.5 million beatings yields 1 murder per 1,746 beatings, whereas using the higher estimate of 4 million beatings per year yields 1 murder per 2,793 beatings.  Dershowitz apparently chose 2,500 as a round number in between these extremes.
   What’s unclear is what proportion of the murdered women had been previously beaten by these men. It seems that Dershowitz was assuming that nearly all the victims were beaten, presumably to make the point that even when the rate is overestimated in this way, it’s still “infinitesimal.”
8. Good’s estimated murder rate of 1 per 20,000 women per year includes battered women, so it was not strictly correct to assume (as he did, and as we did above) that 5 women out of 100,000 would be killed by someone other than the batterer.  But correcting for this doesn’t alter the conclusion significantly, as the following calculation shows.
   According to the FBI Uniform Crime Reports, 4,936 women were murdered in 1992.  Of these murder victims, 1,432 (about 29 percent) were killed by their husbands or boyfriends.  The remaining 3,504 were killed by somebody else.  Therefore, considering that the total population of women in the United States at that time was about 125 million, the rate at which women were murdered by someone other than their partners was 3,504 divided by 125,000,000, or 1 murder per 35,673 women, per year.
   Let’s assume that this rate of murder by non-partners was the same for all women, battered or not.  Then in our hypothetical sample of 100,000 battered women, we’d expect about 100,000 divided by 35,673, or 2.8 women to be killed by someone other than their partner.  Although 2.8 is smaller than the 5 that Good and we assumed above, it doesn’t matter much because both are so small compared to 40, the estimated number of cases in which the batterer is the murderer.  With this modification, our new estimate of the probability that the batterer is the murderer would be 40 divided by (40 + 2.8), or about 93 percent.
   A related quibble is that the FBI statistics and population data given above imply that the murder rate for women in 1992 was closer to 1 in 25,000, not 1 in 20,000 as Good assumed.  If he had used that rate in his calculation, an estimated 4 women per 100,000, not 5, would have been murdered by someone other than the partner. But this still wouldn’t affect the basic message — now the batterer would be the murderer 40 times out of 40 + 4 = 44, or 91 percent of the time.

Thanks to Paul Ginsparg, Michael Lewis, Eri Noguchi and Carole Schiffman for their comments and suggestions.

Steven Strogatz, New York Times

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Full article and photo: http://opinionator.blogs.nytimes.com/2010/04/25/chances-are/

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It Slices, It Dices

Mathematical signs and symbols are often cryptic, but the best of them offer visual clues to their own meaning. The symbols for zero, one and infinity aptly resemble an empty hole, a single mark and an endless loop: 0, 1, ∞.  And the equals sign, =, is formed by two parallel lines because, in the words of its originator, Welsh mathematician Robert Recorde in 1557, “no two things can be more equal.”

In calculus the most recognizable icon is the integral sign:

Its graceful lines are evocative of a musical clef or a violin’s f-hole — a fitting coincidence, given that some of the most enchanting harmonies in mathematics are expressed by integrals.  But the real reason that Leibniz chose this symbol is much less poetic.  It’s simply a long-necked S, for “summation.”

As for what’s being summed, that depends on context.  In astronomy, the gravitational pull of the sun on the earth is described by an integral. It represents the collective effect of all the minuscule forces generated by each solar atom at their varying distances from the earth.  In oncology, the growing mass of a solid tumor can be modeled by an integral.  So can the cumulative amount of drug administered during the course of a chemotherapy regimen.

Historically, integrals arose first in geometry, in connection with the problem of finding the areas of curved shapes.  As we saw two weeks ago, the area of a circle can be viewed as the sum of many thin pie slices.  In the limit of infinitely many slices, each of which is infinitesimally thin, those slices could then be cunningly rearranged into a rectangle whose area was much easier to find.  That was a typical use of integrals.  They’re all about taking something complicated and slicing and dicing it to make it easier to add up.

In a 3-D generalization of this method, Archimedes (and before him, Eudoxus, around 400 B.C.) calculated the volumes of spheres, cones, barrels, prisms and various other solid shapes by re-imagining them as stacks of many wafers or discs, like a salami sliced thin.   By computing the volumes of the varying slices, and then ingeniously integrating them — adding them back together — they were able to deduce the volume of the original whole.

Today we still ask budding mathematicians and scientists to sharpen their skills at integration by applying them to these classic geometry problems.  They’re some of the hardest exercises we assign, and a lot of students hate them, but there’s no surer way to hone the facility with integrals needed for advanced work in every quantitative discipline from physics to finance.

One such mind-bender concerns the volume of the 3-D region common to two identical cylinders intersecting in a perpendicular fashion, like stovepipes in a kitchen.  It takes an unusual gift of imagination to visualize this shape clearly.

So there’s no shame in admitting defeat and looking for a way to make this shape more palpable.  To do so, you can resort to a trick my high school calculus teacher used — take a tin can and cut the top off with metal shears to form a cylindrical coring tool.  Then core a large Idaho potato or a piece of Styrofoam from two mutually perpendicular directions.  Inspect the resulting shape at your leisure.

Lacking both potato and Styrofoam, we have to settle for trying to convey on a flat screen what this curious solid looks like:

Remarkably, it has square cross-sections, even though it was created from round cylinders.  It’s a stack of infinitely many layers, each a wafer-thin square, tapering from a large square in the middle to progressively smaller ones, and finally to single points at the top and bottom.

Computer animations now make it possible to reveal the structure of the shape much more easily and vividly.

Still, picturing the shape is merely the first step.  What remains is to determine its volume.

Archimedes managed to find it, but only by virtue of his astounding ingenuity.  He used a mechanical method based on levers and centers of gravity, in effect weighing the shape in his mind by balancing it against others he already understood.  The downside of his approach, besides the prohibitive brilliance it required, was that it applied only to a limited range of shapes.

These conceptual roadblocks stumped the world’s finest mathematicians for the next 19 centuries … until Gregory, Barrow, Newton and Leibniz established what’s now known as the Fundamental Theorem of Calculus in the mid-1600s. The Fundamental Theorem is a powerful link between integrals and the subject of last week’s column, derivatives.  It greatly expands the universe of integrals that can be solved, and it reduces their calculation to grunt work. Nowadays computers can be programmed to use it — and so can students.  With its help, even the stovepipe problem that was once a world-class challenge now becomes an exercise within common reach. (For the details of Archimedes’s approach as well as the modern one, consult the references in the notes.)

It’s not practical to state the Fundamental Theorem here (though see the notes for an intuitive analogy).  Instead I’ll try to convey why it represented such an enormous advance.  It allowed mathematicians to forecast a changing world with much greater precision than had ever been possible.

The simplest kind of change can be handled with algebra.  When something changes steadily, at a constant rate, algebra works beautifully.  This is the domain of  “distance equals rate times time.” For example, a car moving at an unchanging speed of 60 miles per hour will surely travel 60 miles in the first hour, and 120 miles by the end of the second hour.

But what about change that proceeds at a varying rate?  Such changing change is all around us — in the accelerating descent of a penny dropped from a tall building, in the ebb and flow of the tides, in the elliptical orbits of the planets, in the circadian rhythms within us.  And only calculus can cope with the cumulative effects of changes as non-uniform as these.

For nearly two millennia after Archimedes, just one method existed for predicting the net effect of changing change: add up the varying slices, bit by bit.  Most of the time it couldn’t be done.  The infinite sums were too hard.

The Fundamental Theorem enabled a lot of these problems to be solved — not all of them, but many more than before. It often gave a shortcut for solving integrals, at least for the elementary functions (sums and products of powers, exponentials, logarithms and trig functions) that describe so many of the phenomena in the natural world.

From this perspective, the lasting legacy of integral calculus is a Veg-O-Matic view of the universe.  Newton and his successors taught us that nature unfolds in slices.  Virtually all the classical laws of physics discovered in the past 300 years turned out to have this character, whether they describe the motions of particles or the flow of heat, electricity, air or water.  Together with the governing laws, the conditions in each slice of time or space determine what will happen in adjacent slices.

The implications were profound.  For the first time in history, rational prediction became possible… not just one slice at a time, but with the help of the Fundamental Theorem, by leaps and bounds.

So we’re long overdue to update our slogan for integrals — from “It slices, it dices” to “Recalculating.  A better route is available.”

NOTES

1. For more about the ways that integral calculus has been used to help cancer researchers, see:
   D. Mackenzie, “Mathematical modeling of cancer,” SIAM News, Vol. 37, January/February 2004.
   H.P. Greenspan, “Models for the growth of a solid tumor by diffusion,” Studies in Applied Mathematics, December 1972, p. 317.
2. The region common to two identical circular cylinders whose axes intersect at right angles is known variously as a Steinmetz solid or a bicylinder.  Its volume can be calculated straightforwardly but opaquely by modern techniques.  An ancient and much simpler solution was known to both Archimedes and Tsu Ch’ung-Chih.  It uses nothing more than the method of slicing and a comparison between the areas of a square and a circle.  For a marvelously clear exposition, see Martin Gardner’s column:
   M. Gardner, “Mathematical games: Some puzzles based on checkerboards,” Scientific American, Vol. 207 (Nov. 1962), p. 164. And for Archimedes and Tsu Ch’ung-Chih, see:
   Archimedes, “The Method,” English translation by T. L. Heath (1912), reprinted by (Dover 1953).
   T. Kiang, “An old Chinese way of finding the volume of a sphere,” Mathematical Gazette, Vol. 56 (May 1972), pp. 88-91.
3. Moreton Moore points out that the bicylinder also has applications in architecture: “The Romans and Normans, in using the barrel vault to span their buildings, were familiar with the geometry of intersecting cylinders where two such vaults crossed one another to form a cross vault.” For this, as well as applications to crystallography, see:
   M. Moore, “Symmetrical intersections of right circular cylinders,” Mathematical Gazette, Vol. 58 (Oct. 1974), pp. 181-185.
4. For Archimedes’s application of his mechanical method to the problem of finding the volume of the bicylinder, see Proposition 15, p. 48 of  T.L. Heath, “The Method of Archimedes, Recently Discovered by Heiberg” (Cosimo Classics, 2007).
5. It’s interesting that Archimedes viewed his mechanical method as a means for discovering theorems rather than proving them.  As he put it, “… certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the said method did not furnish an actual demonstration. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.”
   That last line offers a timeless lesson about problem solving — when you’re trying to prove something, it helps to know it’s true.
6. For a popular account of Archimedes’s work, see:
   R. Netz and W. Noel, “The Archimedes Codex: How a Medieval Prayer Book Is Revealing the True Genius of Antiquity’s Greatest Scientist” (Da Capo Press, 2009).
7. Interactive demonstrations of the bicylinder and other problems in integral calculus are available online.  You’ll need to download the free Mathematica Player, which will then allow you to explore hundreds of other interactive demonstrations in all parts of mathematics.
8. Mamikon Mnatsakanian at Caltech has produced a series of animations that illustrate the Archimedean spirit and the power of slicing.   My favorite is this visualization of a beautiful relationship among the volumes of a sphere and a certain double-cone and cylinder whose height and radius match those of the sphere.  He also shows the same thing more physically by draining an imaginary volume of liquid from the cylinder and pouring it into the other two shapes.
9. Similarly elegant mechanical arguments in the service of math are given in:
   M. Levi, “The Mathematical Mechanic: Using Physical Reasoning to Solve Problems” (Princeton University Press, 2009).
10. Michael Starbird has created and filmed a fine series of lectures on the basics of calculus.
11. Here’s an analogy that I hope will shed some light on what the Fundamental Theorem is, and why it’s so helpful.  (My colleague Charlie Peskin at NYU suggested it.)  Imagine a staircase.  The total change in height from the top to the bottom is the sum of the rises of all the steps in between.  That’s true even if some of them rise more than others, and no matter how many steps there are.
    The Fundamental Theorem of Calculus says something similar for functions — if you integrate the derivative of a function from one point to another, you get the net change in the function between the two points. In this analogy, the function is like the elevation of each step compared to ground level.  The rises of individual steps are like the derivative.  Integrating the derivative is like summing the rises.  And the two points are the top and the bottom.
    Why is this so helpful?  Suppose you’re given an enormous list of numbers to sum, as occurs whenever you’re calculating an integral by slices.  If you can somehow manage to find the corresponding staircase — in other words, if you can find an elevation function for which those numbers are the rises — then computing the integral would be a snap.  It’s just the top minus the bottom.
    That’s the great speed-up made possible by the Fundamental Theorem.  And it’s why we torture all beginning calculus students with months of learning how to find elevation functions, technically called “antiderivatives.”

Thanks to Charlie Peskin, for the staircase analogy; Margaret Nelson, for preparing the line drawing; and Paul Ginsparg, Tim Novikoff, Andy Ruina and Carole Schiffman, for their comments and suggestions.

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Full article and photos: http://opinionator.blogs.nytimes.com/2010/04/18/it-slices-it-dices/

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Change We Can Believe In

Long before I knew what calculus was, I sensed there was something special about it.  My dad had spoken about it in reverential tones. He hadn’t been able to go to college, being a child of the Depression, but somewhere along the line, maybe during his time in the South Pacific repairing B-24 bomber engines, he’d gotten a feel for what calculus could do.  Imagine a mechanically controlled bank of anti-aircraft guns automatically firing at an incoming fighter plane.  Calculus, he supposed, could be used to tell the guns where to aim.

Every year about a million American students take calculus.  But far fewer really understand what the subject is about or could tell you why they were learning it.  It’s not their fault.  There are so many techniques to master and so many new ideas to absorb that the overall framework is easy to miss.

Calculus is the mathematics of change.  It describes everything from the spread of epidemics to the zigs and zags of a well-thrown curveball.  The subject is gargantuan — and so are its textbooks.  Many exceed 1,000 pages and work nicely as doorstops.

But within that bulk you’ll find two ideas shining through.  All the rest, as Rabbi Hillel said of the Golden Rule, is just commentary.  Those two ideas are the “derivative” and the “integral.”  Each dominates its own half of the subject, named in their honor as differential and integral calculus.

Roughly speaking, the derivative tells you how fast something is changing; the integral tells you how much it’s accumulating.  They were born in separate times and places: integrals, in Greece around 250 B.C.; derivatives, in England and Germany in the mid-1600s.  Yet in a twist straight out of a Dickens novel, they’ve turned out to be blood relatives — though it took almost two millennia to see the family resemblance.

Next week’s column will explore that astonishing connection, as well as the meaning of integrals.  But first, to lay the groundwork, let’s look at derivatives.

Derivatives are all around us, even if we don’t recognize them as such.  For example, the slope of a ramp is a derivative.  Like all derivatives, it measures a rate of change — in this case, how far you’re going up or down for every step you take.  A steep ramp has a large derivative.  A wheelchair-accessible ramp, with its gentle gradient, has a small derivative.

Every field has its own version of a derivative.  Whether it goes by “marginal return” or “growth rate” or “velocity” or “slope,” a derivative by any other name still smells as sweet.  Unfortunately, many students seem to come away from calculus with a much narrower interpretation, regarding the derivative as synonymous with the slope of a curve.

Their confusion is understandable.  It’s caused by our reliance on graphs to express quantitative relationships.  By plotting y versus x to visualize how one variable affects another, all scientists translate their problems into the common language of mathematics.  The rate of change that really concerns them — a viral growth rate, a jet’s velocity, or whatever — then gets converted into something much more abstract but easier to picture: a slope on a graph.

Like slopes, derivatives can be positive, negative or zero, indicating whether something is rising, falling or leveling off.  Watch Michael Jordan in action making his top-10 dunks.

Just after lift-off, his vertical velocity (the rate at which his elevation changes in time, and thus, another derivative) is positive, because he’s going up.  His elevation is increasing.  On the way down, this derivative is negative.  And at the highest point of his jump, where he seems to hang in the air, his elevation is momentarily unchanging and his derivative is zero.   In that sense he truly is hanging.

There’s a more general principle at work here — things always change slowest at the top or the bottom.  It’s especially noticeable here in Ithaca.  During the darkest depths of winter, the days are not just unmercifully short; they barely improve from one to the next.  Whereas now that spring is popping, the days are lengthening rapidly.  All of this makes sense.  Change is most sluggish at the extremes precisely because the derivative is zero there.  Things stand still, momentarily.

This zero-derivative property of peaks and troughs underlies some of the most practical applications of calculus.  It allows us to use derivatives to figure out where a function reaches its maximum or minimum, an issue that arises whenever we’re looking for the best or cheapest or fastest way to do something.

My high school calculus teacher, Mr. Joffray, had a knack for making such “max-min” questions come alive.  One day he came bounding into class and began telling us about his hike through a snow-covered field.   The wind had apparently blown a lot of snow across part of the field, blanketing it heavily and forcing him to walk much more slowly there, while the rest of the field was clear, allowing him to stride through it easily.  In a situation like that, he wondered what path a hiker should take to get from point A to point B as quickly as possible.

One thought would be to trudge straight across the deep snow, to cut down on the slowest part of the hike.  The downside, though, is the rest of the trip will take longer than it would otherwise.

Another strategy is to head straight from A to B.  That’s certainly the shortest distance, but it does cost extra time in the most arduous part of the trip.

With differential calculus you can find the best path.  It’s a certain specific compromise between the two paths considered above.

The analysis involves four main steps.  (For those who’d like to see the details, references are given in the notes.)

First, notice that that the total time of travel — which is what we’re trying to minimize — depends on just one number, the distance x where the hiker emerges from the snow.

Second, given a choice of x and the known locations of the starting point A and the destination B, we can calculate how much time the hiker spends walking through the fast and slow parts of the field.  For each leg of the trip, this calculation requires the Pythagorean theorem and the old algebra mantra, “distance equals rate times time.”  Adding the times for both legs together then yields a formula for the total travel time, T, as a function of x.   (See the Notes for details.)

Third, we graph T versus x.  The bottom of the curve is the point we’re seeking — it corresponds to the least time of travel and hence the fastest trip.

Fourth, to find this lowest point, we invoke the zero-derivative principle mentioned above.  We calculate the derivative of T, set it equal to zero, and solve for x.

These four steps require a command of geometry, algebra and various derivative formulas from calculus — skills equivalent to fluency in a foreign language and, therefore, stumbling blocks for many students.

But the final answer is worth the struggle.  It reveals that the fastest path obeys a relationship known as Snell’s law.   What’s spooky is that nature obeys it, too.

Snell’s law describes how light rays bend when they pass from air into water, as they do when shining into a swimming pool.   Light moves more slowly in water, much like the hiker in the snow, and it bends accordingly to minimize its travel time.  Similarly, light also bends when it travels from air into glass or plastic as it refracts through your eyeglass lenses.

The eerie point is that light behaves as if it were considering all possible paths and automatically taking the best one.   Nature — cue the theme from “The Twilight Zone” — somehow knows calculus.

NOTES

1. In an online article for the Mathematical Association of America, David Bressoud presents data on the number of American students taking calculus each year.
2. For a collection of Mr. Joffray’s calculus problems, both classic and original, see: S. Strogatz, “The Calculus of Friendship: What a Teacher and a Student Learned about Life While Corresponding About Math” (Princeton University Press, 2009).
3. Several videos and websites present the details of Snell’s law and its derivation from Fermat’s principle (which states that light takes the path of least time).   Others provide historical accounts.
4. Fermat’s principle was an early forerunner to the more general principle of least action.  For an entertaining and deeply enlightening discussion of this principle, including its basis in quantum mechanics, see:  R. P. Feynman, R. B. Leighton and M. Sands, “The principle of least action,” The Feynman Lectures on Physics, Volume 2, Chapter 19 (Addison-Wesley, 1964).R. Feynman, “QED: The Strange Theory of Light and Matter” (Princeton University Press, 1988). In a nutshell, Feynman’s astonishing proposition is that nature actually does try all paths.  But nearly all of them cancel out with their neighboring paths, through a quantum analog of destructive interference — except for those very close to the classical path where the action is minimized (or more precisely, made stationary).  There the quantum interference becomes constructive, rendering those paths exceedingly more likely to be observed.   This, in Feynman’s account, is why nature obeys minimum principles.  The key is that we live in the macroscopic world of everyday experience, where the actions are enormous compared to Planck’s constant.  In that classical limit, quantum destructive interference becomes extremely strong and obliterates nearly everything that could otherwise happen.

Thanks to Paul Ginsparg and Carole Schiffman for their comments and suggestions, and Margaret Nelson for preparing the illustrations.

Steven Strogatz, New York Times

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Full article and photos: http://opinionator.blogs.nytimes.com/2010/04/11/change-we-can-believe-in/

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Take It to the Limit

In middle school my friends and I enjoyed chewing on the classic conundrums.   What happens when an irresistible force meets an immovable object?  Easy — they both explode.  Philosophy’s trivial when you’re 13.

But one puzzle bothered us: if you keep moving halfway to the wall, will you ever get there?  Something about this one was deeply frustrating, the thought of getting closer and closer and yet never quite making it.  (There’s probably a metaphor for teenage angst in there somewhere.)  Another concern was the thinly veiled presence of infinity.  To reach the wall you’d need to take an infinite number of steps, and by the end they’d become infinitesimally small.  Whoa.

Questions like this have always caused headaches.  Around 500 B.C., Zeno of Elea posed a set of paradoxes about infinity that puzzled generations of philosophers, and that may have been partly to blame for its banishment from mathematics for centuries to come.  In Euclidean geometry, for example, the only constructions allowed were those that involved a finite number of steps.  The infinite was considered too ineffable, too unfathomable, and too hard to make logically rigorous.

But Archimedes, the greatest mathematician of antiquity, realized the power of the infinite.  He harnessed it to solve problems that were otherwise intractable, and in the process came close to inventing calculus — nearly 2,000 years before Newton and Leibniz.

In the coming weeks we’ll delve into the great ideas at the heart of calculus.  But for now I’d like to begin with the first beautiful hints of them, visible in ancient calculations about circles and pi.

Let’s recall what we mean by pi.  It’s a ratio of two distances.  One of them is the diameter, the distance across the circle through its center.  The other is the circumference, the distance around the circle.   Pi is defined as their ratio, the circumference divided by the diameter.

If you’re a careful thinker, you might be worried about something already.  How do we know that pi is the same number for all circles?  Could it be different for big circles and little circles?  The answer is no, but the proof isn’t trivial.  Here’s an intuitive argument.

Imagine using a photocopier to reduce an image of a circle by, say, 50 percent.  Then all distances in the picture — including the circumference and the diameter — would shrink in proportion by 50 percent.  So when you divide the new circumference by the new diameter, that 50 percent change would cancel out, leaving the ratio between them unaltered.  That ratio is pi.

Of course, this doesn’t tell us how big pi is.  Simple experiments with strings and dishes are good enough to yield a value near 3, or if you’re more meticulous, 3 and 1/7^th.  But suppose we want to find pi exactly or at least approximate it to any desired accuracy.  What then?  This was the problem that confounded the ancients.

Before turning to Archimedes’s brilliant solution, we should mention one other place where pi appears in connection with circles.  The area of a circle (the amount of space inside it) is given by the formula

Here A is the area, π is the Greek letter pi, and r is the radius of the circle, defined as half the diameter.

All of us memorized this formula in high school, but where does it come from?  It’s not usually proven in geometry class.  If you went on to take calculus, you probably saw a proof of it there, but is it really necessary to use calculus to obtain something so basic?

Yes, it is.

What makes the problem difficult is that circles are round.  If they were made of straight lines, there’d be no issue.  Finding the areas of triangles, squares and pentagons is easy.  But curved shapes like circles are hard.

The key to thinking mathematically about curved shapes is to pretend they’re made up of lots of little straight pieces. That’s not really true, but it works … as long as you take it to the limit and imagine infinitely many pieces, each infinitesimally small.  That’s the crucial idea behind all of calculus.

Here’s one way to use it to find the area of a circle.  Begin by chopping the area into four equal quarters, and rearrange them like so.

The strange scalloped shape on the bottom has the same area as the circle, though that might seem pretty uninformative since we don’t know its area either.  But at least we know two important facts about it.  First, the two arcs along its bottom have a combined length of πr, exactly half the circumference of the original circle (because the other half of the circumference is accounted for by the two arcs on top).  Second, the straight sides of the slices have a length of r, since each of them was originally a radius of the circle.

Next, repeat the process, but this time with eight slices, stacked alternately as before.

The scalloped shape looks a bit less bizarre now.  The arcs on the top and the bottom are still there, but they’re not as pronounced.  Another improvement is the left and right sides of the scalloped shape don’t tilt as much as they used to.  Despite these changes, the two facts above continue to hold: the arcs on the bottom still have a net length of πr, and each side still has a length of r.  And of course the scalloped shape still has the same area as before — the area of the circle we’re seeking — since it’s just a rearrangement of the circle’s eight slices.

As we take more and more slices, something marvelous happens: the scalloped shape approaches a rectangle.  The arcs become flatter and the sides become almost vertical.

In the limit of infinitely many slices, the shape is a rectangle.  Just as before, the two facts still hold, which means this rectangle has a bottom of width πr and a side of height r.

But now the problem is easy.  The area of a rectangle equals its width times its height, so multiplying πr times r yields an area of πr² for the rectangle.  And since the rearranged shape always has the same area as the circle, that’s the answer for the circle too!

What’s so charming about this calculation is the way infinity comes to the rescue.  At every finite stage, the scalloped shape looks weird and unpromising.  But when you take it to the limit — when you finally “get to the wall” — it becomes simple and beautiful, and everything becomes clear.  That’s how calculus works at its best.

Archimedes used a similar strategy to approximate pi.  He replaced a circle by a polygon with many straight sides, and then kept doubling the number of sides to get closer to perfect roundness.  But rather than settling for an approximation of uncertain accuracy, he methodically bounded pi by sandwiching the circle between “inscribed” and “circumscribed” polygons, as shown below for 6-, 12- and 24-sided figures.

Then he used the Pythagorean theorem to work out the perimeters of these inner and outer polygons, starting with the hexagon and bootstrapping his way up to 12, 24, 48 and ultimately 96 sides.  The results for the 96-gons enabled him to prove that

In decimal notation (which Archimedes didn’t have), this means pi is between 3.1408 and 3.1429.

This approach is known as the “method of exhaustion” because of the way it traps the unknown number pi between two known numbers that squeeze it from either side.  The bounds tighten with each doubling, thus exhausting the wiggle room for pi.

In the limit of infinitely many sides, both the upper and lower bounds would converge to pi.  Unfortunately, this limit isn’t as simple as the earlier one, where the scalloped shape morphed into a rectangle.  So pi remains as elusive as ever.  We can discover more and more of its digits — the current record is over 2.7 trillion decimal places — but we’ll never know it completely.

Aside from laying the groundwork for calculus, Archimedes taught us the power of approximation and iteration.  He bootstrapped a good estimate into a better one, using more and more straight pieces to approximate a curved object with increasing accuracy.

More than two millennia later, this strategy matured into the modern field of “numerical analysis.”  When engineers use computers to design cars to be optimally streamlined, or when biophysicists simulate how a new chemotherapy drug latches onto a cancer cell, they are using numerical analysis.  The mathematicians and computer scientists who pioneered this field have created highly efficient, repetitive algorithms, running billions of times per second, that enable computers to solve problems in every aspect of modern life, from biotech to Wall Street to the Internet.  In each case, the strategy is to find a series of approximations that converge to the correct answer as a limit.

And there’s no limit to where that’ll take us.

NOTES:

1. The history and intellectual legacy of Zeno’s paradoxes are discussed in: J. Mazur, Zeno’s Paradox (Blume, 2008).
2. For a delightfully opinionated and witty history of pi, see: P. Beckmann, A History of Pi (St. Martin’s Press, 1976).
3. Bill Willis at Worsley School OnLine has given a very clear explanation of how to find the area of the circle, using the same argument as above but fleshed out in more detail. The school website contains many other excellent math and science resources for students, teachers and parents.
4. The PBS television series “Nova” ran a wonderful episode about Archimedes, infinity and limits called “Infinite Secrets.” It originally aired on Sept. 30, 2003. The program website includes many online resources, including the program transcript and interactive demonstrations.
5. For readers wishing to see the mathematical details of Archimedes’s method of exhaustion, Neal Carothers has used trigonometry (equivalent to the Pythagorean gymnastics that Archimedes relied on) to derive the perimeters of the inscribed and circumscribed polygons between which the circle is trapped. Peter Alfeld’s website features an interactive Java applet that lets you change the number of sides in the polygons.
6. The individual steps in Archimedes’s original argument are of historical interest but you might find them disappointingly obscure.
7. Anyone curious about the heroic computations of pi to enormous numbers of digits should enjoy Richard Preston’s profile of the Chudnovsky brothers. Entitled “The Mountains of Pi,” this affectionate and surprisingly comical piece appeared in the Mar. 2, 1992, issue of The New Yorker, and more recently as a chapter in: R. Preston, Panic in Level Four (Random House, 2008).
8. For a textbook introduction to the basics of numerical analysis, see: W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University Press, 2007).

Thanks to Tim Novikoff and Carole Schiffman for their comments and suggestions, and to Margaret Nelson for preparing the illustrations.

Steven Strogatz, New York Times

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Full article and photos: http://opinionator.blogs.nytimes.com/2010/04/04/take-it-to-the-limit/

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Gathering for Gardner

Homage to the iconic author of Scientific American’s “Mathematical Games” column.

Last Saturday afternoon, on a Japanese-landscaped hillside at the outskirts of Atlanta, several clusters of people were constructing mathematically inspired sculptures of metal, bamboo and balloons. Nearby, a magician showed a mathematician how to “throw” a knot. Others had their photographs taken in an optical illusion they had built, an “impossible box” that from one perspective made people look simultaneously behind and inside it. Around a goldfish pond, groups did puzzles, origami, juggling and card tricks. A magician, a philosopher and a software engineer argued about Wittgenstein.

It was the high point of a four-day conference in honor of Martin Gardner, 95, a public intellectual whose most famous pulpit was “Mathematical Games,” written for Scientific American between 1956 and 1981. Mr. Gardner’s column illuminated the beauty of math and logic in discussions of fractals, origami, optical illusions, puzzles and pseudoscience. It challenged readers to discover how finely math and logic are interwoven through the world.

In Mr. Gardner’s quarter-century as a columnist he corresponded extensively with hundreds of people, including many innovators in math, magic, psychology, philosophy and logic. He retired from the column to become an author—and he’s been prolific, publishing more than 65 books at last count. Despite his public acclaim, though, Mr. Gardner is famously shy and has declined many honors because they’d require public appearances.

For a long time, few of those whom he inspired had met him in person. Finally, in 1993, an Atlanta puzzle-collector named Tom Rodgers persuaded Mr. Gardner to attend an event in his honor, and a “Gathering for Gardner” has been held every two years since 1996. Each is named G4Gn, with n replaced by the number in the series. Last week’s, the ninth, was thus G4G9.

Mr. Gardner, who lives outside Norman, Okla., no longer travels and attended only the first two G4Gs. Still, his influence was everywhere at G4G9. Participants included luminaries from eclectic fields. Lennart Green—a legendary Swedish magician who in 1988 was disqualified from the “Olympics of the magic world” because judges erroneously decided he must have cheated to produce his astounding tricks—gave frequent card-trick demonstrations. Famed Princeton mathematician John Conway, designer of the cellular automaton “Game of Life,” spoke of his latest mathematical work. British physicist Stephen Wolfram, author of “A New Kind of Science,” demonstrated his new and ambitious computational search engine, Wolfram Alpha.

One of Mr. Gardner’s most inspired creations was “Dr. Matrix,” who believed that numbers, and especially coincidences of numbers, have special powers and properties. Dr. Matrix’s pronouncements echoed those of numerologists and astrologers, and Mr. Gardner used the character to expose and ridicule them. A high point of each G4Gn is a heated exchange between two Dr. Matrix impersonators about the meaning of n, the conference number. This year, the pro-9 Dr. Matrix rattled off the number’s powerful role in the universe—witness the number of candles on a menorah, the orders of angels, the lives of a cat, and the spikes in Bart Simpson’s hair—and declared it no coincidence that 9 was on the jersey of the winning quarterback in this year’s Superbowl. “Crap!” declared the anti-9 Dr. Matrix, who pointed out that the Christian hell has 9 circles—Satan inhabits the 9th—while the Greek Hades is guarded by a 9-headed beast, encircled 9 times by the river Styx, and presided over by Pluto, until recently the 9th planet.

Mr. Gardner’s following includes many hacking experts, and among those at G4G9 were mathematician and programmer Bill Gosper, considered the founder of the hacker community, and Pablos Holman, from Intellectual Ventures lab, whose public demonstrations are notorious. His G4G9 one was no exception. “Most people ask: ‘What does a thing do?’” Mr. Holman said. “A hacker asks, ‘What can I make it do?’” To illustrate, he pulled out of his pocket an electronic device he had purchased on eBay for a few dollars, used it to hack into the cellphone and change the voice-mail greeting of one audience member—then displayed the information of a supposedly “safe” credit card in the wallet in the pants of another. The crowd went wild. “Pay no attention to the numbers on the screen,” Mr. Holman said, dryly.

Mr. Gardner’s fans also include psychologists and cognition researchers interested in discovering why people regularly and seemingly inexorably fall victim to optical illusions, faulty logic and pseudoscience. These, too, were represented at G4G9. Al Seckel, a former Caltech cognitive neuroscientist, used sensory illusions to demonstrate how humans “map” incoming information to support pre-existing organizational perceptual frameworks, even if the incoming information is contradictory or false. As an example, Mr. Seckel noted that global-warming skeptics who lack training in science yet appear to argue on a “technical level” tend to be libertarians. If global warming is correct, that suggests large-scale governmental regulation is needed, contrary to the core beliefs of a libertarian. “It is easier for a libertarian to attack the science of global warming,” Mr. Seckel said, “than to alter one’s core libertarian beliefs.”

Puzzles are instructive, Mr. Gardner found, for they teach us to appreciate hidden structures of the world that are not owned by any particular discipline and are potentially useful to all. He saw the world as resembling not a magazine, where the subject of each section bears little relation to that of the next, but a well-written novel, where ideas introduced in one chapter are apt to reappear—transformed, modulated and extended—in others. He taught his readers to see the world in the same way, inculcating in them an openness and alertness to the often surprising possibilities of the world, and the desire to seek them out.

“To get seeded,” a Google engineer told me in Atlanta, after I asked him why he’d come to G4G9. “The atmosphere is like that of a group of hobbyists, but you learn from almost everyone something you can apply to your own field.”

Mr. Crease is chairman of the philosophy department at the State University of New York, Stony Brook.

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Full article: http://online.wsj.com/article/SB10001424052702304370304575151970094262604.html

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Power Tools

If you were an avid television watcher in the 1980s, you may remember a clever show called “Moonlighting.”  Known for its snappy dialogue and the romantic chemistry between its co-stars, it featured Cybill Shepherd and Bruce Willis as a couple of wisecracking private detectives named Maddie Hayes and David Addison.  While investigating one particularly tough case, David asks a coroner’s assistant for his best guess about possible suspects.  “Beats me,” says the assistant.  “But you know what I don’t understand?”  To which David replies, “Logarithms?”  Then, reacting to Maddie’s look: “What?  You understood those?”

That pretty well sums up how many people feel about logarithms.  Their peculiar name is just part of their image problem.  Most folks never use them again after high school, at least not consciously, and are oblivious to the logarithms hiding behind the scenes of their daily lives.

The same is true of many of the other functions discussed in algebra II and pre-calculus.  Power functions, exponential functions — what was the point of all that?  My goal in this week’s column is to help you appreciate the function of all those functions, even if you never have occasion to press their buttons on your calculator.

A mathematician needs functions for the same reason that a builder needs hammers and drills.  Tools transform things.  So do functions.  In fact, mathematicians often refer to them as “transformations” because of this.  But instead of wood or steel, functions pound away on numbers and shapes and, sometimes, even on other functions.

To show you what I mean, let’s plot the graph of the equation

You may remember how this sort of activity goes: you draw a picture of the xy plane with the x-axis running horizontally and the y-axis vertically.  Then for each x you compute the corresponding y and plot them together as a single point in the xy plane.  For example, when x is 1, the equation says y equals 4 minus 1 squared, which is 4 minus 1, or 3.  So (x,y) = (1, 3) is a point on the graph.  After calculating and plotting a few more points, the following picture emerges.

The droopy shape of the curve is due to the action of mathematical pliers.  In the equation for y, the function that transforms x into x² behaves a lot like the common tool for bending and pulling things.  When it’s applied to every point on a piece of the x-axis (which you could visualize as a straight piece of wire), the pliers bend and elongate that piece into the downward-curving arch shown above.

And what role does the 4 play in the equation y = 4 – x²?  It acts like a nail for hanging a picture on a wall.  It lifts the bent wire arch up by 4 units.  Since it raises all points by the same amount, it’s known as a “constant function.”

This example illustrates the dual nature of functions.  On the one hand, they’re tools:  the x² bends the piece of the x-axis and the 4 lifts it.  On the other hand, they’re building blocks: the 4 and the –x² can be regarded as component parts of a more complicated function, 4 – x², just as wires, batteries and transistors are component parts of a radio.

Once you start to look at things this way, you’ll notice functions everywhere.  The arching curve above — technically known as a “parabola”— is the signature of the squaring function x² operating behind the scenes.   Look for it when you’re taking a sip from a water fountain or watching a basketball arc toward the hoop.  And if you ever have a few minutes to spare on a layover in Detroit’s International Airport, be sure to stop by the Delta terminal to enjoy the world’s most breathtaking parabolas at play.

Parabolas and constants are associated with a wider class of functions — “power functions” of the form xⁿ, in which a variable x is raised to a fixed power n.  For a parabola, n = 2; for a constant, n = 0.

Changing the value of n yields other handy tools.  For example, raising x to the first power (n = 1) gives a function that works like a ramp, a steady incline of growth or decay.  It’s called a “linear function” because its xy graph is a line.  If you leave a bucket out in a steady rain, the water collecting at the bottom rises linearly in time.

Another useful tool is the inverse square function 1/x², corresponding to the case n = –2.  It’s good for describing how waves and forces attenuate as they spread out in three dimensions — for instance, how a sound softens as it moves away from its source.

Power functions like these are the building blocks that scientists and engineers use to describe growth and decay in their mildest forms.

But when you need mathematical dynamite, it’s time to unpack the exponential functions.  They describe all sorts of explosive growth, from nuclear chain reactions to the proliferation of bacteria in a Petri dish.  The most familiar example is the function 10^x, in which 10 is raised to the power x.   Make sure not to confuse this with the earlier power functions.  Here the exponent (the power x) is a variable, and the base (the number 10) is a constant — whereas in a power function like x², it’s the other way around.  This switch makes a huge difference.  Exponential growth is almost unimaginably rapid.

That’s why it’s so hard to fold a piece of paper in half more than 7 or 8 times. Each folding approximately doubles the thickness of the wad, causing it to grow exponentially.   Meanwhile, the wad’s length shrinks in half every time, and thus decreases exponentially fast.  For a standard sheet of notebook paper, after 7 folds the wad becomes thicker than it is long, so it can’t be folded again.  It’s not a matter of the folder’s strength; for a sheet to be considered legitimately folded n times, the resulting wad is required to have 2n layers in a straight line, and this can’t happen if the wad is thicker than it is long.

The challenge was thought to be impossible until Britney Gallivan, then a junior in high school, solved it in 2002.  She began by deriving a formula

that predicted the maximum number of times, n, that paper of a given thickness T and length L could be folded in one direction.  Notice the forbidding presence of the exponential function 2ⁿ in two places — once to account for the doubling of the wad’s thickness at each fold, and another time to account for the halving of its length.

Using her formula, Britney concluded that she would need to use a special roll of toilet paper nearly three quarters of a mile long.   In January 2002, she went to a shopping mall in her hometown of Pomona, Calif., and unrolled the paper.  Seven hours later, and with the help of her parents, she smashed the world record by folding the paper in half 12 times!

In theory, exponential growth is also supposed to grace your bank account.  If your money grows at an annual interest rate of r, after one year it will be worth (1 + r) times more; after two years, (1 + r) squared; and after x years, (1 + r)^x times more than your initial deposit.  Thus the miracle of compounding that we so often hear about is caused by exponential growth in action.

Which brings back to logarithms.  We need them because it’s always useful to have tools that can undo one another.  Just as every office worker needs both a stapler and a staple remover, every mathematician needs exponential functions and logarithms.  They’re “inverses.”    This means that if you type a number x into your calculator, and then punch the 10^x button followed by the log x button, you’ll get back to the number you started with.

Logarithms are compressors.  They’re ideal for taking numbers that vary over a wide range and squeezing them together so they become more manageable. For instance, 100 and 100 million differ a million-fold, a gulf that most of us find incomprehensible.  But their logarithms differ only fourfold (they are 2 and 8, because 100 = 10² and 100 million = 10⁸).  In conversation, we all use a crude version of logarithmic shorthand when we refer to any salary between $100,000 and $999,999 as being “six figures.”  That “six” is roughly the logarithm of these salaries, which in fact span the range from 5 to 6.

As impressive as all these functions may be, a mathematician’s toolbox can only do so much — which is why I still haven’t assembled my Ikea bookcases.

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NOTES:

1. The excerpt from “Moonlighting” is from the episode “In God We Strongly Suspect.” It originally aired on Feb. 11, 1986, during the show’s second season.
2. Will Hoffman and Derek Paul Boyle have filmed an intriguing video of the parabolas all around us in the everyday world (along with their exponential cousins, curves called “catenaries,” so-named for the shape of hanging chains). Full disclosure: the filmmakers say this video was inspired by a story I told on an episode of RadioLab.
3. For simplicity, I’ve referred to expressions like x² as functions, through to be more precise I should speak of  “the function that maps x into x².”  I hope this sort of abbreviation won’t cause confusion, since we’ve all seen it on calculator buttons.
4. For the story of Britney Gallivan’s adventures in paper folding, see: Gallivan, B. C. “How to Fold Paper in Half Twelve Times: An ‘Impossible Challenge’ Solved and Explained.” Pomona, CA: Historical Society of Pomona Valley, 2002. For a journalist’s account, aimed at children, see Ivars Peterson, “Champion paper-folder,” Muse (July/August 2004), p. 33. The Mythbusters have also attempted to replicate Britney’s experiment on their television show.
5. For evidence that our innate number sense is logarithmic, see: Stanislas Dehaene, Véronique Izard, Elizabeth Spelke, and Pierre Pica, “Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures,” Science, Vol. 320 (2008), p. 1217. Popular accounts of this study are available at ScienceDaily and in this episode of RadioLab.

Thanks to David Field, Paul Ginsparg, Jon Kleinberg, Andy Ruina and Carole Schiffman for their comments and suggestions; Diane Hopkins, Cindy Klauss and Brian Madsen for their help in finding and obtaining the “Moonlighting” clip; and Margaret Nelson, for preparing the illustration.

Steven Strogatz, New York Times

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Full article and photos: http://opinionator.blogs.nytimes.com/2010/03/28/power-tools/

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Think Globally

The most familiar ideas of geometry were inspired by an ancient vision — a vision of the world as flat. From parallel lines that never meet, to the Pythagorean theorem discussed in last week’s column, these are eternal truths about an imaginary place, the two-dimensional landscape of plane geometry.

Conceived in India, China, Egypt and Babylonia more than 2,500 years ago, and codified and refined by Euclid and the Greeks, this flat-earth geometry is the main one (and often the only one) being taught in high schools today. But things have changed in the past few millennia.

In an era of globalization, Google Earth and transcontinental air travel, all of us should try to learn a little about spherical geometry and its modern generalization, differential geometry. The basic ideas here are only about 200 years old. Pioneered by Carl Friedrich Gauss and Bernhard Riemann, differential geometry underpins such imposing intellectual edifices as Einstein’s general theory of relativity. At its heart, however, are beautiful concepts that can be grasped by anyone who’s ever ridden a bicycle, looked at a globe or stretched a rubber band. And understanding them will help you make sense of a few curiosities you may have noticed in your travels.

For example, when I was little, my dad used to enjoy quizzing me about geography. Which is farther north, he’d ask, Rome or New York City? Most people would guess New York, but surprisingly they’re at almost the same latitude, with Rome being just a bit farther north. On the usual map of the world (the misleading Mercator projection, where Greenland appears gigantic) it looks like you could go straight from New York to Rome by heading due east.

Yet airline pilots never take that route. They always fly northeast out of New York, hugging the coast of Canada. I used to think they were staying close to land for safety’s sake, but that’s not the reason. It’s simply the most direct route, if you take the earth’s curvature into account. The shortest path from New York to Rome goes past Nova Scotia and Newfoundland, then heads out over the Atlantic, and finally veers south of Ireland and across France for arrival in sunny Italy.

This kind of path on the globe is called an arc of a “great circle.” Like straight lines in ordinary space, great circles on a sphere contain the shortest paths between any two points. They’re called “great” because they’re the largest circles you can have on a sphere. Conspicuous examples include the equator and the longitudinal circles that pass through the north and south poles.

Another property that lines and great circles share is that they’re the straightest paths. That might sound strange — all paths on a globe are curved, so what do we mean by “straightest”? Well, some paths are more curved than others. The great circles don’t do any additional curving, above and beyond what they’re forced to do by following the surface of the sphere.

Here’s a way to visualize this. Imagine you’re riding a tiny bicycle on the surface of a globe, and you’re trying to stay on a certain path. If it’s part of a great circle, you won’t ever need to steer. That’s the sense in which great circles are “straight.” In contrast, if you try to ride along a line of latitude near one of the poles, you’ll have to keep turning the handlebars.

Of course, as surfaces go, the plane and the sphere are abnormally simple. The surface of a human body, or a tin can, or a bagel would be more typical — they all have far less symmetry, as well as various kinds of holes and passageways that make them more confusing to navigate. In this more general setting, finding the shortest path between any two points becomes a lot trickier. So rather than delving into technicalities, let’s stick to an intuitive approach. This is where rubber bands come in handy.

Specifically, imagine a slippery elastic string that always contracts as far as it can, while remaining confined to the surface. With its help, we can easily determine the shortest path between New York and Rome, or for that matter, between any two points on any surface. Tie the ends of the string to the points of departure and arrival and let the string pull itself tight, while clinging to the surface’s contours. When the string is as taut as these constraints allow, voila! It traces the shortest path.

On surfaces just a little more complicated than planes or spheres, something strange and new can happen: many locally shortest paths can exist between the same two points. For example, consider the surface of a soup can, with one point lying directly below the other.

Then the shortest path between them is clearly a line segment, as shown above, and our elastic string would find that solution. So what’s new here? The cylindrical shape of the can opens up new possibilities for all kinds of contortions. Suppose we require that the string encircles the cylinder once before connecting to the second point. Now when the string pulls itself taut, it forms a helix, like the curves on old barbershop poles.

This helical path qualifies as another solution to the shortest path problem, in the sense that it’s the shortest of the candidate paths nearby. If you nudge the string a little, it would necessarily get longer and then contract back to the helix. You could say it’s the “locally” shortest path — the regional champion of all those that wrap once around the cylinder. (By the way, this is why the subject is called “differential” geometry; it studies the effects of small local differences on various kinds of shapes, such as the difference in length between the helical path and its neighbors.)

But that’s not all. There’s another champ that winds around twice, and another that goes around three times, and so on. There are infinitely many locally shortest paths on a cylinder! Of course, none of these helices is the “globally” shortest path. The straight-line path is shorter than all of them.

Likewise, surfaces with holes and handles permit many locally shortest paths, distinguished by their pattern of weaving around various parts of the surface. The following video by the mathematician Konrad Polthier of the Free University of Berlin illustrates the non-uniqueness of these locally shortest paths, or “geodesics,” on the surface of an imaginary planet shaped like a figure-8, a surface known in the trade as a “two-holed torus”:

The red, yellow and green geodesics all visit very different parts of the planet, thereby executing different loop patterns. But what they all have in common is their superior directness compared to the paths nearby. And just like lines on a plane or great circles on a sphere, these geodesics are the straightest possible curves on the surface. They bend to conform with the surface, but don’t bend within it. To make this clear, Polthier has produced another illuminating video.

Here, a motorcycle rides along a geodesic highway on a two-holed torus, following the lay of the land. The remarkable thing is that the motorcycle’s handlebars are locked. It doesn’t need to steer to stay on the road. This underscores the earlier intuition that geodesics, like great circles, are the natural generalization of straight lines.

With all these flights of fancy, you may be wondering if geodesics have anything to do with reality. Of course they do. Einstein showed that light beams follow geodesics as they sail through the universe. The famous bending of starlight around the sun, detected in the eclipse observations of 1919, confirmed that light travels on geodesics through curved space-time, with the warping being caused by the sun’s gravity.

At a more down-to-earth level, the mathematics of finding shortest paths is critical in everything from the GPS navigation systems in our cars to the routing of traffic on the Internet. In these situations, however, the relevant space is a gargantuan maze of addresses and links, as opposed to the smooth surfaces considered above, and the mathematical issues have to do with the speed of algorithms — what’s the most efficient way to find the shortest path through a network? Given the myriad of potential routes, the task would be overwhelming, were it not for the ingenuity of the mathematicians and computer scientists who cracked it.

Sometimes when people say the shortest distance between two points is a straight line, they mean it figuratively, as a way of ridiculing nuance and affirming common sense. In other words, keep it simple. But battling obstacles can give rise to great beauty — so much so that in art, and in math, it’s often more fruitful to impose constraints on ourselves. Think of haiku, or sonnets, or telling the story of your life in six words. The same is true of all the math that’s been created to find the shortest way from here to there when you can’t take the easy way out.

Two points. Many paths. Mathematical bliss.

NOTES:

1. By referring to plane geometry as “flat-earth” geometry, I might seem to be disparaging the subject, but that’s not my intent. The tactic of locally approximating a curved shape by a flat one has often turned out to be a useful simplification in many parts of mathematics and physics, from calculus to relativity theory. Plane geometry is the first instance of this great idea.
2. Nor do I mean to suggest that all the ancients thought the world was flat. For an engaging account of Eratosthenes’s measurement of the distance around the globe, see: N. Nicastro, Circumference (St. Martin’s Press, 2008).
3. For a more contemporary approach that you might like to try on your own, Robert Vanderbei at Princeton University recently gave a presentation to his daughter’s high school geometry class in which he used a photograph of a sunset to show that the earth is not flat, and to estimate its diameter. His slides are posted here.
4. An interactive online demonstration that lets you plot the shortest route between any two points on the surface of the earth is available here. (You’ll need to download the free Mathematica Player, which will then allow you to explore hundreds of other interactive demonstrations in all parts of mathematics.)
5. A superb introduction to modern geometry was co-authored by David Hilbert, one of the greatest mathematicians of the 20^th century. This classic, originally published in 1952, has been reissued as: D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination (American Mathematical Society, 1999).
6. Several good textbooks and online courses in differential geometry are listed here.
7. Konrad Polthier has produced a number of fascinating educational videos about mathematical topics. Excerpts can be found online here.
   Award-winning videos by Polthier and his colleagues appear in the VideoMath Festival collection, available as a DVD from Springer Verlag.
8. The classic algorithm for shortest path problems on networks is due to Edsger Dijkstra. A PDF version of his 1959 paper is available here.
9. Textbook treatments of related routing problems on networks are given online here and here.
10. Steven Skiena has posted an instructive animation of Dijkstra’s algorithm.
11. Nature can solve certain shortest path problems by decentralized processes akin to analog computation. For chemical waves that solve mazes, see: O. Steinbock, A. Toth, and K. Showalter, “Navigating complex labyrinths: Optimal paths from chemical waves,” Science 267, p.868 (1995).
    Not to be outdone, slime molds can solve them too: T. Nakagaki, H. Yamada, and A. Toth, “Maze-solving by an amoeboid organism,” Nature 407, p.470 (2000).
    This slimy creature can even make networks as efficient as the Tokyo rail system: A. Tero et al., “Rules for biologically inspired adaptive network design,” Science 327, p.439 (2010).
    
12. For an introduction to the mathematics of GPS navigation systems, see: S. Robinson, “Mapping magic,” SIAM News (Sep. 26, 2004), available online here.
13. Delightful examples of six-word memoirs are given here and here.

Thanks to Robert Vanderbei, for sending the link to his presentation about estimating the earth’s diameter from a photograph of a sunset; Margaret Nelson, for preparing the line drawings; Doug Arnold, Bob Connelly, Paul Ginsparg, Jon Kleinberg, Andy Ruina and Carole Schiffman, for their comments and suggestions; and Konrad Polthier, for generously sharing his videos of geodesics.

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Full article and photos: http://opinionator.blogs.nytimes.com/2010/03/21/think-globally/

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Square Dancing

I bet I can guess your favorite math subject in high school.

It was geometry.

So many people I’ve met over the years have expressed affection for that subject.  Arithmetic and algebra — not many takers there.  But geometry, well, there’s something about it that brings a twinkle to the eye.

Is it because geometry draws on the right side of the brain, and that appeals to visual thinkers who might otherwise cringe at its cold logic?   Maybe.  But other people tell me they loved geometry precisely because it was so logical.  The step-by-step reasoning, with each new theorem resting firmly on those already established — that’s the source of satisfaction for many.

But my best hunch (and, full disclosure, I personally love geometry) is that people enjoy it because it marries logic and intuition.  It feels good to use both halves of our brain.

To illustrate the pleasures of geometry, let’s revisit the Pythagorean theorem, which you probably remember as a² + b² = c².   Part of the goal here is to see why it’s true and appreciate why it matters.  Beyond that, by proving the theorem in two different ways, we’ll come to see how one proof can be more “elegant” than another, even though both are correct.

The Pythagorean theorem is concerned with “right triangles” — meaning those with a right (90-degree) angle at one of the corners.  Right triangles are important because they’re what you get if you cut a rectangle in half along its diagonal:

And since rectangles come up often in all sorts of settings, so do right triangles.

They arise, for instance, in surveying.  If you’re measuring a rectangular field, you might want to know how far it is from one corner to the diagonally opposite corner.  (By the way, this is where geometry started, historically — in problems of land measurement, or measuring the earth: geo = “earth” + metry = “measurement.”)

The Pythagorean theorem tells you how long the diagonal is, compared to the sides of the rectangle.  If one side has length a and the other has length b, the theorem says the diagonal has length c, where

For some reason, the diagonal is traditionally called the “hypotenuse,” though I’ve never met anyone who knows why.  (Any Latin or Greek scholars?)  It must have something to do with the diagonal “subtending” a right angle, but as jargon goes, “subtending” is about as opaque as “hypotenuse.”

Anyway, here’s how the theorem works.  To keep the numbers simple, let’s say a = 3 yards and b = 4 yards.  Then to figure out the unknown length c, we don our black hoods and intone that c² is the sum of 3² plus 4², which is 9 plus 16. (Keep in mind that all of these quantities are now measured in square yards, since we squared the yards as well as the numbers themselves.)  Finally, since 9 + 16 = 25, we get c² = 25 square yards, and then taking square roots of both sides yields c = 5 yards as the length of the hypotenuse.

This way of looking at the Pythagorean theorem makes it seem like a statement about lengths.  But traditionally it was viewed as a statement about areas.  That becomes clearer when you hear how they used to say it:

“The square on the hypotenuse is the sum of the squares on the other two sides.”

Notice the word “on.”  We’re not speaking of the square “of” the hypotenuse — that’s a newfangled algebraic concept about multiplying a number (the length of the hypotenuse) by itself.  No, we’re literally referring here to a square sitting on the hypotenuse, like this:

Let’s call this the large square, to distinguish it from the small and medium-sized squares we can build on the other two sides:

Then the theorem says that the large square has the same area as the small and medium squares combined.

For thousands of years, this marvelous fact has been expressed in a diagram, an iconic mnemonic of dancing squares:

Viewing the theorem in terms of areas also makes it a lot more fun to think about.  For example, you can test it — and then eat it — by building the squares out of many little crackers. Or you can treat the theorem like a child’s puzzle, with pieces of different shapes and sizes. By rearranging these puzzle pieces, we can prove the theorem very simply, as follows.

Let’s go back to the tilted square sitting on the hypotenuse.

At an instinctive level, this image should make you feel a bit uncomfortable.  The square looks potentially unstable, like it might topple or slide down the ramp.  And there’s also an unpleasant arbitrariness about which of the four sides of the square gets to touch the triangle.

Guided by these intuitive feelings, let’s add three more copies of the triangle around the square to make a more solid and symmetrical picture:

Now recall what we’re trying to prove: that the tilted white square in the picture above (which is just our earlier “large square”— it’s still sitting right there on the hypotenuse) has the same area as the small and medium squares put together.  But where are those other squares?  Well, we have to shift some triangles around to find them.

Think of the picture above as literally depicting a puzzle, with four triangular pieces wedged into the corners of a rigid puzzle frame.

In this interpretation, the tilted square is the empty space in the middle of the puzzle.  The rest of the area inside the frame is occupied by the puzzle pieces.

Now let’s try moving the pieces around in various ways.  Of course, nothing we do can ever change the total amount of empty space inside the frame — it’s always whatever area lies outside the pieces.

The brainstorm, then, is to rearrange the pieces like this:

All of a sudden the empty space has changed into the two shapes we’re looking for — the small square and the medium square.  And since the total area of empty space always stays the same, we’ve just proven the Pythagorean theorem!

This proof does far more than convince; it illuminates.  That’s what makes it “elegant.”

For comparison, here’s another proof.  It’s equally famous, and it’s perhaps the simplest proof that avoids using areas.

As before, consider a right triangle with sides of length a and b and hypotenuse of length c, as shown below on the left.

Now, by divine inspiration or a stroke of genius, something tells us to draw a line segment perpendicular to the hypotenuse and down to the opposite corner, as shown above on the right.

This clever little construction creates two smaller triangles inside the original one.  It’s easy to prove that all these triangles are “similar” — which means they have identical shapes but different sizes.  That in turn implies that the lengths of their corresponding parts have the same proportions, which translates into the following set of equations:

We also know that

because our construction merely split the original hypotenuse of length c into two smaller sides of length d and e.

At this point you might be feeling a bit lost, or at least unsure of what to do next.   There’s a morass of five equations above, and we’re trying to whittle them down to deduce that

Try it for a few minutes.  You’ll discover that two of the equations are irrelevant.  That’s ugly; an elegant proof should involve nothing superfluous.  With hindsight, of course, you wouldn’t have listed those equations to begin with.  But that would just be putting lipstick on a p… (the missing word here is “proof”).

Nevertheless, by manipulating the right three equations, you can get the theorem to pop out.  See the notes below for the missing steps.

Would you agree with me that, on aesthetic grounds, this proof is inferior to the first one?  For one thing, it drags near the end.  And who invited all that algebra to the party?  This is supposed to be a geometry event.

But a more serious defect is the proof’s murkiness.  By the time you’re done slogging through it, you might believe the theorem (grudgingly), but you still might not see why it’s true.

Leaving proofs aside, why does the Pythagorean theorem even matter?  Because it reveals a fundamental truth about the nature of space.  It implies that space is flat, as opposed to curved.  On the surface of a globe or a bagel, for example, the theorem needs to be modified.  Einstein confronted this challenge in his general theory of relativity (where gravity is no longer viewed as a force, but rather as a manifestation of the curvature of space), and so did Riemann and others before him, when laying the foundations of non-Euclidean geometry.

It’s a long road from Pythagoras to Einstein.  But at least it’s a straight line… for most of the way.

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NOTES:

• The ancient Babylonians, Indians and Chinese appear to have been aware of the Pythagorean theorem several centuries before Pythagoras and the Greeks.  For more about the history and significance of the theorem, as well as a survey of the many ingenious ways to prove it, see:

E. Maor, The Pythagorean Theorem: A 4,000-Year History (Princeton University Press, 2007).

• Incidentally, on p. xiii of his book, Maor explains that the word “hypotenuse” means “stretched beneath,” and points out that this makes sense if the right triangle is viewed with its hypotenuse at the bottom, as depicted in Euclid’s Elements.  He also notes that this interpretation fits well with the Chinese word for hypotenuse: “hsien, a string stretched between two points (as in a lute).”

• If you enjoy seeing different proofs, a nicely annotated and extensive collection of 84 of them — with creators ranging from Euclid to Leonardo da Vinci to President James Garfield — is available here.

• With any luck, the first proof above should have given you an Aha! sensation.  But to make the argument completely airtight, we also need to prove that the pictures aren’t deceiving us — in other words, they truly have the properties they appear to have.  A more rigorous proof would establish, for example, that the outer frame is truly a square, and that the medium and small squares meet at a single point, as shown.  Checking these details is fun and not too difficult.

• Here are the missing steps in the second proof above.   Take this equation:

and multiply it by a on both sides to get

Similarly massaging another of the equations yields

Finally, substituting the expressions above for d and e into the equation c = d + e yields

Then multiplying both sides by c gives the desired formula:

• Thanks to George Hart of the Museum of Mathematics for sharing his hands-on demo using Pythagorean crackers, Carole Schiffman for her comments and suggestions, and Margaret Nelson for preparing the illustrations.

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Full article and photos: http://opinionator.blogs.nytimes.com/2010/03/14/square-dancing/

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Algebra in Wonderland

SINCE “Alice’s Adventures in Wonderland” was published, in 1865, scholars have noted how its characters are based on real people in the life of its author, Charles Dodgson, who wrote under the name Lewis Carroll. Alice is Alice Pleasance Liddell, the daughter of an Oxford dean; the Lory and Eaglet are Alice’s sisters Lorina and Edith; Dodgson himself, a stutterer, is the Dodo (“Do-Do-Dodgson”).

But Alice’s adventures with the Caterpillar, the Mad Hatter, the Cheshire Cat and so on have often been assumed to be based purely on wild imagination. Just fantastical tales for children — and, as such, ideal material for the fanciful movie director Tim Burton, whose “Alice in Wonderland” opened on Friday.

Yet Dodgson most likely had real models for the strange happenings in Wonderland, too. He was a tutor in mathematics at Christ Church, Oxford, and Alice’s search for a beautiful garden can be neatly interpreted as a mishmash of satire directed at the advances taking place in Dodgson’s field.

In the mid-19th century, mathematics was rapidly blossoming into what it is today: a finely honed language for describing the conceptual relations between things. Dodgson found the radical new math illogical and lacking in intellectual rigor. In “Alice,” he attacked some of the new ideas as nonsense — using a technique familiar from Euclid’s proofs, reductio ad absurdum, where the validity of an idea is tested by taking its premises to their logical extreme.

Early in the story, for instance, Alice’s exchange with the Caterpillar parodies the first purely symbolic system of algebra, proposed in the mid-19th century by Augustus De Morgan, a London math professor. De Morgan had proposed a more modern approach to algebra, which held that any procedure was valid as long as it followed an internal logic. This allowed for results like the square root of a negative number, which even De Morgan himself called “unintelligible” and “absurd” (because all numbers when squared give positive results).

The word “algebra,” De Morgan said in one of his footnotes, comes from an Arabic phrase he transliterated as “al jebr e al mokabala,” meaning restoration and reduction. He explained that even though algebra had been reduced to a seemingly absurd but logical set of operations, eventually some sort of meaning would be restored.

Such loose mathematical reasoning would have riled a punctilious logician like Dodgson. And so, the Caterpillar is sitting on a mushroom and smoking a hookah — suggesting that something has mushroomed up from nowhere, and is dulling the thoughts of its followers — and Alice is subjected to a monstrous form of “al jebr e al mokabala.” She first tries to “restore” herself to her original (larger) size, but ends up “reducing” so rapidly that her chin hits her foot.

Alice has slid down from a world governed by the logic of universal arithmetic to one where her size can vary from nine feet to three inches. She thinks this is the root of her problem: “Being so many different sizes in a day is very confusing.” No, it isn’t, replies the Caterpillar, who comes from the mad world of symbolic algebra. He advises Alice to “Keep your temper.”

In Dodgson’s day, intellectuals still understood “temper” to mean the proportions in which qualities were mixed — as in “tempered steel” — so the Caterpillar is telling Alice not to avoid getting angry but to stay in proportion, even if she can’t “keep the same size for 10 minutes together!” Proportion, rather than absolute length, was what mattered in Alice’s above-ground world of Euclidean geometry.

In an algebraic world, of course, this isn’t easy. Alice eats a bit of mushroom and her neck elongates like a serpent, annoying a nesting pigeon. Eventually, though, she finds a way to nibble herself down to nine inches, and enters a little house where she finds the Duchess, her baby, the Cook and the Cheshire Cat.

Chapter 6, “Pig and Pepper,” parodies the principle of continuity, a bizarre concept from projective geometry, which was introduced in the mid-19th century from France. This principle (now an important aspect of modern topology) involves the idea that one shape can bend and stretch into another, provided it retains the same basic properties — a circle is the same as an ellipse or a parabola (the curve of the Cheshire cat’s grin).

Taking the notion to its extreme, what works for a circle should also work for a baby. So, when Alice takes the Duchess’s baby outside, it turns into a pig. The Cheshire Cat says, “I thought it would.”

The Cheshire Cat provides the voice of traditional geometric logic — say where you want to go if you want to find out how to get there, he tells Alice after she’s let the pig run off into the wood. He points Alice toward the Mad Hatter and the March Hare. “Visit either you like,” he says, “they’re both mad.”

The Mad Hatter and the March Hare champion the mathematics of William Rowan Hamilton, one of the great innovators in Victorian algebra. Hamilton decided that manipulations of numbers like adding and subtracting should be thought of as steps in what he called “pure time.” This was a Kantian notion that had more to do with sequence than with real time, and it seems to have captivated Dodgson. In the title of Chapter 7, “A Mad Tea-Party,” we should read tea-party as t-party, with t being the mathematical symbol for time.

Dodgson has the Hatter, the Hare and the Dormouse stuck going round and round the tea table to reflect the way in which Hamilton used what he called quaternions — a number system based on four terms. In the 1860s, quaternions were hailed as the last great step in calculating motion. Even Dodgson may have considered them an ingenious tool for advanced mathematicians, though he would have thought them maddeningly confusing for the likes of Alice (and perhaps for many of his math students).

At the mad tea party, time is the absent fourth presence at the table. The Hatter tells Alice that he quarreled with Time last March, and now “he won’t do a thing I ask.” So the Hatter, the Hare and the Dormouse (the third “term”) are forced to rotate forever in a plane around the tea table.

When Alice leaves the tea partiers, they are trying to stuff the Dormouse into the teapot so they can exist as an independent pair of numbers — complex, still mad, but at least free to leave the party.

Alice will go on to meet the Queen of Hearts, a “blind and aimless Fury,” who probably represents an irrational number. (Her keenness to execute everyone comes from a ghastly pun on axes — the plural of axis on a graph.)

How do we know for sure that “Alice” was making fun of the new math? The author never explained the symbolism in his story. But Dodgson rarely wrote amusing nonsense for children: his best humor was directed at adults. In addition to the “Alice” stories, he produced two hilarious pamphlets for colleagues, both in the style of mathematical papers, ridiculing life at Oxford.

Without math, “Alice” might have been more like Dodgson’s later book, “Sylvie and Bruno” — a dull and sentimental fairy tale. Math gave “Alice” a darker side, and made it the kind of puzzle that could entertain people of every age, for centuries.

Melanie Bayley is a doctoral candidate in English literature at Oxford.

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Full article and photo: http://www.nytimes.com/2010/03/07/opinion/07bayley.html

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Finding Your Roots

For more than 2,500 years, mathematicians have been obsessed with solving for x.  The story of their struggle to find the “roots” — the solutions — of increasingly complicated equations is one of the great epics in the history of human thought.

And yet, through it all, there’s been an irritant, a nagging little thing that won’t go away: the solutions often involve square roots of negative numbers.  Such solutions were long derided as “sophistic” or “fictitious” because they seemed nonsensical on their face.

Until the 1700s or so, mathematicians believed that square roots of negative numbers simply couldn’t exist.

They couldn’t be positive numbers, after all, since a positive times a positive is always positive, and we’re looking for numbers whose square is negative.  Nor could negative numbers work, since a negative times a negative is, again, positive.  There seemed to be no hope of finding numbers which, when multiplied by themselves, would give negative answers.

We’ve seen crises like this before.  They occur whenever an existing operation is pushed too far, into a domain where it no longer seems sensible.  Just as subtracting bigger numbers from smaller ones gave rise to negative numbers and division spawned fractions and decimals, the free-wheeling use of square roots eventually forced the universe of numbers to expand…again.

Historically, this step was the most painful of all.  The square root of –1 still goes by the demeaning name of i, this scarlet letter serving as a constant reminder of its “imaginary” status.

This new kind of number (or if you’d rather be agnostic, call it a symbol, not a number) is defined by the property that

i² = –1.

It’s true that i can’t be found anywhere on the number line.  In that respect it’s much stranger than zero, negative numbers, fractions or even irrational numbers, all of which — weird as they are — still have their place in line.

But with enough imagination, our minds can make room for i as well.  It lives off the number line, at right angles to it, on its own imaginary axis.  And when you fuse that imaginary axis to the ordinary “real” number line, you create a 2-D space — a plane — where a new species of numbers lives.

These are the “complex numbers.”  Here complex doesn’t mean complicated; it means that two types of numbers, real and imaginary, have bonded together to form a complex, a hybrid number like 2 + 3i.

Complex numbers are magnificent, the pinnacle of number systems.  They enjoy all the same properties as real numbers — you can add and subtract them, multiply and divide them — but they are better than real numbers because they always have roots.  You can take the square root or cube root or any root of a complex number and the result will still be a complex number.

Better yet, a grand statement called The Fundamental Theorem of Algebra says that the roots of any polynomial are always complex numbers.  In that sense they’re the end of the quest, the holy grail.  They are the culmination of the journey that began with 1.

You can appreciate the utility of complex numbers (or find it more plausible) if you know how to visualize them.  The key is to understand what multiplying by i looks like.

Suppose we multiply an arbitrary positive number, say 3, by i.  The result is the imaginary number 3i.

So multiplying by i produces a rotation counterclockwise by a quarter turn.  It takes an arrow of length 3 pointing east, and changes it into a new arrow of the same length but now pointing north.

Electrical engineers love complex numbers for exactly this reason.  Having such a compact way to represent 90-degree rotations is very useful to them when working with alternating currents and voltages, or with electric and magnetic fields, because these often involve oscillations or waves that are a quarter cycle (i.e., 90 degrees) out of phase.

In fact, complex numbers are indispensable to all engineers.  In aerospace engineering they eased the first calculations of the lift on an airplane wing.  Civil and mechanical engineers use them routinely to analyze the vibrations of footbridges, skyscrapers and cars driving on bumpy roads.

The 90-degree rotation property also sheds light on what i² = –1 really means.  If we multiply a positive number by i², the corresponding arrow rotates 180 degrees, flipping from east to west, because the two 90-degree rotations (one for each factor of i) combine to make a 180-degree rotation.

But multiplying by –1 produces the very same 180-degree flip.  That’s the sense in which i² = –1.

Computers have breathed new life into complex numbers and the age-old problem of root finding.  When they’re not being used for Web surfing or e-mail, the machines on our desks can reveal things the ancients could never have dreamed of.

In 1976, my Cornell colleague John Hubbard began looking at the dynamics of Newton’s method, a powerful algorithm for finding roots of equations in the complex plane.  The method takes a starting point (an approximation to the root) and does a certain computation that improves it.  By doing this repeatedly, always using the previous point to generate a better one, the method bootstraps its way forward and rapidly homes in on a root.

Hubbard was interested in problems with multiple roots.  In that case, which root would the method find?  He proved that if there were just two roots, the closer one would always win.  But if there were three or more roots, he was baffled.  His earlier proof no longer applied.

So Hubbard did an experiment.  A numerical experiment.

He programmed a computer to run Newton’s method, and told it to color-code millions of different starting points according to which root they approached, and to shade them according to how fast they got there.

Before he peeked at the results, he anticipated that the roots would most quickly attract the points nearby, and thus should appear as bright spots in a solid patch of color.  But what about the boundaries between the patches?   Those he couldn’t picture, at least not in his mind’s eye.

The computer’s answer was astonishing.

The borderlands looked like psychedelic hallucinations.  The colors intermingled there in an almost impossibly promiscuous manner, touching each other at infinitely many points, and always in a three-way.  In other words, wherever two colors met, the third would always insert itself and join them.

Magnifying the boundaries revealed patterns within patterns.

The structure was a “fractal” — an intricate shape whose inner structure repeated at finer and finer scales, as shown in this continuous zoom:

http://vimeo.com/9770779

Furthermore, chaos reigned near the boundary.  Two points might start very close together, bouncing side by side for a while, and then veer off to different roots. The winning root was as unpredictable as a game of roulette.  Little things — tiny, imperceptible changes in the initial conditions — could make all the difference.

Hubbard’s work was an early foray into what’s now called “complex dynamics,” a vibrant blend of chaos theory, complex analysis and fractal geometry.  In a way it brought geometry back to its roots.  In 600 B.C. a manual written in Sanskrit for temple builders in India gave detailed geometric instructions for computing square roots, needed in the design of ritual altars.   More than 2,500 years later, mathematicians were still searching for roots, but now the instructions were written in binary code.

Some imaginary friends you never outgrow.

NOTES:

• The story of the search for solutions to increasingly complicated equations, from quadratic to quintic, is recounted in vivid detail in:

M. Livio, The Equation That Couldn’t Be Solved (Simon and Schuster, 2005).

• To learn more about imaginary and complex numbers, their applications and their checkered history, see:

P.J. Nahin, An Imaginary Tale (Princeton University Press, 1998);

B. Mazur, Imagining Numbers (Farrar, Straus and Giroux, 2003).

• For a superb journalistic account of John Hubbard’s work, see:

J. Gleick, Chaos: Making a New Science (Viking, 1987), p. 217.

• Hubbard’s own take on Newton’s method appears in Section 2.8 of:
J. Hubbard and B.B. Hubbard, Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, 4^th edition (Matrix Editions, 2009).

• For readers who want to delve into the mathematics of Newton’s method, a more sophisticated but still readable introduction is given in:
H.-O. Peitgen and P.H. Richter, The Beauty of Fractals (Springer, 1986), chapter 6, and also see the article by A. Douady (Hubbard’s collaborator) entitled “Julia sets and the Mandelbrot set,” starting on p.161 of the same book.

• The snapshots and animations shown here were computed using Newton’s method applied to the polynomial z³ – 1.  The roots are the three cube roots of 1.  For this case, Newton’s algorithm takes a point z in the complex plane and maps it to a new point

z – (z³ – 1)/(3z²).

That point then becomes the next z.  This process is repeated until z comes sufficiently close to a root, or equivalently, until z³ – 1 comes sufficiently close to zero, where “sufficiently close” is a very small distance, arbitrarily chosen by the person who programmed the computer.  All initial points that lead to a particular root are then assigned the same color.  Thus red labels all the points that converge to one root, green labels another, and blue labels the third.

• The snapshots of the resulting “Newton fractal” were kindly provided by Simon Tatham.  For more on his work, see his web site.

• The video animation of the Newton fractal was created by Teamfresh.  Stunningly deep zooms into other fractals, including the famous Mandelbrot set, are available here: http://www.hd-fractals.com.

• Hubbard was not the first mathematician to ask questions about Newton’s method in the complex plane; Arthur Cayley had wondered about the same things in 1879.  He too looked at both quadratic and cubic polynomials, and realized that the first case was easy and the second was hard.  Although he couldn’t have known about the fractals discovered a century later, he clearly understood that something nasty could happen when there were more than two roots.  The final sentence of his one-page article in the American Journal of Mathematics (reprinted here) is a marvel of understatement:  “The solution is easy and elegant in the case of a quadric equation, but the next succeeding case of the cubic equation appears to present considerable difficulty.”

• For an introduction to the ancient Indian methods for finding square roots, see:
D. W. Henderson, Experiencing Geometry on Plane and Sphere, (Prentice Hall, 1996).

• Thanks to Carole Schiffman and John Smillie for their comments and suggestions, and to Margaret Nelson for preparing the line drawings.

• I am especially grateful to Teamfresh for creating the animation of the Newton fractal, and to Simon Tatham for computing the snapshots of it.  Both of them generously provided their expert help on very short notice.

Steven Strogatz, New York Times

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Full article and photos: http://opinionator.blogs.nytimes.com/2010/03/07/finding-your-roots/

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The Joy of X

At this stage in the series it’s time to shift gears, moving on from grade school arithmetic to high school math.

Over the next few weeks we’ll be revisiting algebra, geometry and trig.  Don’t worry if you’ve forgotten them all — there won’t be any tests this time around, so instead of worrying about details, we have the luxury of concentrating on the most beautiful, important and far-reaching ideas.

Algebra, for example, may have once struck you as a dizzying mix of symbols, definitions and procedures, but in the end they all boil down to just two activities — solving for x and working with formulas.

Solving for x is detective work.  You’re searching for an unknown number, x.  You’ve been handed a few clues about it, either in the form of an equation like 2x + 3 = 7, or, less conveniently, in a convoluted verbal description of it (as in those scary “word problems”).  In either case, the goal is to identify x from the information given.

Working with formulas, on the other hand, is a bit like art and science.  Instead of dwelling on a particular x, you’re manipulating and massaging relationships that continue to hold, even as the numbers in them change.  These changing numbers are called “variables,” and they are what truly distinguishes algebra from arithmetic.

The formulas in question might express elegant patterns about numbers for their own sake.  This is where algebra meets art.  Or they might express relationships between numbers in the real world, as they do in the laws of nature for falling objects or planetary orbits or genetic frequencies in a population.  This is where algebra meets science.

This division of algebra into two grand activities is not standard (in fact, I just made it up), but it seems to work pretty well.   In next week’s column I’ll have more to say about solving for x, so for now let’s focus on formulas, starting with some easy examples to clarify the ideas.

One day last year, my daughter Jo realized something about her big sister Leah.  “Dad, there’s always a number between my age and Leah’s.  Right now I’m 6 and Leah’s 8, and 7 is in the middle.  And even when we’re old, like when I’m 20 and she’s 22, there will still be a number in the middle!”

Jo’s observation qualifies as algebra (though no one but a proud father would see it that way) because she was noticing a relationship between two ever-changing variables: her age x and Leah’s age y.  No matter how old each of them would get, Leah would always be two years older: y = x + 2.

Algebra is the language in which such patterns are most naturally phrased. It takes some practice to become fluent in algebra, because it’s loaded with what the French call “faux amis” — false friends that sound right in one language (in this case, English) but mean something horribly different when translated into another (here, the symbols of algebra).

For example, suppose the length of a hallway is y when measured in yards, and f when measured in feet.  Write an equation that relates y to f.

My friend Grant Wiggins, an education consultant, has been posing this problem to students and faculty for years.  He says that in his experience, students get it wrong more than half the time, even if they have recently taken and passed an algebra course.

If you think the answer is y = 3f, welcome to the club.

It seems like such a straightforward translation of the sentence, “One yard equals three feet.”  But as soon as you try a few numbers, you’ll see that this formula gets everything backwards.  Say the hallway is 10 yards long; everyone knows that’s 30 feet.  Yet when you plug in y = 10 and f = 30, the formula doesn’t work!

The correct formula is f = 3y.  Here 3 really means 3 feet/yard.  When you multiply it by y in yards, the units of yards cancel out and you’re left with units of feet, as you should be.

Checking that the units cancel properly helps avoid this kind of blunder.  For example, it could have saved the Verizon customer service reps (discussed in last week’s column) from confusing dollars and cents.

Another kind of formula is known as an “identity.”  Whenever you factored or multiplied polynomials in algebra class, you were working with identities.  You can use them now to impress your friends with numerical parlor tricks.  Here’s one that impressed the physicist Richard Feynman, no slouch himself at mental math:

“When I was at Los Alamos I found out that Hans Bethe was absolutely topnotch at calculating.  For example, one time we were putting some numbers into a formula, and got to 48 squared.  I reach for the Marchant calculator, and he says, ‘That’s 2300.’  I begin to push the buttons, and he says, ‘If you want it exactly, it’s 2304.’

The machine says 2304.  ‘Gee! That’s pretty remarkable!’ I say.

‘Don’t you know how to square numbers near 50?’ he says.  ‘You square 50 — that’s 2500 — and subtract 100 times the difference of your number from 50 (in this case it’s 2), so you have 2300.  If you want the correction, square the difference and add it on.  That makes 2304.’ ”

Bethe’s trick is based on the identity

(50 + x)² = 2500 + 100x + x².

He had memorized that equation and was applying it for the case where x is –2, corresponding to the number 48 = 50 – 2.

For an intuitive proof of this formula, imagine a square patch of carpet that measures 50 + x on each side.

Then its area is (50 + x) squared, which is what we’re looking for.  But the diagram above shows that this area is made of a 50 by 50 square (this contributes the 2500 to the formula), two rectangles of dimensions 50 by x (each contributes an area of 50x, for a combined total of 100x), and finally the little x by x square gives an area of x squared, the final term in Bethe’s formula.

Relationships like these are not just for theoretical physicists.  An identity similar to Bethe’s is relevant to anyone who has money invested in the stock market.  Suppose your portfolio drops catastrophically by 50 percent one year and then gains 50 percent the next.  Even after that dramatic recovery you’d still be down 25 percent, because 0.5 times 1.5 equals 0.75.

In fact, you never get back to even when you lose and gain by the same percentage in consecutive years. With algebra we can understand why.  It follows from the identity

(1 – x)(1 + x) = 1 – x².

In the down year the portfolio shrinks by a factor 1 – x (where x = 0.5 in the example above), and then grows by a factor 1 + x the following year.  So the net change is a factor of

(1 – x)(1 + x)

and according to the formula above, this equals

1 – x².

The point is that this expression is always less than 1, for any x other than 0.  So you never get back to even.

Needless to say, not every relationship between variables is as straightforward as those above. Yet the allure of algebra is seductive, and in gullible hands it spawns such silliness as a formula for the socially acceptable age difference in a romance.  According to some sites on the Internet, if your age is x, polite society will disapprove if you date someone younger than (x/2) + 7.

In other words, it would be creepy for anyone over 82 to eye my 48 year-old wife, even if she were available.  But 81?  No problem.

Ick.  Ick.  Ick…

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NOTES:

For sticklers, Leah is actually 21 months older than Jo.  Hence Jo’s formula is only an approximation.  Obviously!

Feynman tells the story of Bethe’s trick for squaring numbers close to 50, in:

R. P. Feynman, “Surely You’re Joking, Mr. Feynman!” (W.W. Norton and Company, 1985), p. 193.

The identity about the effect of equal up and down percentage swings in the stock market can be proven symbolically, by multiplying 1 + x by 1 – x, or geometrically, by drawing a diagram similar to that shown above.  If you’re in the mood, try both approaches as an exercise.

The “half your age plus seven” rule about the acceptable age gap in a romantic relationship is discussed here.

Thanks to Carole Schiffman and Grant Wiggins for their comments and suggestions, and to Margaret Nelson for preparing the illustration.

Steven Strogatz, New York Times

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Full article and photo: http://opinionator.blogs.nytimes.com/2010/02/28/the-joy-of-x/

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Division and its Discontents

There’s a narrative line that runs through arithmetic, but many of us missed it in the haze of long division and common denominators. It’s the story of the quest for ever-more versatile numbers.

The “natural numbers” 1, 2, 3 and so on are good enough if all we want to do is count, add and multiply. But once we ask how much remains when everything is taken away, we are forced to create a new kind of number — zero — and since debts can be owed, we need negative numbers too. This enlarged universe of numbers called “integers” is every bit as self-contained as the natural numbers, but much more powerful because it embraces subtraction as well.

A new crisis comes when we try to work out the mathematics of sharing. Dividing a whole number evenly is not always possible … unless we expand the universe once more, now by inventing fractions. These are ratios of integers — hence their technical name, “rational numbers.” Sadly, this is the place where many students hit the mathematical wall.

There are many confusing things about division and its consequences, but perhaps the most maddening is that there are so many different ways to describe a part of a whole.

If you cut a chocolate layer cake right down the middle into two equal pieces, you could certainly say that each piece is “half” the cake. Or you might express the same idea with the fraction 1/2, meaning 1 of 2 equal pieces. (When you write it this way, the slash between the 1 and the 2 is a visual reminder that something is being sliced.) A third way is to say each piece is 50 percent of the whole, meaning literally 50 parts out of 100. As if that weren’t enough, you could also invoke decimal notation and describe each piece as 0.5 of the entire cake.

This profusion of choices may be partly to blame for the bewilderment many of us feel when confronted with fractions, percentages and decimals. A vivid example appears in the movie “My Left Foot,” the true story of the Irish writer, painter and poet Christy Brown. Born into a large working-class family, he suffered from cerebral palsy that made it almost impossible for him to speak or control any of his limbs, except his left foot. As a boy he was often dismissed as mentally disabled, especially by his father, who resented him and treated him cruelly.

A pivotal scene in the movie takes place around the kitchen table. One of Christy’s older sisters is quietly doing her math homework, seated next to her father, while Christy, as usual, is shunted off in the corner of the room, twisted in his chair. His sister breaks the silence: “What’s 25 percent of a quarter?” she asks. Father mulls it over. “Twenty-five percent of a quarter? Uhhh … That’s a stupid question, eh? I mean, 25 percent is a quarter. You can’t have a quarter of a quarter.” Sister responds, “You can. Can’t you, Christy?” Father: “Ha! What would he know?”

Writhing, Christy struggles to pick up a piece of chalk with his left foot. Positioning it over a slate on the floor, he manages to scrawl a 1, then a slash, then something unrecognizable. It’s the number 16, but the 6 comes out backwards. Frustrated, he erases the 6 with his heel and tries again, but this time the chalk moves too far, crossing through the 6, rendering it indecipherable. “That’s only a nervous squiggle,” snorts his father, turning away. Christy closes his eyes and slumps back, exhausted.

Aside from the dramatic power of the scene, what’s striking is the father’s conceptual rigidity. What makes him insist you can’t have a quarter of a quarter? Maybe he thinks you can only take a quarter of a whole or of something made of four equal parts. But what he fails to realize is that everything is made of four equal parts. In the case of something that’s already a quarter, its four equal parts look like this:

Since 16 of these thin slices make the original whole, each slice is 1/16 of the whole — the answer Christy was trying to scratch out.

A version of the same kind of mental rigidity, updated for the digital age, made the rounds on the Internet a few years ago when a frustrated customer named George Vaccaro recorded and posted his phone conversation with two service representatives at Verizon Wireless. Vaccaro’s complaint was that he’d been quoted a data usage rate of .002 cents per kilobyte, but his bill showed he’d been charged .002 dollars per kilobyte, a hundredfold higher rate. The ensuing conversation climbed to the top 50 in YouTube’s comedy section.

About halfway through the recording, a highlight occurs in the exchange between Vaccaro and Andrea, the Verizon floor manager:

V: “Do you recognize that there’s a difference between one dollar and one cent?”
A: “Definitely.”
V: “Do you recognize there’s a difference between half a dollar and half a cent?”
A: “Definitely.”
V: “Then, do you therefore recognize there’s a difference between .002 dollars and .002 cents?”
A: “No.”
V: “No?”
A: “I mean there’s … there’s no .002 dollars.”

A few moments later Andrea says, “Obviously a dollar is 1.00, right? So what would .002 dollars look like? I’ve never heard of .002 dollars… It’s just not a full cent.”

The challenge of converting between dollars and cents is only part of the problem for Andrea. The real barrier is her inability to envision a portion of either.

From first-hand experience I can tell you what it’s like to be mystified by decimals. In 8th grade Ms. Stanton began teaching us how to convert a fraction into a decimal. Using long division we found that some fractions give decimals that terminate in all zeroes. For example, 1/4 = .2500…, which can be rewritten as .25, since all those zeroes amount to nothing. Other fractions give decimals that eventually repeat, like

5/6 = .8333…

My favorite was 1/7, whose decimal counterpart repeats every six digits:

1/7 = .142857142857….

The bafflement began when Ms. Stanton pointed out that if you triple both sides of the simple equation

1/3 = .3333…,

you’re forced to conclude that 1 must equal .9999…

At the time I protested that they couldn’t be equal. No matter how many 9’s she wrote, I could write just as many 0’s in 1.0000… and then if we subtracted her number from mine, there would be a teeny bit left over, something like .0000…01.

Like Christy’s father and the Verizon service reps, my gut couldn’t accept something that had just been proven to me. I saw it but refused to believe it. (This might remind you of some people you know.)

But it gets worse — or better, if you like to feel your neurons sizzle. Back in Ms. Stanton’s class, what stopped us from looking at decimals that neither terminate nor repeat periodically? It’s easy to cook up such stomach-churners. Here’s an example:

0.12122122212222…

By design, the blocks of 2 get progressively longer as we move to the right. There’s no way to express this decimal as a fraction. Fractions always yield decimals that terminate or eventually repeat periodically — that can be proven — and since this decimal does neither, it can’t be equal to the ratio of any whole numbers. It’s “irrational.”

Given how contrived this decimal is, you might suppose irrationality is rare. On the contrary, it is typical. In a certain sense that can be made precise, almost all decimals are irrational. And their digits look statistically random.

Once you accept these astonishing facts, everything turns topsy-turvy. Whole numbers and fractions, so beloved and familiar, now appear scarce and exotic. And that innocuous number line pinned to the molding of your grade school classroom? No one ever told you, but it’s chaos up there.

NOTES:

George Vaccaro’s blog provides the exasperating details of his encounters with Verizon.

The transcript of his conversation with customer service is available here.

For readers who may still find it hard to accept that 1 = .9999…, the argument that eventually convinced me was this. They must be equal, because there’s no room for any other decimal to fit between them. (Whereas if two decimals are unequal, their average is between them, as are infinitely many other decimals.)

The amazing properties of irrational numbers are discussed at a higher mathematical level here.

The sense in which their digits are random is clarified here.

Thanks to Carole Schiffman for her comments and suggestions, and to Margaret Nelson for preparing the illustrations.

Steven Strogatz, New York Times

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Full article and photo: http://opinionator.blogs.nytimes.com/2010/02/21/division-and-its-discontents/

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Rock Groups

Like anything else, arithmetic has its serious side and its playful side.

The serious side is what we all learned in school: how to work with columns of numbers, adding them, subtracting them, grinding them through the spreadsheet calculations needed for tax returns and year-end reports. This side of arithmetic is important, practical and — for many people — joyless.

The playful side of arithmetic is a lot less familiar, unless you were trained in the ways of advanced mathematics. Yet there’s nothing inherently advanced about it. It’s as natural as a child’s curiosity.

In his book “A Mathematician’s Lament,” Paul Lockhart advocates an educational approach in which numbers are treated more concretely than usual: he asks us to imagine them as groups of rocks. For example, six corresponds to a group of rocks like this:

You probably don’t see anything striking here, and that’s right — unless we make further demands on numbers, they all look pretty much the same. Our chance to be creative comes in what we ask of them.

For instance, let’s focus on groups having between 1 and 10 rocks in them, and ask which of these can be rearranged into square patterns. Only two of them can: 4 and 9. And that’s because 4 = 2 × 2 and 9 = 3 × 3; we get these numbers by “squaring” some other number (actually making a square shape).

A less stringent challenge is to identify groups of rocks that can be neatly organized into rectangles with two rows that come out even. That’s possible as long as there are 2, 4, 6, 8 or 10 rocks; the number has to be “even.” All the other numbers from 1 to 10 — the “odd” numbers — always leave an odd bit sticking out.

Still, all is not lost for these misfit numbers. If we add two of them together, their protuberances match up and their sum comes out even; Odd + Odd = Even.

Yet when it comes to rectangles, some numbers, like 2, 3, 5 and 7, truly are hopeless. They can’t form any sort of rectangles at all, other than a simple line of rocks. These strangely inflexible numbers are the famous “prime” numbers.

So we see that numbers have quirks of structure that endow them with personalities. But to see the full range of their behavior, we need to go beyond individual numbers and watch what happens when they interact.

For example, instead of adding just two odd numbers together, suppose we add all the consecutive odd numbers, starting from 1:

1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25

The sums above, remarkably, always turn out to be perfect squares. (We saw 4 and 9 in the square patterns discussed earlier, and 16 = 4 × 4, and 25 = 5 × 5.) A quick check shows that this rule keeps working for larger and larger odd numbers; it apparently holds all the way out to infinity. But what possible connection could there be between odd numbers, with their ungainly appendages, and the classically symmetrical numbers that form squares? By arranging our rocks in the right way, we can make this surprising link seem obvious — the hallmark of an elegant proof.

The key is to recognize that odd numbers can make L-shapes, with their protuberances cast off into the corner. And when you stack successive L-shapes together, you get a square!

This style of thinking appears in another recent book, though for altogether different literary reasons. In Yoko Ogawa’s charming novel “The Housekeeper and the Professor,” an astute but uneducated young woman with a 10-year-old son is hired to take care of the Professor, an elderly mathematician who has suffered a traumatic brain injury that leaves him with only 80 minutes of short-term memory. Adrift in the present, and alone in his shabby cottage with nothing but his numbers, the Professor tries to connect with the Housekeeper the only way he knows how: by inquiring about her shoe size or birthday and making mathematical small talk about her statistics. The Professor also takes a special liking to the Housekeeper’s son, whom he calls Root, because the flat top of the boy’s head reminds him of the square root symbol, .

One day the Professor gives Root a little puzzle: Can he find the sum of all the numbers from 1 to 10? After Root carefully adds the numbers and returns with the answer (55), the Professor asks him to find a better way. Can he find the answer without adding the numbers? Root kicks the chair and shouts, “That’s not fair!”

But little by little the Housekeeper gets drawn into the world of numbers, and she secretly starts exploring the puzzle herself. “I’m not sure why I became so absorbed in a child’s math problem with no practical value,” she says. “At first I was conscious of wanting to please the Professor, but gradually that feeling faded and it had become a battle between the problem and me. When I woke in the morning the equation was waiting:

1 + 2 + 3 + … + 9 + 10 = 55

and it followed me all through the day, as though it had burned itself into my retina and could not be ignored.”

There are several ways to solve the Professor’s problem (see how many you can find). The Professor himself gives an argument along the lines we developed above. He interprets the sum from 1 to 10 as a triangle of rocks, with 1 rock in the first row, 2 in the second and so on, up to 10 rocks in the 10th row:

By its very appearance this picture gives a clear sense of negative space. It seems only half complete. And that suggests a creative leap. If you copy the triangle, flip it upside down and add it as the missing half to what’s already there, you get something much simpler: a rectangle with 10 rows of 11 rocks each, for a total of 110.

Since the original triangle is half of this rectangle, the desired sum must be half of 110, or 55.

Looking at numbers as groups of rocks may seem unusual, but actually it’s as old as math itself. The word “calculate” reflects that legacy — it comes from the Latin word “calculus,” meaning a pebble used for counting. To enjoy working with numbers you don’t have to be Einstein (German for “one stone”), but it might help to have rocks in your head.

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NOTES:

As I hope I’ve made clear, this piece owes much to two books — one a polemic, the other a novel, both of them brilliant.

The rock metaphor and many of the other ideas and examples above have been borrowed from: Paul Lockhart, “A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form” (Bellevue Literary Press, 2009).

The final example is from: Yoko Ogawa, “The Housekeeper and the Professor” (Picador, 2009).

For young readers who like exploring numbers and the patterns they make, see:

Hans Magnus Enzensberger, “The Number Devil: A Mathematical Adventure” (Holt Paperbacks, 2000).

For elegant but more advanced examples of visualization in mathematics, see:

Roger B. Nelsen, “Proofs without Words: Exercises in Visual Thinking” (Mathematical Association of America, 1997).

Thanks to Carole Schiffman and Tim Novikoff for their comments and suggestions, and to Margaret Nelson for preparing the illustrations.

Steven Strogatz, New York Times

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Full article and photos: http://opinionator.blogs.nytimes.com/2010/02/07/rock-groups/

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The Enemy of My Enemy

It’s traditional to teach kids subtraction right after addition.  That makes sense — the same facts about numbers get used in both, though in reverse.  And the black art of “borrowing,” so crucial to successful subtraction, is only a little more baroque than that of “carrying,” its counterpart for addition.  If you can cope with calculating 23 + 9, you’ll be ready for 23 – 9 soon enough.

At a deeper level, however, subtraction raises a much more disturbing issue, one that never arises with addition.  Subtraction can generate negative numbers.  If I try to take 6 cookies away from you but you only have 2, I can’t do it — except in my mind, where you now have negative 4 cookies, whatever that means.

Subtraction forces us to expand our conception of what numbers are.  Negative numbers are a lot more abstract than positive numbers — you can’t see negative 4 cookies and certainly can’t eat them — but you can think about them, and you have to, in all aspects of daily life, from debts and overdrafts to contending with freezing temperatures and parking garages.

Still, many of us haven’t quite made peace with negative numbers.  As my colleague Andy Ruina has pointed out, people have concocted all sorts of funny little mental strategies to sidestep the dreaded negative sign.  On mutual fund statements, losses (negative numbers) are printed in red or nestled in parentheses, without a negative sign to be found.   The history books tell us that Julius Caesar was born in 100 B.C., not –100.  The subterranean levels in a parking garage often have names like B1 and B2.  Temperatures are one of the few exceptions: folks do say, especially here in Ithaca, that it’s –5 degrees outside, though even then, many prefer to say 5 below zero.  There’s something about that negative sign that just looks so unpleasant, so … negative.

Perhaps the most unsettling thing is that a negative times a negative is a positive.  So let me try to explain the thinking behind it.

How should we define something like –1 × 3, where we’re multiplying a negative number by a positive number?  Well, just as 1 × 3 means 1 + 1 + 1, the natural definition for –1 × 3 is (–1) + (–1) + (–1), which equals –3.  This should be obvious in terms of money: if you owe me $1 a week, after three weeks you’re $3 in the hole.

From there it’s a short hop to see why a negative times a negative should be a positive.  Take a look at the following string of equations:

–1 × 3 = –3

–1 × 2 = –2

–1 × 1 = –1

–1 × 0 = 0

–1 × –1 = ?

Now look at the numbers on the far right and notice their orderly progression:

–3, –2, –1, 0, ?

At each step, we’re adding 1 to the number before it.  So wouldn’t you agree the next number should logically be 1?

That’s one argument for why (–1) × (–1) = 1.  The appeal of this definition is that it preserves the rules of ordinary arithmetic; what works for positive numbers also works for negative numbers.

But if you’re a hard-boiled pragmatist, you may be wondering if these abstractions have any parallels in the real world.  Admittedly, life sometimes seems to play by different rules.  In conventional morality, two wrongs don’t make a right.  Likewise, double negatives don’t always amount to positives; they can make negatives more intense, as in “I can’t get no satisfaction.”  (Actually, languages can be very tricky in this respect.  The eminent linguistic philosopher J. L. Austin of Oxford once gave a lecture in which he asserted that there are many languages in which a double negative makes a positive, but none in which a double positive makes a negative — to which the Columbia philosopher Sidney Morgenbesser, sitting in the audience, sarcastically replied, “Yeah, yeah.”)

Still, there are plenty of cases where the real world does mirror the rules of negative numbers.  When a nerve cell inhibits the firing of another that in turn inhibits a third, the indirect action of the first cell on the third is tantamount to excitation; a chain of two negatives makes a positive.  Similar effects occur in gene regulation: a protein can turn a gene on by blocking another molecule that was repressing that stretch of DNA.

Perhaps the most familiar parallel occurs in the social and political realms, as summed up by the adage, “The enemy of my enemy is my friend.”  This truism, and related ones about the friend of my enemy, and so on, can be depicted in relationship triangles.

The corners signify people, companies or countries, and the sides connecting them signify their relationships, which can be positive (friendly, shown here as solid lines) or negative (hostile, shown as dashed lines).

Social scientists refer to triangles like the one on the left, with all sides positive, as “balanced” — there’s no reason for anyone to change how they feel, since it’s reasonable to like your friend’s friends.  Similarly, the triangle on the right, with two negatives and a positive, is considered balanced because it causes no dissonance; even though it allows for hostility, nothing cements a friendship like hating the same person.

Of course triangles can also be unbalanced.  When three mutual enemies size up the situation, two of them — often the two with the least animosity — may be tempted to join forces and gang up on the third.

Even more unbalanced is a triangle with a single negative relationship.  For instance, suppose Carol is friendly with both Alice and Bob, but Bob and Alice despise each other.  Perhaps they were once a couple but have now suffered a nasty break-up, and each is badmouthing the other to ever-loyal Carol.  This causes psychological stress all around.  To restore balance, either Alice and Bob have to reconcile, or Carol has to choose a side.

Leaving aside the verisimilitude of the model, there are interesting questions here of a purely mathematical flavor.  For example, in a close-knit network where everyone knows everyone, what’s the most stable state?  One possibility is a nirvana of goodwill, where all relationships are positive and all triangles are balanced.  But surprisingly, there are other states that are equally stable.  These are states of intractable conflict, with the network split into two hostile factions.  All members of a given faction are friendly with one another, but antagonistic toward everybody in the other faction.  (Sound familiar?)  Perhaps even more surprisingly, these polarized states are the only states as stable as nirvana.  In particular, no three-party split can have all its triangles balanced.

Scholars have used these ideas, for example, to analyze the run-up to World War I.  The diagram below shows the shifting alliances among Great Britain, France, Russia, Italy, Germany and Austria-Hungary between 1872 and 1907.

The first five configurations were all unbalanced, in the sense that they each contained at least one unbalanced triangle. The resultant dissonance tended to push these nations to realign themselves, triggering reverberations elsewhere in the network.  In the final stage, Europe had split into two implacably opposed blocs — technically “balanced” but on the brink of war.

The point is not that this theory is powerfully predictive.  It isn’t.  It’s too simple to account for all the subtleties of geopolitical dynamics.  The point is that some part of what we observe is due to nothing more than the primitive logic of “the enemy of my enemy,” and this part is captured perfectly by the multiplication of negative numbers.  By sifting the generic from the meaningful, the arithmetic of negative numbers can help us see where the real puzzles lie.

NOTES:

For more of Sidney Morgenbesser’s witticisms and academic one-liners, see the sampling here.

Balance theory was first proposed by the social psychologist Fritz Heider, and has since been developed and applied by social network theorists, political scientists, anthropologists, mathematicians and physicists.  For the original formulation, see:

F. Heider, Journal of Psychology, Vol. 21 (1946), p.107;

F. Heider, The Psychology of Interpersonal Relations (John Wiley and Sons, 1958).

For a review of balance theory from a social network perspective, see:

S. Wasserman and K. Faust, Social Network Analysis: Methods and Applications (Cambridge University Press, 1994), chapter 6.

The theorem that a balanced state in a fully connected network consists of either a nirvana of all friends, or two mutually antagonistic factions, was first proven in:

D. Cartwright and F. Harary, Psychological Review, Vol. 63 (1956), p.277.

A very readable version of that proof, and a gentle introduction to the mathematics of balance theory, has been given by two of my colleagues at Cornell:

D. Easley and J. Kleinberg, Networks, Crowds and Markets: Reasoning about a Highly Connected World (Cambridge University Press, 2010).

In much of the early work on balance theory, a triangle of three mutual enemies (and hence three negative sides) was considered unbalanced.  I assumed this implicitly when quoting the results about nirvana and the two-bloc state as being the only configurations of a fully connected network in which all triangles are balanced.  However, some researchers have challenged this assumption, and have explored the implications of treating a triangle of three negatives as balanced.  For more on this and other generalizations of balance theory, see the books by Wasserman and Faust and by Easley and Kleinberg, cited above.

The example and graphical depiction of the shifting alliances before World War I are from:

T. Antal, P.L. Krapivsky and S. Redner, Physica D 224 (2006), p.130, available online here and here.

This paper, written by three statistical physicists, is notable for recasting balance theory in a dynamic framework, thus extending it beyond the earlier static approaches.

For the historical details of the European alliances, see:

W.L. Langer, European Alliances and Alignments, 1871-1890, 2nd ed. (Knopf, 1956);

B.E. Schmitt, Triple Alliance and Triple Entente (Henry Holt and Company, 1934).

Thanks to Carole Schiffman, Andy Ruina and Jon Kleinberg for their comments and suggestions, and to Margaret Nelson for preparing the illustrations.

Steven Strogatz, New York Times

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Full article and photos: http://opinionator.blogs.nytimes.com/2010/02/14/the-enemy-of-my-enemy

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From Fish to Infinity

I have a friend who gets a tremendous kick out of science, even though he’s an artist. Whenever we get together all he wants to do is chat about the latest thing in evolution or quantum mechanics. But when it comes to math, he feels at sea, and it saddens him. The strange symbols keep him out. He says he doesn’t even know how to pronounce them.

In fact, his alienation runs a lot deeper. He’s not sure what mathematicians do all day, or what they mean when they say a proof is elegant. Sometimes we joke that I just should sit him down and teach him everything, starting with 1 + 1 = 2 and going as far as we can.

Crazy as it sounds, over the next several weeks I’m going to try to do something close to that. I’ll be writing about the elements of mathematics, from pre-school to grad school, for anyone out there who’d like to have a second chance at the subject — but this time from an adult perspective. It’s not intended to be remedial. The goal is to give you a better feeling for what math is all about and why it’s so enthralling to those who get it.

So, let’s begin with pre-school. 

The best introduction to numbers I’ve ever seen — the clearest and funniest explanation of what they are and why we need them — appears in a “Sesame Street” video called “123 Count With Me.” Humphrey, an amiable but dim-witted fellow with pink fur and a green nose, is working the lunch shift at The Furry Arms hotel, when he takes a call from a room full of penguins. Humphrey listens carefully and then calls out their order to the kitchen: “Fish, fish, fish, fish, fish, fish.” This prompts Ernie to enlighten him about the virtues of the number six.

Children learn from this that numbers are wonderful shortcuts. Instead of saying the word “fish” exactly as many times as there are penguins, Humphrey could use the more powerful concept of “six.”

As adults, however, we might notice a potential downside to numbers. Sure, they are great time savers, but at a serious cost in abstraction. Six is more ethereal than six fish, precisely because it’s more general. It applies to six of anything: six plates, six penguins, six utterances of the word “fish.” It’s the ineffable thing they all have in common.

Viewed in this light, numbers start to seem a bit mysterious. They apparently exist in some sort of Platonic realm, a level above reality. In that respect they are more like other lofty concepts (e.g., truth and justice), and less like the ordinary objects of daily life. Upon further reflection, their philosophical status becomes even murkier. Where exactly do numbers come from? Did humanity invent them? Or discover them?

A further subtlety is that numbers (and all mathematical ideas, for that matter) have lives of their own. We can’t control them. Even though they exist in our minds, once we decide what we mean by them we have no say in how they behave. They obey certain laws and have certain properties, personalities, and ways of combining with one another, and there’s nothing we can do about it except watch and try to understand. In that sense they are eerily reminiscent of atoms and stars, the things of this world, which are likewise subject to laws beyond our control … except that those things exist outside our heads.

This dual aspect of numbers — as part- heaven, and part- earth — is perhaps the most paradoxical thing about them, and the feature that makes them so useful. It is what the physicist Eugene Wigner had in mind when he wrote of “the unreasonable effectiveness of mathematics in the natural sciences.”

In case it’s not clear what I mean about the lives of numbers and their uncontrollable behavior, let’s go back to the The Furry Arms. Suppose that Humphrey suddenly gets a call on another line, from a room occupied by as many penguins as before, also clamoring for fish. After taking both calls, what should Humphrey yell out to the kitchen? If he hasn’t learned anything, he could shout “fish” once for each penguin. Or, using his numbers, he could tell the cook he needs six orders of fish for the first room and six more for the second room. But what he really needs is a new concept: addition. Once he’s mastered it, he’ll proudly say he needs six plus six (or, if he’s a show-off, 12) fish.

The creative process here is the same as the one that gave us numbers in the first place. Just as numbers are a shortcut for counting by ones, addition is a shortcut for counting by any amount. This is how mathematics grows. The right abstraction leads to new insight, and new power.

Before long, even Humphrey might realize he can keep counting forever.

Yet despite this infinite vista, there are always constraints on our creativity. We can decide what we mean by things like 6 and +, but once we do, the results of equations like 6 + 6 are beyond our control. In mathematics, we’ll see in the coming weeks, our freedom lies in the questions we ask — and in how we pursue them — but not in the answers awaiting us.

Notes: 

For the Sesame Street video, see “Sesame Street – 123 Count With Me (1997).” It is available for purchase online in either VHS or DVD format.

For the famous essay on the unreasonable effectiveness of mathematics, see:
E. Wigner, “The unreasonable effectiveness of mathematics in the natural sciences,” Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960), pp. 1-14. A pdf version is here.

For a passionate presentation of the ideas that numbers have lives of their own and that mathematics can be viewed as a form of art, see: P. Lockhart, “A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form” (Bellevue Literary Press, 2009).

Steven Strogatz, New York Times

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Full article: http://opinionator.blogs.nytimes.com/2010/01/31/from-fish-to-infinity/

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Russia’s Conquering Zeros

The strength of post-Soviet math stems from decades of lonely productivity

It may be no accident that, while some of the best American mathematical minds worked to solve one of the century’s hardest problems—the Poincaré Conjecture—it was a Russian mathematician working in Russia who, early in this decade, finally triumphed.

Decades before, in the Soviet Union, math placed a premium on logic and consistency in a culture that thrived on rhetoric and fear; it required highly specialized knowledge to understand; and, worst of all, mathematics lay claim to singular and knowable truths—when the regime had staked its own legitimacy on its own singular truth. All this made mathematicians suspect. Still, math escaped the purges, show trials and rule by decree that decimated other Soviet sciences.

Three factors saved math. First, Russian math happened to be uncommonly strong right when it might have suffered the most, in the 1930s. Second, math proved too obscure for the sort of meddling Joseph Stalin most liked to exercise: It was simply too difficult to ignite a passionate debate about something as inaccessible as the objective nature of natural numbers (although just such a campaign was attempted). And third, at a critical moment math proved immensely useful to the state.

Three weeks after Nazi Germany invaded the Soviet Union in June 1941, the Soviet air force had been bombed out of existence. The Russian military set about retrofitting civilian airplanes for use as bombers. The problem was, the civilian airplanes were much slower than the military ones, rendering moot everything the military knew about aim.

What was needed was a small army of mathematicians to recalculate speeds and distances to let the air force hit its targets.

The greatest Russian mathematician of the 20th century, Andrei Kolmogorov, led a classroom of students, armed with adding machines, in recalculating the Red Army’s bombing and artillery tables. Then he set about creating a new system of statistical control and prediction for the Soviet military.

Following the war, the Soviets invested heavily in high-tech military research, building over 40 cities where scientists and mathematicians worked in secret. The urgency of the mobilization recalled the Manhattan Project—only much bigger and lasting much longer. Estimates of the number of people engaged in the Soviet arms effort in the second half of the century range up to 12 million people, with a couple million of them employed by military-research institutions.

These jobs spelled nearly total scientific isolation: For defense employees, any contact with foreigners would be considered treasonous rather than simply suspect. In addition, research towns provided comfortably cloistered social environments but no possibility for outside intellectual contact. The Soviet Union managed to hide some of its best mathematical minds away in plain sight.

In the years following Stalin’s death in 1953, the Iron Curtain began to open a tiny crack—not quite enough to facilitate much-needed conversation with non-Soviet mathematicians but enough to show off some of Soviet mathematics’ proudest achievements.

By the 1970s, a Soviet math establishment had taken shape. A totalitarian system within a totalitarian system, it provided its members not only with work and money but also with apartments, food, and transportation. It determined where they lived and when, where, and how they traveled for work or pleasure. To those in the fold, it was a controlling and strict but caring mother: Her children were undeniably privileged.

Even for members of the math establishment, though, there were always too few good apartments, too many people wanting to travel to a conference. So it was a vicious, back-stabbing little world, shaped by intrigue, denunciations and unfair competition.

Then there were those who could never join the establishment: those who happened to be born Jewish or female, those who had had the wrong advisers at university or those who could not force themselves to join the Party. For these people, “the most they could hope for was being able to defend their doctoral dissertation at some institute in Minsk, if they could secure connections there,” says Sergei Gelfand, publisher of the American Mathematical Society—who also happens to be the son of one of Russia’s top 20th-century mathematicians, Israel Gelfand, a student of Mr. Kolmogorov. Some Western mathematicians, Sergei Gelfand adds, “even came for an extended stay because they realized there were a lot of talented people. This was unofficial mathematics.”

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Math Stars

Besides Grigory Perelman and the Poincaré Conjecture, there are numerous other famous math solvers, and there are still problems to solve.

Andrew Wiles (1953-)

This Princeton mathematician resolved the most famous problem in numbers—Fermat’s Last Theorem—in 1995.

Leonhard Euler (1707–1783)

A Swiss mathematician who made so many contributions, particularly in the early foundations of calculus, that it gets hard to keep track of all that’s named for him.

Kurt Gödel (1906–1978)

This Austrian logician demonstrated that any reasonably powerful system of math contains true statements that can’t be proven.

The Riemann Hypothesis

To the enduring befuddlement of mathematicians, prime numbers—numbers divisible only by themselves and 1—exhibit no pattern at all: 2, 3, 5, 7, 11, 13 are the first few. They aren’t evenly spaced but get scarcer the further out you go. No formula can tell you what the next one will be. In 1859, the German mathematician Bernhard Riemann discovered that a function—known now as the Riemann zeta function (expressed in the graphic above)—appeared to give signposts to where primes lie in the great field of numbers. It provided some order to the mystery. Riemann conjectured that these key signposts—”zeros” of the function—all lie on a single straight line out to infinity, that none are flung off in strange places. In the 150 years since, no one has proved his hypothesis. To a mathematician, the hypothesis looks like this: All non-trivial zeros of the Riemann zeta function have a real part equal to ½.

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One such visitor was Dusa McDuff, then a British algebraist and now a professor emerita at the State University of New York at Stony Brook. She studied with the older Mr. Gelfand for six months, and credits this experience to opening her eyes both to what mathematics really is: “It was a wonderful education… Gelfand amazed me by talking of mathematics as though it were poetry.”

In the mathematical counterculture, math “was almost a hobby,” recalls Sergei Gelfand. “So you could spend your time doing things that would not be useful to anyone for the nearest decade.” Mathematicians called it “math for math’s sake.” There was no material reward in this—no tenure, no money, no apartments, no foreign travel; all they stood to gain was the respect of their peers.

Math not only held out the promise of intellectual work without state interference (if also without its support) but also something found nowhere else in late-Soviet society: a knowable singular truth. “If I had been free to choose any profession, I would have become a literary critic,” says Georgii Shabat, a well-known Moscow mathematician. “But I wanted to work, not spend my life fighting the censors.” The search for that truth could take long years—but in the late Soviet Union, time seemed to stand still.

When it all collapsed, the state stopped investing in math and holding its mathematicians hostage. It’s hard to say which of these two factors did more to send Russian mathematicians to the West, primarily the U.S., but leave they did, in what was probably one of the biggest outflows of brainpower the world has ever known. Even the older Mr. Gelfand moved to the U.S. and taught at Rutgers University for nearly 20 years, almost until his death in October at the age of 96. The flow is probably unstoppable by now: A promising graduate student in Moscow or St. Petersburg, unable to find a suitable academic adviser at home, is most likely to follow the trail to the U.S.

But the math culture they find in America, while less back-stabbing than that of the Soviet math establishment, is far from the meritocratic ideal that Russia’s unofficial math world had taught them to expect. American math culture has intellectual rigor but also suffers from allegations of favoritism, small-time competitiveness, occasional plagiarism scandals, as well as the usual tenure battles, funding pressures and administrative chores that characterize American academic life. This culture offers the kinds of opportunities for professional communication that a Soviet mathematician could hardly have dreamed of, but it doesn’t foster the sort of luxurious, timeless creative work that was typical of the Soviet math counterculture.

For example, the American model may not be able to produce a breakthrough like the proof of the Poincaré Conjecture, carried out by the St. Petersburg mathematician Grigory Perelman.

Mr. Perelman came to the United States as a young postdoctoral student in the early 1990s and immediately decided that America was math heaven; he wrote home demanding that his mother and his younger sister, a budding mathematician, move here. But three years later, when his postdoc hiatus was over and he was faced with the pressures of securing an academic position, he returned home, disillusioned.

In St. Petersburg he went on the (admittedly modest) payroll of the math research institute, where he showed up infrequently and generally kept to himself for almost seven years, one of the greatest mathematical discoveries of at least the last hundred years. It’s all but impossible to imagine an American institution that could have provided Mr. Perelman with this kind of near-solitary existence, free of teaching and publishing obligations.

After posting his proof on the Web, Mr. Perelman traveled to the U.S. in the spring of 2003, to lecture at a couple of East Coast universities. He was immediately showered with offers of professorial appointments and research money, and, by all accounts, he found these offers gravely insulting, as he believes the monetization of achievement is the ultimate insult to mathematics. So profound was his disappointment with the rewards he was offered that, I believe, it contributed a great deal to his subsequent decision to quit mathematics altogether, along with the people who practice it. (He now lives with his mother on the outskirts of St. Petersburg.)

A child of the Soviet math counterculture, he still held a singular truth to be self-evident: Math as it ought to be practiced, math as the ultimate flight of the imagination, is something money can’t buy.

Masha Gessen’s latest book is “Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century,” a story of Grigory Perelman and the Poincaré Conjecture. She lives in Moscow and is the author of three previous books.

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Full article and photos: http://online.wsj.com/article/SB10001424052748703740004574513870490836470.html

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Number-Crushing: When Figures Get Personal

Real-Estate Developers Factor In Love of 6 and 8, Fear of Unlucky 4 and 13; What Happened to Floors 40 Through 59?

Everyone can agree that 1+1=2. But the idea that 7 is greater than 13 — that some numbers are luckier than others — makes no sense to some people. Such numerical biases can cause deep divisions.

And that is what happened earlier this month in Hong Kong. Property developer Henderson Land Development Co. made news for selling a condominium for $56.6 million, a price the developer called a residential record in Asia. But after that sale was announced, the property began making news for other unusual numbers. Henderson is labeling the floors of its property at 39 Conduit Road with numbers that increase, but not in the conventional 1-then-2 way. The floor above 39, for example, is 60. And the top three floors are consecutively labeled 66, 68 and 88.

In Las Vegas, where lucky numbers such as 7 are always welcome, couples gathered at Mandalay Bay to wed on July 7, 2007.

This offended some people’s sense of order. At a protest Sunday against high housing prices, Hong Kong Democratic Party legislators expressed dissatisfaction with the numbering scheme’s tenuous relationship to reality. “You could call the ground floor the 88th floor, but it’s meaningless,” says Emily Lau. “When you say you live on the 88th floor, people expect you to be on the 88th floor, not the 10th floor or something.”

Numerology, a belief that certain digits have greater meaning beyond merely their quantity, has long been been viewed as a kind of loony uncle to mathematics. Numerologists favor or fear certain numbers depending on factors such as the sound of the words for those numbers or the letter in the alphabet they correspond to. That kind of reasoning leads some mathematicians, who are governed by numerical laws and properties, to believe they have one up on numerologists.

But many mathematicians have their own emotional attachments to numbers that drove them to enter the field in the first place. Some will cop to having numerical crushes that might not look that different from numerologists’.

“The idea that numbers are somehow pure and immune to superstitious thinking, because they’re somehow more ‘objective’ than words, doesn’t take into account the fact that every concept exists (in our minds) in an interconnected tapestry of emotionally and culturally charged signifiers,” Golan Levin, designer of the interactive project The Secret Lives of Numbers, which tracks the popularity of every whole number between one and one million, writes in an email. He considers most numerical superstitions harmless.

Thomas Garrity, a mathematician at Williams College, has always had a particular fondness for the number 9. The number 51, however, doesn’t make his favorites list.

“This might stem from childhood, when I regularly thought that 51 should be prime, even though 51=3×17,” he says, taking a trip down mathematical memory lane. But he doesn’t base decisions on his preferences, for instance by avoiding the 51st floor of buildings, he says. “I can understand people having slightly irrational feelings about particular numbers,” Prof. Garrity says. “I don’t get, though, people making real decisions based on such feelings.”

And yet some numerical superstitions do spread, especially when profits are involved. A Las Vegas casino that caters to Hong Kong high rollers also skips floors from 40 to 59, while Henderson’s Hong Kong development omits the 13th floor to cater to Western tastes.

A Henderson spokeswoman says customers “don’t want the fours and the unlucky numbers. These numbers are more interesting.”

Henderson chose to name the floors as it did because of positive associations with 6 and 8, and negative ones with 4. In Cantonese and Mandarin, the word for eight sounds like “faat,” which means prosperity. Hence the Beijing Olympics starting time of 8 p.m. on Aug. 8, 2008. The word for four, meanwhile, “sounds very much like ‘death,’ and is therefore avoided at all costs,” says Hung-Hsi Wu, professor emeritus of mathematics at University of California, Berkeley, who was born in Hong Kong. Six is also considered lucky.

A preference for six over four also guided developers of the 42-floor Mandalay Bay casino in Las Vegas. There, penthouses are on the 60th, 61st and 62nd floor because Mandalay Bay skips the numbers 40 to 59.

Gordon Absher, spokesman for Mandalay owner MGM Mirage, says that decision was shaped by possible perceptions of high rollers when they are assigned to those floors. “You could think that we are trying to, as the casino, give you bad luck,” Mr. Absher says.

Similarly, developers who would assuage fears of 13 can’t avoid the existence of a 13th floor in buildings with 13 or more stories. But they can rename it out of existence. When a 13th floor was added to the Skirvin Hotel in Oklahoma City, in the 1930s, it was named the 14th floor. The hotel was shuttered in 1988 and reopened and renamed in 2007 by Hilton, which nonetheless kept the name for the top floor.

The 22-story headquarters of Chicago-based Marc Realty avoids throwing off the numbers in higher floors by labeling the 13th floor “14A.” It labels the 14th floor “14B.”

“That arrangement keeps the elevations of the upper floors straight in a physical sense,” says Marc marketing coordinator Dan Krc. He adds that triskaidekaphobia, or fear of 13, appears to be fading, with floors labeled 13 in Marc properties showing occupancy rates are no lower than other floors.

The negative associations with 13 have been traced to the number of diners at the Last Supper, before the betrayal of Jesus. Some believed it went back to prehistoric times — the lowest number that couldn’t be counted on ten fingers and two feet. (Apparently, individual toes couldn’t be counted).

But Underwood Dudley, retired professor of mathematics at Depauw University and author of “Numerology,” says he wasn’t able to verify any of these. “As far as I can tell, some number had to be unlucky, and it was 13,” Dr. Dudley says.

Beverly Kay, a numerologist in Mequon, Wisc., doesn’t buy fears of 13. However, she says her work reading meaning into clients’ birth dates and names is consistent with math. “This is scientific,” Ms. Kay says.

Psychologists and historians generally have tied such beliefs to the broader human tendency to seek patterns and systems where none exist. At its extreme, an emotional relationship to a number can creep into obsessive-compulsive behavior. In his book “Strange Brains and Genius,” Clifford Pickover dug through case studies of numerical obsessive-compulsive disorder, and found that it could be tied to just about any numeral. Electricity pioneer Nikola Tesla demanded precisely 18 clean towels a day and showed an intense preference for multiples of three.

While mathematicians generally don’t go to Tesla-like extremes, they possess a generally positive outlook about all numbers and that distinguishes them from numerologists, they say.

For example, Kenneth Ribet, a professor of mathematics at Berkeley, considers some prime numbers “friends,” he says. One is 144,169, which reads like 12 squared followed by 13 squared; another the easily remembered number of 1,234,567,891.

“Mathematicians don’t have numbers that they’re afraid of or shy away from because we do really like all of the numbers,” says Prof. Ribet. “On the other hand, some of us have favorites.”

Corrections & Amplifications

The Book of Revelation identifies 666 as the number of the beast. The  graphic accompanying the Numbers Guy column on October 28 incorrectly called it the Book of Revelations.

Carl Bialik, Wall Street Journal

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Full article and photos: http://online.wsj.com/article/SB125668948820711987.html

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Mathematicians Solve ‘Trillion Triangle’ Problem

Mathematicians from North America, Europe, Australia, and South America have resolved the first one trillion cases of an ancient mathematics problem. The advance was made possible by a clever technique for multiplying large numbers. The numbers involved are so enormous that if their digits were written out by hand they would stretch to the moon and back. The biggest challenge was that these numbers could not even fit into the main memory of the available computers, so the researchers had to make extensive use of the computers’ hard drives.

According to Brian Conrey, Director of the American Institute of Mathematics, “Old problems like this may seem obscure, but they generate a lot of interesting and useful research as people develop new ways to attack them.”

The problem, which was first posed more than a thousand years ago, concerns the areas of right-angled triangles. The surprisingly difficult problem is to determine which whole numbers can be the area of a right-angled triangle whose sides are whole numbers or fractions. The area of such a triangle is called a “congruent number.”

For example, the 3-4-5 right triangle which students see in geometry has area 1/2 × 3 × 4 = 6, so 6 is a congruent number. The smallest congruent number is 5, which is the area of the right triangle with sides 3/2, 20/3, and 41/6.

The first few congruent numbers are 5, 6, 7, 13, 14, 15, 20, and 21. Many congruent numbers were known prior to the new calculation. For example, every number in the sequence 5, 13, 21, 29, 37, …, is a congruent number. But other similar looking sequences, like 3, 11, 19, 27, 35, …., are more mysterious and each number has to be checked individually.

The calculation found 3,148,379,694 of these more mysterious congruent numbers up to a trillion.

Consequences, and future plans

Team member Bill Hart noted, “The difficult part was developing a fast general library of computer code for doing these kinds of calculations. Once we had that, it didn’t take long to write the specialized program needed for this particular computation.” The software used for the calculation is freely available, and anyone with a larger computer can use it to break the team’s record or do other similar calculations.

In addition to the practical advances required for this result, the answer also has theoretical implications. According to mathematician Michael Rubinstein from the University of Waterloo, “A few years ago we combined ideas from number theory and physics to predict how congruent numbers behave statistically. I was very pleased to see that our prediction was quite accurate.” It was Rubinstein who challenged the team to attempt this calculation. Rubinstein’s method predicts around 800 billion more congruent numbers up to a quadrillion, a prediction that could be checked if computers with a sufficiently large hard drive were available.

History of the problem

The congruent number problem was first stated by the Persian mathematician al-Karaji (c.953 – c.1029). His version did not involve triangles, but instead was stated in terms of the square numbers, the numbers that are squares of integers: 1, 4, 9, 16, 25, 36, 49, …, or squares of rational numbers: 25/9, 49/100, 144/25, etc. He asked: for which whole numbers n does there exist a square a2 so that a2-n and a2+n are also squares? When this happens, n is called a congruent number. The name comes from the fact that there are three squares which are congruent modulo n. A major influence on al-Karaji was the Arabic translations of the works of the Greek mathematician Diophantus (c.210 – c.290) who posed similar problems.

A small amount of progress was made in the next thousand years. In 1225, Fibonacci (of “Fibonacci numbers” fame) showed that 5 and 7 were congruent numbers, and he stated, but did not prove, that 1 is not a congruent number. That proof was supplied by Fermat (of “Fermat’s last theorem” fame) in 1659. By 1915 the congruent numbers less than 100 had been determined, and in 1952 Kurt Heegner introduced deep mathematical techniques into the subject and proved that all the prime numbers in the sequence 5, 13, 21, 29,… are congruent. But by 1980 there were still cases smaller than 1000 that had not been resolved.

Modern results

In 1982 Jerrold Tunnell of Rutgers University made significant progress by exploiting the connection (first used by Heegner) between congruent numbers and elliptic curves, mathematical objects for which there is a well-established theory. He found a simple formula for determining whether or not a number is a congruent number. This allowed the first several thousand cases to be resolved very quickly. One issue is that the complete validity of his formula (therefore also the new computational result) depends on the truth of a particular case of one of the outstanding problems in mathematics known as the Birch and Swinnerton-Dyer Conjecture. That conjecture is one of the seven Millennium Prize Problems posed by the Clay Math Institute with a prize of one million dollars.

The computations

Results such as these are sometimes viewed with skepticism because of the complexity of carrying out such a large calculation and the potential for bugs in either the computer or the programming. The researchers took particular care to verify their results, doing the calculation twice, on different computers, using different algorithms, written by two independent groups.

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Full article and photo: http://www.sciencedaily.com/releases/2009/09/090922095651.htm

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Google trick tracks extinctions

 

The large blue butterfly faced extinction when the ant it feeds on began dying off

 

Google’s algorithm for ranking web pages can be adapted to determine which species are critical for sustaining ecosystems, say researchers.

According to a paper in PLoS Computational Biology, “PageRank” can be applied to the study of food webs.

These are the complex networks of who eats whom in an ecosystem.

The scientists say their version of PageRank could be a simple way of working out which extinctions would lead to ecosystem collapse.

Every species is embedded in a complex network of relationships with others. So a single extinction can cascade into the loss of seemingly unrelated species.

Investigating when this might happen using more conventional methods is complicated as even in simple ecosystems, the number of combinations exceeds the number of atoms in the universe. So it would be impossible to try them all.

Co-author Dr Stefano Allesina realised he could apply PageRank to the problem when he stumbled across an article in a journal of applied mathematics describing the Google algorithm.

The researchers say they had to make minor changes to it to adapt it for ecology.

Dr Allesina, of the University of Chicago’s department of ecology and Evolution, told BBC News: “First of all we had to reverse the definition of the algorithm.

“In PageRank, a web page is important if important pages point to it. In our approach a species is important if it points to important species.”

Cyclical element

They also had to design in a cyclical element into the food web system in order to make it applicable to the algorithm.

They did this by including what Dr Allesina terms the “detritus pool”. He said: “When an organism dies it goes into the detritus pool and in turn gets cycled back into the food web through the primary producers, the plants.

“Each species points to the detritus and the detritus points only to the plants. This makes the web circular and therefore leads to the application of the algorithm.”

Dr Allesina and co-author Dr Mercedes Pascual of University of Michigan have tested their method against published food webs, using it to rank species according to the damage they would cause if they were removed from the ecosystem.

They also tested algorithms already in use in computational biology to find a solution to the same problem.

They found that PageRank gave them exactly the same solution as these much more complicated algorithms.

Dr Glyn Davies, director of programmes at WWF-UK, welcomed the work. He said: “As the rate of species extinction increases, conservation organisations strive to build political support for maintaining healthy and productive ecosystems which hold a full complement of species.

“Any research that strengthens our understanding of the complex web of ecological processes that bind us all is welcome.”

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Full article and photo: http://news.bbc.co.uk/2/hi/science/nature/8238462.stm

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Figured out

We don’t understand the math, but can we get the mathematicians?

“Figure,” as a noun, has multiple meanings. It can be a number: “I’ll write up those figures for you.” A person: “an eminent figure in the field.” A shape: “the figure of a triangle.” Those meanings intersect in Mariana Cook’s new book, “Mathematicians: An Outer View of the Inner World.” It consists of 92 black-and-white portraits of just what its title says: figures who, at the highest level of their profession, work with figures.

The idea of a collection of portraits of mathematicians seems on the face of it as irrational as the square root of two. Intellectually opaque, the practice of higher mathematics is visually null. It can be understood, at least by a few people, but that doesn’t mean it can be seen. Mathematics is like theology or poetry that way. Mathematics is a kind of poetry, actually. But poets have a long history of being photogenic. Mathematicians most emphatically do not. There has yet to be a mathematician maudit, or a Byronic mathematician (other, that is, than Byron’s daughter, Ada).

Yet there are ways in which the idea of mathematician portraits makes sense. Celebrity in this culture has long been its own justification, and some of Cook’s sitters wouldn’t look amiss as boldface names: John Nash of “A Beautiful Mind” fame; Sir Roger Penrose, the cosmologist; Benoit Mandelbrot, the father of fractal geometry (who, in fact, has a memoir coming out in October).

Nor does an argument have to be made for the intrinsic intellectual fascination of mathematics. It’s a fascination that transcends understanding – or even renders it irrelevant. If anything, mathematics’s being so esoteric to so many people can work to enhance its allure. “Conformal dynamics”? “Galois cohomology”? “Unipotent flows on quotients of Lie groups”? There’s verbal magic in such abstruseness (see, mathematics really is a kind of poetry). Precisely because we can’t glimpse the world mathematicians see, the prospect of glimpsing the faces of those who can becomes all the more intriguing.

John Horton Conway of Princeton has the shaggy, Bagginsy look of an anorak-wearing hobbit. With his moony face and high forehead, Andrei Okounkov, also of Princeton, seems so preposterously young (even though he’s 40) and very Russian. You don’t have to know that Michèle Vergne is director of research at the Centre National de la Recherche Scientifique to assume she’s French. Such cool appraising eyes, such unemphatic unflappability: Her membership in the sorority of Marguerite Duras and Simone de Beauvoir and Marguerite Yourcenar is as plain as the nose on Nicolas Sarkozy’s face.

Cook’s work extends photography’s tradition of vocational portraits. What may well be the single greatest discrete project in the history of the medium, August Sander’s “People of the 20th Century,” consists in large part of individuals presented as professional archetypes. “Irving Penn: Small Trades,” with more than 250 photographs by Penn of tradespeople, opens next month at the Getty Museum. There are numerous lesser examples: serial portraits of lifeguards, supermodels, firefighters, cowboys, soldiers, athletes, tenant farmers, and so on.

It’s pretty obvious with each of those occupations how a photographer might go about presenting them visually: tools, uniforms, workplaces, even physical types. Walker Evans’s photographs in “Let Us Now Praise Famous Men” don’t have captions. They don’t need captions. Everything about his pictures of the Gudger, Ricketts, and Woods families locates them in a socioeconomic context. Yet conveying the professional identity of a mathematician, someone whose work is so utterly interior and veiled, doesn’t faze Cook.

Sets of numerals and symbols on a board – equations, in other words – that’s about it for mathematics made visible. Cook manages the considerable feat of limiting the number of blackboards and whiteboards to four. “What’s the ontology of mathematical things?” asks Conway. (Each portrait has an accompanying autobiographical text.) “In what sense do they exist? There’s no doubt that they do exist, but you can’t poke and prod them except by thinking about them.” You can’t photograph them, either.

So what about the ontology of mathematical thinkers? Although Cook’s 92 sitters can’t necessarily be taken as a representative sample, they’re as extensive a survey as there’s likely to be. And looking at them we see a profession that largely defeats stereotyping.

Four sitters wear sweater vests, and two wear socks with sandals. But two others wear leather jackets – and one (French, bien sur) sports a pair of quite nifty socks. There’s only one pipe smoker, one bow-tie wearer, and one turtleneck wearer. Canceling out the latter, perhaps, is the turtle that can be seen crawling across a table in the portrait of Kenneth Ribet of the University of California at Berkeley.

Twenty-two of the mathematicians have beards, and two have mustaches. There are more women than you’d likely think, though: a dozen. Two of them are daughters of mathematicians with their own portrait in the book. Further evidence for a predisposition to mathematics is the presence of two sets of brothers: the Feffermans (of Princeton and the University of Chicago), and the Browders (of Princeton, Rutgers, and Brown). Their father was Earl Browder, the onetime head of the American Communist Party. Where’s Joe McCarthy when you need him: Are you now or were you ever a mathematician?

Expectations keep getting defeated. Dennis Parnell Sullivan of the City University of New York and Stony Brook University looks like Rupert Murdoch. Nicholas Michael Katz of Princeton looks like William Faulkner. Richard Ewen Borcherds of Cal-Berkeley looks like Red Sox third baseman Mike Lowell. There’s even a viscount, Pierre Deligne, of the Institute for Advanced Study, but you’d never know it to look at him.

Then again, what exactly do viscounts look like? Do they differ much in appearance from counts, for example? It’s all rather perplexing, this business of trying to infer activity from appearance. Maybe the biggest difference between mathematics and photography is just this: So much of the intellectual appeal of the former lies in its being one of the very few aspects of life in which no value derives from distinguishing appearance from reality. The two are effectively the same. That’s definitely not true of photography. Conversely, this may be the one way in which looking at photographs of mathematicians resembles doing mathematics. “I think,” Okounkov says, “mathematics requires imagination more than any other ingredient.”

Mark Feeney, Boston Globe 

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Full article and photo: http://www.boston.com/bostonglobe/ideas/articles/2009/08/23/figured_out/

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Can Game Theory Predict When Iran Will Get the Bomb?

 

Is Iran going to build a bomb?

Many people wonder, but Bruce Bueno de Mesquita claims to have the answer.

Bueno de Mesquita is one of the world’s most prominent applied game theorists. A professor at New York University and a senior fellow at the Hoover Institution at Stanford, he is well known academically for his work on “political survival,” or how leaders build coalitions to stay in power. But among national-security types and corporate decision makers, he is even better known for his prognostications. For 29 years, Bueno de Mesquita has been developing and honing a computer model that predicts the outcome of any situation in which parties can be described as trying to persuade or coerce one another. Since the early 1980s, C.I.A. officials have hired him to perform more than a thousand predictions; a study by the C.I.A., now declassified, found that Bueno de Mesquita’s predictions “hit the bull’s-eye” twice as often as its own analysts did.

Last year, Bueno de Mesquita decided to forecast whether Iran would build a nuclear bomb. With the help of his undergraduate class at N.Y.U., he researched the primary power brokers inside and outside the country — anyone with a stake in Iran’s nuclear future. Once he had the information he needed, he fed it into his computer model and had an answer in a few minutes.

In June, I visited Bueno de Mesquita at his San Francisco home to see the results. A tall man with a slab of gray hair, Bueno de Mesquita, who is 62, welcomed me with painstakingly prepared cups of espresso. Then he pulled out his beat-up I.B.M. laptop — so old that the lettering on the A, S, D and E keys was worn off — and showed me a spreadsheet that summarized Iran’s future.

The spreadsheet included almost 90 players. Some were people, like the Iranian president, Mahmoud Ahmadinejad, and Supreme Leader Ali Khamenei; others were groups, like the U.N. Security Council and Iran’s “religious radicals.” Next to each player, a number represented one variable in Bueno de Mesquita’s model: the extent to which a player wanted Iran to have the ability to make nuclear weapons. The scale went from 0 to 200, with 0 being “no nuclear capacity at all” and 200 representing a test of a nuclear missile.

At the beginning of the simulation, the positions were what you would expect. The United States and Israel and most of Europe wanted Iran to have virtually no nuclear capacity, so their preferred outcomes were close to zero. In contrast, the Iranian hard-liners were aggressive. “This is not only ‘Build a bomb,’ ” Bueno de Mesquita said, characterizing their position. “It’s probably: ‘We should test a bomb.’ ”

But as the computer model ran forward in time, through 2009 and into 2010, positions shifted. American and Israeli national-security players grudgingly accepted that they could tolerate Iran having some civilian nuclear-energy capacity. Ahmadinejad, Khamenei and the religious radicals wavered; then, as the model reached our present day, their power — another variable in Bueno de Mesquita’s model — sagged significantly.

Amid the thousands of rows on the spreadsheet, there’s one called Forecast. It consists of a single number that represents the most likely consensus of all the players. It begins at 160 — bomb-making territory — but by next year settles at 118, where it doesn’t move much. “That’s the outcome,” Bueno de Mesquita said confidently, tapping the screen.

What does 118 mean? It means that Iran won’t make a nuclear bomb. By early 2010, according to the forecast, Iran will be at the brink of developing one, but then it will stop and go no further. If this computer model is right, all the dire portents we’ve seen in recent months — the brutal crackdown on protesters, the dubious confessions, Khamenei’s accusations of American subterfuge — are masking a tectonic shift. The moderates are winning, even if we cannot see that yet.

Could this possibly be what will happen? Certainly Bueno de Mesquita has his critics, who argue that the proprietary software he uses can’t be trusted and may cast doubt on the larger enterprise of making predictions. But he has published a large number of startlingly precise predictions that turned out to be accurate, many of them in peer-reviewed academic journals. For example, five years before Ayatollah Khomeini died in 1989, Bueno de Mesquita predicted in the journal PS that Khomeini would be succeeded by Ali Khamenei (which he was), who himself would be succeeded by a then-less-well-known cleric named Akbar Hashemi Rafsanjani (which he may well be). Last year, he forecast when President Pervez Musharraf of Pakistan would be forced out of office and was accurate to within a month. In “The Predictioneer’s Game,” a book coming out next month that was written for a popular audience, Bueno de Mesquita offers dozens more stories of his forecasts. And as for Iran’s bomb?

In a year, he said with a wide grin, we’ll know if he’s right.

“I’m not an Iran expert,” Bueno de Mesquita told me cheerfully as we walked down his tree-lined street on our way to grab some Burmese food. Indeed, his career has been built on a peculiar concept: If you want to predict political events, wisdom and expertise, deep knowledge of a country’s culture and history, aren’t enough. To forecast the future, you need to be an expert not in statecraft but in the way individual people make decisions. You need “rational actor” game theory.

Bueno de Mesquita began studying political science in the 1960s. While working on his dissertation at the University of Michigan on parliamentary politics in India, a professor assigned him William H. Riker’s book “The Theory of Political Coalitions,” one of the first works to apply game theory to politics. Game theory is a branch of mathematics that studies the way people will behave in strategic situations — that is to say, when they’re making decisions based on how they think other people will make decisions. Generally, game theory assumes that people are always rational and selfish; they’re always angling to get what’s best for them, which means their behavior can often be predicted. One famous application of rational-choice theory that particularly intrigued Bueno de Mesquita was Duncan Black’s analysis of “committee voting,” which argues that if two rival candidates are trying to get elected on a single issue — say, taxes — they’ll inevitably shift their positions toward the median voter.

Bueno de Mesquita was enthralled by the idea of rendering the messy business of politics and history into precise, logical equations. He began his signature academic work on “the selectorate,” or the group of actors who run a country. In Bueno de Mesquita’s worldview, there is no such thing as a “national interest” (or “state”). There are just leaders trying desperately to stay in power by building coalitions within their selectorate — buying off cronies in the case of a dictatorship, for example, or producing enough good works to keep hoi polloi happy in a democracy.

When Bueno de Mesquita spotted a logical error in one of Riker’s books, he wrote the author a letter; Riker offered Bueno de Mesquita a job in 1972 at the University of Rochester, where a new generation of political scientists was starting to apply formal mathematical models to political analysis.

That’s where Bueno de Mesquita began programming his computer model. It is based loosely on Black’s voter theory, and it works like this: To predict how leaders will behave in a conflict, Bueno de Mesquita starts with a specific prediction he wants to make, then interviews four or five experts who know the situation well. He identifies the stakeholders who will exert pressure on the outcome (typically 20 or 30 players) and gets the experts to assign values to the stakeholders in four categories: What outcome do the players want? How hard will they work to get it? How much clout can they exert on others? How firm is their resolve? Each value is expressed as a number on its own arbitrary scale, like 0 to 200. (Sometimes Bueno de Mesquita skips the experts, simply reads newspaper and journal articles and generates his own list of players and numbers.) For example, in the case of Iran’s bomb, Bueno de Mesquita set Ahmadinejad’s preferred outcome at 180 and, on a scale of 0 to 100, his desire to get it at 90, his power at 5 and his resolve at 90.

Then the math begins, some of which is surprisingly simple. If you merely sort the players according to how badly they want a bomb and how much support they have among others, you will end up with a reasonably good prediction. But the other variables enable the computer model to perform much more complicated assessments. In essence, it looks for possible groupings of players who would be willing to shift their positions toward one another if they thought that doing so would be to their advantage. The model begins by working out the average position of all the players — the “middle ground” that exerts a gravitational force on the whole negotiation. Then it compares each player with every other player, estimating whether one will be able to persuade or coerce the others to move toward its position, based on the power, resolve and positioning of everyone else. (Power isn’t everything. If the most powerful player is on the fringe of an issue, and a cluster of less-powerful players are closer to the middle, they might exert greater influence.) After estimating how much or how little each player might budge, the software recalculates the middle ground, which shifts as the players move. A “round” is over; the software repeats the process, round after round. The game ends when players no longer move very much from round to round — this indicates they have compromised as much as they ever will. At that point, assuming no player with veto power had refused to compromise, the final average middle-ground position of all the players is the result — the official prediction of how the issue will resolve itself. (Bueno de Mesquita does not express his forecasts in probabilistic terms; he says an event will transpire or it won’t.)

The computer model, in short, predicts coalitions. And computers are much better at doing this than humans, because with more than a few players the number of possible coalitions quickly multiplies. With 40 players, the typical size of one of Bueno de Mesquita’s forecasts, there are 1,560 possible pairs to consider just for starters. This is why, he says, his model often produces surprising results. It’s not that it is smarter than humans. But it methodically works through not only the obvious coalitions we know about and expect but also the invisible ones that we don’t.

For Bueno de Mesquita, the first prominent use of the model came in 1979, when the State Department was canvassing academics with expertise on India, including Bueno de Mesquita, to see how some parliamentary maneuverings would unfold. Bueno de Mesquita decided to use his first version of the software (which was, as he puts it, “barely working”) and his own knowledge of India to determine the power players and each of their numbers. Then the university’s mainframe computer worked on the data all night.

In the morning, Bueno de Mesquita said, he was astonished: the predicted victor was a seemingly minor figure, someone discounted by the experts. Bueno de Mesquita shared their opinion, he told me, but he accepted the computer’s verdict anyway. “So I called the person back at the State Department, and told him what I had concluded,” Bueno de Mesquita went on. “And there was a long, quiet period and some laughing. He said: ‘How did you arrive at that? Nobody’s saying that.’ So I told him I had a little computer model. He just guffawed. He said, ‘I wouldn’t repeat that if I were you.’ ”

Three months later, according to Bueno de Mesquita, his prediction turned out to be right.

The son of Jewish immigrants who arrived from Brussels during World War II, Bueno de Mesquita grew up in Manhattan, where his father ran a small publishing company and his mother managed a women’s clothing shop. He went to Queens College when he was 16 — “way too young,” he says — and read history and literature voraciously. (Bueno de Mesquita spent years researching and writing a short novel that defends Ebenezer Scrooge as a kindhearted man.) “He is one the most remarkably intelligent human beings I’ve met in my life, and Bruce does not hesitate to tell you that,” Kevin Gaynor, an environmental lawyer who has twice hired Bueno de Mesquita to advise his corporate clients on “extremely sensitive” government negotiations, told me half-jokingly. “He’s not self-effacing. But he’s not self-effacing in a charming way.” Bueno de Mesquita’s voluminous academic work — he has published 16 books and more than 100 papers — is credited with helping to move game theory and mathematical modeling into the mainstream of political science; according to one count, by 1999 fully 40 percent of papers in the American Political Science Review used modeling. (The figure was so high it prompted deep consternation among non-game-theory political scientists.) While few perform the consulting work he does, other game theorists have produced models very similar to Bueno de Mesquita’s, and he actively promotes his technique, including training N.Y.U. undergraduates to do similar predictions.) He spends half the year at N.Y.U., where he recently finished a four-year stint as the chairman of the political-science department, and half the year at the Hoover Institution at Stanford. Under the terms of his academic contracts, he is permitted to spend one day per week during the academic year doing outside consulting.

It is this consulting, more than his academic work, that has made Bueno de Mesquita both well off and controversial. He began offering predictions to the private sector in 1982, when A.F.K. Organski, a former professor of his, suggested they go into business using Bueno de Mesquita’s model. Business negotiations, they reasoned, were like international relations in that they involved players trying to wheedle and coerce one another. Soon Bueno de Mesquita and Organski (who died in 1998) acquired clients ranging from Arthur Andersen to Union Carbide, which tapped them for advice on placating the Indian government after the Bhopal chemical spill. Today Bueno de Mesquita’s firm essentially consists of himself and Harry Roundell, a former banker at J. P. Morgan who met Bueno de Mesquita when Roundell hired him in 1995 to help the bank figure out how to push for new, favorable regulations in the U.S. They charge $50,000 and up to do a prediction and offer negotiating tips, and they take on 18 to 20 of these assignments a year. Beyond saying it was “a reasonable amount of money,” Bueno de Mesquita would not describe his income from the company.

To produce a corporate prediction, Roundell and Bueno de Mesquita determine the numerical values of the players in a negotiation by interviewing a firm’s executives. This can take anywhere from a few hours on the phone to two days of face-to-face conversations. Both men conduct the interviews, and Bueno de Mesquita enters the information into a spreadsheet.

The real value of Bueno de Mesquita’s work, several clients told me, is not only in his predicting how a corporate event might unfold. It is also in figuring out how to influence that event. Because Bueno de Mesquita’s model forecasts the future by calculating the impact every player has on every other player, round by round, Bueno de Mesquita can go back and see when some players suddenly become more flexible midway through a negotiation. He can thus perform “what if” experiments: What if that person could be persuaded to change his mind? He’ll enter new values into the model, manually changing that player’s position, then run it again to see if this change recasts the future to his client’s advantage. If it does, Bueno de Mesquita now has a piece of advice: focus on that player in real life, and try to influence him. If there are dozens of players and dozens of rounds, the number of possible “what if” scenarios becomes enormous: it can take Bueno de Mesquita days of peering at his spreadsheets to identify useful pressure points.

One of Bueno de Mesquita’s most prominent public consultations occurred in 1999, when Richard Lapthorne, then the vice chairman of British Aerospace, asked him to help engineer a $10 billion acquisition. The British government wanted British Aerospace to form a pan-European firm by merging with the German firm DASA and the French giant Aérospatiale; British Aerospace, however, was more interested in trying to buy the British electronics giant Marconi Electronic Systems. To persuade the British government to approve the Marconi deal, Lapthorne asked Bueno de Mesquita to predict the viability of mergers between the German and French firms. The model forecast that the three firms would never be able to agree on terms, and that the Marconi deal was the better option; when Bueno de Mesquita showed his analysis to the government heads, they agreed to permit the Marconi acquisition. “There’s nothing shimmy shammy or flip-flop about it,” Lapthorne says of the logical nature of Bueno de Mesquita’s prediction. “It’s very clear where the information came from. It has intellectual rigor.” Lapthorne is now chairman of the U.K. telecommunications company Cable and Wireless; he has used Bueno de Mesquita for seven predictions since, though he would not disclose the subjects.

It is difficult to verify how accurately Bueno de Mesquita’s model performs in corporate settings because most firms are loath to discuss his work for them. For most of the cases we discussed, Bueno de Mesquita would disclose details of the negotiation but wouldn’t name the firms in question. In other cases, clients would talk to me and praise Bueno de Mesquita’s work for them, but they would not disclose verifiable details of specific negotiations. There were a few exceptions: Robert F. Kelley, a retired former partner of Arthur Andersen, described using Bueno de Mesquita for “60 or 70” cases, ranging from internal firing decisions to figuring out how to persuade the U.S. to support China’s entry into the World Trade Organization. (Bueno de Mesquita also offered to use his software to predict which of Arthur Andersen’s clients — including, at the time, Enron — were likely to engage in financial fraud. But the firm’s lawyers, Bueno de Mesquita says, didn’t want to use the tool for fear it would put them in awkward legal positions. “Had I been able to convince the firm” to use the model, Kelley says, “I think that Andersen would be alive today.”)

Bueno de Mesquita’s most regular client by far has been the C.I.A. He says he has performed more than 1,200 predictions for the agency, tackling questions like “How fully will France participate in the Strategic Defense Initiative?” and “What policy will Beijing adopt toward Taiwan’s role in the Asian Development Bank?” In 1987, Stanley Feder, a research political scientist for the C.I.A., published a report analyzing forecasts that Bueno de Mesquita’s firm did of political events in 27 countries; he found that the success rate of its predictions was the same as that of the C.I.A.’s own analysts, only more precise. (He “got the bull’s-eye twice as often,” Feder wrote in his report, which was declassified in 1993. No other reports have been declassified since.) Feder noted, for example, that Bueno de Mesquita’s model predicted in a forecast done of Italy’s budget one year a specific figure that turned out to be off by only 1 percent; the C.I.A. method would predict just a deficit.

Those who have watched Bueno de Mesquita in action call him an extremely astute observer of people. He needs to be: when conducting his fact-gathering interviews, he must detect when the experts know what they’re talking about and when they don’t. The computer’s advantage over humans is its ability to spy unseen coalitions, but this works only when the relative positions of each player are described accurately in the first place. “Garbage in, garbage out,” Bueno de Mesquita notes. Bueno de Mesquita begins each interview by sitting quietly — “in a slightly closed-up manner,” as Lapthorne told me — but as soon as an interviewee expresses doubt or contradicts himself, Bueno de Mesquita instantly asks for clarification.

“His ability to pick up on body language, to pick up on vocal intonation, to remember what people said and challenge them in nonthreatening ways — he’s a master at it,” says Rose McDermott, a political-science professor at Brown who has watched Bueno de Mesquita conduct interviews. She says she thinks his emotional intelligence, along with his ability to listen, is his true gift, not his mathematical smarts. “The thing is, he doesn’t think that’s his gift,” McDermott says. “He thinks it’s the model. I think the model is, I’m sure, brilliant. But lots of other people are good at math. His gift is in interviewing. I’ve said that flat out to him, and he’s said, ‘Well, anyone can do interviews.’ But they can’t.”

You might expect Bueno de Mesquita to be the toast of both Washington and Wall Street, constantly in demand for prognostications. Yet he and Roundell have found that it is not so easy to attract clients. This is partly because most of their clients — especially the C.I.A. — swear them to secrecy. (And perhaps also because, as Roundell says, “Bruce and I are . . . terrible salespeople.”) But they have also faced a barrier that’s almost existential, a skepticism that computer models can truly predict the outcome of negotiations. The C.I.A., for example, built its own replica of Bueno de Mesquita’s original forecast model, but as Feder noted in his report, “the vast majority of analysts” didn’t use it because it seemed too rigid. They thought of analysis as reading and pondering until they had an aha! moment — not feeding data points into a computer model and waiting to see what comes up.

When we spoke, Bueno de Mesquita often seemed irritated by resistance to his work. For all his gifts of intuition, he has a Spocklike disdain for gut instinct. When he occasionally hires colleagues to help him with a complex bit of corporate work, he sternly warns them that they must refrain from expressing any personal opinions and describe only what they see in the spreadsheets. Bueno de Mesquita habitually and hissingly disparages traditional political analysis. He is savvy enough to know that nobody likes a scold, yet he can’t help himself; he sheepishly admits to becoming “confrontational” when people think mathematical reasoning can’t be used. At the C.I.A., Feder told me, “there were some people who found him arrogant, which was maybe a reasonable reaction.”

Donald Green, a political scientist at Yale, questions whether Bueno de Mesquita is serving the discipline well. “When I see clips of Bruce at the TED conference,” he says, referring to the annual conference promoting ideas in technology, entertainment and design, “I watch the video and I think, Wow, this is so far from the typical way in which political scientists of any stripe behave.” Some political scientists are openly dubious about the accuracy of Bueno de Mesquita’s model. Stephen Walt, a Harvard professor of international affairs, says that Bueno de Mesquita’s nonprediction work — like his theory of the “political survival” of heads of state — make him a “respected scholar, deservedly so.” It’s the predictions that Walt doesn’t trust, because Bueno de Mesquita does not publish the actual computer code of his model. (Bueno de Mesquita cannot do so because his former firm owns the actual code, but he counters that he has outlined the math behind his model in enough academic papers and books for anyone to replicate something close to his work.) While Bueno de Mesquita has published many predictions in academic journals, the vast majority of his forecasts have been done in secret for corporate or government clients, where no independent academics can verify them. “We have no idea if he’s right 9 times out of 10, or 9 times out of a hundred, or 9 times out of a thousand,” Walt says. Walt also isn’t impressed by Stanley Feder’s C.I.A. study showing Bueno de Mesquita’s 90 percent hit rate. “It’s one midlevel C.I.A. bureaucrat saying, ‘This was a useful tool,’ ” Walt says. “It’s not like he’s got Brent Scowcroft saying, ‘Back in the Bush administration, we didn’t make a decision without consulting Bueno de Mesquita.’ ” Other academics point out that rational-actor theory has come under increasing criticism in recent years, as more evidence accumulates that people make many decisions irrationally.

And it’s true that there have been cases when Bueno de Mesquita’s model has gone awry. In his 1996 book, “Red Flag Over Hong Kong,” he predicted that the press in Hong Kong “will become largely a tool of the state” — a highly debatable claim today. (In 2006, Reporters Without Borders noted concerns about self-censorship but said that “journalists remain free in Hong Kong.”) In early 1993, a corporate client asked him to forecast whether the Clinton administration’s health care plan would pass, and he said it would.

What’s more, with corporate clients in particular, there’s always the potential problem of reflexivity, of the prediction itself influencing events and making it hard to evaluate the prediction’s value. Suppose a firm is told a merger will fail, for example, and abandons its merger efforts. Was the prediction accurate or a self-fulfilling prophecy?

Spending time with Bueno de Mesquita is alternately alarming and reassuring, because he has such confidence in his own predictions about our global fate. Like many, he believes the future of Pakistan is “incredibly distressing” right now, but he has reached this conclusion in his own way: when he and his students modeled its future last year, the power of Al Qaeda and the Taliban in that region grew quickly throughout 2009, far outstripping that of the government and military. Global warming is another area where politics are doomed to fail. World governments are set to meet this December in Copenhagen to commit to firm CO2-reduction levels — but when Bueno de Mesquita modeled the future of these targets, most countries renege on them. No democratic government will seriously limit CO2 if it will hurt its citizens economically.

“When people are asked to make personal sacrifices for the greater good in the longer term, they seem to find 1,001 reasons why their particular behavior is so virtuous that this one particular deviation is really O.K.,” Bueno de Mesquita told me recently as we talked in his home office. “ ‘I have to drive an S.U.V. because I want to protect my little children from a car accident!’ ”

Yet Bueno de Mesquita remains cheerful, almost unnervingly so. Years of peering at his model have shown him that conflicts almost always have hidden solutions — places where the computer illuminates the sort of leverage that could be employed to create a sudden, useful countercoalition. For example, with Pakistan, his model showed that if the U.S. merely doubled its annual aid from $700 million to $1.5 billion, America’s influence in the country would significantly jump, while the militants’ would drop drastically. Why? Because with that sort of financial flow, corrupt rural officials would suddenly profit more from helping the U.S. than from helping the Taliban.

In the short term, though, Bueno de Mesquita’s reputation will be colored by Iran. The last time we met, it was two weeks after the Iranian election, and the opposition protests had been quashed. The hard-liners, I noted, seemed to be winning — did this mean that the prediction was wrong? “The street movement is running out of steam,” Bueno de Mesquita agreed. “Shooting people does act as an effective deterrent.” But he still maintained that his model was likely to prevail, and that domestic coalitions we might not detect from abroad are gathering to overwhelm the religious conservatives.

He spent that morning looking over his Iranian data, and he generated a new chart predicting how the dissidents’ power would grow over the next few months. In terms of power, one category — students — would surpass Ahmadinejad during the summer, and by September or October their clout would rival that of Khamenei, the supreme leader. “And that’s huge!” Bueno de Mesquita said excitedly. “If that’s right, it’s huge!” He said he believed that Iran’s domestic politics would remain quiet over the summer, then he thought they’d “really perk up again” by the fall.

Bueno de Mesquita also approved of Obama’s hands-off approach. Bueno de Mesquita ran an experimental version of his Iranian model without the U.S. in it as a player at all, and the coalitions that oppose Ahmadinejad and the bomb emerge a few months more quickly. In other words, American meddling is indeed counterproductive; the less America tries to influence Iran, the more quickly Iran will abandon nuclear weapons, if the logic of the computer is correct.

It’s a fascinating analysis, but, I wonder, has he given it to anyone in the State Department? He laughed. “I’m working on access.”

Clive Thompson, a contributing writer for the magazine, writes frequently about technology.

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Full article and photo: http://www.nytimes.com/2009/08/16/magazine/16Bruce-t.html

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Huddled maths

An academic journal provides haven for rejected work

PAUL LAUTERBUR, the father of magnetic-resonance imaging, had his seminal paper rejected when he first submitted it to Nature. Peter Higgs, eponymous predictor of physics’s missing boson, faced similar trouble with Physics Letters. But Lauterbur went on to win a Nobel prize for his work, and Dr Higgs is an odds-on favourite to get one soon. A good, rejected paper, then, is by no means an oxymoron.

And that observation is the basis of Rejecta Mathematica, an open-source academic journal that recently went online. As its name suggests, the new journal publishes only papers that, like Lauterbur’s and Dr Higgs’s, have been previously submitted to, and rejected by, others. With Annals of Mathematics, one of the best, denying entry to more than 300 last year alone, Rejecta could be busy.

Rejecta was conceived three years ago by four graduate students at Rice University, in Houston, Texas. Two of its founders, Michael Wakin and Christopher Rozell, had just had a paper on card counting in blackjack rejected. Good work, said the reviewers, but find some other place for it. When they could not, they, along with Mark Davenport and Jason Laska, decided to cut out the middle man and found their own journal.

If Rejecta is a joke, it is a well-executed one. The serious aim is to highlight papers that, although perhaps flawed, may still be interesting. It manages that well. The inaugural issue includes topics ranging from image enhancement to condition numbers of matrices (don’t ask). All come with an “open letter” in which the paper’s author outlines in lay terms why the work was rejected (extra points awarded for bitterness), what has been done since and why it still has merit.

Rejecta’s larger purpose, then, may be a light jab at academia’s bureaucracy and the rigmarole to which it is necessary to submit in order to get published. Whether conventional journals are necessary in the internet age is a matter of active debate. Refereeing maths papers, in particular, requires serious expertise that few have. Those who do, usually receive no pay for their refereeing services. Mistakes can be made. Academia as a whole, some say, could do a better job. But peer review is still necessary. And yes, the editors claim that they too have had to reject some submissions.

The Economist

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See also:

Rejecta Mathematica

 http://math.rejecta.org/about-rejecta-mathematica

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Full article: http://www.economist.com/sciencetechnology/displayStory.cfm?story_id=14119761&source=hptextfeature

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The Triumph of the Random

From banking to baseball, winning streaks owe much to the laws of chance

It was the summer of 1945, and World War II had ended. Former soldiers, including famous baseball stars, streamed back into America and into American life. Yankee slugger Joe DiMaggio was trying to be Yankee fan Joe DiMaggio, sneaking into a mezzanine seat with his 4-year-old son, Joe Jr., before rejoining his team. A fan noticed him, then another. Soon throughout the stadium people were chanting “Joe, Joe, Joe DiMaggio!” DiMaggio, moved, gazed down to see if his son had noticed the tribute. He had. “See, Daddy,” said the little DiMaggio, “everybody knows me!”

What role did chance play in Joe DiMaggio’s epic 56-game hitting streak?

We all interpret the events around us according to our own worldview. By adulthood we’ve either gotten beyond the me-me-me context of 4-year-olds, or gone into politics. But drawing conclusions about data we encounter in sports, business, medicine and even our personal lives, we often make errors as significant as that of Joe Jr.

This holiday weekend—the Fourth of July—kicks off the home stretch of a two-month period that made Joe DiMaggio Sr. an icon of American culture. In 1941, a few months before Joe Jr. was born, and sandwiched between the day Hitler’s insane deputy Rudolf Hess parachuted into Scotland on an unauthorized peace mission and the day a secret British report concluded that the Allies could complete an atomic bomb ahead of Germany, there was a period of 56 successive Yankee games in which Joltin’ Joe had at least one hit.

DiMaggio’s hitting streak was an inspiration in troubled times. The pursuit of any record comes with pressure—Roger Maris lost some of his hair during his attempt to break Babe Ruth’s home-run record in 1961—but most records forgive you an off day as long as you compensate at other times. Not so with a streak, which demands unwavering performance. And so DiMaggio’s streak has been interpreted as a feat of mythic proportion, seen as a heroic, even miraculous, spurt of unrivaled effort and concentration.

But was it? Or was this epic moment simply a fluke?

Kobe Bryant of the Los Angeles Lakers scored 81 points in a 2006 game.

Recent academic studies have questioned whether DiMaggio’s streak is unambiguous evidence of a spurt of ability that exceeded his everyday talent, rather than an anomaly to be expected from some highly talented player, in some year, by chance, something like the occasional 150-yard drive in golf that culminates in a hole in one. No one is saying that talent doesn’t matter. They are just asking whether a similar streak would have happened sometime in the history of baseball even if each player hit with the unheroic and unmiraculous—but steady—ability of an emotionless robot.

That randomness naturally leads to streaks contradicts people’s intuition. If we were to picture randomness, we might think of a graph that looks jerky, not smooth like a straight line. But random processes do display periods of order. In a toss of 100 coins, for example, the chances are more than 75% that you will see a streak of six or more heads or tails, and almost 10% that you’ll produce a streak of 10 or more. As a result a streak can look quite impressive even if it is due to nothing more than chance.

A few years ago Bill Miller of the Legg Mason Value Trust Fund was the most celebrated fund manager on Wall Street because his fund outperformed the broad market for 15 years straight. It was a feat compared regularly to DiMaggio’s, but if all the comparable fund managers over the past 40 years had been doing nothing but flipping coins, the chances are 75% that one of them would have matched or exceeded Mr. Miller’s streak. If Mr. Miller was really merely the goddess of Fortune’s lucky beneficiary, then one would expect that once the streak ended there would be no carryover of his apparent golden touch. In that expectation Mr. Miller did not disappoint: in recent years his fund has significantly lagged the market as he bet on duds like AIG, Bear Stearns, Merrill Lynch & Co. and Freddie Mac.

Of course a 10- or 15-game streak is a far cry from one of 56 games. This is where DiMaggio’s great ability plays a role, for if we are to compare DiMaggio’s performance to a coin, it must be a weighted coin. With a lifetime batting average of .325, DiMaggio had a better-than-75% chance of getting a hit in a game, while a balanced coin has but a 50% chance of success. Moreover, each year for over a century, hundreds of players have sought to achieve a streak such as DiMaggio’s. All those factors increase the odds that such a streak could have occurred by chance alone.

It’s not just the statisticians who wonder whether our heroes achieve records more often than coins. Psychologists, and, increasingly, economists, also puzzle over the seemingly discrete worlds of chance and perception. The fusion of those worlds was sanctified when half of the 2002 Nobel Prize in economics was awarded to psychologist Daniel Kahneman “for having integrated insights from psychological research into economic science, especially concerning human judgment and decision-making under uncertainty.”

In 1988 the Baltimore Orioles lost their first 21 games.

Research into why people misinterpret streaks dates to 1985, and a paper co-authored by Mr. Kahneman’s regular collaborator, the late Amos Tversky, in the journal Cognitive Psychology. (No one doubts that, had he lived, Mr. Tversky would have shared in the prize). The paper was titled “The hot hand in basketball: On the misperception of random sequences.” Everyone who has ever played basketball knows the feeling of being “in the zone.” Your hand is on fire. You can’t miss. But are you feeling a true increase in ability, or is your mind inferring it because you just took a bunch of shots that, for whatever reason, went in?

If a person tossing a coin weighted to land on heads 80% of the time produces a streak of 10 heads in a row, few people would see that as a sign of increased skill. Yet when an 80% free throw shooter in the NBA has that level of success people have a hard time accepting that it isn’t. The Cognitive Psychology paper, and the many that followed, showed that despite appearances, the “hot hand” is a mirage. Such hot and cold streaks are identical to those you would obtain from a properly weighted coin.

Why do people have a hard time accepting the slings and arrows of outrageous fortune? One reason is that we expect the outcomes of a process to reflect the underlying qualities of the process itself. For example, if an initiative has a 60% chance of success, we expect that six out of every 10 times such an initiative is undertaken, it will succeed. That, however, is false. In order to warrant confidence that results reflect a deeper truth, you need many more trials than 10. In fact, one of the most counterintuitive features of randomness is that for a small number of trials, the results of a random process typically do not reflect the underlying probabilities.

For example, suppose we undertake an analysis of the resources, effort and ability of all the companies in the Fortune 500 and determine that every company has the same 60% chance of success in any given year. If we observe all the companies over a period of five years and the underlying probability of success were reflected in each company’s results, then over the five-year period every company would have three good years.

The mathematics of chance indeed dictate that in this situation the odds of a company having zero, one, two, four or five good years are lower than the odds of having three. Nevertheless it is not likely that a company will have three out of five good years—because there are so many of those misleading outcomes, their combined odds add up to twice the odds of having exactly three. That means that of the 500 companies, two-thirds will experience results that belie their underlying potential. In fact, according to the rules of randomness, nearly 50 of the companies will have a streak of either five good years, or five bad years, even if their corporate capacities were no better or worse than their counterparts’. And so if you judged the companies by their five-year results alone, you would probably over- or underestimate their true value.

In sports, the championship contenders are usually pretty evenly matched. But in baseball, even if one assumes that the better team has a lopsided 55/45 edge over the inferior one, the lesser team will win the seven-game World Series 40% of the time. That might seem counterintuitive, but you can look at it as follows. If you play a best-of-one game series, then, by our assumption, the lesser team will win 45% of the time. Playing a longer series will cut down that probability. The problem is that playing a seven-game series only cuts it down to 40%, which isn’t cutting it down by much. What if you demand that the lesser team win no more than 5% of the time—a constraint called statistical significance? The World Series would have to be the best of 269 games, and probably draw an audience the size of that for Olympic curling. Swap baseball for marketing, and you find a mistake often made by marketing departments: assuming that the results of small focus groups reflect a trend in the general population.

We find false meaning in the patterns of randomness for good reason: we are animals built to do just that. Suppose, for example, that you sit a subject in front of a light which flashes red twice as often as green, but otherwise without pattern. After the subject watches for a while, you offer the subject a reward for each future flash correctly predicted. What is the best strategy?

A nonhuman animal in this situation will always guess red, the more frequent color. A different strategy is to match your proportion of red and green guesses to the proportion you observed in the past, that is, two reds for every green. If the colors come in some pattern that you can figure out, this strategy will enable you to be right every time. But if the colors come without pattern you will do worse. Most humans try to guess the pattern, and in the process allow themselves to be outsmarted by a rat. (Those trying to time the market lately might wish they had let the rat take charge.) Looking for order in patterns has allowed us to understand the patterns of the universe, and hence to create modern physics and technology; but it also sometimes compels us to submit bids on eBay because we see the face of Jesus in a slice of toast.

Another reason we reject the power of randomness is our need for control. DiMaggio’s streak affects us because we all appreciate struggle and effort, triumphing despite huge odds. The notion that we might not have control over our environment, on the other hand, causes us to shudder.

Many studies illustrate how this basic aspect of human nature translates to a misperception of chance. For example, a group of Yale students were asked to predict the result of a series of coin tosses. The tosses were secretly rigged so that each student would have some success initially, but end up with a 50% success rate. The students were obviously aware of the random nature of their task. Yet when asked whether their performance would be hampered by distraction, and whether it would improve with practice, a significant number indicated that it would. Their deep-seated need for control trumped their intellectual understanding of the situation.

What about DiMaggio’s streak? There are many subtleties in randomness. For example, do you model a player as having a fixed batting average—that is, probability of a hit—or do you allow for the average to vary within the season, or even game to game? How do you treat the variation in at-bats, walks, etc.? The analyses can get long and the number of data needed unwieldy, so the jury is still out, but one of the studies, by Samuel Arbesman and Steven H. Strogatz of Cornell University, attacked the problem by having a computer generate a mock version of each year in baseball from 1871 to 2005, based on the players’ actual statistics from that year. The scientists had the computer repeat the process 10,000 times, generating in essence 10,000 parallel histories of the sport.

The researchers found that 42% of the simulated histories had a streak of DiMaggio’s length or longer. The longest record streak was 109 games, the shortest, 39. In those 10,000 universes, many other players held the record more often than DiMaggio. Ty Cobb, for example, held it nearly 300 times.

DiMaggio’s streak, for better or worse, defined his life. Decades later, constantly hounded by autograph seekers, he wrote “If I thought this would be taking place due to the streak, I would have stopped hitting at 40 games.” He died just 10 years ago, at age 84. Joe Jr., who had trouble coping with his father’s fame, fell to a history of drug and alcohol abuse. He died five months after his father.

People are remembered—and often rewarded—not for their usual level of talent or hard work, but for their singular achievements, the ones that stand out in memory. It does no harm to view those achievements as heroic. But it does harm us to make investments or other decisions on a basis of misunderstanding. And it can be sad or even tragic when we interpret as failures plans or people simply because they did not succeed. Extraordinary events, both good and bad, can happen without extraordinary causes, and so it is best to always remember the other factor that is always present—the factor of chance.

Leonard Mlodinow teaches randomness at Caltech. His most recent book is “The Drunkard’s Walk: How Randomness Rules Our Lives.”

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Full article and photos: http://online.wsj.com/article/SB10001424052970204556804574261942466979118.html

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Loves Me, Loves Me Not (Do the Math)

“In the spring,” wrote Tennyson, “a young man’s fancy lightly turns to thoughts of love.” And so in keeping with the spirit of the season, this week’s column looks at love affairs — mathematically. The analysis is offered tongue in cheek, but it does touch on a serious point: that the laws of nature are written as differential equations. It also helps explain why, in the words of another poet, “the course of true love never did run smooth.”

To illustrate the approach, suppose Romeo is in love with Juliet, but in our version of the story, Juliet is a fickle lover. The more Romeo loves her, the more she wants to run away and hide. But when he takes the hint and backs off, she begins to find him strangely attractive. He, on the other hand, tends to echo her: he warms up when she loves him and cools down when she hates him.

What happens to our star-crossed lovers? How does their love ebb and flow over time? That’s where the math comes in. By writing equations that summarize how Romeo and Juliet respond to each other’s affections and then solving those equations with calculus, we can predict the course of their affair. The resulting forecast for this couple is, tragically, a never-ending cycle of love and hate. At least they manage to achieve simultaneous love a quarter of the time.

The model can be made more realistic in various ways. For instance, Romeo might react to his own feelings as well as to Juliet’s. He might be the type of guy who is so worried about throwing himself at her that he slows himself down as his love for her grows. Or he might be the other type, one who loves feeling in love so much that he loves her all the more for it.

Add to those possibilities the two ways Romeo could react to Juliet’s affections — either increasing or decreasing his own — and you see that there are four personality types, each corresponding to a different romantic style.

My students and those in Peter Christopher’s class at Worcester Polytechnic Institute have suggested such descriptive names as Hermit and Malevolent Misanthrope for the particular kind of Romeo who damps out his own love and also recoils from Juliet’s. Whereas the sort of Romeo who gets pumped by his own ardor but turned off by Juliet’s has been called a Narcissistic Nerd, Better Latent Than Never, and a Flirting Fink. (Feel free to post your own suggested names for these two types and the other two possibilities.)

Although these examples are whimsical, the equations that arise in them are of the far-reaching kind known as differential equations. They represent the most powerful tool humanity has ever created for making sense of the material world. Sir Isaac Newton used them to solve the ancient mystery of planetary motion. In so doing, he unified the heavens and the earth, showing that the same laws of motion applied to both.

In the 300 years since Newton, mankind has come to realize that the laws of physics are always expressed in the language of differential equations. This is true for the equations governing the flow of heat, air and water; for the laws of electricity and magnetism; even for the unfamiliar and often counterintuitive atomic realm where quantum mechanics reigns.

In all cases, the business of theoretical physics boils down to finding the right differential equations and solving them. When Newton discovered this key to the secrets of the universe, he felt it was so precious that he published it only as an anagram in Latin. Loosely translated, it reads: “It is useful to solve differential equations.”

The silly idea that love affairs might progress in a similar way occurred to me when I was in love for the first time, trying to understand my girlfriend’s baffling behavior. It was a summer romance at the end of my sophomore year in college. I was a lot like the first Romeo above, and she was even more like the first Juliet. The cycling of our relationship was driving me crazy until I realized that we were both acting mechanically, following simple rules of push and pull. But by the end of the summer my equations started to break down, and I was even more mystified than ever. As it turned out, the explanation was simple. There was an important variable that I’d left out of the equations — her old boyfriend wanted her back.

In mathematics we call this a three-body problem. It’s notoriously intractable, especially in the astronomical context where it first arose. After Newton solved the differential equations for the two-body problem (thus explaining why the planets move in elliptical orbits around the sun), he turned his attention to the three-body problem for the sun, earth and moon. He couldn’t solve it, and neither could anyone else. It later turned out that the three-body problem contains the seeds of chaos, rendering its behavior unpredictable in the long run.

Newton knew nothing about chaotic dynamics, but he did tell his friend Edmund Halley that the three-body problem had “made his head ache, and kept him awake so often, that he would think of it no more.”

I’m with you there, Sir Isaac.

NOTES:

For models of love affairs based on differential equations, see Section 5.3 in Strogatz, S. H. (1994) “Nonlinear Dynamics and Chaos.” Perseus, Cambridge, MA.

For Newton’s anagram, see page vii in Arnol’d, V. I. (1988) “Geometrical Methods in the Theory of Ordinary Differential Equations.” Springer, New York.

Chaos in the three-body problem is discussed in Peterson, I. (1993) “Newton’s Clock: Chaos in the Solar System.” W.H. Freeman, San Francisco.

For the quote about how the three-body problem made Newton’s head ache, see page 158 in Volume II of Brewster, D. (1855) “Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton.” Thomas Constable and Company, Edinburgh.

For readers who are curious about the math used here:
In the first story above, Romeo’s behavior was modeled by the differential equation dR/dt = aJ. This equation describes how Romeo’s love (represented by R) changes in the next instant (represented by dt). The amount of change (dR) is just a multiple (a) of Juliet’s current love (J) for him. This equation idealizes what we already know – that Romeo’s love goes up when Juliet loves him – by assuming something much stronger. It says that Romeo’s love increases in direct linear proportion to how much Juliet loves him. This assumption of linearity is not emotionally realistic, but it makes the subsequent analysis much easier. Juliet’s behavior, on the other hand, was modeled by the equation dJ/dt = -bR. The negative sign in front of the constant b reflects her tendency to cool off when Romeo is hot for her. Given these equations and an assumption about how the lovers felt about each other initially (R and J at time t = 0), one can use calculus to inch R and J forward, instant by instant. In this way, we can figure out how much Romeo and Juliet love (or hate) each other at any future time. For this elementary model, the equations should be familiar to students of math and physics: Romeo and Juliet behave like simple harmonic oscillators.

Steven Strogatz, New York Times

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Full article: http://judson.blogs.nytimes.com/2009/05/26/guest-column-loves-me-loves-me-not-do-the-math/?ref=opinion

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What Are The Odds?

As a physicist at California Institute of Technology and the author of many books and articles, a number of them on the science of probability, Leonard Mlodinow spends a lot of time considering the question, What are the odds?

For instance, what are the odds that a single person might write — as Dr. Mlodinow has — a paper titled “Pseudo-spin Structure and Large N Expansions for a Class of Generalized Helium Hamiltonians,” a best-selling book with Stephen Hawking and at least one episode of “McGyver.” (Answer: Pretty low.)

Dr. Mlodinow has a particular, and personal, interest how the most painful events can sometimes yield unexpected results. In the first chapter of his best-seller, “The Drunkard’s Walk: How Randomness Rules Our Lives,” he writes of a conversation with his father, who tells him of how he came to survive his time in the concentration camp in Buchenwald:

It struck me then that I have Hitler to thank for my existence, for the Germans had killed my father’s wife and two young children, erasing his prior life. And so were it not for the war, my father would never have emigrated to New York, never have met my mother, also a refugee, and never have produced me and my two brothers … The outline of our lives, like the candle’s flame, is continuously coaxed in new directions by a variety of random events that, along with our responses to them, determine our fate.

He answered the following questions via e-mail:

An earlier post by the psychologist Daniel Gilbert makes the argument that uncertainty — not knowing what misfortune will come — makes people more unhappy than misfortune itself. Do you find that to be true?

It does seem to be true of my own psychology. Also, I find that what’s most important, whatever happens, is how you deal with it. And once something bad actually happens, you can start that process, and bad can eventually even turn into good.

Many people are understandably anxious and depressed about their present and future situations. Based on what you know about predictions and human behavior, should they be?

I find that predicting the course of our lives is like predicting the weather. You might be able to predict your future in the short term, but the longer you look ahead, the less likely you are to be correct. In my own life, many things that seemed to be very bad at first actually had good consequences. For example, just as I had begun making a living writing in Hollywood many years ago, the Writer’s Guild called a strike. I thought it was awful for my fledgling career, not to mention financially ruinous. But as it turned out, the strike gave busy producers a chance to catch up on their reading, and the day the strike ended I got a call from the show runner at “Star Trek: the Next Generation,” who said he read a “McGyver” script of mine he had found lying around, loved it, and wanted to hire me on the show’s writing staff. It was a plum job and a boost to my career, and it would have never happened if not for the strike.

So I try to relax, and work on making the best of whatever develops, rather than worrying about how awful it is.

In times of crisis, such as this one, are most people able to accurately predict the outcomes of situations? Or do they tend toward too much optimism or too much pessimism?

I don’t think complex situations like this one can be predicted. There are too many uncontrollable or unmeasurable factors. Afterwards, of course, it will appear that some people had gotten it just right: since there are many people making many predictions, no doubt some of them will get it right, if only by chance. But that doesn’t mean that, if not for some unforeseen random turn, things wouldn’t have gone the other way.

The social historian (and socialist) Richard Henry Tawney, wrote, “Historians give an appearance of inevitability… by dragging into prominence the forces which have triumphed and thrusting into the background those which they have swallowed up.” And the (neo)conservative historian Albert Wohlstetter said it this way: “After the event, of course, a signal is always crystal clear. We can now see what event the disaster was signaling … but before the event it is obscure and pregnant with conflicting meanings.”

In some sense this idea is encapsulated in the cliché that “hindsight is always 20/20,” but people often behave as if the adage weren’t true. In government, for example, a “should-have-known-it” blame game is played after every tragedy. In the case of Pearl Harbor, for example, seven committees of the United States Congress delved into discovering why the American military had missed all the “signs” of a coming attack. One of the pieces of evidence cited as a harbinger recklessly ignored by the U.S. Navy was a request, intercepted and sent to the Office of Naval Intelligence in Washington, that a Japanese agent in Honolulu divide Pearl Harbor into five areas and make future reports on ships in harbor with reference to those areas. Of special interest were battleships, destroyers and carriers, as well as information regarding the anchoring of more than one ship at a single dock. In hindsight , that sounds ominous, but at other times similar requests had gone to Japanese agents in Panama, Vancouver, Portland and San Francisco. [The analysis is most famously laid out in the 1963 book, “Pearl Harbor: Warning and Decision,” by Roberta Wohlstetter, who was married to Albert, noted above.]

In addition to the intelligence reports that in hindsight seem to point toward a specific attack, there is also a huge background of useless intelligence, each week bringing new reams of sometimes alarming or mysterious messages and transcripts that would later prove misleading or insignificant. In advance of the event, you can’t tell one sort from the other.

It is hard to say whether people are too optimistic or too pessimistic. That depends on the person. But we should keep in mind that it is easy to concoct stories explaining the past, or to become confident about dubious scenarios of the future. We should view both explanations and prophecies with skepticism.

Should emotions — despair, anger, happiness — play a part in the decisions people make in their lives? In other words, should our feelings matter?

Of course our feelings matter. But emotional decisions are usually not the best ones. On the other hand, your emotions can affect your decisions whether you like it or not because the effects can occur on the unconscious level. One study even showed that subjects holding a hot cup of coffee judged people differently than subjects holding a cold cup. In my case, that effect wouldn’t have been unconscious, though. I know that cold coffee makes me grumpy.

A recent news story about treating cancer told the story of one woman — a non-smoking vegetarian who exercised and had little incidence of cancer in her family — who was shocked by her cancer diagnosis. Was her reaction — and others like it — reasonable? Does “living right” work against the odds that illness or misfortune will strike us?

Assuming one is correct about the proper way to “live right” — and I’m not convinced that a straight vegetarian diet is the healthiest — it is possible to decrease the odds of bad outcomes, but that doesn’t mean they won’t occur. Anything that is possible eventually will occur, which means that some healthy-living people will get cancer, and some chain smokers won’t. I once read a story about a church group that was supposed to meet at a certain time. Ten minutes after the appointed time, due to a gas leak, the church blew up. If they had not showed up late, all 10 would have been killed. Some see that as evidence that God was watching over them. Others might conclude that you should always show up to church late. All I learn from that is that it is a big country, and if you ask around enough, you’ll hear some pretty improbable stories.

Another example, which I analyze in “The Drunkard’s Walk,” is the time Roger Maris, a very good but not great player, broke Babe Ruth’s beloved record, hitting 61 home runs in 1961. Maris had never came close to that output before, nor did he after. What happened?

We all know that players will hit a few more home runs than usual in some years, and a few less in others. But the mathematics of chance also predicts that some years they’ll hit a lot more, and some years a lot less. Those large fluctuations are rare, and wouldn’t be record-breaking for most players, in any case. But the historical statistics of baseball show that there were enough players with excellent, but sub-Ruthian, ability that over the years that it was probable that, by chance alone, one of them would have a single standout year in which they tie or break Ruth’s record. In fact, every stand-out record in any sport that has ever been analyzed has always been found to be consistent with the patterns produced by random fluctuations. Performance over time comes mainly from talent and practice. But achievements that stand out from an athlete’s usual performance — hot streaks or record years — happen with patterns that match the patterns of chance. Just wait long enough, and strange things will happen.

Can a full understanding of the probability of certain outcomes help reduce anxiety? For instance: would knowing the statistical frequency (or infrequency) of plane crashes help someone overcome a fear of flying? Would a smoker knowing the actual odds that he will get cancer make him less fearful of that outcome? In short, do we worry too much, or too little?

My mother worries too much. Some say I worry too little. I guess that shows a) that one cannot say “we” worry too much or too little, and b) that whether an individual worries too much or too little is not 100 percent inherited from your mother.

I was once on a plane that experienced so much turbulence that when I looked out the window, the wings seemed to flap up and down like a bird’s. I noticed, also, that the woman in the window seat next to me looked pale and terrified. Personally, I took comfort in knowing how many miles planes fly through heavy turbulence without any problems at all. So I explained to the woman how planes were designed to withstand such conditions, and told her the slim odds of anything bad happening. When I finished, she turned away and reached for the barf bag.

Some people take solace in an understanding of their environment, others don’t. For me, an understanding of the role played by chance has taught me that one important factor in success is under our control: the number of at-bats, the number of chances taken, the number of opportunities seized. As someone who has taken risks in life I find it a comfort to know that even a coin weighted toward failure will sometimes land on success. Or, as I.B.M. pioneer Thomas Watson said, “If you want to succeed, double your failure rate.”

Leonard Mlodinow, New York Times

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Full article: http://happydays.blogs.nytimes.com/2009/05/22/what-are-the-odds/?ref=opinion

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Math and the City

As one of Olivia Judson’s biggest fans, I feel honored and a bit giddy to be filling in for her. But maybe I should confess up front that, unlike Olivia and the previous guest writers, I’m not a biologist, evolutionary or otherwise. In fact, I’m (gasp!) a mathematician.

One of the pleasures of looking at the world through mathematical eyes is that you can see certain patterns that would otherwise be hidden. This week’s column is about one such pattern. It’s a beautiful law of collective organization that links urban studies to zoology. It reveals Manhattan and a mouse to be variations on a single structural theme.

The mathematics of cities was launched in 1949 when George Zipf, a linguist working at Harvard, reported a striking regularity in the size distribution of cities. He noticed that if you tabulate the biggest cities in a given country and rank them according to their populations, the largest city is always about twice as big as the second largest, and three times as big as the third largest, and so on. In other words, the population of a city is, to a good approximation, inversely proportional to its rank. Why this should be true, no one knows.

Even more amazingly, Zipf’s law has apparently held for at least 100 years. Given the different social conditions from country to country, the different patterns of migration a century ago and many other variables that you’d think would make a difference, the generality of Zipf’s law is astonishing.

Keep in mind that this pattern emerged on its own. No city planner imposed it, and no citizens conspired to make it happen. Something is enforcing this invisible law, but we’re still in the dark about what that something might be.

Many inventive theorists working in disciplines ranging from economics to physics have taken a whack at explaining Zipf’s law, but no one has completely solved it. Paul Krugman, who has tackled the problem himself, wryly noted that “the usual complaint about economic theory is that our models are oversimplified — that they offer excessively neat views of complex, messy reality. [In the case of Zipf’s law] the reverse is true: we have complex, messy models, yet reality is startlingly neat and simple.”

After being stuck for a long time, the mathematics of cities has suddenly begun to take off again. Around 2006, scientists started discovering new mathematical laws about cities that are nearly as stunning as Zipf’s. But instead of focusing on the sizes of cities themselves, the new questions have to do with how city size affects other things we care about, like the amount of infrastructure needed to keep a city going.

For instance, if one city is 10 times as populous as another one, does it need 10 times as many gas stations? No. Bigger cities have more gas stations than smaller ones (of course), but not nearly in direct proportion to their size. The number of gas stations grows only in proportion to the 0.77 power of population. The crucial thing is that 0.77 is less than 1. This implies that the bigger a city is, the fewer gas stations it has per person. Put simply, bigger cities enjoy economies of scale. In this sense, bigger is greener.

The same pattern holds for other measures of infrastructure. Whether you measure miles of roadway or length of electrical cables, you find that all of these also decrease, per person, as city size increases. And all show an exponent between 0.7 and 0.9.

Now comes the spooky part. The same law is true for living things. That is, if you mentally replace cities by organisms and city size by body weight, the mathematical pattern remains the same.

For example, suppose you measure how many calories a mouse burns per day, compared to an elephant. Both are mammals, so at the cellular level you might expect they shouldn’t be too different. And indeed, when the cells of 10 different mammalian species were grown outside their host organisms, in a laboratory tissue culture, they all displayed the same metabolic rate. It was as if they didn’t know where they’d come from; they had no genetic memory of how big their donor was.

But now consider the elephant or the mouse as an intact animal, a functioning agglomeration of billions of cells. Then, on a pound for pound basis, the cells of an elephant consume far less energy than those of a mouse. The relevant law of metabolism, called Kleiber’s law, states that the metabolic needs of a mammal grow in proportion to its body weight raised to the 0.74 power.

This 0.74 power is uncannily close to the 0.77 observed for the law governing gas stations in cities. Coincidence? Maybe, but probably not. There are theoretical grounds to expect a power close to 3/4. Geoffrey West of the Santa Fe Institute and his colleagues Jim Brown and Brian Enquist have argued that a 3/4-power law is exactly what you’d expect if natural selection has evolved a transport system for conveying energy and nutrients as efficiently and rapidly as possible to all points of a three-dimensional body, using a fractal network built from a series of branching tubes — precisely the architecture seen in the circulatory system and the airways of the lung, and not too different from the roads and cables and pipes that keep a city alive.

These numerical coincidences seem to be telling us something profound. It appears that Aristotle’s metaphor of a city as a living thing is more than merely poetic. There may be deep laws of collective organization at work here, the same laws for aggregates of people and cells.

The numerology above would seem totally fortuitous if we hadn’t viewed cities and organisms through the lens of mathematics. By abstracting away nearly all the details involved in powering a mouse or a city, math exposes their underlying unity. In that way (and with apologies to Picasso), math is the lie that makes us realize the truth.

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NOTES:

For Zipf’s law see:

Zipf, G. K. (1949) “Human Behavior and the Principle of Least Effort.” Addison-Wesley, Cambridge, MA.

Gabaix, X. (1999) “Zipf’s law for cities: An explanation.” The Quarterly Journal of Economics 114, 739–767.

For Paul Krugman quote:

Krugman, P. (1996) “Confronting the mystery of urban hierarchy.” Journal of the Japanese and International Economies 10, 399–418.

The new laws of infrastructure for cities are reported in:

Bettencourt, L. M.A., Lobo, J., Helbing, D., Kühnert, C, and West, G. B. (2007) “Growth, innovation, and the pace of life in cities.” Proceedings of the National Academy of Sciences 104, 7301–7306.

For an overview of Kleiber’s law and the theory of West, Brown and Enquist, see:

Whitfield, J. (2006) “In the Beat of a Heart: Life, Energy, and the Unity of Nature.” Joseph Henry Press, Washington DC.

For the tissue culture results about mammalian cells, see:

Brown, M. F., Gratton, T. P., and Stuart, J. A. (2007) “Metabolic rate does not scale with body mass in cultured mammalian cells.” Am J Physiol Regul Integr Comp Physiol 292, R2115–R2121.

Steven Strogatz, New York Times

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Full article: http://judson.blogs.nytimes.com/2009/05/19/math-and-the-city/?ref=opinion

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