“Nobody knew math could be so complicated.”
— nobody ever
The most truthful — and, to me, the most infuriating — thing a certain public figure1 uttered over the course of his ongoing career was “It is what it is.”
Given that he was referring to the deaths of over a hundred thousand people — deaths that, mere months earlier, he had declared with utter confidence would not occur, thereby causing unjustified optimism and imprudent risk-taking — I was angry. I felt that the moment called for an apology, not a tautology.
And yet, who can argue with the proposition that things are what they are? Isn’t this insight — in particular, the insight that our wishing that the truth were different doesn’t change what’s true — the bedrock on which all of science rests?
But belief in the existence of one Truth, in and of itself, is no guarantee that you won’t wind up in some silo of wrong-headedness, believing in wrong answers to important questions, listening only to the people who agree with you. If you want to stay out of such an epistemic prison, you should also believe that Truth is hard to get at, that no one person or group of people is likely to arrive at much of it, and that at any given moment you probably grasp less of it than you think you do.
I’m an agnostic, but I strongly sympathize with an idea that I’ve read is a key tenet of some evangelicals: that Discernment, a process that requires listening, reasoning, observing, reflecting, and considering new perspectives, is as important as the traits we normally classify as virtues. Indeed, Discernment might be the most important virtuous practice of all, because if we don’t rightly discern what’s true, our noblest impulses can be put to terrible uses. (“He who can make you believe the absurd can make you commit the unjust,” wrote Voltaire.) Think of the way the desire to protect children gets grotesquely twisted when it leads people to falsely accuse others of satanic ritual abuse and similar horrors. Credulity ceases to be a private failing when it leads to harmful actions. We have a duty to see the world as clearly as we can.
I don’t believe in a Day of Reckoning, but I believe that if there is one, and if during our lives we wrongfully brought harm to others because of incorrect beliefs about the world, the Reckoner won’t be swayed by the excuse “I did these things because people I trusted misled me.” The Reckoner will reply “And why did you choose to place your trust in these people to begin with?”
Likewise, I don’t believe in a Devil, but if there is one, surely the Devil’s greatest pleasure lies in cunningly perverting noble impulses — such as compassion for the weak and indignation in the face of injustice — to evil ends, simply by distorting good but undiscerning people’s perceptions of reality.
Which brings us back to the undiscerning public figure who said “It is what it is.” A few years earlier, he’d announced “Nobody knew x could be so complicated” (never mind for the purposes of this essay what x was), even though plenty of smart people, including the previous holder of his job, had said often and loud that x was complicated. The fact is, most things are complicated. Math can help us learn this, and at the same time, the right kind of math teaching can help us learn how to cope with this complexity — not by giving us tools like calculus (helpful as they are in solving technical problems) but by giving us practice in respectful and productive debate, or as we might say, practice in Discernment.
This side of math teaching may be less familiar to you than the other, more infamous side — the side of rigidity and pedagogical authoritarianism — so let me explain.
Two of my favorite courses to teach have been honors calculus and probability theory. Lots of the ideas students encounter in an honors calculus course are unintuitive, and some of the ideas they encounter in a probability course can seem downright preposterous. I’m thinking of the Monty Hall problem, but there are plenty of other results in probability that most of us get wrong the first time we encounter them. Often students will split into factions that hold two different views about a problem, and that’s when the real learning can happen.
I let the two sides advocate for their respective opinions, getting them to listen to each other. “Don’t look at me to see what I think,” I say (when I catch them looking at me for “tells” that could reveal which side I agree with); “look at your classmates, the ones you’re trying to convince.” My role as moderator is to get the two sides to explain their beliefs, clarify their assumptions, and devise convincing arguments. From time to time, the class re-votes. I focus on hearing from the people who’ve changed sides. I ask them what swayed them. Sometimes people who’ve switched to the correct view were swayed by invalid arguments; in that situation, I’ll advocate for the incorrect view to force holders of the correct view to come up with better arguments. But on the whole I try to join the fray as little as possible.
Almost always, one side completely wins over the other side. What started as a 15-15 or 20-10 split becomes a 30-0 consensus. The rift between competing realities heals, and we move on.
What’s been learned here goes beyond the mathematical proposition in question; it’s fresh confirmation that, in the right circumstances, people who start out disagreeing can arrive at unanimity. They can overcome their prior beliefs and, by thinking together honestly, discern what’s true.
Of course math is special. Mathematical debates can and usually do converge on a single correct answer because, while perspectives vary, mathematical truth is usually singular and discoverable. Mathematical debates are also special in another way: it’s easy for people to stay respectful and calm when the stakes are so low. Who really cares, in an emotional way, that (for instance) the derivative of an everywhere-differentiable function isn’t necessarily continuous?2 If you were inclined to argue with me about it, I wouldn’t lose my cool, and neither would you. On the other hand, if you and I were debating a real-world subject we cared about deeply, we wouldn’t bring our math-argument bodies to the debate. Our amygdalas and adrenal glands would tell us we’re under attack, as our hearts revved up and our blood pressure rose. We might sincerely believe we were fully listening to each other, but even as the other person spoke, our hyped-up brains would likely be planning rebuttals instead of digging deeper into what we’re hearing.
So, learning to argue discerningly about math may or may not prepare you to argue discerningly about, say, global warming or vaccination mandates. And yet, properly understood, the messiness of mathematical truth has one more lesson to offer us.
Let’s go back to the surprising fact (which you can treat as a black box if you wish) that derivatives of everywhere-differentiable functions aren’t always continuous. This doesn’t correspond to any fact in the real world as far as I know. It’s a purely mathematical issue. Yet it’s rock-solid. If you interpret the words “continuous” and “derivative” in the standard way, then you might believe that derivatives of everywhere-differentiable functions must be continuous, and you might even believe it was obvious, but you’d be wrong. The fact that most people who learn calculus uncritically arrive at the same belief you did doesn’t make your mistaken belief any truer.
But who came up with the definitions of “continuous” and “derivative”? People did! So you would think that we people wouldn’t be surprised by the facts and non-facts latent in our definitions. I mean, we chose them, so how can we surprised by the conclusions they lead us to? We uttered those definitions and thereby spoke a mini-world into being. How then can that world surprise us, its creators?
And that’s where the humility one learns from mathematical research and teaching can be carried over, perhaps even in amplified form, into real life. Because (and here, finally, is my point:) if our instincts can so mislead us about the creations of our own mind, how much more should we expect to be misled by our instincts regarding a world we did not create?
This is not to say that mathematicians are in any way immune to various forms of pig-headed idiocy and evil. The brilliant mathematician Oswald Teichmüller was also an ardent Nazi (see my essay “Jewish Mathematics?”). R.L. Moore, whose “Moore method” of advanced instruction in mathematics was very influential, refused to let women and people of color takes his classes. I could list many other such examples of people whose souls were not saved by math. But I think that in my own case at least, learning math, and then learning more math, and even now continuing to learn math, has made me more humble, not less. My occupation provides me with abundant opportunities to say “I don’t know,” “I’m confused,” and (above all) “I was wrong.” Being wrong so often, even on matters related to my area of expertise, means that even when I’m sure of myself, a little voice in my head reminds me of all the times I’ve been mistaken. (See also my essay “How to Be Wrong”.)
While I’m advocating an appreciation of how complex our world is and of how counterintuitive it can be, it’s important not to be paralyzed by awareness of our cognitive inadequacies as individuals and as a species. Yes, most knowledge is provisional, but we don’t always have the luxury of postponing decisions. Humility should not paralyze action; it should encourage us to act with great care, informed by the best discernment we can muster.
Albert Einstein once said “Imagination is more important than knowledge.” This is a deep truth whose opposite is also a deep truth, and the latter truth is unfortunately becoming increasingly relevant to the world we live in, as disrespect for hard-won expertise becomes ever more rampant. Some good liars have wonderful imaginations, but we have to live in the real world, not their imaginary ones. People tend to disparage facts as dry and inert, but like cement (which is also dry and inert), shared facts hold things together. I worry about societal cohesion if people are convinced of different facts and cease to revere the human race’s long project of discerning what’s true and what isn’t.
Things are what they are, and what they are, above all, is: complicated. A respect for that complexity should be required of everyone who wishes to preside over our world. Alas, it is not. And mere wishing won’t make it so.
Thanks to Sandi Gubin, David Jacobi, and Eliana Propp-Gubin.
ENDNOTES
#1. While this essay focuses on one particular public figure, the tendency to dismiss complexity isn’t unique to any one individual or political affiliation; it reflects a widespread problem in civic discourse.
#2. When students first study calculus, all the examples of everywhere-differentiable functions that they meet have everywhere-continuous derivatives, so it’s natural for them to arrive at the belief (dare I call it a stereotype?) that all of them do. But if they take more advanced calculus classes, they’ll learn that it just ain’t so.
The standard example of an everywhere-differentiable function whose derivative isn’t everywhere-continuous is the function defined by f(x) = x2 sin 1/x for x ≠ 0, with f(0) = 0. Here’s more or less what f looks like:
And here’s more or less what its derivative f ‘ looks like:
The function f’ is not continuous at 0, though it is continuous everywhere else.
One important property of f’ not conveyed by the second picture is that, even though f’(x) fluctuates ever-more rapidly between +1 and –1 in the vicinity of x=0, f’(x) equals 0 at x=0. This corresponds to the fact that in the first picture, if you draw a line between the point (0, f(0)) and the point (x, f(x)), the slope of that line is guaranteed to be closer and closer to 0, the closer x gets to 0, on account of the way f(x) always stays between x2 and –x2 — two functions that both have derivative 0 at x=0.
Calculus turns out to be even more complicated than you thought it was the first time around, if you really dig into it on a second or third pass. Most things are like that, and not just in math.
