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A Numerate Life


John Allen Paulos

Skeptical Inquirer Volume 40.2, March/April 2016

Whether because of my natural temperament, my training as a mathematician, or a late midlife reckoning and reconsideration, I look on the whole biographical endeavor, my own included, as a dubious one. Even George Washington’s signature line about cutting down the cherry tree, “I cannot tell a lie,” is probably flapdoodle. More likely he said, “No comment” or “I don’t recall the incident” or maybe “The tree was rotten anyway.” I tend to scoff when reading that a new biography has revealed that the great So-And-So always did X because 
(s)he secretly believed Y. I’m not particularly ornery, but I often react to such statements about the alleged actions or beliefs of well-known people with a silent That’s B.S. A more likely reaction if someone makes the claim directly to me is a polite, but pointed “How do you know that?” or even “How could anyone know that?” or, in the case of autobiographies, “How could anyone remember that?”

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Memories are often inaccurate or fabricated, perspectives biased, “laws” and assumptions unfounded, contingencies unpredictable; even the very notion of a self is suspect. (But like the nutritionist who secretly enjoys candy and donuts, I’ve always enjoyed reading [auto]biographies, ranging from James Boswell’s The Life of Samuel Johnson, LL.D. to Mary Karr’s Liars’ Club.)

Given my skepticism of the biographical enterprise, it might seem I’ve taken a bold and/or foolhardy step to write a quasi-memoir of my own, but quasi- here means “not so much.”

True to my doubts, what I’ve written is a meta-memoir, even an anti-memoir. Employing ideas from mathematics (quite broadly and non-technically construed) as well as analytic philosophy and related realms, but requiring no special background in mathematics, I’ve tried to convey some of the concerns and questions most of us don’t, but arguably should, have when reading biographies and memoirs or even when just thinking about our own lives. The “arguably” is the burden of this book; imparting a certain modicum of mathematical understanding and biographical numeracy is its presumptuous goal.

One of the first questions that comes to mind when considering a life is an abstract “What is its average length?” or perhaps a more visceral “How long have I got?” Quite relevant is evolutionary biologist Stephen Jay Gould’s article “The Median Isn’t the Message,” in which he describes his cancer diagnosis and the associated median life span of eight months that it allowed. But the median, of course, is not the mean or simple average of patients’ life spans; it is the life span shorter than which half the patients survive and longer than which the other half do. Moreover, the statistical distribution of life spans is right-skewed, meaning many people live considerably longer than the median as did Gould (twenty years). Knowledge of statistics and distributions allayed his anxieties and, more generally, as I’ll try to show, mathematical knowledge can shed much-needed light on many other life situations and life stories.

Let me illustrate with a somewhat disguised statistical point. Whatever else a biography may be, it is usually considered to be a story, the story of a person’s life. And probably people’s most common response to a story is a tendency to suspend disbelief when reading, hearing, or viewing one in order not to spoil its enjoyment. “Let’s pretend. It’ll be fun.” This mindset is quite opposed to that prevailing in mathematics and science where people typically suspend belief in order not to jump to conclusions until they have compelling evidence. “Wait. Why should we believe that?” These two different approaches are not unrelated to different tolerances for false-positive and false-negative conclusions, which I’ll elaborate on later. Not surprisingly, perhaps, the latter show-me tentativeness is the approach I will adopt here. It’s in line with the bumper sticker that counsels: Don’t believe everything you think.

I’ve always liked the idea of rubbing together incongruous subjects, which seems to me almost a necessary condition for generating creative ideas. At times this habit of rubbing together has earned me a good number of eye rolls, sometimes even a bit of vituperation. People don’t always like it when notions or relations they hold dear have reflections in domains such as mathematics that they consider reductive or somehow inappropriate.

That’s too bad considering that mathematics is a most productive way of looking at the world. The philosopher Ludwig Wittgenstein once remarked that he looked forward to the day when philosophy disappeared as a subject but all other subjects were approached philosophically. I have a related but weaker wish for mathematics. I certainly don’t wish for it to disappear as a subject, but I do wish that it, too, was more widely seen to be an adverb and that its insights and ideas could inform the approach to all other subjects, including biography. With this as a partial motivation, I have over the years written about the connections between mathematics and humor, journalism, the stock market, analytic philosophy, religion, and a number of other topics (but not fish and bicycles). Nonobvious but significant points of correspondence almost always arise if one searches for them.

Here I hope to show that the points of correspondence between mathematics and biography are, despite superficial appearances, quite profound. Carl Sagan, the astronomer, skeptic, and science writer, wrote that we—our DNA, our teeth, our blood—are starstuff, made out of the very same material as the stars. As naturally occurring entities in the universe, we are, in a sense, also “mathstuff”—changing and developing according to mathematically expressible relations, instantiating mathematical notions of all sorts, and illustrating mathematical principles from diverse fields. “Mathstuff,” I maintain, is a defensible neologism since patterns are, at least to mathematicians, nonmaterial stuff. It’s thus eminently reasonable to try to obtain an understanding of this mathstuff of which, it can be maintained, we and everything else are made. In particular, how do these mathematical patterns express themselves in our life stories?

Bias and Mindsets, 
Statistics and Biography

I had a number of professors at University of Wisconsin who, I was warned, were horrible; some were said to mumble, others were given to excessive abstraction, a few constantly digressed. Many just lectured. When I enrolled in their courses despite these warnings, I was often surprised. Turns out I usually liked abstract, mumbling digressers. And I much preferred hearing a lecture from someone knowledgeable than listening to fellow students getting together in study groups and giving their usually uninformed perspectives on the topic. Contrariwise, I was often disappointed by those “fun” professors whom most deemed wonderful. The same phenomenon holds for people about whom I’ve heard only bad (or good) assessments that I find to be baseless after meeting the people in question. I sometimes still get annoyed at my own trustingness.

This, of course, is not a particularly unique realization. Everyone has experienced variants of it. One doesn’t have to be too statistically savvy to know that comments and surveys that derive from only a dozen or so people are not very reliable. Neither is it arcane mathematical knowledge that self-selected samples are not very likely to be representative of the population at large. A small self-selected sample of people responding to a television “poll” about more stringent gun control, for example, may very well arouse a disproportionate number of passionate NRA members and significantly skew the results.

Another example of wayward statistics in academe: Like many universities, mine requires that students take a core survey course in mathematics if they’re not going on in the subject. Passing the course requires that a student’s grade be at least a C-. Suspecting that the number of C-’s would be much larger than the number of D+’s because of how crucial this small difference is, I decided to examine the number of C-’s and D+’s given in this course over the years for which I could find the records.

Sure enough, I found that approximately eight hundred C-’s and one hundred D+’s were awarded. Someone might point out that the number of D+’s should be lower than the number of C-’s simply as a result of the normal bell-shaped distribution with an average of C or so. This, however, cannot be the explanation, since the drop-off was so precipitous, eight times as many C-’s as D+’s. (The four hundred or so plain D’s and roughly seven hundred F’s given out during this period indicate that general grade inflation was not the issue.) The reason was probably that, at the crucial cutoff between C- and D+, the faculty were likely to give students a bit of a break. Assigning grades is not a cut-and-dried activity, and many professors seemed to have given students the benefit of a doubt in these close calls rather than being blindly bound to rigid grading that is inevitably somewhat arbitrary.

To vary the examples a bit, consider the museum guard who claimed that a dinosaur on exhibit was 70,000,009 years old. Asked how he knew that, he said that he had been told it was 70,000,000 years old when he’d been hired nine years before. The precision would be laughable, but shouldn’t we find it almost as laughable when someone claims to be relating someone else’s verbatim (precise) conversations as well as their dates, locations, and contexts?

Why are such elementary understandings and explanations seldom invoked in our reading of personal profiles or full-scale biographies? Biographers (and, of course, autobiographers) select themselves in part because they resonate in one way or another with the subject. They may interview many people about their subject, but even their choice of interviewees is likely to be influenced by their biases. So are the questions they ask, rephrasing them over and over if they don’t get the answer they want. Pick almost any potential biographical subject and ask ten people who know him or her what they think of the person, and the responses will certainly be quite varied. Picking only eyewitnesses to events in a subject’s life certainly doesn’t guarantee truth either. Recall the trial lawyer’s quip “The only thing worse than one eyewitness is two eyewitnesses.”

Here’s a simple thought experiment: consider just yourself as a subject and think of who in your background would write the most scathing biography of you, who would write the most sympathetic account (C- rather than D+ on the scale of a whole life), and who would write the most clueless one. Or, if it doesn’t hurt too much, imagine a biography of Stephen Hawking written by Kim Kardashian and one of her written by him. Or come up with your own incongruous pairs of reciprocal biographers. The difficulty of adopting the perspective of a biographical subject’s perspective is suggested by the story of two strangers walking on opposite sides of a river. One of them yells across, “How do I get to the other side of the river?” The second one answers, “You are on the other side of the river.”

Any story of adultery, to cite one last example, will read quite differently depending on which of four natural biographers write it: the injured spouse, the wandering spouse, the outside lover, or a “neutral” observer. It’s interesting to imagine Madame Bovary from Charles Bovary’s point of view. The fake 2007 newspaper headline in the Onion, a satirical magazine, makes the same point: “Majority of Parents Abuse Children, Children Report.” Despite these obvious concerns, most people assume biographies or magazine profiles or even informal spoken descriptions of a person are more or less accurate. You might have inferred that this annoys me.

Phrasing the issue in the jargon of mathematical logic, I note that “is a biography of X” is a so-called unary predicate, and it would be preferable if it were replaced by the binary predicate “is a biography of X written by Y.” Perhaps it’s even more prudent to consider ternary predicates: “is a biography of X written by Y at time Z.” An “autobiography of X,” by contrast, really is a unary predicate unless you happen to be a schizophrenic. Comedian Steven Wright’s quip that he was writing an unauthorized autobiography also comes to mind.

Ideally nasty biographies should give a rough measure of the ratio of research undertaken to secrets uncovered. Laudatory ones as well should provide a ratio of the time taken to the positive tidbits found. Autobiographies should be scrutinized for any traces of the Lake Wobegon effect, whereby the author and everyone closely associated with him or her is above average. The problem, of course, is that if one looks hard enough, one will likely find what one is looking for. We’re all subject to confirmation bias, the tendency to look largely for confirmation of our hunches and beliefs and rarely for disconfirmation, but perhaps few more so than biographers, who are often either in thrall to their subjects or else detest them.

Whether reading a life story or just listening to a neighbor, we should be aware that very many of our beliefs and attitudes are likely to be a consequence of probabilistic misunderstanding and statistical failings, bad sampling in particular. Most people, for example, become somewhat more reclusive when depressed or otherwise behaving “abnormally,” so these behaviors will be under-sampled and thus likely will play a smaller role in their biographies than they do in their lives. Likewise, successful people (as well as their biographers) will tend to see a strong connection between their personal qualities and their success even if they self-effacingly say how lucky they’ve been; conversely, less successful people will tend to see a weak connection. Neither viewpoint is statistically robust.

One other quite significant, albeit underappreciated, point about statistics in the presentation of biographies bears repeating. As I mentioned earlier, an important aspect of a story is that there is a tendency to suspend disbelief while seeing, reading, or hearing it so as to not spoil its enjoyment. “Let’s pretend there really is a monster like this.” Loosely paraphrased in statistical parlance, this means that one risks a so-called Type I error (a false-positive); that is, saying an important incident or phenomenon occurred that really did not. This is not the situation in statistical or scientific contexts where one typically suspends belief so as to not be fooled. “How do we know that?” In statistical parlance, one risks making a so-called Type II error (a false-negative); that is, saying that an important incident or phenomenon that really did occur did not.

Both biographers (storytellers generally) and statisticians (scientists generally) wish to avoid both sorts of errors, but biographers are a bit more careful about not ruling things out, scientists a bit more careful before admitting them. What exists for the two sorts, their ontology, is different; for storytellers it’s generally more baroque, for scientists more bare-bones. To parody this difference in mindset, we might say that out of five crises, storytellers predict seven of them, and scientists predict three of them—hyperbolic versus hypobolic, which should be a word.

Type I and Type II errors are part of a complex of notions surrounding Bayes’s theorem, an extremely seminal result in probability theory that tells us how to update our probabilities in the light of new evidence. For example, if a fair coin (heads-tails, H-T) and a two-headed coin (H-H) are on a table and we choose one of them, the probability we choose the fair coin is 1/2. But if after we’ve chosen a coin, we flip it three times and obtain three consecutive heads, Bayes’s theorem tells us the probability that we chose the fair coin shrinks to 1/9.

This is, of course, much harder to do with more nebulous stories and biographies, but do biographers make much of an attempt to indicate how they come to their initial evaluations of a subject or how they’ve changed their minds about him or her in the light of new documentary evidence? I doubt it. Certainly excusable, but why make so little use of possibly relevant mathematical and scientific tools?

One proto-Bayesian, the empiricist Scottish philosopher David Hume, underlined the importance of considering the probability of supporting evidence when he questioned the authority of religious hearsay: one shouldn’t trust the supposed evidence for a miracle, he argued, unless it would be even more miraculous if the report were untrue. In ancient times, biographies of saints and kings were replete with miracles. Contemporary biographies are devoid of miracles but still contain too many exploits and adventures that seem considerably less likely than their nonoccurrence. It’s the same impulse, but attenuated.

John Allen Paulos

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John Allen Paulos is a professor of mathematics at Temple University and the author of eight previous books, including the bestselling Innumeracy and A Mathematician Reads the Newspaper. He is a fellow of the Committee for Skeptical Inquiry. For further information, visit