Sunday, May 04, 2014

Unexpected mathematics

kw: book reviews, nonfiction, essays, mathematics, savants, savant syndrome

There once was a young man who learned Icelandic in a week. He was then interviewed, in Icelandic, on one of the national TV channels there. What most people remember best from that interview is that he could properly pronounce "Eyjafjallajokull", the volcano that so disrupted European air travel in 2010. The Icelandic J and double L are variations on the Y sound, which are hard for Anglophones to hear, let alone pronounce correctly.

This young man, Daniel Tammet, is sometimes called an Autistic Savant, and scientists who studied him some years back diagnosed "high-functioning autistic savant syndrome". Daniel describes himself as painfully shy and hypersensitive. Seeing him interviewed, it is also evident that he is friendly and open, and very, very focused.

It is a pity that someone such as he would be called "autistic" in any measure. Even the term Asperger's Syndrome for the "high-functioning" end of the "autistic spectrum" can be a bit derogatory. Where does that spectrum end and simple, introverted shyness begin? If there is a scale from 0 (bold as brass) to 1,000 (essentially nonfunctional autism), do you start calling someone abnormal at, say, 600? What about a person who would measure 599?

I am sensitive to such considerations because of psychotherapy I endured about age 12, when I was diagnosed as "withdrawn", and urged by the therapist (a charming Austrian woman my parents' age) to "break out of my shell." Over time, I learned instead to build a doorway so I can come and go as I please. At times I still need my shell. I describe the process as very deliberately allocating about 10 IQ points to constructing a more pleasant and personable personality. The IQ I have left was measured at 160 on the Skyscraper test a couple decades ago, so I feel lucky to have had some brain power to spare.

Mr. Tammet is not only a linguistic genius, he is also a mathematical whiz and a synesthete. He sees numbers as having different colors. He introduces his brand of numerical thinking to us in Thinking in Numbers: On Life, Love, Meaning and Math. The 25 essays therein reveal that we all think mathematically more than we might imagine.

Early essays plumb the cultural use of number as revealed in language. Perhaps you've heard of certain tribes who count, "One, Two, Many" and no further. He describes a language that has no numerical terms at all, for a people who also live without consciousness of either history or future. But he dwells more on the way adjectives are added to numbers, such as the way we always say "four pairs of pants". Were someone to say "four pants" we might consider for a moment he is talking about heavy breathing. The Icelandic and Japanese languages reveal a near obsession with such terms, so in Japanese one does not say "four dinner plates" but "four flat things of dinner plates", which is just 3 words in Japanese.

Another essay grapples with the concept of numbers so large they lose meaning, even for him. These are not just numbers that would be tedious to count, but which are so large that there are more digits than the number of atoms in the visible Universe. It used to be, people would talk about "astronomical numbers", because astronomers bandy about millions and billions of light years as though describing a walk to the corner store. The numbers in "On Big Numbers" are, if anything, hyper-astronomical, requiring a hyper-universe for their quantities to make sense.

We are social creatures, even the extra-shy among us. That means we are in the business of prediction. To learn to cope with a parent, friend or lover, we need to create an interior model of their behavior. How many times has a misbehaving boy said, "If I tell my Mom, she'll KILL me!", exaggerating the trouble he knows he is in already. An entirely nonsocial creature would have no idea whether Mom objects or not. In "A Model Mother" we learn just how mathematical this model-construction behavior really is. We also learn the frailties of such models, because a person's will is more complex than any model. Heck, I can't tell from one moment to the next whether my house cat will enjoy a stroke or run away.

Think of it: your brain uses most of its capacity just running your own personality, which it "knows from the inside". It has much less capacity available to model another person, and since we all have from a dozen to a hundred or so people that we see frequently, we can't take up too much territory with any one model.

Light bulb alert: right in that last pair of sentences you can see the primary reason we have stereotypes. We group people because we must, to save room in our overcrowded brains! To pigeonhole someone into a group, and them remember that person as "Group R plus traits X and Y", simply ties up fewer resources.

In recent years Checkers has become a known game, meaning that every possible play has been enumerated and a strategy for either winning, or forcing a draw, can be computed for any starting configuration. Thus a computer Checkers player can never be beaten by a fallible human. Not so for chess. The essay "Talking Chess" makes it clear that to totally enumerate the game of Chess would produce a database that won't fit in the Universe. So while a mechanism such as Deep Blue may beat a human grandmaster from time to time, there will always be room for a heuristic mind such as ours to jigger a way around the algorithmic mind of a computer.

Another side point: we are very far from understanding the heuristic mind, and thus equally far from "artificial intelligence". I just gotta harp on that point. My wife watched "Transformers" on ABC last evening. Having seen it once, I declined a second viewing, because it is charmless without the surprises of the first viewing, and I know enough physics to be irritated by the repeated impossibilities. Those robots were way too human to have come to Earth from elsewhere, even the evil one.

I was particularly taken by a discussion of "income inequality" as it is currently called in political circles, as illuminated by the Pareto distribution. One version of the Pareto principle is sometimes called the 80-20 rule, meaning 80% of everything is owned (or earned) by 20% of the people. If you square the numbers, you find that 0.8x0.8 = 0.64 and 0.2x0.2 = 0.04, meaning that 64% is owned by 4%. Go to the next power, the cube, and 0.8% owns just over half of everything. That is the "one percent" of the Occupy Protest movement, those who own half of everything.

The author doesn't go into it, but to investigate the poor end of the scale requires taking roots, which most people don't understand. But your calculator can do it. Taking square roots I get 0.894 and 0.447 (rounded off). We need to subtract from 1.0 when going in this direction, so that we find 10.6% is owned by just over half of us. Cube roots? 0.928 and 0.585, or that 7.7% of stuff is owned by the bottom 41.5%. Not much different than the square root, so lets take the tenth root: 0.978 and 0.851, meaning that 2.2% is owned by just under 15%, and that is about the poverty threshold. Put real numbers on that. Assuming the 80-20 rule holds for the U.S., and everything in the country is valued at about 100 trillion dollars. 2.2% of that is $2.2 trillion, but divided among 15% of 320 million of us, it comes to about $46,000. That is an average for 48 million people. It would require yet higher math to figure out the expected net worth of the bottom 1%. Considering the real wealth distribution, 80-20 is probably a bit too egalitarian.

If we change the rule to a 90-10 rule (which changes an exponent in the actual Pareto equation), where 10% own 90%, we find that 1% own 81%, and that may be closer to the truth. Taking tenth roots to examine the poor end, we find 0.9895 and 0.7943, or 20.67% owning 0.0105%, for a net worth near $16,000, the value of a halfway decent car…that is average for the poorest 1/5 of us. Yeah, that is America today, I reckon.

The final chapter is about art. Mathematics is not really all humorless rigor. Mathematicians judge a proof or method by its beauty. My brother offered this analogy: in college where he double-majored in math and art history he had two friends that I'll call Bob and John. "We enter a room, one by one. There is a large mechanism there, and a very evident gear on one side. Bob goes in first, and finds that by pushing the gear with his shoulder he can make the mechanism work. Then I go in, and rummaging around, I find a crank. It fits into the gear, making it much easier to turn. John goes in last, pokes around some more, and finds a plug and a switch. Plug in and switch on, and the machine runs." Three ways of doing math. Can you guess which you'll find described in most journals on the subject? Finding that plug seems to be rather elusive! Those who find the plug and switch get Nobel prizes.

It is a delight to read the unexpected places Mr. Tammet's mathematical mind can take us. If you think you are bad at math, read this book. You'll find that you do more math than you realized, and that the writing of a gifted explainer can illuminate corners of the subject most of us never knew were there.

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