Thursday, August 23, 2007

On Mansfield's 2007 Jefferson Lecture

Later this fall, I'll be giving a lecture as part of the celebration surrounding Bennington College's 75th Anniversary. (Listening to lectures is considered to be a celebratory activity in academia.) I'm still considering what to talk about. The temptation is always there to do something with "the two cultures," or (particularly in the Bennington environment) to address art/science connections. But it's hard to avoid tepid generalities in that sphere. As a reminder of what not to do, I clipped the following paragraph from Harvey Mansfield's 2007 Jefferson Lecture, "How to Understand Politics: What the Humanities Can Say to Science". (See also the 05/09/07 article in the Chronicle of Higher Education.) Mansfield is a famous Harvard political scientist, but the first time I heard of him was in connection with his amusing system for assigning "ironic grades" in his courses, which made the papers a few years ago.

In this passage from the lecture, Mansfield is contrasting literature with science:

Literature, to repeat, besides seeking truth, also seeks to entertain - and why is this? The reason is not so much that some people have a base talent for telling stories and can't keep quiet. The reason, fundamentally, is that literature knows something that science does not: the human resistance to hearing the truth. Science does not inform scientists of this basic fact, and most of them are too consistent in devotion to science to learn it from any source outside science such as common sense. The wisdom of literature arises mainly from its attention to this point. To overcome the resistance to truth, literature makes use of fictions that are images of truth. To understand the fictions requires interpretation, an operation that literature welcomes and science hates for the same reason: that interpreters disagree. Literature is open to different degrees of understanding from a child's to a philosopher's, and yet somehow has something for everyone, whereas science achieves universality by speaking without rhetoric in a monotone, and succeeds in addressing only the company of scientists. Science is unable to reach the major part of humanity except by providing us with its obvious benefits. Literature takes on the big questions of human life that science ignores - what to do about a boring husband, for example. Science studies the very small and the very large, surely material for drama but not exploited by science because in its view the measure of small and large is merely human. Literature offers evidence for its insights from the observations of writers, above all from the judgment of great writers. These insights are replicable to readers according to their competence without the guarantee of scientific method that what one scientist sends is the same as what another receives. While science aims at agreement among scientists, in literature as in philosophy the greatest names disagree with one another.


One clue that this is gibberish is that you can interchange the words "science" and "literature" everywhere in this paragraph, and many of the resulting sentences remain true. Here are a few truisms resulting from this search-and-replace operation:


1. Science knows something that literature does not: the human resistance to hearing the truth.

Galileo knew this well. (So do you, if you've ever argued with a Creationist.)


2. Science is open to different degrees of understanding from a child's to a philosopher's.

We are obviously teaching children something in fifth-grade science class. And I'm guessing that that something is...science.


3. Literature is unable to reach the major part of humanity.

This truism holds because the major part of humanity doesn't read literature.

Well, I should qualify that: the major part of American humanity doesn't read literature. The 2004 NEA study Reading at Risk concluded that only slightly more than a third of adult males now read literature. Overall, less than half of adults in the study read any literature during 2002, the year covered by the questionnaire. "Reading literature" here means reading any novels, plays, short stories, or poetry. For example, reading a Left Behind novel would count as reading literature, for the purposes of the study.

It is a sad thing to realize that literature is unable to reach the major part of humanity, just as it is a sad thing to realize that science is unable to reach the major part of humanity. But I don't think that the explanations for either circumstance are to be found within science and literature themselves.


4. Science takes on the big questions of human life that literature ignores.

Where did it all come from...what is it all made of...where is it all going...what control can we exert over the forces that buffet us...why cannot my father raise himself from his bed?

These, I would argue, are some very big questions of human life, which literature can take up, but not take on.


5. In science as in philosophy the greatest names disagree with one another.

I think of the Einstein-Bohr debates...or of the debate between Einstein and Newton, which unfolded over centuries.


6. Science, besides seeking truth, seeks to entertain.

Guess what: science is fun. And maybe I'm a rarity for trying to use a little rhetoric and trying to avoid monotone in my scientific papers, but I don't think so. We all compete for each other's attention, and a little zip in the sauce never hurts. It's no different in the marketplace of scientific advances. For that matter, I think that excellent writing can be the clearest and most effective writing, even in a purely scientific context.


We could go on like this, but I have probably already become boring, so let me try to wrap this up quickly. (Also the semester is about to start, so I'd better get hopping on that.)

The question I came away with after reading this paragraph is, What does Mansfield think science is? By "science" he seems to mean some corpus of settled material, with its dry prose and careful skirting of controversy. He seems never to have witnessed the liveliness of scientists in their actual milieu: their daily bushwhacking along a dark and bewildering research frontier, as well as their Friday afternoon gab sessions, when everybody kicks back and debates the big picture. Listen in on one of these conversations, and you will realize that even two coauthors on the same journal article can differ importantly in the way they view their results.

Mansfield seems to view scientists as dispassionate masters of a circumscribed and uncontroversial text. But scientists disagree constantly with one another and argue heatedly amongst themselves, sometimes even about textbook material. Scientists also spend 90% of their day confused and off-balance: at a loss to understand their data; crumpling up the fiftieth attempt to get a calculation right. But no one in Mansfield's part of the campus seems to see their struggle and confusion; no one registers the rise and fall of their anxieties and ambitions. Perhaps this is because the scientists run academia now, and it is not in their interests to let their humanity show.

Sunday, August 19, 2007

Where Credit is Due

I remember the night Visa found me. It was 1993 and I was living in England. I was working late at the Mathematical Institute in Oxford. The phone rang. I was used to getting calls from my friends at all hours of the night - for one thing because of the time difference with the United States, and for another thing because my friends are the kind of people who are awake at all hours of the night no matter what time zone they're in. But there was no friend on the other end of this line. Visa had found me.

I left college in 1991 with more debt than I could handle and a bad habit of letting myself forget about it for months at a time. During my senior year of college I had fought a pretty good bout against American Express. I won on points: they wrote off the debt, and I gave back the card. But with Visa I got on my bicycle. I thought if I rode it all the way to England, I'd be safe. I didn't count on my mother turning me in.

***

My grad school years were lean, but they were also fat, because in Berkeley we lived like paupers but ate like kings. If one of us got a fellowship check, all of us ate Roquefort. I liked to say back then that my friends and I would be the first graduate students in the history of graduate school to come down with gout.

None of this lent itself particularly well to paying bills. I remember when I was first beginning to date a particular young woman, and the two of us went back to my apartment for the first time. We walked in, and I flicked the light switch. Nothing. PG&E had shut me down. Gamely, she lit some candles, then said she'd be just a minute in the bathroom. Soon, word came through the closed door: no toilet paper. Gamely, she accepted the paper coffee filters I passed through to her. Gamely, she paid my electric bill the next day. Surprisingly after such a beginning, our relationship lasted another two and a half years.

But to return to the issue of credit. In my fourth year at Berkeley, Capital One, an unknown but clearly hungry new company, offered me my first credit card in years, with a thousand-dollar limit. I took them up on the offer. On the very same day the card arrived, I maxed it out with a plane ticket to Amsterdam. Somebody over at Capital One lost their job that day, I'm pretty sure. But it was legitimate! I was scheduled to give a paper at a conference on quantum mechanics. But I didn't have funding for the airfare, so the card arrived just in time. My relationship with Capital One has lasted to this day, and it's had all the ups and downs of any relationship, though in this case mostly to do with APRs.

***

People ask me mathematical questions all the time. A friend once sent me the following cryptic email:

situation: test.
pool of possible questions: 5
possible amount on test to choose from: 2 or 3.
number of questions to answer: 1.

If I just choose two questions to study, what are the odds of having one of those show up on the test as a posed question to answer?


More recently, a friend asked how best to pay off a high-interest credit card. She'd been paying $600 per month on a $10,000 balance at 21.9% interest, and she was now considering cashing in a 403(b) account to eliminate the debt. After finding out some more about the person's situation, I advised taking out a home equity line of credit to pay off the debt. Assuming she stops using the card completely, then in three years, she can have the home equity loan paid off, and by that time she can also have $10,000 cash reserves in the bank - all for the same $600 per month she's paying now on the credit card. (The caveat of course is that you really do have to stop using the credit card, otherwise in three years you'll just find yourself maxed out again.)

In my own life, I have often wondered, if I owe X amount on a credit card at Y% interest, and I pay Z per month, then how long will it take me to pay off the card, assuming I don't charge anything else on it? The mathematics required to answer this is not trivial. There are online calculators that will do the computation for you - see this one for example. But the downside of using a calculator is that you don't get any insight into the problem you're facing. So I sat down one day several years ago and derived the answer once and for all. Here it is:



In this formula, b_0 is the initial balance, p is what you have decided to pay every month, and i is the daily interest rate, that is, your APR divided by 365. Note that the payoff time N turns out to be a function not of b_0 and p separately, but rather a function of the ratio p/b_0. Physicists will have seen this coming, thanks to their penchant for dimensional analysis. (The answer to the problem - a number of months - has no dollar signs on it; so the dollar signs on p and b_0 have to be got rid of. The only way to do that is to work with the ratio p/b_0.)

The derivation is here.

Getting into the details of this problem gives you a gut-level feel for an important financial fact: payments to the credit card company are mathematically the same as investments that grow with a guaranteed rate of return. Millionaires pay ludicrous fees to hedge funds in exchange for a guaranteed rate of return. Schmucks like us can do the same thing just by overpaying our credit card bill.

For convenience, I have put together the following table, suitable for printing out and stashing away in the utility drawer (click to enlarge):



For example, suppose your starting balance is $10,000, your interest rate is 12.9%, and you figure you can afford to pay $400 per month. Your monthly payment equals 4% of the starting balance. So look down the 4% column until you get to the row for an APR of 13%. You see that it will take you 29 months to pay off the card, or about two and a half years. (After a few months of making payments, call the company every so often to request a lower APR.)

Another example. Your starting balance is $10,000, your interest rate is 12.9%, and you really want to have this card paid off in a year, because you're planning to refinance your house in 18 months, and you want your credit score to be as high as possible. [Can somebody tell me when I suddenly became this middle-aged??] Looking at the table in the 13% row, you see that in order to have a 12-month payoff time, you're going to have to pay 9% of the starting balance, or $900, every month.

***

I liked Jean Chatzky's advice on her financial webpage. But the trouble with Chatzky's program is that it has nine steps. Who can remember nine steps? Who can carry them out? The people who can follow nine steps are not the people with maxed-out credit cards. So, as a recovered credit-card delinquent, I thought I would share my own program, which is so simple it has only one step:

Tithe 10 percent.

Tithing 10 percent means that any time any money whatsoever comes into your household - be it a paycheck, tax refund, honorarium, royalty, or even a good night's poker winnings - sit down that very night and send a check to your credit card company for 10 percent of whatever amount you brought in that day. Don't even wait for your account statement; just keep those checks moving out the door.

(If you have more than one credit card bill, divide your 10% tithe among the various cards in a ratio that makes the most sense to you.)

If your annual net income is, say, $30,000 per year, then tithing will divert $3,000 a year to your credit card, and what's more that'll be on a continual basis, hammering away at the debt before interest can pile up. They say interest never sleeps; well, don't let your payments sleep either. Think of your steady stream of checks as a flurry of jabs that will keep the credit card companies constantly on their heels.

If your tithe isn't bringing the balances down, then you can increase the percentage, make an extra payment when the account statement comes in the mail, and, most important of all, STOP USING THE CARD.

The system works, because in all honesty you can probably spare 10% of whatever you make, especially if you part with it immediately. After all, with the money gone, you can't very well blow it on Roquefort, can you?

Saturday, August 4, 2007

Flop on Pop

My new baby daughter loves to sleep on papa.





Unfortunately, this tends to pin papa down. Here's a quick review of some of the things that go through your mind when even a slight shift of position could throw your entire household into screaming chaos.

1. The best way to stay sane when you can't move a muscle is to invent word games on the spot and then try to play them. I have created dozens of these (see Word Puzzles for the Seriously Smart), and I'm constantly thinking of more. One game I'm playing lately is to think of what I call "unambiguous words." These are words that have only one meaning. By that I mean that the word has only one definition in your dictionary of choice. So far I have thought of the following examples, which have only one meaning in the American Heritage Dictionary, 4th Edition:

pub
schadenfreude
anyone
mattock
logy
vim
withered
bagel
inveigh
inure
iterate
venerate
emanate


I think it would be fun to assemble hundreds of these words and then use them to write poetry or short fiction. Would the paucity of meaning and the poverty of connotation lead to flat writing? Or would every word appear to be, in virtue of its specificity, "le mot juste"?

2. SLEW strikes me as an interesting word, because it can be interpreted as a noun - as in, "a slew of examples" - or as a past-tense verb (Cain slew Abel). So another game I'm playing is trying to find more words like this. The solutions tend to be rather choice. Here's what I have so far:

slew
dove
rung
spoke
felt
dug
spelt
moped
stove

I love these examples because (1) as verbs, they can only be past-tense; and (2) the noun sense and the verb sense have absolutely no conceptual connection to each other. A word like CUT is a past-tense verb and a noun, but CUT is also a present-tense verb, so it violates (1). A word like THOUGHT is a past-tense verb and a noun, but the verb sense and the noun sense are obviously related conceptually. Would love to see more examples satisfying (1) and (2), feel free to add more in the comments section.

3. As we all know, there is no such thing as "up" or "down." Better to think of it as "away from the center of the earth" and "towards the center of the earth." A propos of nothing, I wonder if you could raise a child in such a way that she understood this from the beginning. For example, you would never allow yourself to say things in front of the child like "What goes up, must come down." Instead you'd say, "What goes out, must come back in." Or when a song came on the radio like "Love Lift Us Up Where We Belong," you'd say, "What they really mean, honey, is Love Push Us Out Where We Belong."

Eh. Probably wouldn't work.

4. Although you can't move when you're sitting in that rocking chair, the upside is that you have lots of time to think about moving. Here is something I devised while thinking about locomotion and how it works on planet Earth.

***

The cosmic speed limit of 300,000,000 meters per second imposed by Einstein's theory of relativity is well-known. But it was only recently that I realized there's a way in which ordinary Newtonian physics also places some practical limits on your ability to move quickly from point A to point B on this planet.

To see how this comes about, let's suppose you plan to travel a distance D, beginning in a state of rest and arriving at your destination in a state of rest. Suppose also that your mode of travel relies on friction with the ground to make it work.

I should say, restricting yourself to a friction-based form of locomotion is not as limiting as it may seem. If you plan to run, walk, cartwheel, ride a bike, drive a car, or piggyback on the shoulders of a friendly robot, you'll be using friction to get where you're going. Among animals also, friction underlies the hopping of a toad, the inching of an inchworm, and the slithering of a sidewinder. For eons of evolutionary time, friction has been the basic engine used by man and beast for traveling on land.

An acceleration generated by friction will always scale as b*g, where g = 9.8 is the strength of the earth's gravitational field in standard units, and the dimensionless coefficient b characterizes the roughness of the two surfaces involved - say, the pavement and the soles of your shoes. The presence here of a material property such as the coefficient of friction needs no explanation. The reason for the presence of g is that, as it turns out, the maximum friction force attainable between two surfaces scales directly with the strength of the contact force pressing the two surfaces together. This is why you use "elbow grease" to get out a stain: by pressing harder as you scrub, you are making available a larger friction force to pull the dirt loose. In the case of locomotion, it's the earth's gravity that applies the elbow grease, pressing you to the ground. Hence, the maximum friction force you can use to push yourself forwards ultimately scales with the gravitational field strength g.

The implication of the acceleration-scale b*g is that, for a journey powered solely by friction, the time required to complete a journey of distance D will scale as (D/b g)^(1/2). Turning the reasoning around, we find that the greatest distance D you can cover under "friction power" within a fixed time T is given by D ~ b g T^2.

Plugging in some numbers, we find that in a lifetime of threescore and ten, the greatest possible distance you can cover - and still come to rest when the good Lord says you must - will be on the order of 10^(19) meters. This is a hundred times further than the earth-sun distance, meaning that the friction limit is not one that we'd actually bump up against in practice!

Here I've taken the coefficient of friction b to be of order unity, which is typically the case. Of course, the coefficient of friction does vary in value, depending on what sort of surface you're walking on (rough or slippery); and likewise it depends on what sort of material your bootsoles are made of.

So, one moral of the story: If you want to go far in life, wear track spikes.

***

The Newtonian limit of 10^(19) meters is actually not as strict as the Einsteinian limit, which is c*T ~ 7*10^(17) meters. But the two limits coincide when b g T^2 ~ c T, or when b ~ c/gT ~ 0.01. On a highly polished planet, the Newtonian limit would actually be the sharper of the two.

***

The friction limit is not fundamental physics. For example, it doesn't apply to travel by jet (or, what is much the same, travel by rocket). Rockets move by harnessing a controlled explosion near the rear of the craft. Essentially, the debris from the explosion bangs against the backside of the ship, knocking it forwards. Using a jet engine, you can lift gently off the ground, accelerate forwards at a great rate, coast at high speed, and then reverse the thrusters to bring you back to a state of rest, touching down gently at your destination. The material properties of the intervening land will have nothing to do with your trip time.

An ordinary propeller plane is a closer analogy to the friction limit. This is because a propeller plane depends for its propulsion on the viscosity of the air - and viscosity is more or less the fluid equivalent of friction between solids. A similar mechanism would be an oceangoing ship's propeller screw: it depends for its effectiveness on the viscosity of water. As Rayleigh observed in the 19th Century, if water had no viscosity - today we can produce such "superfluids" in the laboratory - then a ship's propeller would be useless; the ship would merely agitate in place, going nowhere, as the propeller blades slipped through the water without generating any thrust. Of course, as Rayleigh probably also observed, if water had no viscosity, ships would hardly need propeller screws; you could just give the ship a good shove, and it would glide through the water for miles.

***

In a watery environment such as the sea, organisms have a choice between using friction on the seafloor to drive them forwards, or using the viscosity of the water to generate thrust. Creatures who swim leave creeping things in their wake; but on land, where the parameter values are vastly different, the best horizontal runners and the best horizontal flyers enjoy similar speeds, topping out in the 70 mph range.

***

Earlier, I argued qualitatively that a friction-powered trip cannot be completed in a time less than about T ~ (D/bg)^(1/2). A detailed analysis shows that the precise lower limit is T_min = 2(D/bg)^(1/2). The analysis leading to this bound assumes that during the trip, your center of mass remains within a plane parallel to the ground. However, if you can launch yourself through the air Incredible-Hulk-style, then you can actually get to your destination faster. This is because, during the liftoff phase, you'll be pressing into the ground with a force greater than your own weight, and this extra "elbow grease" will allow you to generate a larger horizontal friction force than you otherwise could. Of course, you'll waste some time rising into the air and coming back down; but if you launch yourself at the shallowest attainable angle (ArcTan(1/b)), then the gambit proves worthwhile. In fact, it turns out that a distance D can be leapt in only a time T_min = (2D/bg)^(1/2). The factor of 2 in the ground-based strategy has become a factor of 2^(1/2), which is a time savings of about 30%.

If D is a distance of many meters, then leaping all the way to the destination will be out of the question for a mere human. But thinking of D as being only a meter or two, we see that the strategy of loping is superior to that of a gliding walk. While this observation hardly explains exactly how or exactly why humans spontaneously break into a run to save time - that depends on metabolism and the shapes of our bodies - it may go some ways towards explaining why running as a strategy exists among animals at all.

***

If you find this kind of thing at all interesting, let me recommend a wonderful book, Life's Devices, by Stephen Vogel. This book is the On Growth and Form of our times.

***

My scientific hero, John Bell, once said that impossibility proofs in physics are proof of a lack of imagination. My proof that you can't travel a distance D in a time less than (2D/bg)^(1/2) is no different. For, rather than leaping the distance D, you could simply invent the starting block and get to your destination as fast as you like. (Don't forget the anabolic steroids.)

I sometimes wonder if the cosmic speed limit imposed by Einstein's relativity theory has a similarly simple countermeasure.

***

If you've made it this far, I end with a scrap of creative writing inspired by the foregoing.

***


The Ribenians secrete a thin fluid from their pores, which renders their bodies nearly frictionless. Life in their society is very difficult for this reason.

Efforts to clothe the Ribenians proved fruitless. After many attempts, a shoe was placed on the foot of a child. It slid off as soon as he rested his foot on the ground. Adults, bound in linen cloth, soon faced into their wrappings, or writhed out of them entirely.

The Ribenians organize themselves into clans. Each clan lives in a hollow, where the clan members wriggle over one another in a great tangle of bodies. Life in the clan is a continual struggle to reach the higher levels, where there is more light and air, and where one may drink fresh rainwater instead of ground-seepage. The traditional technique for ascent is to migrate to the bottom of the hollow, then push directly downwards with the feet, with enough force to propel oneself upwards through the layered bodies to reach the surface. The overzealous individual emerges with too much residual momentum, expelling himself from the clan. He glides smoothly along the gently rising grade of the hollow. Slowing continuously, he may eventually fall back towards the clan, or he may slide over a crest and into a neighboring clan's hollow. Such events allow the Ribenians to maintain genetic diversity in their tribes.