Love and Math by Edward Frenkel

Source of book: I own this.

While I didn’t go beyond algebra, trigonometry, and analytical geometry in school, I remain fascinated by math. Had I chosen a different career, I probably would have taken more advanced classes. In any event, I do find certain books about math to be fascinating, and try to read one from time to time.

Probably the one I enjoyed the most was The Nothing That Is by Robert Kaplan, about the history of the use of zero as a placeholder. Truly revolutionary. For more of the books I have reviewed with a mathematical theme, take a look at my index - specifically the section on Science. (I lump them together.) 

Love and Math is a blend of autobiography and an introduction to the area of mathematical theory called the Langlands Program. I’d try to explain it except (1) I’m not sure I really understand it myself and (2) Edward Frenkel, who is pretty good at breaking stuff down for non-mathematicians, took a whole freaking book to do it. So just read the book instead.

Frenkel grew up in Soviet Russia, as a Jew. This was problematic for him, because Jews were not (in practice) permitted to be admitted to the best schools, or even to have careers in certain areas. In his examination to get into Moscow State University, he was given a clearly bogus test, then failed on it. Nevertheless, he persisted in doing what he could. He was taken under the wing by a number of different professors and mathematicians who recognized his genius. Basically, he couldn’t study pure math, but had to look to a career as an engineer or applied math teacher. He could, however, study in his spare time, which is what he did. He eventually was invited to lecture and study at Harvard. He stayed on during the failed coup and the eventual fall of the Soviet Union, and now teaches at UC Berkeley.

The autobiographical parts of the book trace his academic and mathematical career, but don’t really get into his personal life. Instead, the biography serves to connect the mathematical concepts. Frenkel introduces each piece of the puzzle at the point where it fits in his life, and by the end, the reader can (sort of) understand at least the basics of what Frenkel has done in his field.

I must say, Frenkel does a rather outstanding job at simplifying concepts. I checked out a few different sources to try to get a better understanding, and realized that the others were even more confusing to a non-math-major sort. While I can’t say I fully understood everything, I believe I know more than when I started, and enjoyed the process. It took me a while. It was all I could do to read one chapter at at time, and then sit on it for a bit and process the information.

I won’t even attempt to get into the math itself. What is even more interesting is the way that math and the physical world work. Roger Penrose saw reality as a triad: the physical world, the mental world, and mathematics. I think he is right to a large degree. The more we learn about the physical world, the more we understand that it is written in the language of math. As a general rule, math has developed in the mind, in the realm of pure reason. But when we look for ways to describe the behavior of the universe, it turns out that the pure math we have ends up being the necessary language to describe what we see. The DNA of the universe does turn out to be math. This is fascinating and awe inspiring. While Plato’s idea of the “form” may not hold true as a general description of the world around us, math does fit that idea. There is indeed a world of ideal mathematical truth that we don’t so much invent as discover.

Einstein put it thus:

“How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably appropriate to the objects of reality?”

It seems hard to believe - and this may be one of the reasons that laymen often distrust science - but it is true.

There are a few other highlights of the book. One was Frenkel’s description of the way the Soviet system worked.

In the stagnant life of the Soviet period, talented youth could not apply their energy in business; the economy had no private sector. Instead, it was under tight government control. Likewise, communist ideology controlled intellectual pursuit in the spheres of humanities, economics, and social sciences. Every book or scholarly article in these areas had to start with quotations of Marx, Engels, and Lenin and unequivocally support the Marxist point of view of the subject.

It was hard not to laugh at this. After all, this idea was nothing new to the Soviets, and it hasn’t disappeared. You find it in many areas. Just to pick one of which I am all too familiar, within Evangelicalism here in America, most areas of study are off limits as far as intellectual honesty is concerned. You have to start with a quote from the Bible, and end up unequivocally supporting the “theologically correct” view of the subject, actual facts be damned. Likewise for [insert your favorite political dogma here.] (Maybe, say, climate science for conservatives…alternative facts, people. Or, the economic theories of Murray Rothbard - widely embraced by the Paul Ryans of the world - which dismiss empirical evidence as irrelevant to economic policy.) No matter where it happens, ossification of ideas and refusal to embrace the possibility that our beloved dogma may not be, in fact, true, leads to an inability to make further progress.

There were two exceptions to the general anti-intellectualism of Soviet academia were math and physics. It isn’t difficult to see why. They were useful. Useful for inventing and building things that blew up the enemy. And this is where many of the most talented Soviets found their outlet, even as many had reservations about the way their country was using their work.

Two particular discussions of the connection between math and physics were particularly interesting. The first was on the curved nature of space. This is hard for us to visualize. As Frenkel puts it, “We are used to thinking the space we live in is flat, and so in our everyday experience curved shapes seem to appear only within the confines of that flat space. But this is a misunderstanding, an artifact of our narrow perception of reality. And the irony is that the space we live in isn’t flat to begin with!” Perhaps one way to understand this is to think of the fact that in our everyday life, the earth appears flat. Building furniture or even an house can be done perfectly well based on that assumption. However, build a long bridge and you find very quickly that the earth is curved. Now imagine living on the scale of an ant, or a bacterium. Such a creature will never need to experience a round earth. Likewise, we cannot perceive the curvature of space, and flat space works well for our everyday experiences. But not so much when we explore larger scales.

The other discussion which was fascinating was that of duality, which underlies much of physics at the large and small scale.

It might seem strange to look for a duality in physics, but in a sense this is a concept we are all already familiar with. Take electricity and magnetism. Even though these two forces seem to be quite different, they are actually described by a single mathematical theory, called electromagnetism. This theory possesses a hidden duality that exchanges electric and magnetic forces.

This is a “theory” in the scientific sense, of course. It is abundantly clear that this duality exists. Without it, generators and electric motors, speakers and microphones, and so many other electrical and magnetic devices would be impossible.

One final bit. I have noted before that our society treats mathematics like some sort of foreign language, unnecessary for most to learn. We are math-phobic. It is socially acceptable to be illiterate in math, in a way that it is not socially acceptable to be unable to read. Even as our everyday lives are intertwined with math - any time we use technology - we are willing to accept it without bothering to understand. Frenkel quotes the poet Hans Magnus Enzensberger: Math is
“a blind spot in our culture - alien territory, in which only the elite, the initiated few have managed to entrench themselves.” We do not bother to learn the language. As Goethe put it (poets and math...who knew?): “Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different.” This really is a shame. I am doing my best to make sure that all my children enter college with a solid foundation in math, and (I hope) without a fear of it or a willingness to be illiterate. This book is not easy to read, but it is fascinating. For those with a math phobia, probably better to start with something easier. For those who love math, this book will be fun and enlightening.

Hidden Figures

Last night, the lovely Amanda and I went to see Hidden Figures. For those who don’t know, this film tells the story of three pioneering African American women who worked for NASA during the space race.

Mary Jackson, aerospace engineer, in an era when neither women nor blacks were heard of in that position.

Dorothy Vaughan, mathematician and computer programer. And also the first African American woman to be a supervisor at NASA, and later the first to be head of personnel.

Katherine Johnson, who merely calculated the flight path for Alan Shepherd’s spaceflight, calculated John Glenn’s orbit and reentry, worked on the Apollo and Space Shuttle programs, co-authored 26 scientific papers, and now has a facility at NASA named after her for her decades of brilliant contributions to the space program. Basically as badass as they come.

These names really should be better known than they are. Kudos to NASA for doing its part to recognize them. 

 Mary Jackson (Janelle Monae), Katherine Johnson (Taraji Henson) and Dorothy Vaughan (Octavia Spencer)

We tend to forget it these days, but in those pre-electronics years, all the math for space flight - and it is a LOT - was done by human “computers.” And many of these computers were women. (This didn’t just apply to flight, much of science was conducted this way. See How Old Is The Universe? for a bit more about women who contributed behind the scenes but never got credit.)

Some of these women - an entire unit at NASA in fact - were African American. These were the days of Jim Crow, so they had segregated bathrooms and so on, a fact which plays a major role in the film.

I don’t want to spoil the film (although history is history…), so I won’t go into too much further detail. This is a great film, though, and tells the story well. I was impressed that the focus remained firmly on the African American women, and didn’t become all about the whites who helped them.

A few things really struck me. One was the scene where John Glenn refuses to fly unless Katherine Johnson personally calculates his landing trajectory. This sounded a bit Hollywood, so I looked it up. Well blow me to Bermuda! It really happened. And Glenn really was as shockingly progressive as the film portrays him. Fraternizing with the “coloreds” indeed! (Glenn Powell really nailed the portrayal of Glenn too. Nice casting.) 

I also found the issue that arises between Katherine Johnson and Paul Stafford (portrayed by Jim Parsons). I have no idea exactly how this went in real life, but in the movie, Katherine stubbornly affixes her name as co-author of each report she types up, featuring work she does, but Paul insists that only his name be on there, because “computers don’t author reports.” Again, Hollywood license? Maybe. Or not. What I do know is that this sort of thing goes on all the time in the real world. Woman does work and comes up with ideas, man gets credit. My wife was specifically trained, so to speak, on how to play the nurse/doctor game. Nurse comes up with an idea, but must sell it to the doctor in a way that he thinks it is his idea, and can thus take credit for it. My wife’s generation, though, has been saying screw it to this game, and is more likely to insist on interacting as equals in different roles, not as superior and inferior. (Nurses work for the hospital, not doctors…) So I fully believe that Katherine Johnson had to put up with this from the white males on her team. In what had to have been a satisfying development, Johnson would go on to co-author a good many reports and scientific papers - and get credit for them.

Another powerful scene was where Al Harrison (Kevin Costner) truly realizes the cost that segregation is having on his team - Katherine must walk a half mile each way to relieve herself in the “colored” restroom - he goes ballistic, and violently destroys the sign on the segregated restroom, and proclaims that anyone can use any restroom in any building, white or black. His closing line is great:

At NASA, we all pee the same color.

The other line that was amazing and profound was this: Dorothy Vaughan meets Vivian Mitchell, who is some mucky-muck in Personnel in the newly desegregated bathroom. The two of them have a history, because Vaughan has been doing the work of supervising the “colored computer” unit, but has been refused a promotion. Mitchell claims it is just “how NASA works,” but everyone knows that the problem is that Vaughan is black.

After a bit of an awkward exchange in the bathroom, Mitchell says (more or less), “You know I never held any ill will toward you ladies.”

Vaughan hesitates, then softly says, “I know...I know that’s what you want to think.”

Mic drop.

I can’t get this line out of my mind.

Because this is really what I feel I am up against trying to discuss racial issues with my own tribe. Because everyone keeps saying, “I’m not a racist, but…”

“I don’t have ill will toward brown skinned people.” But you just said that Black Lives Matter is a media creation, and that police brutality is a myth.

“I don’t have ill will toward brown skinned people.” But you just voted to give Steve Bannon a leadership position. (Sorry, he bragged about giving White Supremacy, sorry, the “Alt-Right” a platform at Brietbart. He doesn’t get a pass for that.)

“I don’t have ill will toward brown skinned people.” But you just said that blacks are “more criminal” or “less civilized” than whites.

“I don’t have ill will toward brown skinned people.” But you said racism doesn’t exist, or if it does, it is toward white people.

“I don’t have ill will toward brown skinned people.” But you just reposted that meme mocking Michelle Obama in racial terms.

“I don’t have ill will toward brown skinned people.” But you just referred to protesters as “thugs” and “animals.”

“I’m not racist.”

“I know...I know that’s what you want to think.”

Damn, that’s a good line. It absolutely gets to the heart of the problem. Many people don’t really want to think about the impact their actions or inactions have on people of color. What is most important is that they don’t feel discomfort or have to look at themselves honestly. Mitchell wants to preserve her own sense that she is a good person, without having to actually stick her neck out and advocate for justice for Vaughan.

This perhaps is why I will be making sure my kids see this movie at some point. They need to see and understand that positive change requires more than positive feelings. It requires action. Each of the three women has to at some point demand justice for themselves - and they make the white people in power VERY uncomfortable when they do so. It also requires that those of us who wish to be allies for people of color cannot simply content ourselves by saying that we personally would never discriminate. We have to be like Harrison and refuse to allow segregation, consequences be damned. We have to be like the (unnamed) judge who goes against Virginia law to side with Brown v. Board of Education to allow Mary Jackson to attend engineering classes. We have to be like John Glenn and actively and expressly side with our brown skinned brothers and sisters - and yes sometimes that means politically too - no matter how much disapproval we get from our own tribe.

Otherwise, we are just staying in our little bubble, thinking a little good intention will paper over action and inaction that does evil. No, we’re not racist. Right?

“I know...I know that’s what you want to think.”

It’s very encouraging to me that Hidden Figures is doing well at the box office. Yes, it is well cast and well acted. (Taraji Henson, Janelle Monae, and Octavia Spencer are all outstanding.) But the story is powerful, and comes at a time where there has been an open revival of White Supremacy, and far too many people feel free to say openly racist things. This film is a reminder that, jackasses like Steve King notwithstanding, it wasn’t just white people who have contributed to the world. They just haven’t gotten the degree of credit they deserve. I am thrilled that this story is finding an audience.

Go see this film. Take your children (it has no gratuitous sex, violence, or language, actually). Let them see what our nation was like not very long ago, so they can understand what an appeal to return to the past really means for many Americans. This film should also encourage our daughters, and children of color, to aspire to careers in math and science. These women were nothing short of amazing, and without them, the space race would likely have played out very differently. Our society is better off when we encourage the contributions of all, regardless of gender, race, national origin, or religion. We must never forget that, and we must constantly fight against the ideas that seek to deny this truth. 

***

Update: I took the kids to see this. They loved it. We had some great conversations afterward. 

Nathaniel Bowditch and the Power of Numbers by Tamara Thornton

Source of book: Borrowed from the library

Just about any religious homeschooler from the late 1980s will be familiar with Carry On Mr. Bowditch by Jean Lee Latham. This children’s biography of Nathaniel Bowditch was written in the 1950s and won a Newbery medal in 1956. I gained a second lease on life in the 1980s after it was endorsed by a number of Evangelical and homeschool movement luminaries. It is not difficult to see why. The book is a paean to self directed learning, hard work, and diligence. It praised self-learning, and fit the homeschool ethic well. In this sense, it is a continuation of the legend of Nathaniel Bowditch which arose soon after his death. It fits in well with the Horatio Alger narrative of the young man who succeeds through virtue (and karma - something those who have never read Alger might not realize) and effort. The American bootstrap myth which is perhaps our own national fairy tale.

I don’t want to create a wrong impression, however. I loved and re-read Carry On Mr. Bowditch, and encouraged my kids to read it as well. It is a well written book, full of interesting incidents. It makes math sound exciting. And, unlike many books for children, it doesn’t sugar coat all the death that occurred in those times. Lots of people die of disease. Boats are lost. Tragedy is everywhere, and people just had to survive it. It’s a worthwhile book, and I have recommended it to others.

1835 portrait of Bowditch by Charles Osgood

As Nathaniel Bowditch and the Power of Numbers points out, though, the kids book does take some liberties with the story. The first an exaggeration of Bowditch’s innovation - his genius was less in invention than in meticulousness - and he knew it.  The second is perhaps more problematic, which is that Latham, like other biographers of Bowditch, failed to note that Bowditch’s rise was made possible by extensive family connections. Certainly, his hard work and initiative contributed to his success. But he didn’t rise solely on his merits. At every turn, he received significant opportunities as a result of his social status. This takes nothing away from Bowditch, I’ll hasten to say. What it does is creates the impression that merit is the sole factor in success, something which was patently untrue then, and remains true now to a significant extent. As the author put it:

Bowditch was undoubtedly self-taught, but he was not not self-made...The young Bowditch’s connections made him a fit candidate for sponsored mobility, affording him opportunities, from access to the Philosophical Library to a clerkship on a Derby vessel, which would otherwise not have existed. Merit mattered, of course. Had Bentley not been impressed by the youth’s mathematical talents, had Derby not heard good reports of the boy’s work, they would not have assisted Bowditch, but nor would they have lent this kind of aid to a boy from a less than respectable family.

Also ironic is the fact that Bowditch is now given to home school kids as an example of Evangelical virtue, despite the fact that Bowditch himself had no love for Evangelicalism, being a Unitarian for strong philosophical and moral reasons.

Anyway, the real life story of Nathaniel Bowditch is fascinating in any case. Born in colonial Massachusetts, he came to prominence in the early years of the United States through his publication of The American Practical Navigator, a comprehensive - and far more accurate - set of tables for calculating longitude using lunar observations. (This is an oversimplification, but it consists of measuring the angle of the moon compared to various stars. It allowed precise calculation of position during any clear night, without need for a chronometer, a rarity in those days.)

Bowditch was born to a family that had fallen on hard times, but remained connected to two of the most prominent New England families of the time. He got an apprenticeship in trade, which later led to a shipboard position as Supercargo (basically, director of trade for a merchant vessel), eventually became a captain (despite his lack of practical sea knowledge). Later in life, he would stick to land, managing newfangled organizations by the name of Trusts. (We take them and corporations for granted these days, but they were in their infancy in the early 1800s.) He would go on to publish a well respected translation and annotation of Laplace's mathematical treatise, reform the finances of Harvard University, co-found insurance and investment companies. In all of this, the author argues, he essentially invented modern corporate practice, from careful accounting practices to form documents.

Thornton does an excellent job in telling the story, while also bringing in enough historical context to shed light on the events in Bowditch’s life.

There are a few things that were particularly interesting to me. The first was that we often have a completely erroneous view of the education of the past. Many will note that Harvard was a place you sent your teenagers. Which is true. But what is also true is that the level of education at Harvard at the time wasn’t even up to the level of our high schools today. At least academically speaking. In fact, that wasn’t even the point. The goal of college was to prepare upper class boys to act with social polish. Thus, they would study the “classics,” the Latin and Greek works that formed the cultural literacy that identified them as gentlemen. (The US may have been “class free” in theory, but in practice, well, not so much.) Bowditch, despite his best efforts, never attained this “polish,” and was thus considered uncouth, even though he was intellectually far beyond average in knowledge and quality of thought.

Particularly shocking was the lack of mathematics at Harvard. Eventually, (1787) math was added as an elective. And by “math,” I mean arithmetic - elementary school math. Later, algebra was added, but calculus failed to gain a foothold. In fact, in the English speaking world, one mathematician estimated that on a dozen people total could demonstrate proficiency in Leibniz’s version of calculus.

And America was even worse, perhaps. In what has been a refrain for the past, well, 400 years, Europeans have viewed Americans, as the author puts it, as “too boorish, too materialistic to generate or appreciate art, literature, and science.” I believe this did change in some ways. Clearly we have developed scientific proficiency - dominance in some cases. And we have literature and art, although I think many Europeans still consider us relative savages.

Very interesting to me as a lawyer was the section on the development of Trusts and the law surrounding them. Although Henry VIII can be credited with the creation of this area of law, it was fairly unregulated and nebulous in the United States until Bowditch and his associates brought the Trust into the modern age. In essence, charitable trusts were poorly managed and unregulated. To put things charitably, they were an utter mess by modern standards. Trustees routinely commingled their own funds with trust funds, didn’t keep accounts, and had no responsibility to anyone. The primary reason they were unregulated was an accident of jurisprudence, so to speak. In England, which gave us our legal system, the Courts of Equity regulated Trusts, and they were viewed with suspicion because of the way they functioned in England. (Dickens’ Bleak House was about these courts.) Equity Courts had a reputation of being “pay to play” in practice, an arena in which the powerful abused the weak. In our own time, with the merger of the courts of Law and Equity, and with predictable, just laws governing trust administration, we take for granted that the norm for trust administration is a fiduciary duty to the beneficiaries. We assume the trustee will segregate funds, and seek to prudently manage and invest. Furthermore, we take for granted that beneficiaries are entitled to accountings from trustees. All of this got its start with Bowditch. He applied the same approach he did to his lunar calculations to his accountings. He insisted on rigid separation and clinical impartiality.

Another positive that Bowditch shined was in his reform of Harvard. As proof that the issues we face today aren’t as new as we often think, one of his goals was to reduce the cost of college to the students. As he said, Harvard was never meant to be “an establishment for the rich alone, but rather as a place where persons with a moderate property might have their children educated upon equal terms with the rich without being under the necessity of soliciting pecuniary aid in a manner unpleasant to their feelings.” In other words, nobody should have to sell their future to afford an education. Important words in our own time.

Also surprisingly pertinent to our own time was Bowditch’s response to the social upheaval in the early 1830s. That decade saw riots in the northern cities. Riots against immigrants (sound familiar?), Catholics - the theological bogeymen of that era (again, familiar?), African Americans (ditto), and Abolitionists (who were called...wait for it...elitists...hmm.) Bowditch had a natural love for order, and had no great love for the Catholic faith. (He was Unitarian, and experienced a lot of the most superstitious forms of Catholicism in his travels around the world.) However, he was appalled at the outpouring of “fanaticism” and hate within his own community. I sympathize.

Most interesting, however, must be the fact that Bowditch’s true legacy has slipped into obscurity, not because it failed, but because it was so wildly successful that we take it for granted.

Ironically, our memory of Bowditch’s influence on practical affairs is lost not because his innovations were soon made obsolete but because they became the norm. Filling out forms, classifying and filing documents, separating business from personal records, meeting rigid deadlines - these are so much a part of our existence that we hardly understand they have a history. We take for granted that much of what we call modern life consists of dealings with impersonal institutions. Precisely where his impact was the greatest, Bowditch is a historical cypher. But in a final reckoning of his life we need to number him among those who transformed our world.

It’s an interesting thought. One may or may not prefer the modern to the pre-modern. It’s hard to argue that either the impersonal corporation or the complex networks of patronage that worked to preserve privilege and prevent social mobility by those lower down the ladder are the best possible systems. Which one is best is perhaps best left to the philosophers.

Either way, this is an outstanding book, and the author’s research and writing are excellent. For anyone seeking to fill in a portion of our history which often gets lost between the more exciting wars on each end, this book is worthy entry.

Fluke: The Math and Myth of Coincidence by Joseph Mazur

Source of book: Borrowed from the library

I first borrowed my dad’s copy of The Hitchhiker’s Guide to the Galaxy when I was, well, pretty young. I didn’t ask permission, and was a slight bit shocked at some of the language (which means I must have been young indeed…) but I was very much hooked.

I mention this because, as all fans will know, a key plot point in the book is the “Infinite Improbability Drive” at the heart of the spaceship, The Heart of Gold. As Douglas Adams explains, “finite improbability” drives had been invented long before, so that tricks like removing a woman’s clothing by random chance had been done for centuries. But it was the ability to do things that were not merely improbable - and with known probabilities - but things for which one could not even calculate the many variables, that made the ship so useful. 

Joseph Mazur, a math professor and author, attempts to put the realm of probability into perspective for the layperson. His attempt is partially successful, which is impressive given the vast scope of such an undertaking, and the difficulty in making it useful for those of us without the ability to do higher math and upper level research. (And by that, I mean beyond algebra and statistics - about my comfort level.) To do this, he limits his topic significantly, and focuses on a few specific scenarios that are likely to be relevant to real life and to the stories of remarkable coincidence that make up our folklore.

To start with, Mazur quotes the dictionary definition of “coincidence,” which most of us probably have not actually thought about.

“A surprising concurrence of events or circumstances appropriate to one another or having significance in relation to one another but between which there is no apparent causal connection.”

As the author points out, several of the elements of a true coincidence are often missed when discussing seemingly chance occurrences.

First, it is important to note that these are events which have some significance in relation to each other. In other words, we don’t notice these occurrences unless they have some significance or connection. In practical terms, then, “coincidences” in the colloquial sense happen all the time, all around us, but they are not noticed because they do not trigger a response in us.

As a practical example of this, chances are you and I have both seen the same person in different circumstances. Had that happened with a friend or someone who stood out, we would call it a remarkable coincidence. Instead, because the person had no personal meaning to us, we never noticed.

The second is the phrase “apparent causal connection.” Let me pause for a moment to appreciate the use of “causal,” one of my favorite words from law school. The point the author makes with this is that many things which seem to be mere coincidences actually have causes that make them much more likely.

For instance, if I were to tell you that I would have a case in front of the same judge that presided over Michael Jackson’s estate, you might (if you knew nothing about me) find it to be a striking coincidence. After all, what are the odds that two people would have a case in the same courtroom at about the same time?

Well, for two specific random people, perhaps it would be rare (although, as Mazur points out, on a population basis, it would be inevitable.) But I am not a random person. I am a California attorney, and I take cases that are heard in the Probate division. Furthermore, while I don’t do much in Los Angeles County, I live just one County over, so I do have cases there. So the chances that a person like me might have this “coincidence” are actually higher than one might think. Because there are many causes for the “coincidence”: I am an attorney in the area who practices Probate.

This also leads to another point that Mazur makes. While the odds of a particular occurrence happening to any one individual (like you…), over large populations, the odds of a particular thing happening increase greatly.

The example that Mazur uses is of a woman who won the lottery four times in 18 years. What are the odds of that? Well, the odds of her winning the lottery that many times in that time frame are indeed astronomically low. So don’t bet your life’s savings expecting to do the same.

However, in 18 years, given the total number of lotteries and players, it is actually highly likely that someone would win this many times. It is even more likely, too, given human behavior - and economics. She almost certainly played more once she won the first time. She had motive, and money to spend…

The best part about this section of the book is the math. Okay, at least in my opinion. Mazur makes the basics fairly easy to understand, at least if you can do algebra. (I love algebra, for what that is worth, even though I never took calculus.) Once you have the basics down, you can figure out probabilities. Then, when you start adding those probabilities together, the likelihood of “rare” occurrences starts to approach a value of one.

This is essentially what powers The Heart of Gold. If you can take improbable events, and stretch the time and number event horizons to infinity, the likelihood (in theory) becomes one, and the event occurs.

Let me also note with approval his use of both coin flips and dice rolls to illustrate the concepts. My brother invented a version of “fantasy” baseball (more or less) based on dice probability - back when he was in grade school. While it could have used some refining, he wasn’t too far off in correlating the odds of specific batting outcomes with the odds of rolling certain numbers with a pair of dice.

The book starts with ten stories of “coincidence” that are then analyzed. Some turn out to be indeed unusual, while others turn out to be highly probable, for the reasons above, and only have significance because the persons involved noticed - and told their stories.

There are some other interesting chapters in the second half of the book, which I thought I should mention.

First is one on DNA evidence in criminal trials. As a lawyer, this has been a fascinating topic, not least of which is because I entered law school not long after the O. J. Simpson trial. (As a former Los Angeles resident at the time, I also knew a good bit about the persons and places involved.)

Mazur points out what most of us attorneys know: DNA evidence is not a panacea, and its validity depends very strongly on the way it was processed and handled. Even if this is perfect, DNA evidence is either highly useful, or not particularly useful at all, and the specifics of when and why are poorly understood by legal professionals and law enforcement, to say nothing of the general population.

Just to give some highlights - which is what Mazur does - from a larger field. (I’ve done a bit of research on my own.) First, there is a non-trivial chance of a false positive or a false negative for DNA testing. Mazur points out, however, that the implications of one are different than the other. The chapter opens with Maimonides’ quote that it is better to let some guilty people go free than to have a single innocent executed. (That this is lost on many in our own age is all too true.)

But it gets worse. Mazur cites an independent investigation of the Houston PD crime lab, which found that 32% had major mishandling issues - and the investigators believed that intentional scientific fraud was the cause. (In other words, police workers manipulated results to get convictions.)

And then it gets even worse, at least from the scientific point of view. Fraud is horrible. But fraud can be fixed. Not so much for systemic problems. Let me see if I can summarize this in an understandable manner. Let’s say that we do DNA testing on a person who is already a suspect for legitimate reasons. (Like, say, motive, opportunity, lack of alibi, and so on.) The chances of a false positive for that person are pretty low. DNA might well be useful evidence here. (Although chance contamination is also a possibility, so it isn’t conclusive, just another piece in the puzzle.)

But what if, instead, there are no suspects, but law enforcement runs all their possible DNA samples to look for a match? Well, now we have a problem. Once you invoke the laws of large numbers, the chances of a false positive increase really fast. With enough “suspects,” it becomes almost certain that there will be a match - a false match.

As Mazur points out, this is particularly a problem in the United States. Believe it or not, while we have a mere 4.4 percent of the world population, we have nearly ¼ of the total prisoners. Let that sink in. Our incarceration rate is the highest in the civilized world. Higher than such hotbeds of “freedom” as Russia and Rwanda. And each of those prisoners has given a DNA sample to be checked. And then think about two more things: we disproportionately incarcerate African Americans; and the chances of a false positive increase with consanguinity or membership in an ethnic group. In other words, the system is set up to frame certain groups of people, and the laws of probability demonstrate the risk. The average jury will not hear this, as you might well imagine.

Just for contrast, DNA evidence does appear to be relatively reliable for one thing: exonerating suspects. The fact that there has been so much resistance from law enforcement to organizations who wish to re-evaluate old convictions is a bit disturbing to me as well.

This review has already run long, so let me mention just a few other things. First, I do think a weakness of the book is that it lacks focus. I think the author would have loved to have gone into depth in a few more areas, covered the math in detail, and made this a far longer book. Instead, he had to pick and choose, and the choices show. There are several of the stories he starts with that I would have loved to have examined in more detail, but space permitted only a cursory look at the ideas, rather than the actual math. Most readers might disagree, however, having glazed over after the one chapter dedicated to the math itself.

This is related to a vague dissatisfaction that I myself felt - and which has occurred to other reviewers. While this book can give some clarity about some aspects of chance, so much still boils down to a combination of hard math and the immense difficulty of teasing out the variables which affect chance. One is left with the feeling that, while things like lotteries are mathematically clear, much of what we experience in life defies easy analysis. The variables are just too many, and the math quickly becomes impossible.

The point of the math, though, is that some things may seem unlikely, but in fact are extremely likely. Others may remain more mysterious. And telling the difference is indeed important. What I might draw from this - and from the fact that variables are hard to determine - is that one needs to be careful not to ascribe to fate (or whatever supernatural being you prefer) what may well be chance. This doesn’t mean one cannot see the hand of God in things, but that one needs to be very careful. Yesterday’s coincidence may well turn out to be tomorrow’s foreseeable result.

Mazur clearly loves his topic, and has thought through the problems presented by coincidence more than most of us ever will.

The questions still remain for us to ponder: when we see something as a truly remarkable coincidence, is it truly mathematically rare? Or it is not so much the workings of an inscrutable fate as the inevitable result of known causes working with the laws of mathematics? For that matter, does it matter to us personally? And how should it shape our public policy? (Yes, the lawyer can’t leave that one out…)

It’s an interesting book. I recommend it, but urge readers to sit down and learn the math. It is the most essential part of the issue.