Tuesday, March 03, 2015

Math is a way of thinking

kw: book reviews, nonfiction, mathematics, mathematical thinking, mathematical games

In the realm of the English and Americans being "divided by a common language" (widely attributed to Shaw, but author not known), the abbreviation for "mathematics" is "maths" in England and "math" in the U.S. The term itself can be colloquially translated "learnèd techniques". Note the accent; thus, mathematics are techniques of those who are learnèd.

Matt Parker wants to make math—he writes "maths", being British—enjoyable. For most people, "Math is hard," to quote the talking Barbie doll. The funny thing is, we use math all the time. To make us more aware of our penchant for mathematical thinking, and to show us some ways to play in a mathematical way, he has written Things to Make and Do in the Fourth Dimension. He bills himself as a stand-up comic and mathematician. The book is subtitled "A Mathematical Journey Through Narcissistic Numbers, Optimal Dating Algorithms, at Least Two Kinds of Infinity, and More".

Well, how can mathematics, which encompasses much more than mere number-work, be made enjoyable? Can it be FUN? In my case, Parker is preaching to the choir. I was the kind of geeky kid who did enormous long division problems for fun. The kind who angered a series of calculus instructors by correcting them during class (It took me decades to learn sufficient tact to brace a fellow with his errors in the privacy of his office).

To anyone who has survived the standard American curriculum and graduated from High School, we started with "four banger" arithmetic (add, subtract, multiply, divide), went on to just a bit of exponents and roots (in my day we learned to extract a square root with pencil and paper), then geometry and algebra (in either order), trigonometry, and, if you were a High School senior after about 1966, introductory calculus.

Once you'd been schooled in algebra and plane geometry, did anyone bother to tell you they are equivalent? that one can solve with straight edge and compass the same problems that are presented with X's and Y's and such? I thought not. Probably because they were taught by different teachers; the algebra teacher probably didn't know geometry all that well, and vice versa: nobody told them either!

OK, what's fun about math anyway? Do you remember π? That odd number a bit larger than 3 that has something to do with a circle? For everyday purposes we can use 3.14 or 3 1/7 or 22/7. If you get familiar with it, you can win bar bets and get the occasional free drink. Here's how. You make a bet with someone that the glass he or she is drinking from is bigger around than it is tall. Make sure to use the word "around" not "across". Most people will say, "No way!" If they take the bet, hand them a piece of string. Have the person wrap it around the glass, and mark the length, then hold it next to the glass. The mark will nearly always be above the rim. Why do I say, "nearly always"? Some drinking glasses are quite tall and thin, but not the kind you'll find beer in. So do this for preparation. Get some string and do the comparison using all the different kinds of drinking glasses you find around the house. It is likely that only a really skinny iced-tea glass will be taller than it is around. In a bar, just eyeball that the height is less than three times the width, and you'll be OK.

But fun with math is more than just bar bets. Parker's stand-up routine is based on math, and he writes of a number of card tricks that use mathematical methods. One well-used card trick bases its "clairvoyant" result on the fact that 27 is 3x3x3…and here you thought the deck a stage magician was using had all 52 cards in it! And there are the numbers for lovers (Parker calls them "amicable numbers"). The smallest "loving" pair is 220 and 284. All the factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110. Add those 11 numbers: the sum is 284. All the factors of 284 are 1, 2, 4, 71, and 142. Add those 4, and the sum is 220. You can sometimes buy a little "puzzle heart", in two pieces with 220 on one half and 284 on the other. There are other (mostly much larger) pairs of amicable numbers, if you want a more geeky puzzle heart made to order.

And on the subject of amicability, or better, there is that "optimal dating algorithm" of the subtitle. An algorithm is a recipe, for cooking up the solution to a problem. In this case, the problem is finding a compatible spouse. In an early chapter, Parker refers us to a few gents (very few early mathematicians were female) who showed that the "optimal testing proportion" of a string of dates is the square root of the total number of dates (with different people) you are prepared to embark upon. Thus, if you plan to allow up to two years for the search, and have time for one date weekly (Friday or Saturday, your choice), that is about 100 maximum dates. The square root of 100 is 10, so you use the first 10 dates to gather information, make your lists, compile the strong and weak points, and determine which person you dated is the most compatible potential spouse. Then, you continue dating new people until someone comes along who is better than the best of the first 10. Stop your search and propose marriage. Suppose you get to the end of the two years, and nobody beat "good old #7"? You can't go back, #7 probably already married someone else. And the chances are about 5% that the 2-year search will fail, statistically. Now what? You can shrink the chance of such failure this way. After the next group of 10, you drop your standard a little, say to better than the second-best of the first 10. There's more statistics one can do, but you'll probably get swept off your feet by someone unexpected long before you reach the 100th date anyway!

Number tomfoolery and some mapping stuff (like the 4-color problem) take up 9 chapters, and then we get into higher dimensions. The 4th dimension is just the beginning. Though it takes a while, we eventually read of a conjecture that requires the use of a space with nearly 200,000 dimensions! The fact that we are alive is sufficient proof to me that no 4D space exists, at least not one that can contact our 3D space. An entity who lives in a 4D space could reach inside us and stop our heart, or remove it for our inspection, as Regina and Rumpel do in episodes of "Once Upon a Time". Doing so would be as easy for them as it is for us to touch the middle of a circle drawn on paper. There is a bigger reason, though, that he mentions as an aside. Orbital mechanics won't work in 4D, not even a little bit. You can't get a planet to orbit a star in any dimension higher than 3. And this is why I deny "string theory", which requires either 10 or 11 dimensions for the math to work.

A lot of the ground the book covers is in the field of topology. Mathematical I may be, but topology is an area I have shunned. The author did more to give me at least a glimmer of topological understanding, than shelves of math books by others. But not more than a glimmer. It really depends how your mind works.

Clearly, in dimensional and topological math, Parker is a genius compared to me. I do find that he comes up short in other areas, however. For one, he mentions at one point his computer idling along at 2.7 GHz, and follows with a parenthesis and a footnote:

The parenthesis: "(2.7 gigahertz is a measurement of how many times its logic gates can be run every second).*"

The footnote: * Actually, this is how many times the processor performs commands in a second, each of which could involve more than one calculation. So this is a low estimate for comparison. A more dedicated me would research how many actual calculations it does per second, aka FLOPS.

The italics in the footnote are mine, and point out an error. The original parenthesis is correct. 2.7 GHz is the rate at which the processor's clock runs, and the clock controls the logic gates. Some hardware operations (what he loosely calls "commands") take one clock tick to run, others take more than one, usually two to four, but perhaps even more. So the basic hardware instruction rate is slower than 2.7 GHz, and 2.7 GHz (for the CPU in his computer) is the highest rate, and thus is a high estimate, not a low one as he states. Furthermore, FLOPS refers to FLoating-point Operations Per Second, where floating-point refers to the calculation of numerical quantities. A 2.7 GHz processor includes a special floating-point processor, these days called a math unit, and it tops out at several hundred MFLOPS (millions of FLOPS).

Back to areas in which our author shines. He presents a geometric proof that an infinite series can have a finite sum, using one based on Zeno's Paradox (though he doesn't say so). Zeno asked, if a runner (he called him Achilles) has two miles to run, first he runs a mile, then a half mile, then a quarter mile, and so forth: does he ever arrive? Of course we know that the second mile is run in about the same time as the first. But it is stated as 1 + 1/2 + 1/4 + 1/8 and so forth, a series that goes on forever. We know in our gut that the sum is 2. Here is the geometry:

It is easy to see that you can continue dividing by 2 as long as your patience holds out. The little blue square holds all the pieces I didn't have patience to draw.

This is the most ancient (known) example of a converging series. Most series diverge, and the one that is right on the edge is the sum of all reciprocals: 1 + 1/2 + 1/3 + 1/4 and so forth. The book has a very clear proof on page 289 that this sum grows without bound (I was careful not to use the word "infinity". That is for later).

For those who aren't afraid of exponents, the sum of reciprocal numbers to a power, where the exponent is close to one, has a finite sum as long as the exponent is greater than one, but grows without bound if it is one or less. Thus, 1/1ⁿ + 1/2ⁿ + 1/3ⁿ + 1/4ⁿ is finite even if n is 1.00000000001 (or add as many zeroes as you like, but keep that last 1 ).

OK, let's talk about infinity. A late chapter is called "To Infinity and Beyond" (nods to Buzz Lightyear). Do you recall the different kinds of numbers? For review:
  • Natural numbers: 1, 2, 3, etc. Also called Counting Numbers.
  • Integers: the Natural numbers plus zero and negatives of the Natural numbers.
  • Rational numbers: Ratios of any two integers such as 1/2, 19/14, 32768/4195.
  • Irrational numbers: All non-Integers that have unending, nonrepeating decimal parts. The most familiar examples are √2 and π, and most people remember at least 1.414 for the one and 3.1416 for the other.
As it happens, there are two kinds of Irrational numbers, but not everyone hears of them even in high school math classes. Firstly, Algebraic numbers are also called Computable numbers, because they are the solution to certain computations, primarily involving polynomials, such as square roots. Secondly, Transcendental numbers are a great deal trickier. Some of them such as π are found in trigonometric equations, and others such as e (2.71828...) in logarithmic and exponential expressions. But they are not "computable" the way square roots are.

With that under our belt, Algebraic irrational numbers are abundant and comparatively familiar. Transcendental numbers are difficult to deal with, and the ones that are known to be so are rather few. It is very difficult to prove that a certain quantity is a transcendental number. The odd thing is, it is not hard to prove that there are a lot of them lurking in the number line. In fact, the Transcendental numbers infinitely outnumber all the rest! A paradoxical phrase I learned in graduate school states:
Between any two transcendental numbers, there exists at least one algebraic number. Between any two algebraic numbers, there exists an infinite quantity of transcendental numbers.
Parker demonstrates this with an amusing analogy called the Hilbert Hotel, attributed to Georg Cantor (Hilbert and Cantor were math geniuses of roughly 120 years ago). Infinite busloads of several kinds of "guests", meaning several kinds of algebraic numbers, are accommodated in the hotel and can always be fit in. Then a bus with just the transcendental numbers between 0 and 1 shows up, and the hotel cannot hold them all. The proof is on page 413, and makes sense while I am reading it, but escapes me immediately thereafter!

This shows that there are at least two kinds of infinity, now called Aleph-0 (or -null) and Aleph-1. But it is not known if there is a different Aleph that is "larger" than Aleph-0 but "smaller" than Aleph-1.

I think Matt Parker genuinely believes that anyone could love and enjoy math, given the right approach. I'd agree only if we recognize that mathematical thinking of certain kinds may be universal among us humans, but that a great many branches of the math tree are forever beyond the reach of many people, no matter what kind of schooling or inducement is offered. Certain kinds of minds are required to do certain kinds of thinking. As I get older, I realize more and more the immense diversity of humankind. A political scientist, a journeyman carpenter, and a medical technician, all regularly think thoughts in realms that will forever be beyond my understanding. They can think thoughts I could never learn to think. That's OK. I think I have a few thoughts of my own that many other folks will never comprehend.

I'll go further. Look at your automobile. The days are long gone that a single person can design and build an entire auto, the way Carl Benz did in 1885. It takes about 8 different kinds of engineer to do so now. Even 40-50 years ago I could take out the motor and rebuild it (did so, 3 times). Now I couldn't get it out without a set of tools I can't afford.

But don't let my quibbles and quandaries discourage you from reading the book. Matt Parker writes delightfully, with a clarity that gets around the defenses we might have against allowing any more math to get into our overstuffed head. Reading this book is like looking through a microscope or telescope. It shows a new landscape, and you may not comprehend it all, but the view is worth it anyway.

Saturday, February 21, 2015

Sandwich Generation - the rubber hits the road

kw: book reviews, short fiction, short stories, poetry, caregiving

One of my earliest memories is of leading my grandfather by the hand to take a walk around the front yard of his winter house in the desert. I was about six years old. My last memory of him comes about six years later, shortly before he died; it is my only memory of him speaking. In those days it was called "hardening of the arteries." It was probably Alzheimer Syndrome. He was peaceable in his dotage and my grandmother cared for him until the end.

Take a look at this face. This is his grandmother Elizabeth when she was about sixty. This is an Alzheimer face. He wore a similar face much of the time: disconnected and vaguely unhappy. This woman's daughter, my great-grandmother, died in her fifties. From dementia? We don't know.

My mother and her sister inherited the syndrome. My aunt was more cheerful about the prospect, telling me (before she lost the power of speech), "If I'm going to go crazy, I intend to enjoy it!" My mother coped the best she could, and we felt fortunate she didn't stop speaking.

My last memories of my mother, just a month before her death, began when I walked into my parents' sitting room where her gurney-bed was: she looked up and called me by name. She hadn't spoken my name for five years or more. During that last visit, which lasted just four days, she called me by name a few more times. I am told this is common. In both her case and her father's case there was a rallying and greater clarity in the last month or two of life. Somehow I knew this without being told, and expected she would not long survive my visit. Six weeks later I crossed the country again to attend her funeral.

My father would tell a more arduous story. Until he was 80 he cared for her himself, but he had to hire a nurse to help during the last year or so. He got so burned out. So did the nurse. After Mom died she went into a different line of work.

Now I sometimes wonder whether I'll be next in line. It used to worry me a lot. Here is a poem I wrote to my mother (but never showed her) a few months before that last visit:
I held your father’s hand
When I was just a little boy.
He needed help to find his way around.
He was like a friendly puppy,
And he liked to be with me.
When I’d walk around the block, he’d come along.

The only time I heard him speak,
I was nearly 12.
I was asking for some tweezers for a thorn.
He spoke up, and said, “I have some!”
And he led me down the hall
To his tool bench at the back of the garage.

A retired piano tuner,
He had tools of every kind:
Wrenches, screwdrivers, a tuning hammer, saws.
The tweezers that he handed me
Were longer than my hand.
But I managed to pull out that thorn with them…

More than forty years have passed,
And as we walk around the block
I must hold your hand, so you can find your way.
This is something in our family,
They say it’s in the genes.
When it is my turn, who will hold my hand?
Had I known a certain volume was being prepared, I'd have submitted this, perhaps with some small chance it would be accepted. I've just read Living in the Land of Limbo, edited by Carol Levine. It is subtitled "Fiction and Poetry about Family Caregiving", and is one of the more touching and memorable volumes I've read.

Ms Levine has organized the book well, because "family" means relationships in all directions, up, down and sideways. I noticed that many of the writers are Chinese or other Easterners. The Western way seems to be to warehouse people when we get uncomfortable with them, and then feel virtuous if we happen to visit at least weekly. Folks seem to add it to their list of duties right on a par with "going to church". "OK, an hour for church on Sunday morning. Check. An hour with Dad (or Mom or Aunt Rose…) on Wednesday afternoon. Check." If you've never heard the song by Harry Chapin, "The Cat's in the Cradle", click and listen to it now.

I usually abhor "free verse", but the poems in this book are so touching I didn't mind. The various pieces got me to think about all my relationships. I am so glad I knew all my grandparents, and that our son got to know all four of his. I am glad for "immediate family" of course, but also for aunts (one still living) and uncles and cousins, and those second and even third cousins I've been privileged to meet. One good friend of ours helps out his cousin frequently. He wonders how she goes on, with so many ailments, and a husband in even worse shape! They are fortunate he is available to help out.

Well, it is clear a book like this is hard to write about. It is the best kind of book, one that triggers self-reflection and self-revelation. I have little fear of Alzheimer's Syndrome now. I seem to have inherited my father's brain (he is 93 and still pretty competent) rather than my mother's. And if I do succumb to dementia? It is in God's hands. A wonderful book like this shows we need not feel alone when we need to care for someone, or need care ourselves. We have plenty of company.

Now I must reveal my identity explicitly, which I haven't done before in this blog. The poem above, titled "Memory", is Copyright, 2004 by Larry J. Van Stone. Please contact me if you wish to use it, using a Comment to tell me your e-mail address; I monitor Comments, so the Comment will not appear unless I Publish it to the blog, which I would not do without your permission.

Tuesday, February 17, 2015

Water as the next fuel – for War

kw: book reviews, nonfiction, textbooks, water, hydrology

I have been saying since the 1980s that the wars of the 21st Century will be fought over water. Similar, and quite definite statements are found in the last chapter of a 634-page text, Groundwater for the 21st Century: A Primer for Citizens of Planet Earth by John A. Conners. But water wars are not his focus, knowledge is. The aim of Dr. Conners is to educate the populace, the "Citizens" of his title. 'Tis a pity none can similarly educate national and business leaders whose focus is the next election or quarterly profit/loss statement.

Though the author claims the book was not written as a textbook, it is one. In fourteen chapters it covers the field of groundwater science quite thoroughly at a layman's level, if you don't mind an occasional equation and a little chemistry here and there. Though I spent the past week reading the book, I'd have taken twice as long had I not been able to skim much of the material.

What is groundwater? Simply put, it is all the water beneath our feet. In most parts of America, particularly the rainy East and Southeast, you can dig a foot or two down and see water seep into the hole. That is groundwater trickling out into view.

How much is there? A lot, but the rub is, there are a lot of us and we use a lot of water. At this point, I'll take a brief aside: Units are used quite inconsistently throughout the book. Sometimes we find square miles or cubic miles, and at others square or cubic kilometers. Sometimes a volume is in gallons, then in liters, and larger amounts may be in acre-feet or cubic whatever. Sometimes a conversion between Metric and (mainly) English units is given, sometimes not. Here I will use SI (the "official" Metric set of units, out of 3 flavors of Metric), and convert when necessary. So again, there is a lot of water, but there are billions of us, and the more affluent we are, the more water we use. On average, and American wastes an amount of water weekly that is equal to the entire water budget of a person in a Third World country, for a year.

How is it being used? From quite well to quite abominably. In the rich West we take it for granted unless it is our job to worry about it. This is not always wise. I once lived on a hill high above a flood plain. On this flood plain there were several mobile home communities. The typical setup was this: each trailer/manufactured home sat on about a half acre of ground. In the front yard near each house was a water well. On a flood plain the water table, which is the upper limit of groundwater connected to the nearby river, is at a depth of several feet, so the wells were shallow. Guess what was in every back yard? A septic tank and outflow field. The tank and piping were typically set shallower than the water table, meaning that whatever came out the pipes tricked down into the groundwater. I wonder if anyone living there ever considered that they were drinking their own slightly filtered toilet waste...and their neighbors'!

Most groundwater is extracted for agricultural use. At one point, we find that the minimum requirement of water needed to produce one day's food is about 3,000 liters. You only need a few liters to drink, and a few more to sponge bathe and to clean eating utensils. Depending on what we eat, our agricultural water use can be even higher. It takes 200 liters to produce one hen's egg, more than 15,000 l per kg of beef and 1/3 of that per kg of chicken meat. On the herbivory side, an apple tree consumes 125 l per apple, it takes more than 1,800 l to grow 1 kg of wheat, and once the wheat is milled and made into bread, each slice has 60 l of water use hidden within.

In addition to overt and semi-overt water consumption, a hidden "consumption" of water is contamination and pollution. To pollute water is to render it unusable, or at the very least, risky to use. I once heard a European water policy expert discuss laws—I don't know if they are on the books or only proposed—that mandate every manufacturing plant that wishes to release "used" water into a waterway, must put the outfall upstream of its own inlet. That supposedly gives them incentive to clean the water up before releasing it. But it doesn't address whether a company might do only partial cleanup of effluent, and more thorough cleanup of water it is using, only as needed on a process-by-process basis. A clever enough company might still pollute, but at lower cost than thoroughly cleaning their effluent. So I'd go further. I favor a law requiring that every executive and manager and salaried employee be required to live in a dwelling that has its water supply hooked up directly to the outflow from the plant: to drink the water, to cook with it, to shower or bathe with it, to wash their clothes and water their lawns and gardens with it.

OK, you say, "Dream on, dude!" Yeah, I know. The powerful always find ways to circumvent everything. That's why we need occasional revolutions.

Back to the book. In it we find that we are not using groundwater as fast as it is formed. The trouble is, groundwater varies in its purity and accessibility and in the cost to retrieve and transport it. The "cheapest" groundwater is mainly in underground formations, called aquifers, that are not being replenished very fast. In most of the world, we are extracting water that won't be replaced. It is called water mining. Only once it is gone will we turn to more costly water, and maybe one day we'll learn to use the water that is being replaced the most rapidly. But, give us time, and we'll find a way to outstrip even that supply. We need, not more water, but wiser water use. In the usual case, we do only what we are forced to do. This will most likely continue.

The book has no call to action. It is entirely educational. People need to understand what is actually out there, and what is actually going on. The learning itself will trigger action. That is the author's hope.

Wednesday, February 11, 2015

Some non-essentials are less essential than others

kw: book reviews, collections, literature

I like to read the occasional "Best of" or "Best American" collection, everything except poetry, since so little genuine poetry is being written at present. So when I came across The Best American Non-Essential Reading 2013, edited by Dave Eggers, I brought it home and dived right in. I was momentarily put off by the cover art by Camille Rose Garcia. She specializes in cartoonish illustrations that range from creepy to just plain ugly. This cover is exceptionally ugly. I soon obtained a clue to this.

I hadn't encountered this series before. In the introduction I found that the "editor" brings together a gaggle of high school students from all over the San Francisco bay area to read, debate, and select the pieces for each volume. To the way of thinking of most folks over 35 or so, kids that age prefer ugly stuff. Fortunately, that is not uniformly true. And it bears considering that when we were 16 or so, what we liked came across to our own parents as quite ugly.

Any literary collection strives to present a variety of reading experiences. This collection achieves that, and then some. Compared to this collection's range of voices and viewpoints, other collections are monochromatic. So even I found an item or two to like.

I suppose it is obligatory for me to complain that most of the fiction pieces are about losers who learn nothing. Much of the reportage is similarly lifeless. "All Due Respect" by Peter Hessler is an exception. It reads like fiction, but portrays Jake Adelstein and the Yakuza among whom he moved during more than a decade in Japan. It gets my vote as the best writing in the volume. The phrase "all due respect", as said by a Japanese in Japanese, has overtones of the "offer you can't refuse" of Godfather fame.

About a third of the fiction pieces I'd already read, in other "Best of" volumes. I recalled they hadn't thrilled me the first time around, so skipping them was no loss. One piece of "poetry" was the most non-poetic item I've encountered, "Crazy Horse Boulevard" by Sherman Alexie. A selection of 4 short poems inspired by Kurt Vonnegut ranged from moderately accessible free verse to non-verse (anti-verse?) of the most acidic sort. Free verse is almost exactly a century old. That is time for 3-4 generations to arise who have never read anything with both rhyme and meter, and it shows. You can write almost anything with odd pacing, perhaps break the lines in peculiar places, and call it a poem. I guess the market for rhyming dictionaries has essentially vanished.

One piece that I read all the way through, and shouldn't have, succeeded in disturbing my sleep: "Snake River Gorge" by Alexander Maksik. I think it is fictional. If it isn't, it sheds a very different light on those youngsters that show up on your doorstep selling magazine subscriptions, as a particularly heinous sort of human trafficking. Even if it is fiction, it'll still make whoever has read it get the willies when another kid rings the bell.

I reckon if you're a Millennial and don't know any better, you'll like many of these pieces. If you're a Boomer or an X-er, probably not so much.

Friday, February 06, 2015

Are psychopaths evil, or broken?

kw: book reviews, nonfiction, psychology, psychopaths, autobiographies, fmri

Psychopaths and psychopathy have been of growing interest for about thirty years. Amazon currently lists 96 hardbacks with "psychopath" (singular or plural) in the title, and more than 600 if paperbacks and Kindle editions are counted. Perhaps a quarter of these books delve into the science to some extent. The rest are more sensational treatments or contain advice about dealing with a troublesome boss, co-worker, lover, child or parent, who may or may not actually be psychopathic.

Of books on the subject with a more scientific or investigatory aim, I suspect most are at least partly based on the work of Kent Kiehl, who has just published The Psychopath Whisperer: The Science of Those Without Conscience. Beginning with the work of Drs. Hervey Cleckley and Robert Hare, and based very much on the PCL-R (Psychopathy Check List – Revised), he initiated the study of brain structure and function in psychopaths using fMRI (functional Magnetic Resonance Imaging).

While he was a graduate student Dr. Kiehl began working with prisoners convicted of the most violent crimes, learning to apply the PCL (I'll leave the R off; it is understood these days). To properly use the PCL, interviews lasting a few hours, conducted by a well trained clinician, are needed. The score ranges up to 40, with 30 being the cutoff. The average for all inmates in maximum-security prisons is 11. The average for the general population is 4. The average for serial killers is 35, but not all serial killers have been found to be psychopaths.

Throughout the book the chapters each begin with a mini-fact. The first is
One in four maximum-security inmates is a psychopath
So if you have a bunch of inmates whose average score is 11, but a quarter of them have an average score of about 32 (this assumes that higher scores are more scarce), then the rest will average 4, the same as the general population of the non-incarcerated!

Psychopathy is a primarily male affliction. While about one man in 150 is a psychopath, the rate for women is closer to one in 1500, so about 90% of psychopaths are male. If we confine our concern to those between the ages of 18 and 50, in the U.S. population about half a million men and 50,000 women are psychopaths, as measured by PCL-R.

I wondered about the 30-point cutoff. Its utility depends on the distribution of scores. For example, if someone is rated by a trained clinician, for whatever reason, and is scored a 29, is he considered "almost a psychopath" or a non-psychopath? Having dug around some, I didn't find much on score distributions, and nothing for the "general population". But I did find a few histograms compiled for psychiatric populations. They showed a bimodal distribution with a pronounced low region in the range 20-30. Curiously, among many articles that mention score distribution, most treat the scores as a normal (that is, Gaussian) distribution, which introduces serious errors if the true distribution is bimodal (think of a camel with two humps; the Gaussian curve has one hump only).

It is a terrible pity that so many scientists, psychologists in particular, try to shoehorn all distributions into the Gaussian model, when so few natural phenomena are truly Gaussian! Sure, height in males or females tends to be normally distributed ("normally" meaning "according to the Gaussian model"). So do a small number of other measurable things. But consider this question:
What is the average number of digits (fingers plus thumbs) possessed by persons the day of their birth...or death?
Neither question can be answered "exactly ten". On the day of birth, some babies are born deformed and have fewer than ten, and in rare cases, no digits or even hands at all. Also, ten is not the maximum number because some are born with twelve, and sometimes more. The internet abounds with pictures of babies born with 14 digits or more. And at the end of life, a significant number of folks have lost one digit or more to accident or disease. So while the mode (greatest frequency) of the distribution curve is right at ten, the number ranges from zero to at least 16, and is strongly skewed, numbers smaller than ten predominating. To analyze frequencies of digit quantity using Gaussian statistics would be a serious error.

The difficulty of labeling is also discussed. Young people can also display psychopathic tendencies, and there is a PCL for juveniles, but it is a breach to tell a youngster the result. In one case in the book, a young man with some emotional problems was told by a doctor that he was a psychopath, whereas it turned out later he was not one at all! But he believed the doctor and decided he'd live a life of crime, including killing.

Dr. Kiehl's work has been primarily with serious criminals. A significant focus of his work has been predicting rates of recidivism, or re-offending, after a prisoner is released. Psychopaths are six times as likely as others to re-offend. Does that mean that we ought to give the PCL to a freshly incarcerated person and, if he "fails", lock him up and throw away the key? Not so fast. The author spends a chapter discussing the Mendota Juvenile Treatment Center (MJTC) in Wisconsin, where a different approach has been used to ameliorate the antisocial traits of the least-manageable juveniles, who are termed "callous and unemotional" to avoid labeling them "psychopaths" at too early an age.

Psychopaths in general do not learn from punishment or other negative consequences. They seem immune to correction, and many are proud of it. At MJTC, as I understand it, the juvenile offenders are trained in a way similar to performing animals. Every slightest "good behavior" is rewarded, and while serious misbehavior may be sanctioned for the safety of the staff, most misbehavior is simply ignored. Everyone there is trained in the method, from clinicians to cleaning staff. The results have been spectacular. For example, among juveniles not treated who were released at age 18, a certain number became adult criminals and several committed murder. Among an equal number of those who completed the MJTC program, fewer than half as many committed any crimes, and none were murders. Some went on to get more education and were able to hold jobs. Getting such results is neither quick nor cheap, but considering that crime in America costs at least a trillion dollars yearly, not doing anything is even more costly!

I find it interesting that Dr. James Fallon has studied psychopaths, using tools developed by Dr. Kiehl, and found that he is himself a psychopath, as are a number of people, such as Niel Armstrong, who are not in any way in trouble with the law. It seems one's fMRI scan can show the suppressed emotional brain activity characteristic of a psychopath, and one can score 30 or more on the PCL, while still having respect for law. Dr. Fallon believes such psychopaths outnumber the criminal ones. Let's hope so!

A "horse whisperer" is one who has a special rapport with horses and can train them quickly and effectively. The book's title points not so much to the author as to the originator of the program at MJTC. I hope the work there leads to follow-on programs that can take the fangs out of  the most dangerous young persons and, one might fondly hope, gradually depopulate our prisons. It is a national shame that America has such a large number of prisoners.

Sunday, February 01, 2015

One future diverting another

kw: book reviews, science fiction, science fantasy, near-future, dystopias

Concerning time travel, one would have the same question that Enrico Fermi did about Martians or other aliens visiting Earth: "Where is everybody?" In The Peripheral, William Gibson finesses this in a milieu where contact across time is possible but difficult, and thus limited and easier to conceal. Here we find also a future, late 22nd Century so far as I can determine, with a much lower population, and a very, very small number who can afford to pursue cross-time contact as a hobby. Also, conveniently, early in the novel we read that the earliest date that can be contacted is some time in the 2020s. The mechanism is a new kind of virtual reality system, with a server "somewhere in China". More than this is left a mystery.

What is a "peripheral"? Computer-savvy folks think of printers, external hard drives, game controllers,…all the things you might attach to a computer (or tablet or phone, these days) except the external monitor. Somehow screens aren't thought of as peripherals. In the novel, "peripheral" is reserved for robots and (Gibson doesn't use the word) androids that someone can inhabit virtually, via a special brain-contact headset that wraps around at forehead level.

I have minimal interest in the plot. Suffice it to say that the protagonists are one Flynne Fisher, a young woman in about the year 2110, and Wilf Netherton, a middle-aged man living some 75± years later. In Flynne's era, there is something like an advanced Skype machine on wheels, like a Segway with a screen and cameras on a stalk at head level. Devices like this are currently called telepresence robots, and enable someone a certain limitedly mobile presence at a distance. Presumably they'll be a lot better by 2110 or so. After a further seven decades, the technology of choice is a genetically human animaloid with neither brain nor alimentary canal (they are fed intravenously). In place of a brain, there is an AI that can manage the body when not in use, and interface electronics for its use. For tasks needing great strength or small size, various "homunculi" are used similarly.

Thus, Flynne gets to visit the future by inhabiting a peripheral that is a fully-functional young woman who looks similar to her, but cannot eat (and doesn't eliminate either). Wilf gets to visit the past in a telepresence robot. As it happens, Flynne is very fortunate that her brother is a former Special Forces (or something similar) soldier, with lots and lots of friends who are very, very good with weapons; Wilf is well-connected with an English-born Russian gajillionaire and his "klept". Think of a klept as a crime family with ambitions to grow into a kleptocratic government.

The plot hinges on a murder that Flynne witnessed while monitoring what she thought was a video game. As the only witness, she is targeted by future assassins who can only work by offering millions to folks in her time who will kill her. The future folk who contact her include a mysterious police inspector of great age, and members of the klept, who finance the search for the original killer. Of course, the killer will be fingered and dealt with, along with certain other evil folk. Other than that, there is remarkably little killing.

William Gibson is a master of high-tech future dystopian world-building. With The Peripheral he has crafted two dystopias, with the worse one attempting to avert their own fate for the other. I ought to mention another time-contact concept: first contact between someone in, say, 2085 with someone in 2010 splits off a "stub", which is now affected by that contact and develops differently without changing the future that is doing the contacting. Believe me, this is less of a mental conundrum than we usually find in "time travel" literature! (Most of it seems to be written just to set up, then solve, such conundrums.) Thus, it is possible for Wilf and his group to help Flynne and her group avoid "the jackpot" that led to the world he inhabits.

This "jackpot" is one of the intriguing ideas in the book. It is described not as a single catastrophic event, but a decades-long tangle of co-synergistic "slow disasters" (only global warming is presented as an example) that end up reducing human population to around 10% of what it was before. This is sufficiently plausible to be chilling.

The univere-splitting function of cross-time contact is another. At one point someone muses whether the contact initiates a split, or if quantum universe splits occur frequently and the cross-time function can only occur between worlds on different world-lines. This is in accord with the popular "many worlds" interpretation of quantum theory. The most extreme version has every quantum "choice", everywhere, triggering a split into two or more universes, depending on how many possible outcomes the quantum event could have. If Richard Feynman's "virtual particle sea" interpretation of quantum electrodynamics is true, and quantum universe-splitting is also true, then new universes are created at the rate of about 1024 per second per cubic femtometer of space. That is universe creation at a rate, per second, that is a number with roughly 500 digits. And people think this is more reasonable than my belief in God!!! "Stub" creation by intentional action is tons more reasonable than any "many worlds" theory I've read about. And Gibson nicely pushes off any such splitting into the future by a century or so. I found myself wondering whether the plot would twist into next-level contact, when someone from, say the middle 2200s contacts either Wilf or Flynne. Maybe it is something Gibson will take up later, except he is probably busy building another world instead just now.

I've already discussed the technology of "peripherals" a little. It is very reminiscent of the Avatar technology of the film Avatar. I kept wondering if this novel would end similarly, with Flynne's peripheral being replaced by one that can eat, and her getting stuck in it in the future, say because her body dies in her own time. Gibson had another idea, and a better one.

In the Flynne time frame, the technology of the day is "fabbing" using 3D printers of roughly a century in our future. In the Wilf time frame they use "assemblers", nanotechnology devices by the quadrillions. In one scene, a blocking wall just seems to appear. I had to step back and think about that. Where did the material come from? What about the energy? Even if this kind of "assembling" is not breaking and making chemical bonds, the particles being assembled will still be subject to van der Waals forces. vdW forces are what make glue work, and they facilitate the zipping and unzipping of DNA. But even assembly relying only on vdW forces requires energy. So much energy that, while the wall might be able to "arrive" in a second of time, its temperature would be a few thousand degrees. A different application of assembling, that brings a weapon into Flynne's hand through solid rock, would use at least as much energy as melting the rock. Her hand would be burnt off to the shoulder. This point in particular is why I added the tag "science fantasy" to the metadata.

No matter how hard or soft the science is in a science fiction novel, its enjoyment requires the suspension of disbelief. I thoroughly enjoyed the novel. Then, I enjoyed speculating on the ideas presented here just as much. I am not pointing out errors, but confronting the concepts with physics as we know it today. Many of our devices would seem magical to people of Ben Franklin's time. We know some physics that was not known then. The physics of 200 years to come could advance a similar amount. Maybe there's a way to shift a vdW bond, or even a covalent bond, using much less energy than the break-plus-make procedure we must use today. I love Gibson's writing. His dystopias are more hopeful than most.

Wednesday, January 28, 2015

Are we dumbing ourselves down?

kw: book reviews, nonfiction, technology, automation, surveys, critiques

Six years ago Historian George Dyson wrote on Edge.org, "What if the cost of machines that think is people who don't?", summarizing an article by Nicholas Carr. Writing about 60 years earlier, in "The Feeling of Power" Isaac Asimov presented a future in which small calculating devices had so usurped arithmetical abilities, that a man who rediscovers paper-and-pencil methods of addition and subtraction is a phenomenon (Strangely, I haven't been able to find a date for this story).

Nicholas Carr has continued to think and write about automation and its effects on us. His recent book The Glass Cage: Automation and Us explores mainly the uglier side of current trends in technology. The best quote in the book is, "How far from the world do we want to retreat?" (p. 137)

Every person will have a unique answer to this question. For people like me, the answer would be, "Very far indeed, for long stretches of time, but with the option to return to full engagement at times and for durations of my choosing." For most of my life I have been more comfortable with machines than with people. Yet I need human contact…just not on the constant basis required by extroverts.

Technology is ancient and continuing: Stone tools as old as 3-4 million years; the successive technologies after Stone of Bronze, Iron, Steam and now Electronics; pocket computers we call "phones" for which making calls is now a minor function. The first time I saw a cell phone in use, some 15 years ago, there were two girls about 7 years old, running together through a park, each talking on a phone to someone else. (Note to self. Try making most calls while walking or jogging. Might be a good way to shed that next 5 pounds or so.) I recall predicting that during my lifetime, our "phone" would be installed in the mastoid bone at puberty and be entirely voice operated. Little wire to a microphone embedded in our jaw somewhere, and software filtering to subtract out the effects of flesh-to-bone conduction of our voice.

I am no slouch when it comes to computer use. I've been what was once called a Power User since the 1960s, when computers were too big to fit in most bedrooms. The motto of the Elephant Club: Don't Trust a Computer You Can See Over. Except today, a new club—the Power Tower Club?—might need a new motto: Don't Trust a Computer Smaller Than a Toaster Oven. Sure, my wife and I have a laptop, but my favorite workstation is a tower 18" tall (46 cm) with a pair of screens that gives me about a meter-wide view into cyberspace. For some of the work I do, that much screen real estate is essential. But do you know what one of my favorite activities is? A few times monthly I am a Historical Interpreter at Hagley Museum, in the Machine Shop, demonstrating machine tools (lathes, drill presses, mills, etc.) from the 1860s and 1870s, powered by a water mill in Brandywine Creek.

I wonder, though, if some machine workers of the early 1800s thought it was somehow "cheating" to use a power tool, when they were perfectly capable of making parts using hand tools. Probably not! Particularly for machine work, one needs a peculiar combination of intelligence and patience. I often point out to museum visitors that cutting the teeth on a medium-sized gear (5" thick and a foot in diameter; about 120 mm and 300+ mm) took a week in the 1870s. You set up a machine with eye and hand. You monitor the machine by ear; by the end of apprenticeship a machinist knows the changes in cutting sound that herald trouble on the way. So you need the brains to set the right index for a 17- or 19-tooth gear on a 40:1 indexing attachment, and the patience to listen for trouble for the next 60 working hours of your life…with resetting of the cut and rotation of the piece about 4 times per hour. Fast-forward to the modern era: Such a gear, if needed today, could be produced in a few minutes using electromachining, or in about an hour on a more conventional NC mill. Those old-time machinists would drool!

In many areas we are going through a transition, and Mr. Carr points out several of importance. The airline industry was among the first to automate wayfinding and autopilot aircraft control. If needed, any modern jetliner, and many smaller planes, are capable of taking off, flying themselves, and landing, without the pilot doing a thing. The trouble is, machines break, thunderstorms and solar flares disrupt communications and sometimes damage equipment, and because no program is totally bug-free, a rare combination of factors puts the autopilot's program into a confused state. In all these cases, the "fail safe" provisions immediately turn control over to the pilot. A few times, this has caused crashes, typically with the loss of everyone on board.

This brings to mind another principle that seems to be lost on modern engineers and programmers, "fail soft". Is it really appropriate for all the software to totally cut out so instantly? If the plane is at all still capable of level flight, the autopilot needs to alert the pilot(s) while keeping the plane on some standard course, giving the humans time to get their brain in gear. There may still be cases such as the "standard course" being straight into a mountainside (and I am reminded of the crash of a small plane in Malaysia in 1991, that effectively decapitated the Conoco corporation), but further development of "standard course" back-up routines ought to take care of that.

Such issues multiply when we come to the driverless car. It sounds seductive. Plot your course on a GPS navigator, press "GO" and take a nap, or play cards, or read or whatever. But the "lanes" in which an airplane "drives" are a few miles wide. Highways lanes are 12 feet (3.66 m) or less. For most of a plane flight, course corrections are few and may be hours apart. On the road, course corrections can occur minute-by-minute and even second-by-second. I have read a time or two about auto-driving "road trains", made up of dozens of autos on a superhighway at superspeed, inches apart to take advantage of drafting. Now suppose a solar magnetic storm disables half a dozen GPS satellites, the road-side "driving aid" equipment being relied upon by the cars, and perhaps some of the electronics in the cars. What is the "fail soft" scenario? Is one even possible??

We are in transition, all right. Casualties of all kinds are one price of progress. YouTube abounds with videos of people so engrossed in texting as they walk that they walk into fountains, manholes, lampposts and each other. We can expect the phones to become even smarter, so they would be on the lookout for such events. Maybe blare, "Look up, dope!", and make a red, flashing screen as such incidents approach. Smart phone technology is not yet complete, nor even appropriate for human use. It's why I use a flip phone that can call and make texts. Period.

In a late chapter Mr. Carr writes of the young Robert Frost and his poems about scythe work. The scythe is an extraordinary instrument. Using one creates muscle tone around your rib cage that no other exercise can match. Learn to use a scythe properly, and use it frequently, and you'll never have back problems. It exemplifies the kind of work that keeps a fellow close to the earth. Even as we try to re-educate America for a supposed post-manufacturing economy, there are huge numbers of jobs that remain very, very hands-on. A company may outsource its call center, computer programming, and database management to drones in India or China. You can't outsource construction, electrical work, plumbing and paving, nor landscaping or even repairing (and washing!) your car. Yeah, I know most New Yorkers just look puzzled when asked by a tourist where to buy gas…like they'd know! But deep in the bowels of the city are track workers and subway car maintenance folks that they'd suddenly feel a great need for if there were a month-long strike.

I have major mixed feelings about automated medicine, though. In certain cases, the Caduceus program has been able to make quick diagnoses where medical experts were baffled. But in others it has been embarrassingly off the mark. In medicine as in many other areas, the term "robotic" is being misused, most notably with the Da Vinci Robot for precision surgery. Let us reserve the term "robot" for autonomous devices such as the Roomba vacuum cleaner. The Da Vinci device is actually a tele-operated "Waldo" with vision magnification and down-scaled, feedback-enhanced motions so a surgeon can operate on something half an inch across while feeling like the object is the size of a basketball. I was once trained on a soldering Waldo used for attaching leads to integrated circuits. It worked at 25x, so a millimeter looked and felt like an inch. It greatly simplified the job. By the way, "Waldo" comes from an old story (1942!) by Robert Heinlein, where the concept was first made public.

Do I want a doctor to cede control in an operation to a robot? Probably not. Diagnosis? Not without human review. Only humans have a sense of what is sensible! How about prescribing? Ditto. I prefer the physician to not only make the decision, albeit aided by the computer system, but also to discuss it with me, because in teaching me how and why he chose a certain medication or treatment, he's rethinking it in a way that is useful to him and may cause him to realize something extra he might at first have missed (Feminists out there, I'd have used "she" and "her" if I had a female physician). "Thinking out loud" is often the most useful kind.

Artificial intelligence, once it gets on a par with us as a conversationalist, will still be quite different from us, so it could provide a very useful function: serving as a "straight man" to our musings, asking questions no human would think to ask, and adding a powerful level of synergy. A very neglected area of ergonomics has been to determine what tasks humans will always do better than machinery, and which tasks should be at the top of the list for turning over to machines. The various Zooniverse projects, citizen science at its best, primarily take advantage of our superior visual abilities. We can recognize the difference between spiral and elliptical galaxies at a glance; or different kinds of beetles; or see that a certain black-and-white blob is a rock and its shadow rather than a penguin.

Some might see The Glass Cage as a Luddite polemic. Not at all. Mr. Carr points out that we are a technological species. We can't live without it. Even the prototypical "cave man" was no naked savage killing prey with teeth and fingernails. The tool kit of Paleozoic people included dozens of tools that require skill to produce but reduce either the effort or the danger of doing the work. I, for one, am glad of today's technologies. I am equally glad that I can pick and choose which to use and which to ignore. Looking around the room I am writing in, I see several thousand objects, nearly all artifacts. Only the insect collection and a few shelves of mineral and fossil specimens are not technologically produced (though I used technology to mount and display them!).

Physically we are more "gracile" (that is, thinner and weaker) than the Cro-Magnons of just 20,000 years ago. They are called "modern" in an anthropological sense, but the technologies they inherited from their ancestors, and added to in following millennia, resulted in a modern civilization in which we don't need the great strength they required for day-to-day living. Our teeth are a little smaller, and as our jaws shorten, most of us need our wisdom teeth removed. All this results from technology. Will this age of intelligent machines cause our brains to atrophy? It's not likely. Our descendants will probably think differently than we do, just as we think differently than our grandparents who mostly grew up without radio, television, airplanes or automobiles.

I'm thinking of my own grandparents, here, all of whom were born in the late 1800s. I know most millennials are of an age to be my own grandchildren. We got our first television when I was 8 years old. Black and white, in a console the size of a divan. Our phone was on a party line. The most local of calls were made by clicking the button 1, 2, 3, or 4 times. "In-exchange" calls needed only 4 or 5 digits. All long distance calls went through an operator. So the change in the "thought world" of today's young adults is as different from mine as mine is from my own grandparents. It is another side of progress. So the book is more of a call to enter the future thoughtfully. We are creating it, after all.

Sunday, January 25, 2015

Chemistry for those who don't know any

kw: education, chemistry, basics

Think of a scientist and what do you see in your mind's eye? Probably someone in a white coat mixing chemicals. Chemistry is the bane of humanities majors everywhere, because you have to take Chem 1 with a (barely) passing grade to get on with your major. (Those with a sharp eye will note that the cylinder being poured from is about to dump all its contents at once!)

So let's knock out a few basic concepts to jump-start your education. First the ultra-quick version:
Chemistry studies how atoms share or exchange electrons. Of roughly 100 kinds of atoms, a few—twelve, to be exact—have one or two "loose" electrons that are easy to strip off, while another twelve have room for one or two more, and will easily plunder those loose electrons. Some others can either gain or lose three, four, or even five electrons. The rest typically share electrons. Chemistry is learning all the ways this can happen, and which elements behave in which fashion.
For more, read on. We begin with Electrons.


Electrons are particles that make up the outer "skin" and "flesh" of atoms. What we usually mean when we say "chemistry" is properly "electron chemistry". There is also nucleon chemistry, plus other subdisciplines such as crystal chemistry and organic chemistry. The odd thing is, you first have to know a little about nucleon chemistry to get a framework to learn electron chemistry.

Nucleons and Elements

Perhaps you have heard that there are 92 "natural" elements, or maybe, as I wrote above, that there are "about 100 elements". There are actually 90 elements called "naturally occurring". That is because, although the heaviest natural element is Uranium, #92, the elements numbered 43 and 61 are not found in nature, for reasons we'll soon get into.

Nucleons are the particles that make up the nucleus: Protons and Neutrons. The number of protons in a nucleus determine what element it belongs to. For a nucleus to be stable (and the "what for" about this is a major subject of nucleon chemistry) there need to be neutrons present also. Only one element has no neutrons in its nucleus, Hydrogen. An atom of hydrogen, the simplest and lightest element, has one proton and one electron, and nothing more. Every other kind of nucleus has at least one neutron, and with only one exception, the number of neutrons is at least as large as the number of protons.

The main item of nucleon chemistry that you must know is that the Atomic Number is the number of Protons. The term Atomic Number is used everywhere. It is also extremely useful to understand that radioactivity expresses the tendency for certain combinations of protons and neutrons to break apart in one way or another. A very few kinds of "unstable" nuclei are nearly stable and last for millions or billions of years. Uranium is one of these.

Nuclei of elements #43 (Technetium) and #61 (Promethium) are always unstable, in every variety, no matter how many or how few neutrons are in there with the protons. In this case, "unstable" means having a half-life short enough that every single atom of these elements that may have existed billions of years ago when Earth was formed, has decayed. Half-life is another very useful term, though mainly in nucleon chemistry. For a bunch of any specific, unstable kind of nucleus, the half-life is the time it takes for half of them to decay. Lots of uranium (originally produced when big stars blew up billions of years ago) is still here because its half-life is about 4.7 billion years.

The fundamental tool for understanding electron chemistry is a table in order of Atomic Number, that is arranged according to how electrons pack together in each kind of element: the Periodic Table.

Periodic Table

Get ready for it! I am about to explain this monstrosity:

The columns are arranged the way they are because elements in a column have similar chemical behavior. Down the left side, for example, the six elements Li, Na, K, Rb, Cs, and Fr all have similar chemical behavior because the outermost electron is "loose" and easily lost to more acquisitive elements. Hydrogen is special; though it can both lose and gain an electron, it also participates in a third kind of sharing bond we'll describe later.

Each row represents an electron shell, which fills from left to right. The rightmost column, topped by Helium (element #2) contains all the elements with a completely filled shell. This is the group of elements with the easiest chemistry: They don't participate in chemical reactions! But right next to them we find F, Cl, Br, I, At, and the "artificial" element currently called Uus (Un-Un-Septium, a fake Latin term for 117). They all have an outermost shell that is nearly filled, but is ready to grab an electron from another element that has a "loose" one available.

The rows are different lengths because the shells have different capacities. It takes some learning in quantum physics to comprehend what electrons are doing (as much as that may be possible!). Here is the simple explanation:
  • Electrons come in pairs.
  • The first shell is filled by a single pair, thus Helium has a filled shell. This filled shell is the core of all heavier elements.
  • The shells of all elements other than Hydrogen and Helium have sub-shells.
  • The sub-shells were discovered by spectroscopy, and are called, for historical reasons, s, p, d, and f.
  • Sub-shells increase by odd numbers of electron pairs:
    • p has 3, so s+p = 4 pairs or 8 electrons.
    • d has 5, so s+p+d = 9 pairs or 18 electrons.
    • f has 7, so s+p+d+f = 16 pairs or 32 electrons.
  • Shells 2 and 3 have s+p only; 4 and 5 also have d (thus the lower-middle block); and 6 and 7 also have f (shown as the extra stuff below the main table).
  • The placement of the rows shows that the d sub-shell fills before the p sub-shell, and the f sub-shell fills before d.
Note that the number of electron pairs in a completed shell is a square number: 1, 4, 9, 16. The next square would be 25, though no elements currently existing make any attack on that shell.

All the elements from 93 to 118 have been produced in nuclear reactors and particle accelerators. With element #118, the seventh shell is filled, so once elements #119 and greater are produced, an eighth shell will begin to fill. This is expected to have a new sub-shell, usually called g. It can contain 9 electron pairs. It is likely that the g sub-shell will begin to be filled with element #121, but we will only know this for sure if element #121, or a heavier one, has a long enough half-life so the electron arrangement can be studied before the whole sample decays away.


When one atom takes control of the loose electron given up by a different atom, or when atoms share electrons, we talk of a chemical bond. To discuss this, a version of the Periodic Table with different highlighting will be helpful:

You know that term "alkali"? It refers to substances that neutralize acids. The two columns of elements at the left, in lavender and blue coloring, are called the Alkali Metals (lavender) and the Alkaline Earth Metals (blue). The ones with an odd atomic number have one loose electron, and the even ones have two loose electrons. They participate in compounds that tend to be alkaline; in some cases, the compounds are so caustic they will remove your skin.

Now, at the far right, as I mentioned above, the elements in the last column do not combine chemically with others. A few very extreme experiments have been done to force them into unstable chemical compounds. We call them the Noble Gases. They, and four other elements in which the lettering is dark green colored, are gases at "room temperature", defined for chemists as 25°C or 77°F.

The elements in the next column, with beige coloring, are called Halogens. "Halogen" is from the Latin word for "salt". They like to glom onto loose electrons. Any of these reacted with hydrogen will form a strong acid, but when paired with one of the Alkali Metals or an Alkali Earth Metal, they form stable salts. Two of them are usually gases, one is a liquid (Br, with dark blue letters), and the rest are solids. They are a major part of a group also called Non-Metals.

Hydrogen plus the other elements in orange coloring are the rest of the Non-Metals. In element form, solidified at low temperature in the case of Nitrogen and Oxygen, they are insulating solids that look like soft ceramics. While Oxygen and those below it tend to snatch two loose electrons whenever possible, they also participate in the sharing bond I mentioned earlier.

The elements with brown coloring are called Semi-Metals. In element form, they are semiconductors, and one in particular, Si or Silicon, forms the basis for most electronic circuits. The lime green colored elements are Metals that are either semiconductors by themselves, or form semiconductors when alloyed with Semi-Metals.

All the rest of the elements in the main part of the table are colored light yellow, and are Metals. The top row of them, from Scandium to Zinc, are the Transition Metals. "Transition" refers to their similar chemistry. They all have a filled s sub-shell and an empty p sub-shell, and from 1 to 10 electrons in the d sub-shell, which is "hidden" beneath the filled s sub-shell. However, those two outermost electrons can act as loose electrons to combine with Non-Metals or Oxygen, and frequently one of the d electrons will also do so. Thus, they have more complicated chemistry than those to the extreme right or left. The three pale yellow rows below behave a lot like the Transition Metals, but it is harder and harder to get them to react. In particular, Platinum and Gold (Pt and Au) are very resistant to participating in chemical activity, as are the elements directly beneath them, though those are radioactively unstable and are very short-lived.

The Transition Metals are useful to living things in various amounts, usually quite small amounts. Even Iron (Fe), the most abundant metal in our bodies, is present as 4-6 grams in an adult human, or less than 1/100 of a percent. The heavier metals are called "heavy metals", particularly in medicine, because they are all toxic. Lead (Pb) is the most familiar toxic metal.

Ionic Bonds

The shift of one or more electrons between strong "electron donors" such as Li or Ca, and "electron acceptors" such as Se or Cl, produces an Ionic Bond. This kind of bond is strong in the pure solid, but is pulled apart in water to dissolve salts such as LiCl, CaBr2, or MgSe. However, salts with S or Se are poorly soluble compared to salts with Halogen elements "on the right". In water solution, the elements that have lost electrons are + ions, and those that have accepted electrons are - ions.

Covalent Bonds

Electron sharing in which two atoms form a strong bond to fill their outermost shell produces mainly insoluble compounds held together by Covalent Bonds. The Non-Metals, when in elemental form, usually exist as paired atoms sharing one or more electrons. The simplest example is ordinary Hydrogen:

Here the electrons are shown as dots. The shared electrons satisfy the s sub-shell of both atoms.

Most elements can participate in covalent bonds. The most versatile is Carbon, which has 4 outer electrons, and thus room for 4 more. It prefers to share a covalent bond in 4 directions. This makes it the most versatile in its chemistry, and a huge discipline, Organic Chemistry, is the study of carbon chemistry. Where a chemist who studies inorganic chemistry will become familiar with thousands or tens of thousands of chemical compounds, the number of organic compounds so far known exceeds 50 million.

The Take-Away

So, what do you really need to know to be ready for Chem 1? Or, just to be at least glancingly familiar with the subject? Chemistry is about the ways atoms transfer or share electrons. The outer electron shell of an atom can have between 1 and 8 electrons. The more promiscuous atoms, mainly Carbon, Nitrogen, Sulfur and Oxygen, induce the other elements to form complex molecules. In the absence of these four, most compounds are simple and easier to study.

Tuesday, January 20, 2015

Wisdom is not automatic

kw: book reviews, nonfiction, thinking, psychology

In his late 90's, Art Linkletter was asked the secret of his success interviewing children, most famously on his long-running TV program Art Linkletter's House Party. He said, "It's simple, but you probably can't do it: they must know that you are on the same intellectual level." With this gentle dig at himself he revealed that connecting with anyone is to reflect them. He knew he was just a big kid, and the kids could tell.

On a similar note, if someone could ask Joseph Bell, the inspiration for Sherlock Holmes, or even the author Arthur Conan Doyle, what was the secret of his deductive abilities, I imagine him replying, "It's simple, but you probably can't do it: you must exclude no possibility without a reason to do so."

We are, by habit, quick to close doors and slow to open them. Our everyday language is full of door-closing phrases:
"I can't do that."
"This must be so."
"Why would you think that?"
"That is impossible."
"It won't work."
In the film The Help I found it extremely touching when the nanny holds a small girl and repeats to her, "You is good, You is Kind…" and so forth, and the girl trustingly repeats with her, "I am good, I am kind…" How can this fail to establish a helpful basis for the girl's character?

At this moment, I am less concerned with the things we tell our children than with what we tell ourselves. "What you think is what you get" could be a mantra for Maria Konnikova, author of Mastermind: How to Think Like Sherlock Holmes. Having grown up hearing the Holmes stories read aloud by her father (and a great many other good books, she hints here and there), Ms Konnikova in eight chapters, jam-packed with examples and exhortations, shows us how to re-form our ways of thinking, and problem-solving in particular.

You and I may never need to solve a crime or find a kidnapped prince. We may never cross wits with a purblind and misguided police inspector. But our lives are full of conundrums big and small that a bit of Holmes-style thinking can help us resolve. It is more than just "thinking outside the box," though that is helpful; first we must know what the box is!

Throughout the book the author uses the analogy of an attic. In what state is our memory? Certainly, it contains thousands of things, but how are they stored? We're not talking psychobiology here but mental discipline. Continuing the analogy of an attic, or even better, a vast warehouse, how are its contents arranged? Is everything in piles like in the house of a hoarder, such that you can barely squeeze your way hither and yon to find things? Perhaps things are in boxes, but are things grouped with similar things or just jumbled together, box by box by myriads of boxes? Is anything labeled?

I think of interior views of the shelves in M5 on Mythbusters, such as this image. Jamie and Adam didn't rise to the top of the special-effects field by being sloppy curators of their "stuff". The boxes, bins and jars may exhibit a wondrous diversity of their own, but they are sorted alphabetically. I reckon that beats trying to sort them functionally; Jamie would need a taxonomy of function, and there would inevitably be an "Other" category that would soon grow out of control. Better this way. (But note in the bottom row that "Small Pumps" is misplaced. Would you sort that with S or P? Who knows how it got between T and U!)

Anyway, key #1 to Sherlock Holmes's method is having a mental attic with much of the "stuff" labeled and sorted. He is able to quickly retrieve what he needs.

This doesn't happen by accident. I suppose it will always be true that most of what we take in and retain (and we retain a very small percentage) is quickly strewn helter-skelter, and there is little we can do about that. It is probably one function of sleep to sort through recent new memories and nudge them this way and that into some sort of order. You and I may not consciously be good curators of our memories, but some amount of curation is carried out anyway. We must be thankful for that. But we are all different, and if that curation is too sloppy, we are called "scatterbrained" at best, and probably other, less flattering terms behind our backs.

But we read in Mastermind of observing with intention, of taking in what is most likely to be useful, then curating that properly. Like many others, I collect a number of things. My stamp collection is, for the most part, labeled and sorted. My minerals, not so much. I have a rather small number of minerals on display, a somewhat larger amount stored in boxes, but it is more of an accumulation than a collection. Then the books! There are a few thousand, and I have certain subsets well arranged in special places. The rest simply line the shelves of three rooms. One friend has at least this system: all his books are arranged by the color of the spine, so his main library is a rainbow. Another, now deceased, had a true library, with a Dewey Decimal notation in white ink on every book, and a card catalog in the corner. Now that is a collection!

A second key is the extent to which we allow our emotions free reign. In the Holmes stories, Dr. Watson is a kind of Everyman. He represents nearly all of us, jumping to a premature conclusion and then falling in love with it, which makes it quite impossible to proceed in any useful way. Let us remember the maxim that I foisted on Dr. Bell in my imagination: "Exclude no possibility without a reason to do so." Holmes is a master of the creative back-step. When formulating hypotheses he quite automatically pulls back to take in a wider view and be sure he is excluding nothing that might be useful. He (usually) did not allow his fondness for a neat explanation to deter him from discerning other explanations. Thus, when the first "neat" explanation is found wanting, he would have further avenues to explore. Watson-style thinking far too often confronts us with a blank wall and empty pockets.

Some people are openers, some are closers. Both are needed. More rare are those who can both open and close with equal ability. I am referring to opening up more and more possibilities in the early stages of a project or puzzle, followed by closing off one possibility after another as each is proved impossible or unfeasible, to drive to an appropriate conclusion. Holmes's most familiar dictum is, "When you have excluded everything that is impossible, whatever remains, however improbable, must be the truth." And suppose you have excluded everything you could think of? Time for more opening exercises. Conan Doyle has Holmes make a few mistakes, and they tend to be in this category: closing off possibilities too early or not thinking of them in the first place. If every avenue is blocked, back off and look for others. Oh, how loath we are to retrace our steps! Yet sometimes that is most necessary.

Later in the book Ms Konnikova dwells on the value of getting away. Holmes will sometimes simply go elsewhere for a day, or he might spend an hour playing violin (Einstein did so also, to world-changing effect!). Conscious mental effort is not always, or even usually, the most effective. I built a nearly 40-year career writing scientific software on the following practice: At the end of a period working, I'd focus on the most troubling puzzle (usually some algorithm that was hard to code) and deliberately arrange all the pertinent facts and parameters in my mind (closing my eyes lets me "write" on a mental "screen"), then sort of say, "Away with you, now" as I push it to "somewhere else" in my mind and go do something else. I might get something to eat, or talk to someone or, if it is late in the day, go home and sleep. I frequently awoke at 3 AM or so with a neat package on my mental doorstep, so I would write it all down, in earlier days, or log in and code it all out on the spot in later years.

Here and there in the book we find suggestions for exercising the mind, and it is easy to get overwhelmed and think, "Oh, it is all too much for me." Everything is too much for us if taken all at once. Remember how to eat an elephant: one forkful at a time…and it helps to have a large room full of chest freezers! We can do any number of things to improve the arrangement of our mental attic, to distance ourselves from over-fondness for first ideas, and to improve our skepticism for overly simple solutions. One thing at a time. Pick one, any one, and have a go at it. It is like learning to juggle, which nearly everyone can do with about 3 months of daily practice. It doesn't come in a single day. And once learned, it has to be continued by juggling at least once or twice a week, or the skill diminishes. No matter at what stage we are, we can improve. And that is what this author is telling us. In place of the door-closing statements above, let us tell ourselves,
"I can do that."
"There must be a solution somewhere."
"Why should this not be so?"
"It had to happen somehow."
"If a question is never asked, the answer is always NO. Ask!"

Sunday, January 18, 2015

Owls are cats with wings

kw: book reviews, nonfiction, pets, memoirs, owls

In the early 1980s, on one particular day on the road from London to Kent, a driver who was paying attention might have seen another driver with an owl perched on his shoulder. The owl's name was Mumble, and the driver's, Martin Windrow.

For 15 years, Windrow shared his flat, and later a home in Sussex, with the Tawny Owl he'd obtained with the help of his brother. He writes of those years together in The Owl Who Liked Sitting on Caesar: Living With a Tawny Owl. For this rather lonely young writer and editor, Mumble was a godsend. His brother had persuaded him to try caring for an owl, but a first attempt, with a less congenial species, was humiliating and blessedly brief. If a Tawny Owl is similar to an affectionate tabby, this first owl was more like a fiery Siamese, the kind who either ignores or hates everything you do. Fortunately, he was willing to try again.

When he was introduced to Mumble, egg-raised for the purpose, not wild-caught, it was love at first sight for both. It had to be; as he describes it, living with Mumble was like being a single parent with an infant who never grows beyond a year or two yet becomes an adult in certain ways.

I was particularly taken when he described pet owls as "like cats with wings". Cats I can relate to. However, where a typical house cat might weigh 5 to 12 lbs (2.2 - 5.4 kg), this species of owl weighs at most 1.8 lbs (0.8 kg). But its talons compare to the claws of an Ocelot, so if you encounter even a small owl and it goes for your face, you're in real trouble!

All Mumble ever did with Windrow's face was nuzzle, and a bit of nibbling of his beard, in a similar fashion to her own feather-preening. In fact, Mumble liked what the author calls "necking" on nearly a daily basis.

A few chapters in the book outline the natural history of Tawny Owls, Strix aluco, but most relate the experiences of owl and man carrying on a life together. Mumble was somewhat sociable with others his first year, just as a human child is. After that, she became a one-man owl, and it was not safe to allow others into her presence. Everyone other than Martin Windrow was an intruder in her territory, and even the comparatively gentle Tawny species defends territory quite fiercely! Fortunately, with proper introduction and assimilation, he was able to persuade Mumble to accept one friend's caretaking while he was away once for more than a week.

Her life ended prematurely when someone, probably a misguided and misinformed "environmentalist", entered her outdoor aviary. From evidence on the scene, she apparently took a fierce whack at the intruder before dying of a heart attack. Windrow found her unmarked body in the open enclosure upon returning home. He sincerely hopes she marked the fool for life, and I heartily agree.

The "Caesar" of the title was a bust of Germanicus Caesar, and her bust-sitting is mentioned once in the text. I suppose it makes for a spiffier title, but her favorite perch was the top of the kitchen door. Not great title material.

I guess I'd describe this as a very comfortable book. Just the right book to read on chilly winter evenings.

Tuesday, January 13, 2015

The seductive power of mathematics

kw: book reviews, nonfiction, mathematics, mathematical thinking

We are nearly two weeks into the new year, and this is my first post of the year. It is not because the book was extra-hard to read, but that the year itself has begun extra-busy! Actually, though the book was long (437 pp + 15 pp notes), I spent less time reading it than many shorter ones because math is of great interest to me.

More than 2/3 of those who read that first paragraph will respond, "But not to me", and be tempted to stop right there. I hope you will continue anyway, because the author's design is to show how we all use mathematical thinking and can benefit from a better acquaintance with it. Theoretical mathematician Jordan Ellenberg has written How Not to be Wrong: The Power of Mathematical Thinking.

Contrary to popular thought, Mathematics isn't mainly about numbers. If you break the word down it means "The Studies of Learning". Note the "s" on "mathematics" and on "studies". The field has hundreds of branches, thus where an American would, in our streamlined way, speak of "math", the English speak of "maths". Being American, I'll go the American way. Only two of the many disciplines under the "math" umbrella explicitly involve numbers.

For most of us, our introduction to math began with arithmetic and the "plus table" and "times table". Though even grade schoolers are now permitted to use calculators in class, it is useful to know how to do simple sums and multiplications in one's head. At the very least, when you punch in some numbers and get a result, you are more likely to detect a punching-in error if your mind is at least estimating the result in the background.

The second numerical branch of math is Number Theory, which deals with properties of whole numbers. A big sub-field is Prime numbers, which we will return to later on. But most people who might read this have been exposed to additional branches.

At least in Western and Westernized societies, facility in basic arithmetic was needed to advance through Plane Geometry, Algebra, Trigonometry, Analytical Geometry (sometimes just called Charting), and Calculus. Before the early 1960s Calculus was not introduced to high school students, but the teacher of my senior class in Analytical Geometry was one of the first to finish the school year with a few weeks of instruction in basic Calculus. Now at least half a year is taught to most high school seniors.

So if you had all those courses, think back: most of the work was learning to use certain symbols and sets of symbols in a consistent way. Working out problems using numbers was less important than the proper use of those symbols. That's why the teacher kept harping on "Show Your Work!". Also, particularly in Geometry, formal proof methods were introduced, primarily because visual proofs are easier to comprehend than the symbolic proofs that are the stock in trade of "higher math" (that is, stuff for college juniors and beyond, and only in technical disciplines).

Most of us shudder at that word, "proof". Few understand it. It takes a certain kind of mind to construct a useful proof. My brother, a working mathematician for some years, whose name I shall call Rick, had two friends at college; call them Tom and Harry. They all took some rather gnarly "higher math" courses together, and did lots of formal proofs. Another friend described them thus:
"Send Tom into a room with a mysterious machine in it having several large gears, a big flywheel and other bulky items of unknown import. He is requested to make its wheel turn. By putting a shoulder to the largest gear and pushing very hard, he is able to make it turn, slowly. He leaves and Rick enters. He noses around a bit and finds, behind the machine, a crank with a long handle. Fitting the handle into a convenient socket, he is able to turn the wheel more easily. He leaves and Harry enters. He looks around further, sees the crank, but keeps looking until he finds a button. He presses the button and a motor somewhere makes it all run."
"Pushing the right button" represents concocting a useful proof. I like visual proofs, and you can see one that proves the Pythagorean Theorem here. Remember the Pythagorean Theorem? It pertains to a right triangle, one for which one angle measures 90°. If the two sides that meet at that right angle have lengths represented by a and b, their relationship to the third side, of length c is c² = a² + b². In words, we say that the sum of the squares of the lengths of the two legs of a right triangle equal the square of the length of the hypotenuse (the third side). Pythagorean triples are sets of three whole numbers that can be used to produce a right triangle, such as 3, 4, 5 (3²=9, 4²=16, 5²=25, and 25=9+16). Try with 5, 12, 13 and 8, 15, 17.

So if math isn't primarily about numbers, it sure uses them a lot. But the power of most branches of math lies not in the use of numbers, but in the core concept of math: Operators. To illustrate, when we learn the Plus Table, we are actually learning to use an operator, the +, the addition operator. With a little more thought and practice, we also learn the operator, the subtraction operator. Similarly, the Times Table helps us learn the ×, the multiplication operator, and later the ÷, or division operator. Even later we learn the exponentiation operator, which has several symbols, but the ² is the special one for squaring (multiplying a number by itself). And, we soon learn the , the square root operator, and allied symbols for taking other roots. And on and on it goes. In the middle of learning Algebra, we learn of Polynomials, and how the + and and × seem to attain superpowers to add and subtract and multiply these groups of many symbols, as though they were unitary in themselves. Calculus adds further superpowers, while adding a further set of operators. Sure, these operators work on numbers, but that is baby steps compared to the symbols and sets of symbols (and so forth) that they also work on.

Very few have a mind like Harry's. Most of us don't need one, just as most of us don't need to be an automobile mechanic to be able to drive a car. However, a certain amount of mechanical smarts can make us a better driver. Dr. Ellenberg's notion is to make us a little better at thinking in operational terms, like a mathematician. Then we might be "less wrong" about many things. And the title provides a clue to the author's aim. The kind of mathematical thinking that underlies most of the examples is Statistical thinking.

The book has five sections. First is Linearity. The most amusing example is found in its third chapter, "Everyone is Obese". A soberly-written article came out a few years ago that can be summarized thus:

  • In about 1972 half of Americans had a BMI of 25 or greater. (Body Mass Index over 25 is "overweight" and beyond 30 is classified as "obese", at least in government literature)
  • Twenty years later, the number of overweight Americans was 60%.
  • By 2008 just under 75% had a BMI of 25 or more.
  • At this rate, all Americans will be overweight by 2048.

If you chart these three points and project a straight line through them, it will cross 100% at 2048. But do you see the fault in this reasoning? Firstly, the "line" one wants to project isn't very straight. The percent of overweight first goes up 10 points in 20 years, then another 15 points in 16 years. Do get from 2008 onward, do you project the next 25 points (100% - 75%) over 50 years, or closer to 25 years? The authors of the study projected an average of the two shorter-term rates and got there in 40 years. But why didn't they say, "Well, the rate of obesity increase has nearly doubled more recently. Maybe it will continue to speed up, and double again. Then the (now curved) line will hit 100% in just 12-13 more years, and we'll all by fat by 2020."

The real case is that, while many people are prone to gaining more weight as their prosperity increases, it isn't so for everyone. I seem to be like the majority, easily gaining weight; my wife is not, and has weighed between 98 and 108 pounds for the whole 40+ years I have known her. And she never diets. If my wife and I are still around in 2048 (we'll be over 100), I am pretty certain that she, at least, will not be obese. My BMI stays around 28-29, and is more likely to go down than up as I exceed the age of 85 or so. And our very fit son, who will almost certainly be alive in 2048, is very, very unlikely ever to have a BMI greater than 24.

The Earth is round, but we treat it as flat for most everyday uses. Straight lines serve us well. But look at a survey of Sections in the central plains. A Section is a square mile, very hearly. On a perfectly flat Earth, every Section would have exactly 640 acres. But on U.S. Geodetic Survey maps you'll see a correction every six miles further north you go. Only the southern row of Sections has something close to the full 640 acres. The northern row of a 6-by-6 Section Township has Sections with about 639 acres, because the curvature of the Earth has drawn together the meridians used to lay out the survey, by five feet near 40°N.

The takeaway point of the first section: Very few phenomena in nature proceed in a straight line forever. Keep that as a maxim in your mental bag of tricks.

The second section is titled "Inference". Here is the largest mass of material related to proofs. But it is presented in a much more entertaining way than you'd find in a college math course (or even your Middle School Geometry class). He begins with the legendary Baltimore stock broker, something I call the Binary Scam.

You get a piece of junk mail (these days, spam e-mail) with the bold statement, "Using my special stock evaluation system I predict Apple stock will rise tomorrow." The next day, Apple's stock price indeed rises, and soon another missive arrives: "See it at work. The stock will rise again the next day." It does so, and a third message now predicts a drop, which indeed happens. After a couple of weeks— and the messages now include a "Click here to invest" button—the fellow has been right ten times out of ten. You are ready to invest!!

What don't you know? You don't know that the first message went to more than 100,000,000 people. Half of them got a message saying the stock would go, not up, but down! Those 50 million or so never got the second message, but half of those who did, got one saying the opposite of the one you received. And so it goes. After 10 "predictions", the field has been cut by a factor of about 1,000. (Strictly speaking, by exactly 1,024, the tenth power of two). This leaves 100,000 or so people who tend to think this guy has a system that really, really works. If even 1% of them invest with him, that could be millions of dollars. And on day 11 he might just be in Switzerland or somewhere with those millions, and a "dead" address with no forwarding.

There is a variation of this, in which, even though half the people on day 5 got a "prediction" that "failed", they get a special message: "As you can see, nothing is perfect, but I think you will be pleased when the system continues to produce a high rate of correct calls." Guess what? Our psychology is such that a larger number of those folks will invest!

Inference is all about doing your best to gather more information, and when you have done so, remembering what Donald Rumsfeld said (I paraphrase), that we make decisions based on what we know, and try to take account of what we don't know, which is in two parts: the Known Unknowns and the Unknown Unknowns. The more "wonderful" an opportunity seems, the more likely it is that the unknown unknowns are so much bigger than what you know and what you know you don't know, that you are at best guessing while wearing a blindfold.

He closes the section with a cogent explanation of Bayesian Inference, which is quite a bit different from ordinary statistical thinking. Though it is more powerful than the kind of inference used in a typical scientific journal article, it takes a different kind of thinking, and I confess I can't use it numerically without having a text open to guide me. This is evidently true of scientists in general.

I promised a return to prime numbers. The first several prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Prime numbers have no divisors, no factors (1 doesn't count). You can see that 4 of the first 7 natural numbers are primes. Then they start to thin out: 4 of the next 12, then 2 of the next 10. Something called the Prime Number Theorem states that the number of prime numbers, P, less than some large number N, is equal or less than N/ln(N), where ln refers to the Natural Logarithm. Look it up if it interests you. Here we can test it with the 10,000th prime number, 104,729. P=10,000 and N=104,729. N/ln(N)=9060.28 and some more digits. The millionth prime is 15,485,863, and the calculation on these numbers yields 935,394 (and some decimals), about 6.5% lower than a million. For really big numbers, the theoretical number gets quite close.

What the Prime Number Theorem tells us is that prime numbers thin out steadily, and somewhat predictably, the further out we go on the number line. But they never die away completely. There may not be a high density of primes between 100 quadrillion and 101 quadrillion, but there are still a lot of them, roughly 25 trillion. However, this is very thin indeed, with only one number out of 40 being a prime at this level, on average.

Why should this be useful? Prime numbers are at the core of modern encryption, which is used by your bank to send a secure message or payment to another bank whenever you make a credit card transaction or write a check. Your password is also encrypted. The encryption method uses a long number made up of two or more long prime numbers. The rarity of long primes means there are lots of long numbers to choose from, that are hard for a computer program to figure out whether they are prime or not, and what their factors are. 101 quadrillion is only an 18-digit number, and your bank is using numbers of 85 digits or longer. Just cracking an 18-digit "composite number" (in the industry this means a long number with only two prime factors of roughly equal size) requires doing several million divisions. Today's computers can do that in a few seconds. But an 85-digit composite? No machine yet built can determine its factors in less than a few billion years. And when machines get millions of times faster? We'll just go to 200- or 400-digit encryption.

Well, there are three sections of the book to go. "Expectation" is about using probability methods to figure out how likely something is. The weather forecaster uses an expectation method to say that the chance of rain tomorrow is 40%. But particularly for weather, expectation is not like it is when rolling dice or playing roulette. If a 6-sided die is make properly (most are pretty close), each number will come up 1/6 of the time if you roll it many times. Of course, if you make only 12 trials, you are very likely to find three instances of a particular number and only one or none of another. The 1-out-of-6 expectation starts to get accurate only for a few hundred rolls at least. And here is a key point. If you roll a 2. How likely is it that the next roll will be a 2? The same as the first time, 1 out of 6. But we don't think that way, which leads to all kinds of grief at the craps table! We think a 2 is less likely than it was the first time. Not so.

In weather, expectation works a bit differently. Weather systems are not usually solid lines of rain clouds, but storm cells with space between. If an advancing storm front is made up of storms 3 miles wide with 2 miles between them on average, then the 60% chance of rain really means there is a 100% chance of rain over 60% of the area. (Dr. Ellenberg doesn't state it this way. This is my example)

There is a very entertaining chapter on the lottery, and how certain lotteries can be beaten. But don't expect a how-to on getting rich at your state's expense. When a lottery is ill-conceived enough to be beaten, you still might have to fill out half a million lottery tickets to take advantage of the odd statistics, and thus risk half a million to a million dollars in the process. And there is always a chance that every one of those tickets will be a loser, even though if you play that lottery several times you are certain to come out ahead. There are easier ways to make a buck, for certain! Being the Baltimore stock broker, for example, if you don't mind exile at some point. But lotteries can be thought of primarily as entertainment for imaginative people, and as a tax on folks who can't do math. The state takes 30%-40%, so they only pay out 60-70 cents on each dollar taken in.

Fourth is "Regression", and this word has two meanings. One is a formal process of figuring out the best line to cast through a set of points that are correlated, but not perfectly so. One chapter talks about this kind of regression, but the main point in this section is that extraordinary results are usually not followed by more extraordinary results. The classic example is adult height in a family. Suppose a couple are both extra-tall; the man may be 6'-4" and the woman a 6-footer. Average heights in America are 5'-10" for men and 5'-4" for women. Knowing only this, if the couple has four children, when they are grown, do you expect all four to be extra-tall? While there is some chance that at least one boy might exceed the father's height, it is most likely that the four will be taller than average, but not extremely tall. Conversely, if a man and woman are very short, their children will also probably be shorter than average, but it is unlikely that they will be even shorter than their parents.

This is called Regression to the Mean. Human height is partly driven by genetics, but also partly by dietary factors, and partly by chance such as getting a disease that stunts growth, or conversely over-stimulates the pituitary leading to extreme height. There are numerous factors that influence height, and they are more likely to average against one another than cause additive extreme results. It is the same for sports performance. A basketball player who usually hits 55% of his free throws may hit his first 3. Does that mean he is likely to have a 100% season? Nope. There's that straight line again. We actually see that most ball players do better in the first half of a season than the second half, from a combination of tiredness and injuries coming in later on. Yet a few players will "rise through the months". Bookmakers make a lot of money from bettors who don't think through these things. In fact, a great principle is stated in a chapter on gambling: If you find gambling exciting, you're going about it wrong. Those who do best at gambling actually gamble the least. They find ways to make the largest number of sure bets and the fewest number of risky bets. You might want to read a book by Amarillo Slim on the subject before your next casino visit.

The final section is "Existence". Pundits predict a lot of things. It turns out, and clear numerical examples demonstrate, that such things as "public opinion" seldom exist. Voting seems a straightforward matter. It is, when there are only two candidates in a race, or only a yes/no question to be decided. Add a third choice, and it all goes out the window. Some lawmakers were wise enough to require a run-off election where no candidate gets a clear majority in a race with 3 or more. But even this doesn't guarantee you'll really get "the people's choice", and several entertaining examples, some historical and some theoretical, show what that means. Suffice it to say that, like the 3-body problem in astronomy—which is unsolvable!—3-way political races are impossible to craft into a perfect system. Just ask Al Gore…

The "power of mathematical thinking" is at its root a call to back off and think more broadly than a subject at first appears. For example, recall the tall family mentioned above. Suppose I told you an additional fact, that both the man and his wife were the tallest of several siblings, and the only one in each family who was taller than their parents? Would that change your estimate of their children's heights? If it would, you are thinking in a more Bayesian way, which isn't a bad thing at all!

And I find that I've written so much without looking at a single one of the pages I'd dog-eared. I like it when an article flows. Good way to start the year.