Everyone Can Do Math (Even You) and Math is Interesting
A Different Sort of Essay
Photo by Dan-Cristian Pădureț on Unsplash
I hope this essay is a fun break for readers. It’s a bit out of the way from the sorts of topics I normally write about, but I’ve been toying around with the idea for a month or two now. You can file it under my “Renaissance Person” theme (I’ve written before that I prefer not to specialize – and, in fact, I think everyone should be a bit of a generalist).
This topic was sparked by some conversations I’ve had recently, as well as a couple of articles I read last fall about poor numeracy in the American public. And it’s also partly inspired by a book I read recently by the English mathematician Marcus Du Sautoy.
Last fall I published a series of LinkedIn posts about numeracy. My audience was adults who aren’t terribly good at math but could be if they tried. The posts offered a series of tricks or heuristics that I use – ones that I think self-proclaimed innumerates could pick up quite easily – that make mental calculation easier or that help one grasp the relationships between numbers.
One of my takes on American innumeracy is that it’s at least partly by choice. Not completely. No doubt education plays a factor. As does reliance on information technology (if you always have a calculator handy, you could go months or years without practicing mental calculation). But I do find that quite a number of people who work in non-math-heavy professions sell themselves short. I’ve lost count of the number of times I’ve heard intelligent people say, “I can’t do math,” or “I’m bad with numbers.” To be sure, many of them haven’t taken a math class since high school. But most of our grandparents and great-grandparents could calculate tips in their head and pay with exact change every time without breaking a sweat – many of them without graduating high school, or even eighth grade. So why can’t we? Reliance on calculators plays a role. As does education.
But I think attitude also plays a role. And comfort with numbers. I’ll explain what I mean by that in a moment, but my main point is that the human brain is capable of understanding numbers and performing basic calculations. Which means that yours probably is too.
Attitude and Comfort:
In school, some students take a bad attitude towards learning (or towards learning anything they think they “will never use”).
But perhaps more important than this is ease with numbers. It’s true that humans are hardwired differently. Some have an aptitude for numbers, while others have an aptitude for words (or music, or drawing, etc., etc.). But I was one of those kids who “wasn’t good at math,” and who clearly had to work harder than many of my peers to grasp numbers (I think more easily in words and sentences than I do in pictures or shapes or numbers).
So, it’s true that some level of comfort with numbers is natural and varies from person to person. But it’s also true that attitude plays a role. I “wasn’t good at math,” but because I took math classes up through partial differential equations (in college), I was forced to use numbers until I became more comfortable with them. In other words, I wasn’t naturally comfortable with numbers, but all that meant is that it took a little more practice to grow more comfortable with them. Which is why I think many people who claim to be bad at math could be decent at math if they wanted to be.
One educational factor here is rote memorization of the basics. As a kid, I definitely didn’t understand the value of rote memorization. Why should anyone have to memorize the times tables if we have calculators? But here’s the thing, if you have 6x7=42 memorized – such that you can instantly recall it upon command – all the other, higher-level mathematical operations become easier. You literally expend less mental effort to perform 6x7-92+9(5-4)=? if you have 6x7 memorized, than if you do not (answer: it’s -30). Advanced calculations are easier for people who have the basics down pat, whether those basics came naturally or took effort to learn. Just as someone who has never run before (assuming average natural talent) will have to work harder to run 8 miles at the pace I typically run for 8 miles.
So, the more you use numbers, the more comfortable you’ll be with them and the more readily you’ll be able to perform mental math. Rote memorization might be boring, but it makes the more interesting mathematics more accessible later on. I studied engineering, even though math was one of my worst subjects (worst SAT score, etc.), and a by-product of that was that I became much more comfortable with the average calculations encountered by most people in daily life.[1]
“But You Said Math was ‘Interesting’ and that this Essay would be ‘Fun:’”
Yes. I did. So, at this point, rather than go into some heuristics about performing mental calculations or understanding the slope of a line on a graph, I wanted to give you some reasons why you might want to consider looking into math again, especially if you don’t use it in your job. Reasons beyond simply: “it’s useful to be able to calculate your macros in your head,” or “you can tip the waiter without pulling out your phone.”
Perhaps one of the reasons people often find mathematics boring is that they never really got beyond the basics (including basic algebra, which can be quite dry).[2] I didn’t really enjoy a math class until calculus, which I actually found pretty fascinating. (Although I did have a love/hate relationship with trigonometry.)
What excited me about calculus was the mystery and the paradoxes (all that stuff about infinity and dividing by zero). Suddenly, the old rules of math didn’t completely apply. It helped to have a teacher who focused on the weird quirks and the intriguing questions. He could explain why the derivation was profound instead of just doing the derivation. While I was never good with numbers before that, I like logic and theory and patterns (hence why I liked some of trig). When you think about what a logical leap it must have been for Newton to realize he could divide by zero to get the instantaneous rate of change of any curve, and that he’d actually get an answer and not an error, you realize it must have been an exciting moment.[3] Same for when he figured out he could take a derivative backwards and it was the same as approximating the area under the curve by dividing the area into an infinite number of even rectangles (which gives you the exact answer, not just an approximation).
Similarly, negative numbers are boring until you realize that the Greek mathematicians didn’t believe they existed. In Du Sautoy’s book, I learned that this innovation came from the Chinese, who used negative numbers to calculate the debt a person owed (because you can have less than zero money if you owe more than you have). Negative numbers allow for all kinds of innovations, but one of those innovations is the ability to buy on credit (borrowing money from your future self).
If you have a general attitude towards the world that it’s filled with interesting things, and if you look for these things everywhere (especially outside your area of expertise), you’ll quite often find them. (The reverse is also true.) It helps if you find people who love their fields and can coherently explain why their fields are interesting.
But I still haven’t explained to you why math is interesting. It will depend to some extent on what interests you. For example, some people do not find unanswered questions that have eluded experts for decades or centuries to be interesting. But here are a few of the things you can find in mathematics.
Beauty, Paradox, and Unresolved Questions:
Perhaps a mind-blowing fact like “there are multiple sizes of infinity (not all infinities are created equal)” will pique your interest. But I find one of the most intriguing things about mathematics to be the fact that there is a fundamental beauty in it. Not just symmetry, but beauty.
If there is such a thing as Beauty (the capitalized version, as in Truth, Beauty, Objective Goodness, etc.), it must be said that one of the things that is beautiful about the world is Euler’s Identity.[5]
Euler’s identity is eiπ+1=0. Those are the five most fundamental numbers in the universe. And there is a single equation (identity) that relates all of them without involving a single other number.
What’s so important about these numbers?
Zero and One are obvious. The square root of negative one, i, is the imaginary number (all other imaginary numbers are multiples of i or sums that include it). Most people know about pi. Fewer know just how important it is not just to geometry but to physics, air travel, construction, medicine, music, and all advanced forms of mathematics (among other things). Like pi, e is an irrational number (it’s also known as Euler’s number…) that crops up over and over and over again (to the extent that the natural world contains pi and e everywhere you look). The more you know about these five numbers (again, they are the five most important numbers), the more you’ll understand just how profound it is that all five of them relate in a single equation without any other numbers involved.
Coincidentally, I just began reading the book Lost in Math, about how mathematical beauty actually leads physics astray in attempting to reconcile quantum mechanics with general relativity. Thus far, it must be said that physics as we currently understand it is premised on an almost uncanny level of symmetry, simplicity, and beauty.[6] How odd it is that the concept behind E=mc2 could be expressed in such a simple form. It’s E=mc2, not 54E=13me-4-5c8. The latter (which I made up) is nonsense and is meaningless and the former – which is not just prettier to look at, but easier to understand – explains something fundamental about energy, mass, and the speed of light in a vacuum. How odd it is that the two intrinsic qualities of an object relate to each other simply by multiplying one of them by the square of the most important physical quantity in the universe (the velocity that light particles travel at in a vacuum).
The author of Lost in Math, Sabine Hossenfelder, suggests that we have been lulled into a false sense of security by the beauty of current physical equations, and that to reconcile the greatest questions in modern physics without resorting to unprovable theories (multiverse theory) or logical leaps for which we have no evidence (supersymmetry, multiverse theory),[7] we will have to embrace some ugly math again. But I digress.
Now, perhaps at this point you might be thinking, “Beauty is in the eye of the beholder! I’ll define beauty however I please. There’s nothing beautiful about eiπ+1=0. Actually, I think 4.567383774884848e0.00000001234449393939i-1E103π+33.47=89 is more beautiful.”
You might be thinking that. But I’m betting you’re not.
The Things That Are Beyond Our Grasp:
Perhaps beauty isn’t the right angle from which to approach mathematics. At least not for everyone. Some people might prefer to grapple with ugly contradictions and wrestle with the unsolvable paradoxes of our age. For those of you who like stimulating questions, mathematics also presents many such questions. Du Sautoy and others in his field spend their days wrangling with the boundaries of human knowledge. They attempt to solve unsolved proofs, some of which have gone unsolved for decades or centuries, despite the best work of some of the brightest minds to have ever lived.
This is where the mystery of math lies. In the limits of human knowledge. Perhaps someday future humans will look upon our current mathematicians the way we look upon the ancients who first invented imaginary numbers, or negative numbers, or the concept of zero. Or the ones who first discovered that not all infinities were created equal (just as some animals are more equal than others, some infinities are more infinite than others).
In many ways, the most interesting questions are not the ones we have already solved. The most interesting questions are always the ones that lie just beyond our ken, or beyond the beyond. “For all experience is an arch wherethrough gleams that untraveled world whose margin fades, forever and forever as I move,” as the poet tells us. Beyond our current mathematical horizon, there is always another horizon, always another question to be answered, another proof to be solved. Perhaps the most interesting question Du Sautoy poses in his book is whether or not there are things in math or science which will always remain beyond our grasp, no matter how cleverly humans try to understand them. In other words, are there things we not only will never know, but things we cannot know?
Anyone who thinks math is boring and staid hasn’t fully grasped that there are unassuming men and women laboring in obscurity who are at this very moment probing the very foundations of known mathematical knowledge.
But Does It Matter?
Those of another philosophical bent will scoff at the efforts of these obscure researchers. What if their wrangling with arcane proofs never yields an ounce of useful knowledge? Those of this persuasion set aside Beauty for Beauty’s sake and Knowledge for Knowledge’s sake. Perhaps you hail from this crowd. Perhaps you say, “that’s all very well and good, but what is its utility? Isn’t math only useful insofar as it can advance human scientific understanding in ways that improve human life?”
Throughout the history of math and physics, mathematicians have raced ahead down what some of their peers called rabbit holes, only for physicists to later discover that, “Eureka, this was the math I needed all along to solve my equations.” Newton invented calculus to solve his physical laws (which govern the world you see and touch).[8] Einstein stumbled upon obscure mathematical theories without which he never would have produced Special Relativity or General Relativity (the latter of which he labored at for about a decade while trying to find the right math to make the theory work).
If you’ve ever used a GPS, marveled at human space exploration, considered astronomy (especially that beyond our solar system), looked at the movements of the planets in the night sky, celebrated the U.S. victory in the Persian Gulf War, or really used just any modern technology or lived in modern society for very long, you’ve been a beneficiary of Einstein’s theories of relativity.
In other words, advanced mathematics has been not just instrumental, but crucial to the modern physics and engineering that have created the world in which we live.
But you probably know that.
However, what you may not have considered is just how uncanny it is that obscure and un-useful branches of mathematics often find their applications in advanced physics decades later. It really is quite startingly to realize that math, numbers, and equations describe the natural world at a fundamental level. Perhaps the real reason Euler’s Identity is so profound is that the five numbers that have the greatest bearing on human life (and all life) are also intrinsically related in a way that is at once simple and highly complex.
Lest I lose you, I’ll bring up science fiction. In Carl Sagan’s book, Contact (which was better than the movie and a bit different from it – although I liked the movie), the language of mathematics and numbers is the language the aliens use to communicate with human beings.
Alright, that is all very well, but what is perhaps most interesting is that the aliens also teach human beings about hidden mathematical paradoxes within the world around them, which contain messages for intelligent life to discern and discover. For instance, there is a message that humans had hitherto overlooked in the digits of pi.
Contact is considered “hard sci-fi,” because the book has a lot more math and physics than the movie. (Although it’s not as “hard” as, say, Poul Anderson’s Tau Zero, in which the entire plot revolves around 1/√(1-v2/c2).) However, it’s eminently readable and very thought-provoking. The aliens use math to communicate with humans because numbers describe reality (in other words, math is encoded into nature). And yet, even these highly advanced beings express awe and wonder (and something approaching theology) at this fact. Because for all their technological progress, they recognize that there is something beyond themselves and that there are things about the universe that they still do not understand. And the thing that is most eerie of all is the realization that mathematical puzzles are hidden in the bedrock of reality.
Einstein said that “the most incomprehensible thing about the universe is that it is comprehensible.” When you stop to think about it, perhaps the weirdest thing about the human relationship with mathematics is that any of us could manage to find it not just ordinary, but boring. (Despite claims to the contrary, the physical world is not a social construct. But human boredom surely is.)
To Sum Up:
In the spirit of remaining interested in, and capable of, many different things (i.e., in the spirit of a generalist or a Renaissance Man or Woman), I think people who claim to be “bad at math” or “uninterested in math” should take a second look. Don’t sell yourself short.
I realized much later in life that my favorite part of math class or physics class was doing the derivations that just involved letters. I liked the equations that resolved themselves to get rid of all the numbers. When doing homework in engineering classes, I preferred working with abstract variables to substituting in actual numbers (at the end you substitute in the known quantities and simplify to find your answer). What I realized was that – all my life – I’d had a preference for letters and alphabetical symbols. In other words, I still like words more than numbers.
But I also discovered that – after many years in which they didn’t – numbers started to come naturally to me. And the more I studied, the more I learned that there really are some intriguing aspects of mathematics. Once you get down past the bits you need to learn via rote memorization, you can begin to explore more abstract topics. I’ll never solve the Navier-Stokes existence and smoothness problem (heck, after whole classes on partial differential equations and fluids, I can still barely tell you what the Navier-Stokes equations are – and even then, I had to glance at Wikipedia).[9] But I can marvel at the fact that you can line up two different infinite sets of numbers and find that one of the sets is longer than the other. And, like Terry Pratchett’s Ponder Stibbons, I can take the square root of 27.4 in my head: it’s “5 and a bit.”
In all seriousness, once you get the hang of it, mental math is a very useful and satisfying skill. I could even argue that it’s a prerequisite for good citizenship in a republic: you’ll be able to understand the debt to GDP ratio, for instance. Or you’ll know that 20% of Americans don’t make a million dollars a year.
But perhaps more importantly, once you relearn a few of the basics, a much more varied and exciting world opens up to you. If you haven’t taken a math class since high school, you may never be more than a dilletante when it comes to advanced number theory or partial differential equations. But there’s nothing wrong with that! Because the world is an interesting place, and there are many things about it that are interesting. And if you focus too much on any one thing, you’ll miss out on a lot of that.
[1] In calculus (or in physics) you usually have to do a fair amount of algebra to arrive at your final answer. In algebra, you have to do addition and subtraction and multiplication and division. In this way, the higher-level mathematics force you do get even better at the lower-level mathematics.
[2] Of course, some higher-level mathematics are not, in fact, all that interesting. Even among math-lovers, partial differential equations or multivariable calculus or other classes are sometimes considered mundane or unexciting.
[3] And not just an answer, but the exact answer. As you approximate the slope more and more closely, you approach the exact answer but never quite arrive at it. Technically, Newton wasn’t just dividing by zero, he was getting closer and closer to the exact answer as he got closer and closer to dividing by zero, and then he was making a logical jump from approximation to exact answer. 0.9999999 repeating infinitely isn’t almost equal to 1. It is exactly equal to 1.
[5] Euler was a GOAT among Renaissance Men. If you’ve ever studied engineering, astronomy, physics, or math, you’ve heard of him. Everything named after Euler is named after one guy. He might be one of the individuals in history with the most things named after him. He has multiple Wikipedia pages, including one that is entirely devoted to the list of things that are named after him.
[6] The author of Lost in Math takes issue with the fact that symmetry is used to predict new theories on the basis of no evidence other than that it makes the math “look right.”
[7] Technically, multiverse theory isn’t a single theory, but a variety of different ones (various string theories, M theory, etc.), that have been proposed to reconcile the two great theories in modern physics (Quantum Mechanics and General Relativity). Multiverse theory wasn’t invented out of a hat, but many versions of it include as one premise that it is impossible to ever prove that multiple universes exist. In other words, no evidence of another universe can ever exist in this one. Therefore, the theory is not falsifiable (or “implausifiable,” or disprovable). Which, by the definition of science, is not scientific. (Some philosophers of science think we need to rethink our philosophy of science to include unprovable claims, but that sounds like religion to most people.) Not every theory that has been suggested that involve multiple universes includes this bit about, “you can’t ever prove it,” but the ones that do leave some questions begging.
[8] Leibniz invented calculus simultaneously, but neither one was ripping the other off. Each was unaware of the other’s invention.
[9] And I didn’t do poorly in those classes, either! The Navier-Stokes equations are abstruse.