Those Damn Denominators
Math used to be a comfort zone for me in times of confusion. Not anymore.
When scientists and reporters tell us more people will die if we do this or that, shouldn’t we know: more than what?
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
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-
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-
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
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-
(That’s something else Covid-
The expanding universe and Covid-
And that’s the tricky part. The modelers who are managing to make some sense of Covid-
Consider tests for Covid-
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-
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-
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,
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-
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-
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.