Let’s start with something uncontroversial: a valid mathematical assertion like 3+3+3=9 can be “wrong” if it’s been dragged into a situation in which it just doesn’t belong. Consider the Sufi tale1 of Mullah Nasrudin and his wife.
Three months after Nasrudin married his new wife, she gave birth to a baby girl.
“Now, I’m no expert or anything,” said Nasrudin, “and please don’t take this the wrong way-but tell me this: Doesn’t it take nine months for a woman to go from child conception to childbirth?”
“You men are all alike,” she replied, “so ignorant of womanly matters. Tell me something: how long have I been married to you?”
“Three months,” replied Nasrudin.
“And how long have you been married to me?” she asked.
“Three months,” replied Nasrudin.
“And how long have I been pregnant?” she inquired.
“Three months,” replied Nasrudin.
“So,” she explained, “three plus three plus three equals nine. Are you satisfied now?”
“Yes,” replied Nasrudin, “please forgive me for bringing up the matter.”
A trickier example is an old riddle about a missing dollar:
Three guests check into a hotel room. The manager says the bill is $30, so each guest pays $10. Later the manager realizes the bill should only have been $25. To rectify this, he gives the bellhop $5 as five one-dollar bills to return to the guests.
On the way to the guests’ room to refund the money, the bellhop realizes that he cannot equally divide the five one-dollar bills among the three guests. As the guests aren’t aware of the total of the revised bill, the bellhop decides to just give each guest $1 back and keep $2 as a tip for himself, and proceeds to do so.
As each guest got $1 back, each guest only paid $9, bringing the total paid to $27. The bellhop kept $2, which when added to the $27, comes to $29. So if the guests originally handed over $30, what happened to the remaining $1?
It’s absolutely true that 2+27=29, but in the context of the story, adding the numbers 2 and 27 makes no sense, though it would make sense to subtract 2 from 27.
An accountant friend of mine (call him Lenny) used to be in partnership with another accountant (call him Bob). While the partnership was operating, each partner had a “capital account” in the partnership into which they paid their earnings and out of which they paid their expenses. Over time, Lenny’s practice thrived while Bob’s languished, so Lenny’s capital account had a positive balance and Bob’s had a negative balance. Eventually Bob decided to dissolve the partnership. Both agreed that Bob’s negative balance was a problem, but Bob insisted that to set things right it was Lenny who needed to make up the negative in Bob’s capital account. (And Bob was not joking, which come to think of it may explain why Bob’s accountancy practice was doing so badly to begin with.)
Although these calculations are nonsensical, at least they combine months with months, or dollars with dollars. There are examples of nonsensical calculations in which the units don’t even match up. My favorite examples of such nonsense are the tongue-in-cheek calculations seen on certain road-signs, such as this one from Gold Hill, Colorado:
(Ben Orlin wrote about this sign a few years back.)
Sometimes addition is wrong even when the units match up because the units aren’t the kind of thing that can be added. An amusing example was relayed to me by pre-reader Mark Saul, whose aunt was afraid to use both levels of her double-decker oven at the same time, because “If you have 300 degrees on top, and 300 degrees on bottom, that’s 600 degrees. You could have a fire!”
Sometimes addition is wrong because of what’s left out, as in the practice of breaking down the American electorate into Democrats and Republicans and ignoring the independents (but see Endnote #2). Another mistake happens when you break down a set into smaller sets but ignore overlap between the sets.
Do you have a favorite real-world example of people adding numbers (or more generally subtracting, multiplying, or dividing them) when in fact that calculation isn’t sensible? Please post to the Comments!
The reason I’m bringing this up is that over the summer Twitter saw a lot of discussion of the equation 2+2=4. The equation is seen by some as a touchstone for objectivity and truth and by others as a manifestation of a repressive social system. To read what Keith Devlin has to say about this, read his blog post. To read what Michael Barany has to say about this, read his Twitter thread and the related article. If you want to know what I think, read on!
APPLES AND ORANGES
Some people say you can’t add apples and oranges, but of course you can: two apples plus two oranges equals four pieces of fruit. Pictorially:
In the first picture, we’re distinguishing between pomaceous and citrus fruits; in the second picture, we’re ignoring the differences between them. The curves have been redrawn but no fruits have been harmed or moved.
To me, 2+2=4 isn’t a fact about the physical world; it’s more a window through which I view the world, or a channel for my thoughts about the world. It says that the top and bottom pictures show the same state of the world. All that’s changed is the way I’m compartmentalizing things.
Waxing philosophical, I’ll say that the equation 2+2=4 serves as an emblem of the duality between the analytic and synthetic modes of thought. The left-hand side of the equation represents the way we take the world apart; the right-hand side represents the way we put the world together out of the pieces we’ve divided it into. In our attempts to make sense of the world, we need both the kind of thinking that attends to nuances and distinctions and the kind of thinking that can see past those distinctions.
Of course, in saying that both kinds of thinking are needed to help us make sense of the world, I’m presupposing there’s a world for us to make sense of, and this brings us to another philosophical take on 2+2=4, which is that the formula reminds us that what is is what is, regardless of whether or how we mentally break it into pieces. In other words, reality is real.
I believe that, but I also believe that reality is really hard to know. Our prejudices get in the way, and categories of thought that seem neutral in themselves can subtly affect our interpretations of reality. Self-styled champions of the concept of “objectivity” (you can often recognize them because of the way they hype “2+2=4” and say things that amount to “Ha ha, I’m objective and you’re not”) all too often are championing their convictions about what’s true, ignoring all the ways in which their experience is partial, their interpretations biased, and their statements couched in language that’s vague and subject to multiple interpretations. We may think that truth is a butterfly and language the net that captures it, but too often truth is a rabbit and language is a mound of jello that we throw at the rabbit, and some of the jello sticks to the rabbit but most of it falls off when the rabbit runs away, and we look at where the jello is and argue about what it tells us, but the jello landed mostly where we threw it, as it was inevitably bound to do, and the rabbit is long gone (and doesn’t eat jello anyway).
ABSOLUTE TRUTH, AT A PRICE
Let’s rehash an old line of argument about 2+2=4 and see where it leads us.
It sure seems as if 2+2=4 is saying something about the world. But what if I’m counting animals and two of the four make a baby together? Then 2+2 makes 5, right?
“Clever,” you say, “but you know that’s not what I meant; 2+2=4 applies to situations in which you’re combining collections without adding something new.”
To which I reply “What if I’m counting clouds, and two of the clouds merge? Then 2+2 makes 3.”
“Okay,” you say, “but you know that’s not what I meant either; when you combine collections you also have to make sure you’re not blurring the boundaries between things. … Come to think of it, you’re also not allowed to lose any of the items (because that’s the gotcha you were going to try next, am I right?). If all those conditions are met, then the new collection will contain four objects.”
But then I ask you to explain what you mean by “blurring the boundaries”. When two clouds seem to merge, that may be a trick of perspective. They may be at different altitudes, and only seem to merge because of where I’m viewing them from.
This kind of back-and-forth can go on for a long time. It might seem as sophomoric as dorm-room arguments about solipsism (“How do we really know anything? How do I know I’m not a lonely ghost in a void?”), but 20th century philosophers wrestled with the metaphysics of addition in a serious way. The more you explain what you mean by saying that 2+2 equals 4, and the more I raise pesky objections, and the more you counter my objections by hemming in the assertion with qualifications that remove loopholes, the less 2+2=4 looks like a statement about reality and the more it looks like a statement about how we look at the world, the interpretations we make, and the rules we apply in mentally dissecting and reassembling the world.
Take this to extremes, and you find 2+2=4 to be a statement that’s not about the world at all, but about how the mind perceives and categorizes. Now take one crucial step further and remove any explicit acknowledgment of Mind, so that all that’s left is the possibility of a Mind and the possibility of a World, and the truths of math seem to survive as ghostly sorts of constraints on possible minds — ways of saying “Well, if there were a world, and there were a mind that tried to understand that world by cutting it into pieces and putting the pieces back together again, here are the kinds of experiences that that mind would have.”
This non-place we’ve arrived at is a strange place to be, but it’s where I do most of my work. Some call it Plato’s realm of pure form, and some even think it’s even more real than ordinary reality, though that sounds crazy to me. But I can see the appeal of the conceit. The realm feels like a place to me, with stable features that persist from one visit to the next. It’s something like the ghostly void of the solipsist fantasy, but equipped with furniture to bump into. In this realm, or at least the sub-realm first mapped by Giuseppe Peano, it’s a fact that 2 is 1+1 is (or rather a definition; 2 is defined to be 1+1), it’s a fact that 3 is 2+1 (again, by definition), and it’s a fact that 4 is 3+1 (once again, by definition), and armed with these facts we can prove 2+2 = 4:
2+2 = 2+1+1 = 3+1 = 4
(If you want a longer proof, you can insert an extra step where the associative property of addition is exploited, but if you’re the kind of person who knows about the associative property then you probably didn’t need me to tell you that. And if you know that Peano didn’t actually define 2 as 1+1 but rather defined 2 as “the successor of 1”, and similarly for 3, 4, etc. … well, then you don’t need me to tell you that either.)
So we’ve found a realm in which 2+2 is absolutely 4. What else is there to say?
It turns out there’s a lot more to say. Because even if the Platonic realm is (as some claim) more real than we are, we only know it through our finite and fallible human minds. “2+2 = 4” became human knowledge through historical/social/psychological processes (how else could humans have come to know it?), and whenever people come into a story the story gets complicated.
MATH, COMMERCE, AND CAPITAL
One way to bring people into the story is to consider where the symbols3 in “2+2=4” came from, and how. The 2’s and the 4 are Indian-Arabic numerals, brought into Europe for commercial purposes in the late Middle Ages. One of the original Arabic treatises on the decimal system, Muhammad ibn Musa al-Khwarizmi’s “The Hindu Art of Reckoning”, gave us the word “algorithm” as a latinization of the author’s name. Sometimes it’s claimed that the new algorithms won out in Europe because they were more efficient for calculations than the abacus methods that preceded them, but that’s not entirely clear. What is clear is that calculations done in writing using the new symbols were more auditable than the evanescent motions of sliders on a rack or tokens on a board, and that this auditability helped trading companies expand their operations to ever-larger regions of the globe.
Historians of mathematics agree that the explosion of trade in the late Middle Ages and early Renaissance played a major role in spurring the development of mathematics. Mathematics in turn made industrial capitalism possible, by streamlining the ways in which the flow of capital and labor could be regulated. Regardless of whether you think capitalism is “good” or “bad” (I think it’s both), it’s important to recognize the economic aspect of 2+2=4 as part of its social history.
Let me acknowledge some personal bias: I like money. And I don’t mean that I like to have money (though that’s true too). What I mean is, I’m glad that I live in a society that has a universal medium of exchange, because I’d find barter overwhelmingly complicated. Which brings me to a story about a fateful 19th-century barter between a European and an African whose reverberations are still with us.
THE GENTLEMAN AND THE SHEPHERD
The famous Charles Darwin had a famous (and nowadays also infamous) cousin, Francis Galton. Like Darwin, Galton did a fair bit of traveling. In his 1853 book “Tropical South Africa”, describing his travels among the Damara people in what is now Namibia, Galton wrote:
When bartering is going on, each sheep must be paid for separately. Thus: suppose two sticks of tobacco to be the rate of exchange for one sheep, it would sorely puzzle a Damara to take two sheep and give him four sticks. I have done so, and seen a man first put two of the sticks apart and take a sight over them at one of the sheep he was about to sell. Having satisfied himself that that one was honestly paid for, and finding to his surprise that exactly two sticks remained in hand to settle the account for the other sheep, he would be afflicted with doubts; the transaction seemed to come out too “pat” to be correct, and he would refer back to the first couple of sticks, and then his mind got hazy and confused, and wandered from one sheep to the other, and he broke off the transaction until two sticks were put into his hand and one sheep driven away, and then the other two sticks given him and the second sheep driven away.
Stories like this are part of a long tradition in which Europeans depicted Africans and other foreigners as being ignorant or stupid. Even when Galton acknowledges differences between one African and another, he attributes those differences to genetic causes (“The Damaras were for the most part thieving and murderous, dirty, and of a low type; but their chiefs were more or less highly bred”). Galton went on to found the eugenics movement, a program of conscious breeding among upper class Brits that might have ended up a quaint footnote in the annals of English dottiness but for the fact that its poisonous ideology of racial difference found fertile soil in the post-Civil War United States, Hitler’s Germany, and elsewhere.
I learned this story from an essay by Michael Barany, listed in the References. One of the points Barany makes is that we never get to hear the shepherd’s side of the story. Meanwhile, Galton’s account is suspiciously omniscient; in particular, I’m struck by the phrase “his mind got hazy and confused”. Really? How could Galton know what was going on in the shepherd’s mind? I’m often puzzled in the classroom by things my students say, but I don’t pretend to know what’s going on in their minds. Sometimes I discover that a student’s wrong answer is the right answer to another question, and that my question wasn’t completely clear — and that’s in a classroom in which the teacher and the students are all speaking the same language! Galton had to rely on interpreters and his own guesses, and he may have missed what was really going on.
Consider that, for a shepherd, a flock is a collection of individuals, more like 1+1+1+1 than 2+2 or 4. The value of a sheep as measured in sticks of tobacco might vary from sheep to sheep. More importantly, value might not be what economists call additive. For instance, a female sheep is probably worth more than a male (as we might say in symbols nowadays, F > M), but two female sheep are probably worth less than a breeding pair (F + F < M + F), which is a mathematical contradiction if we assume value is additive.4 Galton’s assumption that any shepherd who’s willing to trade two sticks of tobacco for one sheep must (if he’s rational) be willing to trade four sticks of tobacco for two sheep makes sense in the context of a mercantile economy based on interchangeable goods, but doesn’t fit so well with a barter economy based on goods that are far from identical. And even when goods are identical, modern economists recognize that value can behave nonadditively. Here’s my favorite example: If you’re willing to sell me one of your kidneys for two million dollars, does it follow that you’re willing to sell me both your kidneys for four million dollars?
Galton also derided the shepherd for being unwilling to start the second transaction until the first was completed. But in fact there are pitfalls associated with making two overlapping transactions, notably the “change raising scam”, which gets its punch from cognitive overload; when there’s a lot going on, the person operating a cash register may forget that a bill on the counter is supposed to be on its way from the customer’s pocket to the register and not the other way around.
Even smart people can be fooled by the con, and more importantly, most people are unaware that they can be conned in this way. For all their education, modern urbanites in the English-speaking world usually lack metacognition about their ability to be fooled when their short-term memory is being overtaxed.
In comparison, that shepherd could be viewed as a metacognitive sophisticate, wisely separating two transactions rather than trying to combine them!
The big cons going on these days involving true-but-irrelevant math are hidden from view inside computers, or should I say, inside black-box algorithms. When I was young, the word “algorithm” meant a procedure for operating on numbers (think: long division), and later on when I studied computer science, some of the basic algorithms I got to know were procedures for sorting numbers, with cute names like QuickSort, HeapSort, MergeSort, and BatchSort. You could go into a lecture with no previous knowledge of such an algorithm and emerge an hour later with a clear understanding of why the algorithm always found the right answer and how long it took to find it.
Nowadays the term “algorithm”, when used outside of academia, mostly refers to procedures for compressing information, such as the procedure that takes your entire Netflix viewing history (along with dozens of other facts about you and an enormous number of facts about movies) and distills it down to a recommendation for what movie you’d enjoy watching next. You can’t get an intimate knowledge of these algorithms in an hour, a day, or a year. These algorithms aren’t humanly understandable because they weren’t created by humans; they were created (or at least tuned) by computers through a process called machine learning. It may not make obvious sense to add your zip code to your age, but if a learning algorithm finds that this sum is predictive of what movies you’ll watch, that’s what it’ll use.
Algorithms, in the new sense of the word, aren’t perfect (it’s not even clear what constitutes the “right” answer), but imperfection isn’t always a big problem. If a movie recommendation engine picks a movie you don’t like, you can always switch to another movie. The real problem comes when algorithms of this new kind are used not for recommending movies but for sorting people, deciding who is credit-worthy, college-worthy, job-worthy, or parole-worthy. Guess who tends to benefit from these algorithmically-driven decisions: those who already have lots of social privilege or those who don’t?
When a people-sorting algorithm makes a mistake (that is, outputs an answer that most observers agree is wrong), it isn’t easy to track down where the algorithm went astray because it’s so inhumanly complicated. And that’s assuming that the owners of the algorithm are willing to open the hood, which is usually not the case. It takes a big outcry (like the recent International Baccalaureate scandal) to force the owners of the algorithm to open that hood and let the world peek inside. And even when there’s an outcry, there’s a tendency to view the algorithms as authoritative, because aren’t the algorithms based on math? And how can math be wrong?
I mentioned before that a big advantage of the algorithms introduced in Europe in the late Middle Ages was their auditability. How ironic that 21st century algorithms turn back the clock and return computation to the shadows!
Most victims of algorithmic injustice are powerless individuals, unaware of each other and sometimes unaware of the nature of their victimization. This problem is likely to get worse, not better, in years to come, in education and elsewhere in society. And unlike a miscalculated Netflix recommendation, this is not a movie we can just stop watching.
I talked about how the story of the equation 2+2=4 intersects with the story of commerce and capitalism and then I talked about how it intersects with racism, eugenics, and genocide. So we’re not on purely mathematical turf anymore, and you can imagine how battles over 2+2=4 could happen on social media as a proxy for battles over real-world issues.
Many people I sympathize with politically stake out positions in these battles I don’t find very convincing. Some of them try to find contexts in which 2+2=5, but it always involves subverting the common shared meaning of 2+2=4 (a subversion that they sometimes admit to and sometimes don’t). For instance, it’s absolutely true that if we round the numbers in the equation “2.3+2.4=4.7” to the nearest whole number, we indeed get “2+2=5”. But that doesn’t mean 2+2=5.
I can play this game too. In music theory the interval of a 2nd (a step up from F to G) plus the interval of a 2nd (a step up from G to A) yields the interval of a 3rd (from F to A). But does that imply “In music theory, 2+2=3”? I don’t think so.
Sometime soon I’ll tell you about a context in which mathematicians write 1+1=0, and another context in which they write 1+1=1. But these are different number systems than the ones you learned about in school. In these number systems, the symbols “0”, “1”, “+”, and even “=” can have a different meaning than in the standard context, and mathematicians don’t pretend otherwise. And even in these bizarro arithmetics, “2+2=4” is still true (though in one of the arithmetics 4=5 so 2+2=5 is also true).
Interestingly, math researchers on Twitter were on the whole fairly sympathetic to challenges to 2+2=4. When you spend years training your imagination to make sense of the bizarre and counterintuitive, and someone says “There are contexts in which 2+2 isn’t 4”, part of your mind goes “Hmm, I wonder what such a context might be?”
In my reading on equity, mathematics, education, etc., one comment I found trenchant (from the book “Ethnomathematics: Challenging Eurocentrism in Mathematics Education”) is the juxtaposition of the true assertions “2 apples plus 2 pears equals 4 fruits” and “2 pants plus 2 jackets equals 2 suits”. This example demonstrates how real-world knowledge can invisibly pervade a math problem, making mathematical content accessible to those who have the real-world knowledge and inaccessible to those who lack it. It makes me think harder about what knowledge my students bring into the classroom.
I’m unmoved by slogans like “Western mathematics is a tool of cultural imperialism”, though I’m sympathetic to concrete critiques of specific teaching practices that I believe are the wellspring of the math-as-imperialism ideology voiced by some math educators.5 At the same time, I’m aghast at the sort of sexist and racist tropes that have been flung at proponents of critical theory by self-proclaimed defenders of “objectivity” when the proponents are women or people of color or both. I’ll take ideology over bigotry any day. In a profession that’s still too dominated by white men, I’m willing to round down some overheated rhetoric.
I think the people who voice concern about 2+2=4 being a cultural imposition should be more worried about the algorithms that use valid equations like 2+2=4 in invalid ways to create inequitable outcomes. To be empowered to challenge those algorithms, students need an understanding of the mathematics of technocracy, not the mathematics that their ancestors used.
I said at the beginning that 2+2=4 is seen by some as a touchstone for objectivity, and by others as a manifestation of a repressive social system. And now you know that I think both are right. The mercantile-capitalist-technological civilization that gave us the mathematics of the modern era has done wonderful things and terrible things. Most scientific knowledge was discovered by flawed people, working in flawed institutions that excluded many people from contributing, situated within economic systems that exploited many people at home and abroad. There’s no way to purge this knowledge of its origins. We can and must try to do better, but that doesn’t mean we throw away what we’ve learned.
If the equation 2+2=4 can be viewed as a story of accumulation and wealth-building, its companion 4=2+2 tells a different story, of equitable sharing. Likewise, if 4=2+2 seems to tell a story of divisiveness, of a society coming apart into factions, 2+2=4 tells a story of people coming together. We need to find better ways of dealing with difference, cooperating, and sharing. Properly applied, the mathematical knowledge that our species has acquired, including 2+2=4, can bring us closer to a world in which all can share the fruits (the apples and the oranges) of progress.
Thanks to Michael Barany, Keith Devlin, Sandi Gubin, Brian Hayes, David Jacobi, David Merfeld, Ben Orlin, Evan Romer, and Mark Saul.
Next month: When 1+1 Equals 0.
#1. I first heard this story as a Jewish tale about the citizens of a city of fools called Chelm, and my guess is that versions of the story exist in other cultures as well.
#2. Although a voter can’t be registered with more than one party, a candidate can be endorsed by more than one, as for instance happened when Earl Warren, back before he served on the U.S. Supreme Court, ran for Governor of California and won endorsements from both major parties and some minor ones as well.
#3. The symbols “+” and “=” came centuries after “2” and “4”. “+” was introduced in Europe in the 1400s as an abbreviation of the Latin word “et” (meaning “and”). The equals sign was invented by Robert Recorde in the 1500s as a deliberate coinage, consisting of twin lines of the same length “because no two things can be more equal”.
#4. Thomas Aquinas similarly argued that just because an angel is better than a stone, it doesn’t necessarily follow that two angels are better than one angel and one stone. Perhaps he should be hailed as the patron saint of diversity.
#5: Often you’ll see titles like “Western mathematics as a tool of cultural imperialism”; the “as” can be read either as “is nothing more than” or “is in some respects”, according to one’s audience, and I think the ambiguity is intentional. Maybe someone should write an article called “‘As’ As Multivalent Signifier”?
Marcia Ascher and Robert Ascher, Ethnomathematics, in “Ethnomathematics: Challenging Eurocentrism in Mathematics Education”.
Michael Barany, One, Two, Many: The Prehistory of Counting. https://www.newscientist.com/article/mg21028081-500-one-two-many-the-prehistory-of-counting/
Meredith Broussard, When Algorithms Give Real Students Imaginary Grades, https://www.nytimes.com/2020/09/08/opinion/international-baccalaureate-algorithm-grades.html
Keith Devlin, Of Course, 2+2=4 is Cultural. That Doesn’t Mean the Sum Could be Anything Else. https://www.mathvalues.org/masterblog/of-course-2-2-4-is-cultural-that-doesnt-mean-the-sum-could-be-anything-else
Cathy O’Neil, Mutant Algorithms Are Coming for Your Education, https://www.bloomberg.com/opinion/articles/2020-09-08/mutant-algorithms-are-coming-for-your-education
Cathy O’Neil, Weapons of Math Destruction. https://weaponsofmathdestructionbook.com/
Ben Orlin, The Smartest Dumb Error in the Great State of Colorado. https://mathwithbaddrawings.com/2015/08/19/the-smartest-dumb-error-in-the-great-state-of-colorado/