Posted tagged ‘Mathematics’

Friday xkcd break: When Mathematicans Stray

December 5, 2008

With thanks, as always to xkcd, in whose head it must be passing strange, (though fun) to live.

Integers we have loved — in honor of National Poetry Month

April 15, 2008

This blog talks a lot about the importance of quantifying things — using numbers to abstract from the details of experience. What I rarely add in sentences like that is that there is a purpose to such abstraction: to find meaning deeper than the surface impressions with which we begin.

Well — you take insight where you can get it, and this morning, I got up early to attend my son’s Second Grade class Poetry Cafe. There each of the sixteen kids got up and recited a favorite poem from memory. One of my son’s friends, Sparky, got up and declaimed about four stanzas of Mary Cornish’s “Numbers.”

Soulless thug that I am, I had never come across it. Hearing it in that squeaky seven-year-old voice, I found it captured precisely the idea I have labored to express. Numbers are generous, in that “they are willing to count anything or anyone.” All I’m asking is that we embrace such kindness.

National Poetry Month it is…so enjoy.

Numbers

I like the generosity of numbers.
The way, for example,
they are willing to count
anything or anyone:
two pickles, one door to the room,
eight dancers dressed as swans.

I like the domesticity of addition–
add two cups of milk and stir
the sense of plenty: six plums
on the ground, three more
falling from the tree.

And multiplication’s school
of fish times fish,
whose silver bodies breed
beneath the shadow
of a boat.

Even subtraction is never loss,
just addition somewhere else:
five sparrows take away two,
the two in someone else’s
garden now.

There’s an amplitude to long division,
as it opens Chinese take-out
box by paper box,
inside every folded cookie
a new fortune.

And I never fail to be surprised
by the gift of an odd remainder,
footloose at the end:
forty-seven divided by eleven equals four,
with three remaining.

Three boys beyond their mothers’ call,
two Italians off to the sea,
one sock that isn’t anywhere you look.

–by Mary Cornish.

Originally published in Poetry magazine, Volume CLXXVI, Number 3, June 2000.

Image Person Scott Foresman, “Abacus,” copyright donated to the Wikimedia Foundation.

I don’t know nuthin’ ’bout economics, but…: NPR/Henri Poincaré/Mortgage follies edition

February 25, 2008

Innumeracy is a problem I have and will come back to a lot here. But as I listen to more and more popular presentations of technical subjects, I still get astonished by the intersection of two structural problems in the media.

That is: many reporters — not so high a proportion of self-described science writers, though still plenty there — have trouble with even the most elementary uses of quantitative approaches to their stories because they just don’t think in numbers at all. That’s the negative way of framing the problem; journalists have a lack that inhibits their capacity to do good work in an ever-more technically imbued world.

Then there’s the affirmative problem. Reporters establish stories by anecdote, by individual bits of data, single episodes. They’re called stories for a reason: the goal is to perform one of the most powerful acts of communication humans have figured out, to convey information that compels belief because its hearer can place themselves right into the narrative.

That’s why, to edge closer to the real subject of this post, so much of the reporting on the mortgage crisis (fiasco) centers on some family that’s about to lose a house, and spend little time, on the meaning of the big numbers, like the implications of a repricing of US housing on a large scale.  The point is that not only do many journalists not know a set of ideas that could help them figure out such things;  what they do know leads them away from the kind of approach to their work that more mathematical sophistication would provoke.

But there’s a wonderful passage that bears on this from the great French mathematician Henri Poincaré in a collection of essays that greatly influenced the young Albert Einstein:

We can not know all facts, and it is necessary to chose those which are worthy of being known.

Choose? Worthy? Surely Poincaré is not going prematurely po-mo on us here?

Not really. The notion embedded in his deliberately provocative turn of phrase is that facts need form, some apparatus that can incorporate a given datum into a richer story — one with a meaning larger than that of a single incident. That apparatus is quantitative.

(BTW — I use the word “quantitative” rather than mathematical, because for a great deal of human experience, the math needed to make sense of what’s going on is not that complicated.  It’s often a matter of counting, sorting, and extracting relationships within the formal limits of what you learn by the end of high school.  I have posted on a couple of such examples from great scientists — Freeman Dyson, for one, and J.B.S. Haldane for another.  There are lots more — perhaps readers could be persuaded to post examples of what they think are elegant, simple insights a bit of math can give us ?)

All of this  into mind while I listened to NPR this morning.

This is the story that got me going — a short (1 minute, 10 seconds) reporter-voiced account of what seemed to the Morning Edition team to be something strange: Even though the Fed is cutting interest rates, mortgage rates went up sharply last week. That ain’t how its supposed to be, according to the reporter, Adam Davidson, because when the Fed lowers its rates, other rates are supposed to drop.

The reason Davidson gave for what he saw as weird is not all wrong: he said that lenders are newly afraid of inflation, and hence want to charge a higher price for money that is going to be paid back over time.

But look at the unexamined assumption: that the Fed can control rates in general. That’s not true.

What’s missing here? An understanding of the real importance of time.

The Fed mostly exerts its influence on interest rates through the shortest of short-term instruments, the overnight federal funds rate — which is just the price banks pay for extremely brief loans required to keep their minimum reserves up to snuff.

But real people borrow money for houses on long time scales, most famously through 30 year mortgages. The enormous difference between the types and uncertainties of risk between those two scales of time serve at least partially to decouple the two rates — see the data to be retrieved here for a survey view of this.

So it is true that fear of inflation could keep push term rates up, whether or not the Fed was playing around with short term rates. But so could lots of other things.

Perhaps that the value of US real estate is unclear in a falling market, and thus lenders demand a risk premium before they lend against such difficult-to-value assets. Perhaps the overall credit worthiness rating of American real estate borrowers has dropped in the aggregate.  Perhaps lenders fear that the secondary market for mortgages is going to get a bit less liquid.  Lots of factors play into long term interest rates that have nothing to do with the reasons the Fed makes its interest rate decisions.

In other words: and the NPR story was either meaningless or misleading. And it failed because the reporter glossed over or did not fully understand what the mortgage rate summarizes as a single number — all the complex calculations of risk and profit that underpin the decision of whether or not to make a loan.

What I would have loved to hear instead of a “this fact is strange” report would be that story: how do interest rates express quantitatively our ideas about the future.  It’s still a good, fully human story:  Those numbers tells us a tale about what we think we know about what’s coming down the pike — and how much in dollars and cents we fear changes in our perception of what we don’t know.

Image: Rembrandt van Rijn, “The Money Changer,” 1627. Source: Wikimedia Commons.

On Being The Right Size (Hollywood edition).

January 25, 2008

I can’t believe it, but I am going to link to Gregg Easterbrook twice in one day without (too much) snark.

So, while his TMQ column for Monday (sic) did contain an elementary error (the planets move against a background of the “fixed” stars, not the other way round — which Easterbrook honorably corrected at the top of his next column) he gets something else quite right.

In a ramble through absurdities in the movie Cloverfield, he and his correspondents pause for moment on the issue of the monster’s size:

TMQ’s estimate of 100,000 tons for the Cloverfield monster was based on the Empire State Building weighing 340,000 tons; TMQ assumed a biological object the size of that building might weigh less, containing no steel. Kendal Stitzel of Fort Collins, Colo.,, countered, “Therein lies the rub, for there is no known bony material that could support the weight of something that large without collapsing under the creature’s own mass. This is the famous square-cube problem: when a creature gets larger, its weight (which increases in proportion to volume) increases as the cube of the increased dimensions. The animal’s strength, however, can only increase in proportion to the square of the increase in dimension. Just as the Empire State is not supported by its masonry but by the steel and concrete structures inside, you would need some kind of similarly strong biological material to support any giant monster, be [it] Godzilla, Mothra or Cloverfield. There have been giant critters in the past, but no land mammal larger than the woolly mammoth. Whales are big, but their bodies are supported by water. Dinosaurs grew to be perfectly enormous; some were an order of magnitude larger than any other land creature since. Skeletal adaptations let them do this — but they were near the limit of what is possible for critters on our planet, and the largest dinosaurs reached only a fraction of the size of many movie monsters.”

Readers with a taste for both great science writing and the history of modern biology probably know the ur-form of this idea as expressed by the great British biologist J. B. S. Haldane, in his classic essay, “On Being the Right Size.

Read the whole thing. It’s smart, witty, elegantly written, and it contains one of the earliest popular accounts of perhaps the most important single change in the practice of biology in the last century. Haldane himself was one of the pioneers in the mathematical treatment of natural selection and evolutionary theory, and he introduced the general public to the virtues of applying even the simplest quantitative ideas in “On Being the Right Size,” a simple, virtuouso tour through the implications of scale for everything an organism might want to do.

And in making the point that Easterbrook’s correspondent, Kendal Stitzel picks up, Haldane produced one of the truly great passages in all of science writing — the quotation of which is the reason for this entire post:

You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom, it gets a slight shock and walks away, provided that the ground is fairly soft. A rat is killed, a man is broken, a horse splashes.

Splashes!

That’s real writing. Once read, it is impossible to forget the idea within the image.

Image: The Darley Arabian (one of the three founding horses of English thoroughbred brood stock. After 1704. Source: Wikipedia Commons.