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A Mathematician’s Apology (Modern Library Nonfiction #87)

(This is the fourteenth entry in The Modern Library Nonfiction Challenge, an ambitious project to read and write about the Modern Library Nonfiction books from #100 to #1. There is also The Modern Library Reading Challenge, a fiction-based counterpart to this list. Previous entry: Six Easy Pieces.)

mlnf87Clocking in at a mere ninety pages in very large type, G.H. Hardy’s A Mathematician’s Apology is that rare canapé plucked from a small salver between all the other three-course meals and marathon banquets in the Modern Library series. It is a book so modest that you could probably read it in its entirety while waiting for the latest Windows 10 update to install. And what a bleak and despondent volume it turned out to be! I read the book twice and, each time I finished the book, I wanted to seek out some chalk-scrawling magician and offer a hug.

G.H. Hardy was a robust mathematician just over the age of sixty who had made some serious contributions to number theory and population genetics. He was a cricket-loving man who had brought the Indian autodidact Srinivasa Ramanujan to academic prominence by personally vouching for and mentoring him. You would think that a highly accomplished dude who went about the world with such bountiful and generous energies would be able to ride out his eccentric enthusiasm into his autumn years. But in 1939, Hardy survived a heart attack and felt that he was as useless as an ashtray on a motorcycle, possessing nothing much in the way of nimble acumen or originality. So he decided to memorialize his depressing thoughts about “useful” contributions to knowledge in A Mathematician’s Apology (in one of the book’s most stupendous understatements, Hardy observed that “my apology is bound to be to some extent egotistical”), and asked whether mathematics, the field that he had entered into because he “wanted to beat other boys, and this seemed to be the way in which I could do so most decisively,” was worthwhile.

You can probably guess how it all turned out:

It is indeed rather astonishing how little practical value scientific knowledge has for ordinary man, how dull and commonplace such of it as has value is, and how its value seems almost to vary inversely to reputed utility….We live either by rule of thumb or other people’s professional knowledge.

If only Hardy could have lived about sixty more years to discover the 21st century thinker’s parasitic relationship to Google and Wikipedia! The question is whether Hardy is right to be this cynical. While snidely observing “It is quite true that most people can do nothing well,” he isn’t a total sourpuss. He writes, “A man’s first duty, a young man’s at any rate, is to be ambitious,” and points out that ambition has been “the driving force behind nearly all the best work of the world.” What he fails to see, however, is that youthful ambition, whether in a writer or a scientist, often morphs into a set of routines that become second-nature. At a certain point, a person becomes comfortable enough with himself to simply go on with his work, quietly evolving, where the ambition becomes more covert and subconscious and mysterious.

Hardy never quite confronts what it is about youth that frightens him, but he is driven by a need to justify his work and his existence, pointing to two reasons for why people do what they do: (1) they work at something because they know they can do it well and (2) they work at something because a particular vocation or specialty came their way. But this seems too pat and Gladwellian to be a persuasive dichotomy. It doesn’t really account for the journey we all must face over why one does something, which generally includes the vital people you meet at certain places in your life who point you down certain directions. Either they recognize some talent in you and give you a leg up or they are smart and generous enough to recognize that one essential part of human duty is to help others find their way, to seek out your people — ideally a group of eclectic and vastly differing perspectives — and to work with each other to do the best damn work and live the best damn lives you can. Because what’s the point of geeking out about Fermat’s “two squares” theorem, which really is, as Hardy observes, a nifty mathematical axiom of pure beauty, if you can’t share it with others?

But let’s return to Hardy’s fixation on youth. Hardy makes the claim that “mathematics, more than any other art or science, is a young man’s game,” yet this staggering statement is easily debunked by such late bloomers as prime number ninja Zhang Yitang and Andrew Wiles solving Fermat’s Last Theorem at the age of 41. Even in Hardy’s own time, Henri Poincaré was making innovations to topology and Lorentz transformations well into middle age. (And Hardy explicitly references Poincaré in § 26 of his Apology. So it’s not like he didn’t know!) Perhaps some of the more recent late life contributions have much to do with forty now being the new thirty (or even the new twenty among a certain Jaguar-buying midlife crisis type) and many men in Hardy’s time believing themselves to be superannuated in body and soul around the age of thirty-five, but it does point to the likelihood that Hardy’s sentiments were less the result of serious thinking and more the result of crippling depression.

Where Richard Feynman saw chess as a happy metaphor for the universe, “a great game played by the gods” in which we humans are mere observers who “do not know what the rules of the game are,” merely allowed to watch the playing (and yet find marvel in this all the same), Hardy believed that any chess problem was “simply an exercise in pure mathematics…and everyone who calls a problem ‘beautiful’ is applauding mathematical beauty, even if is a beauty of a comparatively lowly kind.” Hardy was so sour that he compared a chess problem to a newspaper puzzle, claiming that it merely offered an “intellectual kick” for the clueless educated rabble. As someone who enjoys solving the Sunday New York Times crossword in full and a good chess game (it’s the street players I have learned the most from; for they often have the boldest and most original moves), I can’t really argue against Hardy’s claim that such pastimes are “trivial” or “unimportant” in the grand scheme of things. But Hardy seems unable to remember the possibly apocryphal tale of Archimedes discovering gradual displacement while in the bathtub or the more reliable story of Otto Loewi’s dream leading the great Nobel-winning physiologist to discover that nervous impulses arose from electrical transmissions. Great minds often need to be restfully thinking or active on other fronts in order to come up with significant innovations. And while Hardy may claim that “no chess problem has ever affected the development of scientific thought,” I feel compelled to note Pythagoras played the lyre (and even inspired a form of tuning), Newton had his meandering apple moment, and Einstein enjoyed hiking and sailing. These were undoubtedly “trivial” practices by Hardy’s austere standards, but would these great men have given us their contributions if they hadn’t had such downtime?

It’s a bit gobsmacking that Hardy never mentions how Loewi was fired up by his dreams. He seems only to see value in Morpheus’s prophecies if they are dark and melancholic:

I can remember Bertrand Russell telling me of a horrible dream. He was in the top floor of the University Library, about A.D. 2100. A library assistant was going round the shelves carrying an enormous bucket, taking down book after book, glancing at them, restoring them to the shelves or dumping them into the bucket. At last he came to three large volumes which Russell could recognize as the last surviving copy of Principia mathematica. He took down one of the volumes, turned over a few pages, seemed puzzled for a moment by the curious symbolism, closed the volume, balanced it in his hand and hesitated….

One of an author’s worst nightmares is to have his work rendered instantly obsolescent not long after his death, even though there is a very strong likelihood that, in about 150 years, few people will care about the majority of books published today. (Hell, few people care about anything I have to write today, much less this insane Modern Library project. There is a high probability that I will be dead in five decades and that nobody will read the many millions of words or listen to the countless hours of radio I have put out into the universe. It may seem pessimistic to consider this salient truth, but, if anything, it motivates me to make as much as I can in the time I have, which I suppose is an egotistical and foolishly optimistic approach. But what else can one do? Deposit one’s head in the sand, smoke endless bowls of pot, wolf down giant bags of Cheetos, and binge-watch insipid television that will also not be remembered?) You can either accept this reality and reach the few people you can and find happiness and gratitude in doing so. Or you can deny the clear fact that your ego is getting in the way of your achievements, embracing supererogatory anxieties and forcing you to spend too much time feeling needlessly morose.

I suppose that in articulating this common neurosis, Hardy is performing a service. He seems to relish “mathematical fame,” which he calls “one of the soundest and steadiest of investments.” Yet fame is a piss-poor reason to go about making art or formulating theorems. Most of the contributions to human advancement are rendered invisible. These are often small yet subtly influential rivulets that unknowingly pass into the great river that future generations will wade in. We fight for virtues and rigor and intelligence and truth and justice and fairness and equality because this will be the legacy that our children and grandchildren will latch onto. And we often make unknowing waves. Would we, for example, be enjoying Hamilton today if Lin-Manuel Miranda’s school bus driver had not drilled him with Geto Boys lyrics? And if we capitulate those standards, if we gainsay the “trivial” inspirations that cause others to offer their greatness, then we say to the next generation, who are probably not going to be listening to us, that fat, drunk, and stupid is the absolute way to go through life, son.

A chair may be a collection of whirling electrons, or an idea in the mind of God: each of these accounts of it may have its merits, but neither conforms at all closely to the suggestions of common sense.

This is Hardy suggesting some church and state-like separation between pure and applied mathematics. He sees physics as fitting into some idealistic philosophy while identifying pure mathematics as “a rock on which all idealism flounders.” But might not one fully inhabit common sense if the chair exists in some continuum beyond this either-or proposition? Is not the chair’s perceptive totality worth pursuing?

It is at this point in the book where Hardy’s argument really heads south and he makes an astonishingly wrongheaded claim, one that he could not have entirely foreseen, noting that “Real mathematics has no effects on war.” This was only a few years before Los Alamos was to prove him wrong. And that’s not all:

It can be maintained that modern warfare is less horrible than the warfare of pre-scientific times; that bombs are probably more merciful than bayonets; that lachrymatory gas and mustard gas are perhaps the most humane weapons yet devised by military science; and that the orthodox view rests solely on loose-thinking sentimentalism.

Oh Hardy! Hiroshima, Nagasaki, Agent Orange, Nick Ut’s famous napalm girl photo from Vietnam, Saddam Hussein’s chemical gas massacre in Halabja, the use of Sarin-spreading rockets in Syria. Not merciful. Not humane. And nothing to be sentimental about!

Nevertheless, I was grateful to argue with this book on my second read, which occurred a little more than two weeks after the shocking 2016 presidential election. I had thought myself largely divested of hope and optimism, with the barrage of headlines and frightening appointments (and even Trump’s most recent Taiwan call) doing nothing to summon my natural spirits. But Hardy did force me to engage with his points. And his book, while possessing many flawed arguments, is nevertheless a fascinating insight into a man who gave up: a worthwhile and emotionally true Rorschach test you may wish to try if you need to remind yourself why you’re still doing what you’re doing.

Next Up: Tobias Wolff’s This Boy’s Life!

quincunxcrop

The Taming of Chance (Modern Library Nonfiction #98)

(This is the third entry in The Modern Library Nonfiction Challenge, an ambitious project to read and write about the Modern Library Nonfiction books from #100 to #1. There is also The Modern Library Reading Challenge, a fiction-based counterpart to this list. Previous entry: Operating Instructions.)

mlnf98In the bustling beginnings of the twentieth century, the ferociously independent mind who forever altered the way in which we look at the universe was living in poverty.* His name was Charles Sanders Peirce and he’d anticipated Heisenberg’s uncertainty principle by a few decades. In 1892, Peirce examined what he called the doctrine of necessity, which held that every single fact of the universe was determined by law. Because before Peirce came along, there were several social scientists who were determined to find laws in everything — whether it be an explanation for why you parted your hair at a certain angle with a comb, felt disgust towards specific members of the boy band One Direction, or ran into an old friend at a restaurant one hundred miles away from where you both live. Peirce declared that absolute chance — that is, spontaneity or anything we cannot predict before an event, such as the many fish that pelted upon the heads of puzzled citizens in Shasta County, California on a January night in 1903 — is a fundamental part of the universe. He concluded that even the careful rules discovered by scientists only come about because, to paraphrase Autolycus from A Winter’s Tale, although humans are not always naturally honest, chance sometimes makes them so.

The story of how Peirce’s brave stance was summoned from the roiling industry of men with abaci and rulers is adeptly set forth in Ian Hacking’s The Taming of Chance, a pleasantly head-tingling volume that I was compelled to read twice to ken the fine particulars. It’s difficult to articulate how revolutionary this idea was at the time, especially since we now live in an epoch in which much of existence feels preordained by statistics. We have witnessed Nate Silver’s demographic models anticipate election results and, as chronicled in Moneyball, player performance analysis has shifted the way in which professional baseball teams select their roster and steer their lineup into the playoffs, adding a strange computational taint that feels as squirmy as performance enhancing drugs.

But there was a time in human history in which chance was considered a superstition of the vulgar, even as Leibniz, seeing that a number of very smart people were beginning to chatter quite a bit about probability, argued that the true measure of a Prussian state resided in how you tallied the population. Leibniz figured that if Prussia had a central statistic office, it would not only be possible to gauge the nation’s power but perhaps lead to certain laws and theories about the way these resources worked.

This was obviously an idea that appealed to chin-stroking men in power. One does not rule an empire without keeping the possibility of expansion whirling in the mind. It didn’t take long for statistics offices to open and enthusiasts to start counting heads in faraway places. (Indeed, much like the early days of computers, the opening innovations originated from amateurs and enthusiasts.) These early statisticians logged births, deaths, social status, the number of able-bodied men who might be able to take up weapons in a violent conflict, and many other categories suggested by Leibniz (and others that weren’t). And they didn’t just count in Prussia. In 1799, Sir John Sinclair published a 21 volume Statistical Account of Scotland that undoubtedly broke the backs of many of the poor working stiffs who were forced to carry these heavy tomes to the guys determined to count it all. Some of the counters became quite obsessive in their efforts. Hacking reports that Sinclair, in particular, became so sinister in his efforts to get each minister of the Church of Scotland to provide a detailed congregation schedule that he began making threats shrouded in a jocose tone. Perhaps the early counters needed wild-eyed dogged advocates like Sinclair to establish an extremely thorough baseline.

The practice of heavy-duty counting resulted, as Hacking puts it, in a bona-fide “avalanche of numbers.” Yet the intersection of politics and statistics created considerable fracas. Hacking describes the bickering and backbiting that went down in Prussia. What was a statistical office? Should we let the obsessive amateurs run it? Despite all the raging egos, bountiful volumes of data were published. And because there was a great deal of paper being shuffled around, cities were compelled by an altogether different doctrine of necessity to establish central statistical hubs. During the 1860s, statistical administrations were set up in Berlin, New York, Stockholm, Vienna, Rome, Leipzig, Frankfurt-am-Main, and many others. But from these central offices emerged a East/West statistics turf war, with France and England playing the role of Biggie on the West and Prussia as Tupac on the East. The West believed that a combination of individual competition and natural welfare best served society, while the East created the welfare state to solve these problems. And these attitudes, which Hacking is good enough to confess as caricaturish even as he illustrates a large and quite important point, affected the way in which statistics were perceived. If you believe in a welfare state, you’re probably not going to see laws forged from the printed numbers. Because numbers are all about individual action. And if you believe in the Hobbesian notion of free will, you’re going to look for statistical laws in the criminal numbers, because laws are formed by individuals. This created new notions of statistical fatalism. It’s worth observing that science at the time was also expected to account for morality.

Unusual experiments ensued. What, for example, could the chest circumference of a Scotsman tell us about the stability of the universe? (Yes, the measurement of Scottish chests was seriously considered by a Belgian guy named Adolphe Quetelet, who was trying to work out theories about the average man. When we get to Stephen Jay Gould’s The Mismeasure of Man several years from now, #21 in the Modern Library Nonfiction canon, I shall explore more pernicious measurement ideas promulgated as “science.” Stay tuned!) More nefariously, if you could chart the frequency of how often the working classes called in sick, perhaps you could establish laws to determine who was shirking duty, track the unruly elements, and punish the agitators interfering with the natural law. (As we saw with William Lamb Melbourne’s story, the British government was quite keen to crack down on trade unions during the 1830s. So just imagine what a rabid ideologue armed with a set of corrupted and unproven “laws” could do. In fact, we don’t even have to jump that far back in time. Aside from the obvious Hollerith punch card example, one need only observe the flawed radicalization model presently used by the FBI and the DHS to crack down on Muslim “extremists.” Arun Kundnani’s recent book, The Muslims Are Coming, examines this issue further. And a future Bat Segundo episode featuring Kundnani will discuss this dangerous approach at length.)

Throughout all these efforts to establish laws from numbers (Newton’s law of gravity had inspired a league of scientists to seek a value for this new G constant, a process that took more than a century), Charles Babbage, Johann Christian Poggendorf, and many others began publishing tables of constants. It is one thing to publish atomic weights. It is quite another to measure the height, weight, pulse, and breath of humans by gender and ethnicity (along with animals). The latter constant sets are clearly not as objective as Babbage would like to believe. And yet the universe does adhere to certain undeniable principles, especially when you have a large data set.

It took juries for mathematicians to understand how to reconcile large numbers with probability theory. In 1808, Pierre-Simon Laplace became extremely concerned with the French jury system. At the time, twelve-member juries convicted an accused citizen by a simple majority. He calculated that a seven-to-five majority had a chance of error of one in three. The French code had adopted the unusual method of creating a higher court of five judges to step in if there was a disagreement with a majority verdict in the lower court. In other words, if the majority of the judges in the higher court agreed with the minority of jurors in the lower court that an accused person should be acquitted, then the accused person would be acquitted. Well, this complicated system bothered Laplace. Accused men often faced execution in the French courts. So if there was a substantial chance of error, then the system needed to be reformed. Laplace began to consider juries composed of different sizes and verdicts ranging from total majority (12:0) to partial majority (9:3, 8:4), and he computed the following odds (which I have reproduced from a very helpful table in Hacking’s book):

hacking-juryerror

The problems here become self-evident. You can’t have 1,001 people on a jury arguing over the fate of one man. On the other hand, you can’t have a 2/7 chance of error with a jury of twelve. (One of Laplace’s ideas was a 144 member jury delivering a 90:54 verdict. This involved a 1/773 chance of error. But that’s nowhere nearly as extreme as a Russian mathematician named M.V. Ostrogradsky, who wasted much ink arguing that a 212:200 majority was more reliable than a 12:0 verdict. Remember all this the next time you receive a jury duty notice. Had some of Laplace’s understandable concerns been more seriously considered, there’s a small chance that societies could have adopted larger juries in the interest of a fair trial.)

French law eventually changed the minimum conviction from 7:5 to 8:4. But it turned out that there was a better method to allow for a majority jury verdict. It was a principle that extended beyond mere frequency and juror reliability, taking into account Bernoulli’s ideas on drawing black and white balls from an urn to determine a probability value. It was called the law of large numbers. And the great thing is that you can observe this principle in action through a very simple experiment.

Here’s a way of seeing the law of large numbers in action. Take a quarter and flip it. Write down whether the results are heads or tails. Do it again. Keep doing this and keep a running tally of how many times the outcome is heads and how many times the coin comes up tails. For readers who are too lazy to try this at home, I’ve prepared a video and a table of my coin toss results:

edcointoss

The probability of a coin toss is 1:1. On average, the coin will turn up heads 50% of the time and tails 50% of the time. As you can see, while my early tosses leaned heavily towards heads, by the time I had reached the eighteenth toss, the law of large numbers ensured that my results skewed closer to 1:1 (in this case, 5:4) as I continued to toss the coin. Had I continued to toss the coin, I would have come closer to 1:1 with every toss.

galtonbox

The law of large numbers offered the solution to Laplace’s predicament. It also accounts for the mysterious picture at the head of this essay. That image is a working replica of a Galton box (also known as a quincunx). (If you’re ever in Boston, go to the Museum of Science and you can see a very large working replica of a Galton box in action.) Sir Francis Galton needed a very visual method of showing off the central limit theorem. So he designed a box, not unlike a pachinko machine, in which beans are dropped from the top and work their way down through a series of wooden pins, which cause them to fall along a random path. Most of the beans land in the center. Drop more beans and you will see a natural bell curve form, illustrating the law of large numbers and the central limit theorem.

Despite all this, there was still the matter of statistical fatalism to iron out, along with an understandable distrust of statistics among artists and the general population, which went well beyond Disraeli’s infamous “There are three kinds of lies: lies, damned lies, and statistics” quote. Hacking is a rigorous enough scholar to reveal how Dickens, Dostoevsky, and Balzac were skeptical of utilitarian statistics. Balzac, in particular, delved into “conjugal statistics” in his Physiology of Marriage to deduce the number of virtuous women. They had every reason to be, given how heavily philosophers leaned on determinism. (See also William James’s “The Dilemma of Determinism.”) A German philosopher named Ernst Cassirer was a big determinism booster, pinpointing its beginnings in 1872. Hacking challenges Cassierer by pointing out that determinism incorporated the doctrine of necessity earlier in the 1850s, an important distinction in returning back to Peirce’s idea of absolute chance.

I’ve been forced to elide a number of vital contributors to probability and some French investigations into suicide in an attempt to convey Hacking’s intricate narrative. But the one word that made Perice’s contributions so necessary was “normality.” This was the true danger of statistical ideas being applied to the moral sciences. When “normality” became the ideal, it was greatly desirable to extirpate anything “abnormal” or “aberrant” from the grand human garden, even though certain crime rates were indeed quite normal. We see similar zero tolerance measures practiced today by certain regressive members of law enforcement or, more recently, New York Mayor Bill de Blasio’s impossible pledge to rid New York City of all traffic deaths by 2024. As the law of large numbers and Galton’s box observed, some statistics are inevitable. Yet it was also important for Peirce to deny the doctrine of necessity. Again, without chance, Peirce pointed out that we could not have had all these laws in the first place.

It was strangely comforting to learn that, despite all the nineteenth century innovations in mathematics and probability, chance remains very much a part of life. Yet when one begins to consider stock market algorithms (and the concomitant flash crashes), as well as our collective willingness to impart voluminous personal data to social media companies who are sharing these numbers with other data brokers, I cannot help but ponder whether we are willfully submitting to another “law of large numbers.” Chance may favor the prepared mind, as Pasteur once said. So why court predictability?

* Peirce’s attempts to secure academic employment and financial succor were thwarted by a Canadian scientist named Simon Newcomb. (A good overview of the correspondence between the two men can be found at the immensely helpful “Perice Gateway” website.)

Next Up: Janet Malcolm’s The Journalist and the Murderer!

Walken or Shatner? A Philosophical Inquiry

To Carolyn Kellogg: Given the strange question “Walken or Shatner?” I might likewise find myself opting for the latter, purely out of chronological consideration. I would select Shatner because the man is twelve years older than Walken, and there is greater pressure from the elements. From a pragmatic standpoint, Shatner is likely to expire earlier in time than Walken. But this assumes that these two men will die at more or less the same age in their respective lives. There may indeed be twelve more years to see Walken. Then again, there may not. Walken could die in some freak accident next month. Or perhaps the two men could die on the same day, with Shatner’s last words being, “Walken still lives.” This seems to me a sufficient speculative premise that unites these two gentlemen in some hard and inevitable future, suggests mutual respect and consideration of the other’s works, and dovetails this all rather nicely into a notable historical coincidence that occurred on July 4, 1826.

But back to the initial question (“Walken or Shatner?”), we can express this proposition in mathematical terms:

S = W + n
W = S – n

In present time, n = 12. Upon expiration of W or S, n = 12 – m, where m represents the difference between W or S’s final value and the number of years the other variable has to catch up to first expired variable’s final value.

Now this is a cold and morbid formula. I certainly wish both Walken and Shatner long lives. They have both entertained and informed audiences in unexpected ways. But I recuse myself from the equation’s insensitive auxiliaries by impugning the individual who put forth the question in the first place. The question should never be “Walken or Shatner?” There should be an option accounting for both choices. In this way, both Walken and Shatner can both be afforded respect and the person carrying the burden of this question will not have to make a terrible decision.