chapter06

KOVALEVSKY, PREDICTION, AND MY GRANDMOTHER'S PETTY LARCENY

Before I indulge a bit more in reminiscence, I'd like to digress and offer up the story of Sofia Kovalevsky for reasons that will become clear.

Sofia Kovalevsky (Kovalevskaya) was a brilliant Russian mathematician of the nineteenth century who died at forty-one. Her story, like that of mathematicians Évariste Galois, Srinivasa Ramanujan, and Frank Ramsey, all seminal mathematical thinkers who died too early, is a sad but romantic one. Youthful accomplishment and promise cut short usually is. Despite Kovalevsky's cossetted early life followed by the daunting antifemale cultural environment she faced later, she managed to make major contributions to partial differential equations. The field is the multivariable generalization of ordinary differential equations, which relates the rates of change of some quantity with respect to other quantities. This may sound fairly banal, but the subject of differential equations has clarified some of the chief glories of modern civilization, among them Newton's laws of motion, Laplace's heat and wave equations, Maxwell's electromagnetic theory, the Navier–Stokes equations for fluid dynamics, and Volterra's prey-predator systems. (The latter is related to the analysis of the love affair in Gone with the Wind mentioned earlier. The repeated and alternating rise and fall of the prey and predator species—one up, the other down—can be shown to be somewhat analogous to the repeated and alternating love and hate attitudes of Rhett Butler and Scarlett O’Hara.)

Kovalevsky's dissertation, “The Theory of Partial Differential Equations,” helped expand inquiry into differential equations to include nonlinear dynamic phenomena, many independent variables, and what would come to be called chaos theory.1 She was a colleague of such other groundbreaking mathematicians as Karl Weierstrass and Henri Poincaré, who in addition to his work on chaos, topology (Poincaré conjecture), and qualitative differential equations, established before Einstein the mathematical framework for special relativity theory. This late nineteenth-century focus on dynamic phenomena and many complex interacting variables is perhaps a mathematical reflection of the many cultural, economic, and political changes then reshaping the world. Cultural and societal issues do sometimes affect the types of problems mathematicians work on. Note the relatively sudden rise of interest in the mathematics of networks, which is not unrelated to Twitter, Facebook, and other common networks.

The reason I write about Kovalevsky, however, relates to a brief biography of her by Nobel Prize–winning author Alice Munro in her collection of stories titled Too Much Happiness. In her masterful story Munro tries to integrate the mathematical and the personal aspects of Kovalevsky's life. The story itself struck a chord with me because, very roughly speaking, Munro's story attempts to do for Kovalevsky what in a small way I've attempted to do here for myself.²

Kovalevsky's mathematical, personal, and social concerns were interconnected. One example is her work on quantities (functions) dependent on several variables and their dynamic behavior as well as her unhappy love affair and its continuation, which was shaky and contingent on many factors. She herself wrote a memoir, Recollections of Childhood, and an intriguing autobiographical novel, Nihilist Girl, whose titles alone resonated with me. Therein she commented: “In what manner should we act in the future to make our common life happier? Mathematically we could have stated this question in this manner: given a definite function (in this case our happiness), which depends upon many variables (namely our monetary resources, the possibility of living in a pleasant place and society and so forth)—in what manner can the variables be defined so that the given happiness function will reach a maximum? Needless to say we are unable to solve the problem mathematically.”3

I've made no seminal contributions to mathematics, nor have I won a Nobel Prize in literature, nor have I led an inordinately dramatic life. Nevertheless as noted I have also tried to yoke the disparate realms of the personal and the mathematical in an unusual way that I hope will prove interesting and perhaps useful to readers.

To that end, let me leave Kovalevsky, whose childhood bedroom, incidentally, was decorated with mathematical symbols, and resume the story of my own, very different nonfiction nihilist boy. Important components of my or anyone's early life are models, not in the mathematical sense but in the conventional one of parents, grandparents, and their contemporaries. Given rapid social and technological change, these are people from whom you're increasingly likely to differ, and who thus provide a measure of how much you've changed. This musing is prompted by the general considerations above (adult behavior incongruous with childhood experiences) and, in particular, by memories of my relationship with my diminutive Greek grandmother, whose bedroom, incidentally, was decorated with relatives who all appeared to me to be aliens and several hundred years old. She was kind and gentle and I loved her dearly, but just as she would have been surprised, were she alive, at how I “turned out,” I find it slightly shocking to recall some of her traits. One of these was a tendency toward petty larceny.

Because of this trait, going somewhere with my grandmother was often an adventure. She and my grandfather emigrated from Greece soon after their marriage, and after raising their family and owning various restaurants in Chicago (where I consumed gargantuan portions of rice pudding), they retired to Denver, Colorado. For years I would visit them—or just her after he died—during summer vacations. They bought four small apartments in a one-story building on Denver's east side, and my grandmother would go there with obsessive frequency to water the lawn, sometimes mumbling something about the grassy bliss around the apartments reminding her of Greece. The claim seemed far-fetched to me even then.

She would take the bus from her house on Jasmine Street to the apartments on Xenia Street and insist to the driver that I was under twelve and therefore was permitted to ride for free even though I was about fifteen and near my adult height of 5’11”. The force of her personality and threatening glare were enough to intimidate both the bus driver and me. He and I shrugged and sometimes exchanged complicitous smiles, and I tried to hurry down the aisle to get away from the frowns of the people near the front who had heard the embarrassing interchange.

When we got to the apartments she would drag out the hoses and sprinklers and leave me to read the Rocky Mountain News or Denver Post, George Gamow's One, Two, Three…Infinity, or, if it was very hot, to amble down to the motel at the corner of Colfax and Xenia to get grape sodas from the soda dispenser there. The labyrinthine metal path that the soda had to pass through to be retrieved from the machine always made the anticipation of its coldness a little keener. Picking up a soda that's fallen into a trap door on modern dispensing machines is much less enchanting.

The lawn watering accomplished, we'd walk ten blocks to a dilapidated house overgrown with weeds and nearly covered with a large weeping willow tree where my grandmother had found a mysterious old woman even shorter than her 4’10”. The woman sold raw honey, meli in Greek. My grandmother seemed oblivious to the stench of the house and claimed that this was real honey, not the sort they sold at King Super, the local supermarket. She illustrated her distaste for the contemporary version with a characteristic squinching of her nose and a quick upward movement of her head, which was also her normal way of saying no.

Then, if I was lucky, we took the bus back to her house on Jasmine. (Why is the specificity of place names so evocative and redolent even without the prompting of a fragrant name like Jasmine?) If I was not so fortunate, the embarrassment of the bus ride, to which I was accustomed if not inured, was supplemented by a visit to the King Super. It may not have carried the kind of honey she liked, but it did have a lot of other products she wanted and that she thought were exorbitantly expensive. The result was not hard for me to predict, and I usually insisted on waiting outside the supermarket, where she would meet me with her loot. Once she took a large ham out from under her shawl-covered armpit and displayed it with pride.

Even a trip to the local A&W drive-in root beer stand was tinged with my grandmother's lawlessness. There would be six of us in the car, six orders of root beer, but only five mugs when the poor attendant came back to pick up the tray. This only happened a couple of times because she was shamed and badly outnumbered by the five law-abiding citizens in the car.

Looking back on these incidents, I can see elements of the present me in the past me, but lacking a comparable foresight, I doubt if, at the time, I would have been able to fathom much of my present self. What can I trace forward? I tend to be just as absentminded as I was when I would walk home from kindergarten during recess. And my basic honesty I may owe to my father's easygoing decency, as well as, oddly, to my grandmother's petty thefts.

Of course, projecting traits into the future is problematic and seldom falsifiable. Plausible alternative explanations will always abound because the underlying “psychological science” drastically under-determines the traits in question, as philosophers from Pierre Duhem to Willard Van Orman Quine have observed.4 The same is true of “culinary science.” The success or failure of a complicated recipe can be attributed to almost any ingredient(s) in it as well as to variations in the amounts, in the times, in the methods of cooking, in the order of ingredient addition, and so on.

Whatever effect, psychological or culinary, my grandmother may or may not have had on me, whatever her lapses in honesty, I loved her. Still, I didn't always approve of her, and, were she in the habit of making pronouncements, I doubt that I would agree with many of them. But she generally confined herself to cooking for me and telling me what a wonderful kid I was, actions and assertions with which it was difficult to disagree even for an adolescent yearning to be independent.

And I still like honey, soft drinks, and reading newspapers (and their various online descendants) and dislike anything to do with lawns and, in fact, lawns themselves.

TURNING POINTS, ACADIA TO KENYA

Biographies generally focus on “turning points” in a person's life. V was a W-ish fellow until X happened, after which he became Y-ish and obsessed with Z. Lives certainly have turning points like X, but is there ever a turning point in one's life that is largely internally generated, a decision that is not a direct response to outside events but largely due to internal ones alone? The adverb largely is a bit of a weasel word since it can mean just about anything, and all turning points are at least minimally affected by both internal deliberations and the external environment. Nevertheless, are there many such points where the outside effect is tiny, all but invisible? Do people, after long consideration, whether conscious or not, ever decide to make a sharp break to go that way rather than this? This is an impossibly vague question bordering the tricky philosophical terrain surrounding free will and determinism, but my suspicion is that an appropriately qualified “sometimes, but seldom” is the rough informal answer.5

As I've suggested, our belief that we are captains of our ships and largely autonomous beings is quaint, but likely only true if the ship is a raft bobbing about in a raging, stormy sea. This isn't to say that our psychological and physical makeups are irrelevant. They're not. Decisions are filtered through us, but the environment, being vastly more complex than we are, is, in a sense, the primary decider.

But let me sidestep the theoretical issue of free will and its definition and simply note that if we decide something, it has to be, informally speaking, a real choice that might have gone the other way. This, I suppose, is the interpretation of the well-known joke about Jean-Paul Sartre working on his book Being and Nothingness in his favorite Paris café. He tells the waitress that he'd like more coffee, but with no cream. The waitress replies, “Monsieur Sartre, we are out of cream. How about more coffee, but with no milk? We have milk.”6

This isn't necessarily a dumb waitress joke. It may be a brilliant waitress joke. One of the central ideas in Sartre's philosophy is the distinction between real choice and the mere appearance of such. Arguably Sartre can't genuinely choose to have more coffee with no cream, because cream isn't available, but he can genuinely choose to have more coffee with no milk, because milk is available.

That all said, I've had many of the conventional turning points in my life—points on a life's arc where there are sharp twists. Marriage and fatherhood in particular have been life-giving, life-altering, and life-enriching. (Some sentiments can be both saccharine and true. These are certainly not meant to devalue other choices or lifestyles.) I won't elaborate, but the words dad and husband tend to change one irrevocably. So in a far lesser way did stopping for a two-day visit to Bar Harbor and Acadia National Park on our way to Quebec. The fortuitous detour began my family's long, annually renewed love affair with Maine's beautiful Mount Desert Island and the hiking, biking, and climbing it affords us. Even these, however, were quite context-dependent and did not grow ex nihilo but developed naturally out of certain situations (age, location, culture, etcetera).

My experience in the Peace Corps was even more a consequence of external events, namely, my belief in the imminence of being drafted to fight in Vietnam in 1969. Teaching in Kenya seemed a much superior option. I was in graduate school in mathematics at the time and, despite the excitement and turmoil of the ’60s, had lived a rather conventional life. Teaching and traveling around Kenya, Tanzania, and Uganda made many topical issues much more vivid and visceral than they'd been before. These included political and economic matters (an intense exposure to real spirit-sapping poverty), personal ones (risky sex, in several senses, in particular with the wife of a local chieftain with whom I regularly played cards), social mores (horror at the memory of the headmaster of my school using the lower half of a hollowed-out elephant leg as an umbrella stand), and scientific issues (wildlife, conservation, the Great Rift Valley). I even met President Jomo Kenyatta, often considered the founding father of Kenya, who, glancing at my ’60s-era shoulder-length hair, asked me if they had barbers in the United States.

More exciting than Russell's answer was the fact that this illustrious ninety-two-year-old philosopher who at the time resided in Wales would take the time to respond to a fan letter from a college kid in Wisconsin. My letter to him perhaps reached him at a low moment. Whatever the reason I was thrilled. A few years later in graduate school I wandered through the university bookstore and noticed the third volume of Russell's autobiography had just arrived and that a copy of it was splayed open to pages 252–253 on a display stand. I picked it up, and my name caught my eye. Russell had included his letter to me amid letters to a pantheon of twentieth-century personages ranging from Nehru and Khrushchev to T. S. Eliot, D. H. Lawrence, and Ludwig Wittgenstein, inducing in me a strange sense of proximity to these historical figures.

Publication of my book Innumeracy was also a turning point, which I'll get to later, but all this turning is dizzying, so let me remark on a related general phenomenon: sudden and wholesale changes in personality, goals, and outlook brought about by different material circumstances, religious conversion, sexual infatuation, and whatever. In the example of religion such changes often seem to be a sharp movement toward a more fundamentalist version of the person's formerly nominal religion. I know indirectly of people like this and find their conversions somewhat unfathomable and, as recent news stories demonstrate, frightening. In Thailand, where I've lectured several times, I've also seen firsthand a secular phenomenon that is more understandable, but oddly similar: sexual infatuation and the subsequent personality changes it brings about. It's unsettling to witness (or hear about from my friend Christopher G. Moore, a Canadian novelist living in Bangkok) middle-aged (or sometimes much older) pillar-of-the-Peoria-community types become so quickly almost unrecognizable. Quiet, earnest family men, they suddenly act giddy as they fulsomely pile on the adjectives to gush over their thin, little, pretty Thai girlfriends with their flashing white smiles and long, lustrous black hair—young women who, the usually temporary converts aver, are adorable, sweet, spirited, and playful. And, it should be added, generally quite poor. These sudden changes are, it seems to me, a usually futile attempt to choose a different, more vibrant self, more consistent perhaps with their youthful fantasies.

I mentioned above points in a life at which there is sharp twist. Such religious and sexual transformations are more than twists or bends but rather are points where, in math talk, the life arc is no longer differentiable but is a spiky fold. Change and turning points are what biographers and people in general are most interested in. There is not much to say if life proceeds smoothly and uneventfully along.

I'll end with a common set of usually faux turning points: milestone multiple-of-ten birthdays, thirty, forty, fifty, and so on. To underline their artificiality and lessen the dread that often accompanies them, I sometimes point out to people that their age can be expressed less traumatically in a numeral system with a different base. Happy 40th Birthday, for example, becomes Happy 34th Birthday in a base-12 system—3 twelves and 4 ones. For greater reductions in age-angst, we can express Happy 50th Birthday as Happy 32nd Birthday in a base-16 system (hexadecimal)—3 sixteens and 2 ones.

The same holds with seemingly significant dates such as the turn of the century. The year 2000 in a base-2 system is 11111010000, and 2015 turns out in base 2 to be a palindromic year equaling 11111011111, while 2048 turns out to be the epoch-changing 100000000000.

PAST ACCOMPLISHMENTS VERSUS PRESENT POTENTIAL

How much weight do you give to past accomplishments and how much to present abilities when assessing a person's life? Of course, the weights differ in different contexts, but near the extremes are the illustrious person with a prolific and stellar background but who suffers from dementia and the extraordinarily bright college student with magnificent potential but who still toils toward an initial substantial achievement. How does a biographer or an interested onlooker balance these two extremes—the contemporary and the historical—when writing about someone living?

The question occurs to me because I've met a number of very eminent people, some of whom were, at the time I met them, at least, superficially unimpressive. A bit disconcerting, these meetings should nevertheless have been encouraging rather than disillusioning. “Wow. Isn't it wonderful that someone so unprepossessing should have achieved what he or she has.” (One notable exception was Isaac Asimov, who on the spur of the moment composed a funny and suggestive limerick about my wife whom he had just met. He had also recently read my book Mathematics and Humor and riffed extemporaneously and wittily on a couple of its jokes and commented insightfully on my use of the mathematical theory of catastrophes to model certain joke types.)

What counts as a significant accomplishment will vary over the lifetime of a person as well. For example, I invented a variant of the Rubik's Cube that I called About Face, which was suggested by a book I had once seen that consisted of drawings of faces that became other faces when inverted. Moreover, twisting the cube replaced the parts of one face with parts of another face. I thought it was an appealing idea and patented it. Alas, no toy company was interested since the Rubik's Cube fascination was fading. At the time I thought my About Face was no mien accomplishment. Now, not so much.

For a more general example, consider someone who has mastered most of the algorithms in college-level mathematics. He will, if he lives long enough, see his mastery grow increasingly unimpressive as simple programs and apps will be able to accomplish the same calculations more quickly and reliably. At least in this way technology does seem to lead to the devaluing of old people. In the other direction, however, the last practitioners of a dying, formerly quite common art, say, being a sushi chef, will see their accomplishments valued much more highly than when they were young.

Whatever one's reaction to these issues, the distinction between past accomplishments and influence and present abilities and potential is a natural one. It brings to mind the notion of “physical entropy,” which was originally employed by the physicist Wojciech Zurek and others to clarify the problem of Maxwell's demon and related issues in classical thermodynamics. (Science writer George Johnson's Fire in the Mind lays out the notion of physical entropy more accessibly.)

Zurek defined physical entropy to be the sum of the complexity, appropriately quantified, of (what's already been revealed about) an entity and the surprise, appropriately quantified, inherent in (the yet-to-be-revealed aspects of) an entity. Imagine a long but finite sequence of 0s and 1s. As more and more of it is revealed, the complexity of the revealed part grows, while the surprise at the yet-to-be-revealed part shrinks. This technical notion can also be used metaphorically to model the way the weights accorded to past accomplishments and future potential change over time. The entity relevant here is a person's life; complexity of his or her past grows with time, while surprise at his or her future shrinks. (The two notions of complexity above are due to Greg Chaitin and Claude Shannon, respectively.) As a life progresses, the algorithmic complexity of the recorded past (Chaitin's notion, which I will discuss later) grows, while the surprise and potential of the future (Shannon's notion, which is a measure of uncertainty) shrinks.

One aspect of this metaphor that I like is that the notions involved are in the same conceptual ballpark as the second law of thermodynamics, which British chemist and novelist C. P. Snow famously used to illustrate the gulf between scientific elites and literary ones, the latter presumed not to understand the significance of the second law. For this reason I find its historical resonance and its marginal relevance to biography especially satisfying. It's marginal perhaps, but still worth more than yet another account, for example, of Sylvia Plath's life, poetry, and suicide. Incidentally, Plath's interest in astrology suggests she probably didn't understand any thermodynamics.

Finally, let me note that the biographical use of such precise physical analogues is problematic, but even more so is evaluating a potential biographical subject's accomplishments. This is not a mechanical exercise. As with choosing the friendliest colleges or the most lovable cities, we can by an appropriate choice of criteria, measurement protocols, and weightings make someone more or less worthy of a biography. Nothing is wrong with this as long as we are aware of the inevitable element of subjectivity present in every such choice.

A different sense of thermodynamics is the topic of the next chapter.