CONSCIOUSNESS, BIOGRAPHIES, AND SHMATA

As I've expressed frequently herein, for a variety of reasons I find biographies to be somewhat inadequate in getting at what a person is like. I've cited paradoxes, counterfactual assertions, straightforward lies, mistaken assumptions, bizarre interpretations, statistical solecisms, cognitive foibles, and just plain wrongness, but much of what I write in this book can be stated rather succinctly. People in general and biographers in particular are often, to use a common slang expression, full of it (the “sh” is generally omitted in polite company). The phrase is common because we apply it so often to others, but those with any self-awareness at all must realize that the admittedly vague phrase often applies to themselves as well. It definitely applies to me at times. In rereading the autobiographical vignettes of this book, for example, I'm discomfited by the feeling that, though true, parts of them indicate that I'm at least 14.3 percent full of it.

Three quotes attributed to authors Mark Twain, Paul Auster, and Rebecca West, respectively, compellingly express similar sentiments.

What a wee little part of a person's life are his acts and his words! His real life is led in his head, and is known to none but himself…. Biographies are but the clothes and buttons of the man. The biography of the man himself cannot be written.1

Every life is inexplicable, I kept telling myself. No matter how many facts are told, no matter how many details are given, the essential thing resists telling. To say that so and so was born here and went there, that he did this and did that, that he married this woman and had these children, that he lived, that he died, that he left behind these books or this battle or that bridge—none of that tells us very much.2

Just how difficult it is to write biography can be reckoned by anybody who sits down and considers just how many people know the real truth about his or her love affairs.³

Of course, we shouldn't conclude that lives are completely inscrutable or that clothes and buttons aren't sometimes fascinating. It's just that biographies are usually heavy on movement, anecdotes, official acts, and superficial connections among the protagonist and other people, but light on insight into the subject's real attitudes and personal worldview.

This is especially true of the dynamic and chaotic relations (these adjectives are intended in both their everyday and their mathematical senses) among families, friends, and colleagues, the open secrets that people know while remaining ignorant of others’ knowledge of same (the distinction mentioned earlier between mutual knowledge and common knowledge), the sturm und drang (storm and stress) of coming of age, and their associated endearing memories and enduring resentments. Knowing my limitations, I haven't even attempted to limn these topics here, where I've focused largely on myself and mathematics.

The quotes above clearly suggest this and seem to be pointing to the lack of Verstehen in most, if not all, biographies. Verstehen is a German word meaning “to understand,” but in sociology it's often used in a special way to refer to a sort of understanding that requires interpretation and empathetic involvement. The understanding is not completely objective but requires that you appreciate how the person in question sees the world, view the person as a subject and not merely as an object of study, even identify with his or her point of view. (The term also carries a good deal of unconvincing metaphysical baggage that is not relevant here.)

Verstehen is obviously important in biography since how biographers describe an event or person is often critical, and their ability to “see as” the other person does is essential (and at best very rare). Even simple actions, not to mention complex sequences of actions, illustrate this. Here is an example I've used before: If a man touches his hand to his forehead, we may see this simply as an indication that his temple is throbbing. We may also see the gesture as a signal from a football coach to the quarterback. Then again, we may infer that the man is trying to hide his guilt by appearing nonchalant, that it is simply a habit of his, that he is worried about getting dust in his eye, or indefinitely many other things depending on indefinitely many perspectives we might have and on the indefinitely many human contexts in which we might find ourselves. Describing the movement in only physical terms—how fast and at what angle his arm moves, the physiological correlates of the movement, etcetera—obviously does not provide Verstehen. Even young children understand this distinction since they might deny hitting someone and claim they were just swinging their hands when somebody's face walked into them. My son Daniel, now a very good lawyer, once made this claim after he'd hit my daughter Leah. She, ever savvy even as a toddler, wasn't at all taken in by this subterfuge.

People generally have enough Verstehen of their spouses (plural, spice?) to know that how he or she sighs can signal a change in mood or a change in attitude toward someone or something, especially when a verbal acknowledgment is too hard to make immediately. Sometimes my wife responds to the merest twitch of my eyebrow with something like “Why do you think that?” (Of course, she's often wrong about what I was thinking.) Likewise, when one knows a person intimately, that person's facial expressions, interactions, and the incidents they choose to relate can sometimes also hint at immensely complicated, almost incompressible, yet slightly ambiguous stories. This complex indeterminateness can even arise when observing or writing about oneself. Seeing an account of my actions or thoughts on the page sometimes makes me reconsider who this “I” is that I carry around all day and who seems to always be speaking for me. For the record I'm usually quite fond of the guy, but I wish he'd shut up occasionally, since he sometimes gets things a little wrong.

An account of a life that relied almost totally on Verstehen rather than exclusively on clothes and buttons would occupy its subject's consciousness, explore his or her relations with significant others, and would be a rather electric matrix of diversion and digression. Messy—like life (and this book). That is, details, both big (say, political or economic concerns) and small (say, annoyance at the inclusion of nutmeg in a dish or a momentarily all-consuming hangnail), on matters both critical and trivial would follow one another unpredictably through the chronicle.

The story and its multilayered details—like endlessly jagged coastlines, or creased and cracked mountain surfaces, or the whorls and eddies of turbulent water, or a host of other “fractured” phenomena—might suggest to you the mathematical metaphor of a fractal in some sort of “cognitive space.” After all, characteristic of fractals is a kind of relative simplicity that is nevertheless complex-seeming, unlimited branching and diverging, and recursive self-similarity, whereby individual instances are defined in terms of their predecessors and all have a somewhat similar look or feel no matter on what scale one views them. (The latter property explains the following joke about the founder of fractal geometry. What does the B in Benoit B. Mandelbrot stand for? Answer: It stands for Benoit B. Mandelbrot, the “father of fractals,” the sequence of imbedded names constituting a kind of fractal. A Russian doll, another example, is one that is full of ever-smaller Russian dolls.)

By exhibiting—that is, showing as well as telling—some of the inexhaustibility of such self-reflective rambling (some of it truly complex in a sense I'll get to shortly and not at all fractal-like), such a biography would demonstrate the flavor of a particular fractal human consciousness. Many of its little episodes and vignettes would resemble nothing more than teenage intrigues (“I didn't tell Mimi because she would certainly tell Helen, who might tell Dick, and I didn't want him to hear something other than what Ellen and Jerry had told him before they knew the extent of it.”) The protagonist's thoughts would be dense with life and would give the reader a substantial degree of Verstehen for the person. Although they're not biographies, some close literary cousins of this approach might be James Joyce's Ulysses or Marcel Proust's Remembrance of Things Past. In fact novels, though fictional, are often better at imparting Verstehen than are biographies.

But even if it made sense and it was possible to really grok a person, to “get inside the head of” the subject and know and think the way he or she did, we would still have two huge problems. Since Freud, countless psychological and neurological studies have demonstrated repeatedly that people, biographical subjects or not, are not aware of many of their own motivations. They (we) do things that they (we) don't understand. Clothes and buttons certainly aren't sufficient in fashioning a biography, although they are often interesting.

Our common folk psychology, which is indispensable in everyday life, and even Verstehen fall short if the subject is unaware of why he or she did something. (Once again, “did” may be unjustified; people see agents everywhere even when the movements are not at all intentional.) The tired metaphor of the mind being an iceberg most of which is under the surface is misleading in that it suggests that the conscious and unconscious parts are somehow connected, somehow continuous.4 Not so. This unconscious unawareness of many of our actions and motivations would seem to limit the value of both commonsense psychology and Verstehen in understanding a person's state of mind.

Secondly, whether of the clothes and buttons or even of the much more full-bodied and vibrant Verstehen variety, biographies fall short for another quite different reason: we're just too complex to be captured fully. The informal notion of complexity can sometimes be modeled by the formal mathematical notion I mentioned earlier, which is due to both the computer scientist Gregory Chaitin in his book The Limits of Mathematics and to the Russian mathematician Andrey Kolmogorov.

Please indulge me as I digress to establish a few details of the notion. The definition is simple and elegant. The complexity of a sequence of bits (0s or 1s) is roughly the length of the shortest program needed to generate the sequence in question. The repetitive sequence 011011011011…11011011…is, for example, not very complex since a very short program generates it: Print 011 and repeat however many times are needed. More complex sequences require longer programs to generate, and a sequence of bits is said to be random if the shortest program generating it is incompressible and essentially as long as the sequence itself, say, a sequence like 0100100001011010011110010100001011…. A program generating such a sequence can do little more than repeat it: Print 010010000101101001111001010…. It simply spits out the sequence itself since it can't compress it. If I can resort to base reductionism for a second, I note, given that everything—DNA, music, us—can be encoded in such sequences, these definitions apply to us as well.

I mentioned music for a reason. The computer scientist Donald Knuth illustrates this notion by examining common ways to reduce the complexity of songs in order to appeal to children or to ease the burden on memory. Having a refrain that occurs throughout a song is one way of doing so. Consider “Old MacDonald Had a Farm.” The familiar “and on that farm he had an (animal name), E-I-E-I-O, with an (animal noise twice) here and an (animal noise twice) there, here an (animal noise), there an (animal noise), everywhere an (animal noise twice)” followed by “Old MacDonald had a farm, E-I-E-I-O” leads to a considerable reduction in complexity.5 A way to reduce complexity even further is exemplified by songs such as “99 Bottles of Beer on the Wall,” where the song proceeds downward through 98, 97,…3, 2, 1. Finally, consider “That's the way I like it, uh-huh, uh-huh” repeated over and over again.6

Chaitin developed the notion of complexity in part to give an alternative proof of Kurt Gödel's famous incompleteness theorems. Gist of the proof: We can't expect (the complexity) of 5 pounds of axioms to yield (the complexity of) 10 pounds of theorems. In order to generate a certain degree of complexity—10 “pounds” worth, say—what is needed is more than 10 pounds worth of complexity.

This is also why explaining the complexity of the world by invoking the even greater complexity of a deity explains nothing; it replaces one imponderable with an even bigger one. It also suggests that to the extent we ourselves are formal systems, we are subject to the same existential incompleteness.

Not unrelated is the Berry paradox, first published in 1908 by Bertrand Russell. This paradoxical twenty-word sentence asks us to consider the following task: “Find the smallest whole number that requires, in order to be specified, more words than there are in this sentence.”⁷ Examples such as “number of hairs on my head,” “number of different states of a Rubik's Cube” and “speed of light in millimeters per century” each specify, using fewer than twenty words, some particular whole number. The paradoxical nature of the task becomes clear when we realize that the Berry paradox specifies a particular whole number that, by its very definition, the sentence contains too few words to specify.

End of digression. For my purposes here I note that lying between the extremes of compressibility mentioned above is where life resides, where some compression is possible—via fractals, for example, or in the case of songs via refrains—and where some parts are relatively incompressible. This is a big interval. Some people's lives are simpler than others, and their descriptions admit a higher degree of summarization and compression, something captured by philosopher Derek Parfit's description of such lives as containing “only Muzak and potatoes”;⁸ a description of other more complicated lives would of necessity be much longer. Of course, nobody (not even Tristram Shandy) could write or read a full biography that left out no significant facts and events. The bottom line is that the complexity of human lives is too great to be captured in any reasonably full way, and a serviceable approximation of Verstehen is the best we can hope for. The map can never be as complete as the territory.

These considerations and limitations are more salient for authors attempting biographies of people from very different cultural backgrounds (anthropologists also use the term Verstehen) or historical periods (explaining the mythical Achilles’ motivation in the Iliad, for example). Going much, much further afield to “biographies” of other species suggests a more theoretical matter and brings to mind the philosopher Thomas Nagel's famous article “What Is It Like to Be a Bat?”9

In his somewhat polemical article Nagel argues that consciousness has an irreducibly subjective aspect and states that “an organism has conscious mental states if and only if there is something that it is like to be that organism—something it is like for the organism.”¹⁰ Unlike the situation with physical concepts, to even have the notion of a mental concept requires, according to Nagel, that we be directly acquainted with it. Mental states are not best understood from an objective perch. For such states there is no such thing; the person is the perch. Being a person entails having a subjective perch from which to view the world, Nagel argues, and any attempt to objectively account for or explain away this perch, this personal perspective, will of necessity omit what we would like to understand.¹¹

More controversially Nagel adds that a real grasp of consciousness and mental concepts might require a profound change in our understanding of both the mental and the physical.¹² Although they appear to be essentially different, that may be only because of our present state of knowledge. To write a “real” biography, to allow others to understand what it's like, beyond clothes and buttons, to actually be an Abraham Lincoln, a Thomas Edison, or an Aung San Suu Kyi might require an un-anticipatable advance in our understanding of physics and consciousness.

Although no compelling physical evidence exists that consciousness is anything more than a purely neurological phenomenon, I have to grant that it may be. Entering foolishly into nearly (or totally) nonsensical speculation inspired by Nagel, I note that consciousness, the ghost in the physical machine of our brains, might, like the Higgs boson, be almost impossible to detect and yet be a pervasive field of energy. Like the Higgs field, the “consciousness field” might impart consciousness to the right sorts of entities—sufficiently complex, appropriately structured, suitably self-referential physical objects such as human and animal brains.

And these considerations and speculations prompted by Nagel, physicist Freeman Dyson, and others finally bring me to Shmata. One day my wife brought home a small, shaggy dog, a Bichon Frise, from a kill shelter. Someone had for some hard-to-fathom reason taken her there. The dog was white and fluffy but very dirty and matted, so my wife named her Shmata, Yiddish for rag. Gradually Shmata overcame her abusive background and grew into the affectionate, smart, and neurotic dog we now know. She's full of idiosyncrasies and relates to my wife and me in two entirely different ways. She licks me constantly but wants my wife to always be petting her. She barks loudly and jumps up and down like an insane jack-in-the-box if we get out of the car and we're a nanosecond late in letting her out. She knows dozens of words and phrases, and she is omnivorous—and deceptive about it too when she discovers something on the street she knows we'd object to her eating. She also loves matzoh, so the long, formal version of her name is Shmata the Matzoh-Eating Canine-ite. She is as deserving of a biographical sketch (even if it's only clothes and buttons) as any other sentient being. Hence this paragraph that I'm sure falls far short of answering the question: What Is It Like to Be Shmata?

LEAH AND DANIEL, MY GRANDSONS AND I

I do have a vastly better feel for what it's like to be my daughter Leah or my son Daniel than I do about Shmata, but I don't wish to write much about them. They didn't sign up for embarrassment by their writerly father and need to be the authors of their own life stories. I do want to say that they are mensches (or, ignoring etymology, a mensch and a womensch). They're sensitive, kind, smart, and level-headed and are an unalloyed joy (or maybe only exceedingly rarely alloyed).

Since this book is mathematically tinged, I will say something about their relation to the subject. When they were very young, I taught them to answer the questions “What is the derivative of x squared and, secondly, what is the derivative of e to the x?” They would respond in their toddler voices, “2x is the derivative of x squared and e to the x is the derivative of e to the x” and then would, like me, be amused by the momentary surprise their answers elicited from my mathematical colleagues. Leah and Daniel knew it was a joke and, unlike more than a few of my calculus students, were under no illusion that they understood anything. More generally, they liked the mathematical puzzles—both standard (pico, fermi, zilch) and idiosyncratic—that I would pose to them while driving from Philadelphia to their grandparents’ apartment in New York on the long, boring New Jersey Turnpike. They also both excelled at mathematics in school, my son receiving a master's degree in it before opting for law school over a PhD in the subject, and my daughter majoring in engineering in college before switching to English and related endeavors.

I'll limit myself to their math prowess and skip the accounts of family trips in this country as well as to England, France, Greece, Israel, Thailand, Japan, the jokes and code words that living together engender, playing ball against Sandy Run Middle School, the countless conversations—most silly, some serious—and simply repeat that they're great and probably will be embarrassed to read even these few sentences.

Children often have children, and so a few words about my grandson Theo (he of the earlier Cheerios mention), who is too young to complain about my writing of his skill as a reverse Roomba. Like most toddlers he is a fount of either wisdom or unwisdom. I'm not sure which. When he finally started to use the potty, he told Leah that a lot of it got on the floor but that he was “okay with that,” a phrase he had apparently heard somewhere. Some of his little friends in the play group were discussing their pets and asked if he had a pet. He replied earnestly, “No, just Charlie, my brother.” Charlie, who calls himself a very mean pirate (except when he has a cold and then calls himself a sick pirate) despite being ever agreeable, would probably have said “yah” had he heard the exchange.

One viscerally loves one's children and grandchildren for their own sakes (although the feeling necessarily becomes abstract with, say, one's great-great-great-great-great grandchildren). Grandchildren, children, their spouses (Andy and Marie, in my lucky case), other family members, friends, colleagues, and fellow citizens allow us a way to extend the short thread that is our life by weaving it into a larger patch that is in turn part of the whole social fabric. It's not just about us. That's the idea anyway, and it's compelling and appealing, but we might look at the metaphor in a more jaundiced way: individually we're just pieces of lint. Not sure if that's a piece of wisdom or unwisdom. Theo, Charlie, and I are just trying to make sense of things as best we can at our respective stages in life. As I write this their cousin Max, recently arrived on the scene, has just begun his long and complex trajectory.

The thought of eventually fading away and being survived by one's own and the larger social family suggests to me what will happen when the earth's low-lying islands become submerged because of global warming and the melting of the ice caps. This prospect is fairly imminent, as the Maldives and many others surely won't make it for many more decades. The prudent leaders from some of these islands are now buying land on other nearby islands or on the mainland to relocate their people there when their home island goes under. They are also arguing to maintain all the fishing and natural resource rights not only on the home island itself but also in a twelve-mile buffer zone around the soon-to-be-nonexistent island. The less than watertight analogy is to the transference of one's estate or legacy to one's children as we, like these islands, get washed away in the sands of time.

To counter the latter cliché, I'll end with this wise quote from the poet Robert Frost: “In three words I can sum up everything I've learned about life: it goes on.”13

GOMPERTZ'S LAW OF HUMAN MORTALITY AND LIFE SPAN

Becoming a grandparent or simply getting older usually brings about a keener sense of mortality. Few unasked questions are more human than: How much longer do I have? How many more times will I travel here, eat there, do this or that thing I've enjoyed (or simply endured) doing?

Excluding external causes of death, we can give a general answer that is somewhat surprising. Whatever the probability p that you, an adult, will die during the next year, 8 years from now the probability you'll die during the following year is 2p or twice as great. For example, if you're 40 years old and your chances of dying during the next year are about 1 in 800, when you reach 48, your chances of dying during the following year would increase to about 1 in 400, and when you reach 56, your chances of dying during the following year would rise to approximately 1 in 200.

The scope of this relation, which was discovered by Benjamin Gompertz, a nineteenth-century British actuary, extends across borders and time periods. Even if the life span in a given country during a given historical period is much lower, the probability of a person's dying during the next year still doubles every 8 years. (This phenomenon also brings to mind the Kruskal lockstep progressions mentioned earlier.)

Gompertz observed that the annual mortality rate rises exponentially, which implies that the probability of surviving to a given age shrinks rapidly. The phenomenon is not unrelated to the rule of 70 in financial matters. If your money is growing at a rate of r percent, it will double in 70/r years. It's not at all clear why this mortality relation should hold.

As physicist Brian Skinner has observed, Gompertz's law, though somewhat mysterious, suggests that our exponentially increasing mortality is the result of a “built-in expiration date.” He further observed that various theories of the causes of death do not give rise to known mortality tables.14

If, for example, we were to subscribe to a constant mortality rate of 1 in 80, the latter number being the average life span in the United States, this would mean your chances of dying next year would be 1 in 80, the chances the year after that also 1 in 80, and so on. This would result in an average life span of 80, to be sure, but with many people dying in their teens and twenties and a few living to age 300 or more.

Even if one makes this approach more realistic, for example by saying you have a 1 in 16 chance of suffering some injury or disability or deterioration and accumulating 5 of them kills you doesn't give real-life mortality tables. These assumptions also give rise to an average age of 80 (16 × 5), but with many people living past 150. Similar “fixes” also fail to yield, as Gompertz's law does, known mortality tables.

One model described by Skinner that does yield realistic life-span distributions is called the “cops and criminals” theory of the immune system. It is schematic and suggestive in something like the way Hofstadter's theory of simms and simmballs is (see chap. 3). If a sufficient number of circulating cops are patrolling your body, they can wipe out any criminals they happen upon. However, if the criminals are allowed to loiter too long, they build up fortresses that the cops can't penetrate, and disease or cancer develops. As the number of circulating cops decreases for whatever reason, their patrols become less frequent and criminal fortresses more numerous.15

This colorful model dovetails nicely with science writer George Johnson's abstract perspective on the inevitability of cancer.¹⁶ The ultimate cause he argues is simple entropy. Over time DNA deteriorates or is miscopied or otherwise degraded, events whose inevitability is a consequence of the second law of thermodynamics. Once again we behave like mathstuff.

“Cops and criminals” is, of course, quite suggestive of an immune system fighting off assault, a model made all the more plausible by the fact that cancer incidence also seems to double every eight years à la Gompertz. Unlike cancers of the young, the many cancers of the elderly have remained and will remain recalcitrant, making getting rid of “criminal mutations” ultimately impossible. The problem is that mutations, criminal or not, are, as Johnson observes, the engine of evolution and without them, no complex life would exist. Sometimes and somehow whether because of glitches, a reduced police force, or whatever, some mutations begin to grow into cancer. Things fall apart, including us.¹⁷

Finally, I should mention a heuristic device that gives a reasonable estimate for your remaining life expectancy. Based on figures from the US government, it states that you can expect to live another 72 years minus 80 percent of your present age if you are under 85. If you are over 85, you can expect to live another 22 years minus 20 percent of your present age. For example, if you're 60 years old, your remaining life expectancy is 24 years.

Once again, no matter how we slice it, we fall apart.