Life is messy. Numbers are neat.
Or so I’ve liked to think. Math has been a comfort zone in times of confusion, a way to pick out signals in the noise, see patterns in chaos. At the least, it could provide a sense of boundaries, scaffolding, a handhold. With a little help from math, we can begin to grasp even concepts we’ll never really grok.
But now my friends the numbers seem to have deserted me—an insidious side effect of Covid-19. The blinding rain of data the virus unleashed can drown out more than it reveals. Downpours are opaque—a good reason not to drive in them or try to navigate through a crisis.
The worst offenders are the damned denominators—often unknown, unspoken, misused, dubious, deceptive, dopey, even demonic.
To a person of my vintage, denominators are personal. If I’m 73 years old today, what is my denominator? If it’s 74, it’s time to start saying goodbyes, putting my affairs in order, as they say. Anyone facing death is coming to terms with their life’s denominator, something a pandemic has a way of putting in perspective.
I hadn’t been thinking much about denominators until I got an email from Kathleen Hall Jamieson, the director of UPenn’s Annenberg Public Policy Center, where I’m currently a scholar-in-residence. “The number diagnosed is a bad denominator,” she wrote to her staff, way back on March 20, just as testing was ramping up. Scientists and journalists alike were basing fatality rates on “cases,” a number nobody really knows. Recovery rates were just as cloudy. “Recovered” implies a person was infected. But no one knows how many people who “recovered” from flu back in January and February (also) had the virus. They were almost certainly not counted as cases.
Often, denominators are simply disappeared. “US surpasses Italy in total deaths,” The New York Times (and others) all but screamed, conjuring images of corpses in the street. But the US was bound to have more fatalities; it has a lot more people. If you look at deaths per capita (denominator, please!), then the fatality rate in the US looks less bleak.
I have the same sort of “duh” response to continual reminders that, for us elderly and people with preexisting conditions, the fatality rate is much higher. Higher than what, you might (you should) ask? What’s the denominator? Most old folks have preexisting conditions, and age itself is, sooner or later, a cause of death. In a normal, non-Covid year, most people who die are over 80. The major causes of death are heart disease, cancer, stroke, injury, suicide. How do these diseases and conditions (preexisting or not) interplay or overlap?
Among younger people, suicide and “unintentional accidents” (including homicide) are right up there with cancer and upper respiratory disease—both affected by air pollution, which we’re told is decreasing, thanks to people staying at home. At the same time, domestic violence is increasing, along with divorce (no surprise there). What effect does isolation have on mental health? On stress? Yes, people are standing in line to buy bleach and toilet paper, but they’re also buying guns.
Does it make any sense to try to compare such disparate types of risks when so little is known? Probably not a lot. Still, when scientists (and reporters) tell us more people will die if we do this or that, shouldn’t we know: More than what?
To compare anything at all, we need, at the least, some kind of base rate (a denominator): 1.2 percent of 382 million (US population) is a whole lot more than 1.2 percent of 60 million (Italy). If I tell you half of the homes on my street have swimming pools, you could conclude I live in a fancy neighborhood. Or you could ask: How many houses on my street? If the number is two, it certainly changes the perspective.
Denominators make drastic differences. Twitter founder Jack Dorsey recently announced that he’d given $1 billion toward the coronavirus effort—28 percent of his wealth. Mark Zuckerberg and Jeff Bezos each gave millions, but because their denominators are so huge, their seemingly generous gifts amount to less than 0.1 percent of their wealth. As a friend in finance put it: “Bottoms are fundamental.”
Not surprising, then, that denominators are often used deliberately to deceive. Take those college rankings. The more selective a school appears, the more appealing. And yet, puffing up the dominator is as easy as encouraging loads of students to apply. Suddenly, the number of students who get in looks tiny compared to the hordes beating at the door. It’s simple supply and demand. Except the demand, in this case, is what some might call a dirty (certainly dubious) denominator.
Conversely, pruning down a denominator (to nothing, if possible) is an easy way to give yourself a tax break. No need even for a rate cut. Simply make profits, taxable income, and capital gains magically go away—sometimes literally, as in “offshore” or “Ireland.” The financially savvy know all about “shelter in place” when it comes to money.
Denominators behaving badly pop up so often that they often slip right by us. One that irks me in particular is the one used to calculate the percentage of people who say they read science stories in the newspaper. It’s always getting smaller. But wait! What is the denominator? Readers of papers that cover science regularly? At all? Well? It’s hard to work up interest in something you aren’t exposed to or read stories that aren’t there.
Some decisive denominators are set for us by nature itself. The total surface area of the earth, along with most of the resources in and on it, are set, you might say, in stone—something we ignore at our peril when we use up all the space (and stuff) as if the supply of stuff (and space) were limitless.
The size and contents of the observable universe, on the other hand, changes—including some major denominators. It was humbling enough to know that the stuff of “ordinary” matter—people, atoms, and stars—accounted for less than 20 percent of the total stuff in the universe (the denominator). The rest of the matter is “dark,” or, more accurately, transparent. (If it were dark, it might cast shadows.) We still have no idea what it is; we only know that gravity pulls on it.
In the 1990s, that denominator (not quite literally) exploded. Astronomers discovered unmistakable evidence that some unknown stuff was pushing (actually pulling) distant galaxies farther and farther apart. The cosmos contained so much of that stuff, known as dark energy, it diminished everything else; the ordinary stuff we’re made of now makes up a measly 5 percent of the matter/energy of the universe.
The new cosmic denominator doesn’t merely expand space, the expansion goes faster and faster—that is, it accelerates.
That’s a double damned denominator, like the one that tells us how fast things fall. It changes over time. Drop your keys and they fall to earth at 32 feet per second/per second—that means an additional 32 feet per second every second. The farther you fall, the harder you splat. And that second “per,” what a geek might call a second derivative, is easy to miss.
Jamieson recently nailed a particularly troubling double damned denominator in a Covid-19 graphic she saw that compared the spread of the pandemic state to state, city to city. She looked at the projected death rate and asked herself: “How likely is it that people unschooled in these kinds of analysis are going to realize that is a per 100,000 people per day number, not a per capita number? What is the likelihood a person will see a 0.2 percent rate for Philly and say, wow, that’s pretty good, Philly is pretty safe?” In other words, what is the likelihood they’ll miss the second “per” and come to the wrong conclusion?
The answer is: highly likely. As Jamieson explained recently in an interview with Wisconsin Public Radio, “We now have the capacity to generate data that has outstripped the ability to communicate it.” More data doesn’t necessarily lead to better understanding. Indeed, it produces a cornucopia of low-hanging fruit for cherry-picking—sometimes harvested for political gain.
(That’s something else Covid-19 and gravity have in common, even beyond double damned denominators: Neither gives a fig whether you lean left or right. They’ll pull you down just the same.)
The expanding universe and Covid-19 also forced us to redefine what denominator means: When astronomers had to revise the total amount of stuff in the universe, the denominator wasn’t just more of the same; now it included something different, something that wasn’t included before because it was not known to exist before.
And that’s the tricky part. The modelers who are managing to make some sense of Covid-19 for the rest of us—holding umbrellas, in a sense, to keep us from getting soaked—are pretty good at tinkering with unknowns. It’s the unknown unknowns that are the real demons. And a lot of those demons are denominators.
Consider tests for Covid-19, the way we get “confirmed cases,” which is often used as a base rate for calculating fatality. That number depends on who’s been tested, which varies from place to place. If only sick people are tested, the disease will look a lot deadlier than it actually is. If only confirmed cases are counted and very few are tested, then a place can seem deceptively safe. Reporting methods vary so widely that even the most numerically literate can easily get lost.
Numerators are easier, but hardly a slam dunk. How many people died? Do we include those who died at home? How many people who are hospitalized for the virus actually die of other causes (heart attacks, for example)? Does the distinction even matter?
We don’t know what we don’t know.
Modelers take a lot of flak for getting it wrong, but right or wrong misses the point. Models aren’t meant to eliminate uncertainty. Their role is to illuminate uncertainty, quantify it, round it up and rope it in. They give us possible outcomes, attached to probabilities, which is the best you get in the real world—even in good old physics, much less in murkier fields like medicine and epidemiology.
In that sense, epidemiological models are not so different from other kinds of models—i.e, role models, the kind you’ve probably had a lot of by the time you get to 73. Role models, too, tell us about possibilities.
In my twenties, I went with a girlfriend to hear Margaret Mead at the American Museum of Natural History in New York, and because we were fearless, we invited her out for a drink. Over whisky, I decided then and there, when I was old, I wanted to be her. The cape, the cane, the whole megillah. I didn’t exactly follow in her footsteps, but she modeled a kind of spunk and smarts and taste for adventure that gave me an entirely new set of possibilities for what being a woman could mean.
Another role model, a physicist who’d worked on the Manhattan Project, gave me an appreciation of how poorly our human perceptual apparatus equips us for dealing with truly large numbers. He was Frank Oppenheimer, the younger brother of Robert Oppenheimer, the “father” of the atomic bomb (which made Frank, he mused, the “uncle” of the bomb).
Like others who felt the first atomic explosion firsthand, he was sufficiently blown away that he spent much of the rest of his life trying to convey the meaning of “nuclear weapon” to others. He tried repeatedly to find everyday examples that would help people grasp the difference a factor of a thousand could make. It’s the difference between a million and a billion, and also the difference in destructive power of the exponentially more powerful atom bombs compared to other bombs. If you had a dinner party for four people, he said, and a factor of a thousand more people showed up, then you’d have to deal with 4,000 people, in the same house, same dishes, same food.
Or as his pal the physicist Albert Bartlett put it: “The greatest shortcoming of the human race is our inability to understand the exponential function.”
Bartlett, then at the University of Colorado, came up with one of the most compelling stories ever for graphically illustrating how our inability to grasp exponential growth means we so often get blindsided—why we “don’t see it coming.” He asks us to imagine growing a couple of bacteria inside an empty Coke bottle; they start reproducing at 11 am, and double their numbers every minute until noon—at which point the bottle is full. What time would it be, Bartlett asked, when even far-sighted bacteria politicians realized that they were running out of space? 11:59. One doubling time before noon. Still more space left than the bacteria have used in the entire history of their civilization.
Math also helps us interpret probability, something we don’t give proper respect. Intuitively, we think probability means “mere” chance. But a high enough probability is the same as a cause. Play Russian roulette with one bullet in your gun and your chances of dying are one in six. Put five and it’s all but certain. For frontline health care workers, the cause that makes them sicker from Covid-19, and sick more often, is their multiple exposures, multiple chances to get infected, a higher probability—too many bullets.
Aging, also, is caused by probability. My face could lift itself, I suppose, but it’s highly unlikely. Things (including people) fall apart because there are so many avenues toward decay, so the probability becomes a certainty. This inevitable rush toward disorder, or entropy, is so predictable we can use it to tell the direction of time simply by watching faces wrinkle, paint peel, dropped eggs splatter. But it’s “just” probability.
Tightly connected to both probability and exponential growth is the notion that “more is different,” so aptly put by the recently deceased physicist Phil Anderson. Quantity changes quality. More of the same produces entirely new phenomena, known as emergent properties. One neuron can’t have a thought, one person can’t behave like a crowd, Covid-19 is not just more of the seasonal flu.
Being 73 is not just more of being 70, much less 50 or 20. If I look at my age as a denominator, it’s easy to see why the process of aging speeds up. At age one, one year was my whole life; age 20, one year was 1/20th; at age 73, a year is a much thinner slice—1/73rd. Relatively speaking, time goes by much faster.
Even as the slices get smaller, they grow in number, in experiences, in role models. When I feel cooped up staying at home, I think of my friend who spent four years hiding in an attic evading Nazis. I remember the fear of polio.
Fear of Covid-19, for me, is probably most like the fear I felt as a child crouching under my desk at school, all but certain we were about to be vaporized by Khrushchev’s nuclear weapons; he did, after all, vow to bury us. Our daily “duck and cover” drills made this “existential” thread very personal and very real.
I once asked my physicist friend who worked on the bomb why, knowing the horrifying consequences, they didn’t try harder to scare people. “We did try to scare people,” he said. “Scaring people doesn’t work. You have to make them angry.”
Well, now I’m angry. I’m angry that we aren’t testing enough people for Covid-19 to tame those damned denominators and make more sense of the hail of numbers.
Denominators are context, and numbers mean nothing without it. The digits 911 can signify a number to call in case of emergency, a date on which a disaster occurred, the number of beans in a jar. Numbers alone aren’t facts, much less truths. As Bertrand Russell famously put it: “Mathematics may be defined as the subject in which we never know what we are talking about, or whether what we are saying is true.”
I’ll never stop loving my pals the numbers—the whole dang delicious family: natural, unnatural, imaginary, surreal, transcendental, irrational, prime, complex, perfect. But even I have to resist their allure of authority. Absent human brains, they can’t be counted on to tell us much about human problems. When it comes to meaning, that’s still up to us.
Photograph: Miguel Medina/AFP/Getty Images
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