That Whole 2+2=5 Thing
No, 2+2 does not equal 5, but that was also never the point
Back in August 2020, an article showed up in Popular Mechanics called “Why Some People Think 2+2=5,” with the tantalizing subtitle, “…And why they’re right.” Spicy!
It was essentially an article about one Twitter thread from Harvard PhD candidate Kareem Carr, where he explains why he thinks people who say 2+2=5 are making “a tremendously deep point.” Some people got very, very pissed off from this thread.
2+2 is not equal to 5. But I believe the most important part of Carr’s thread is actually not the whole “2+2=5” thing. A lot of people (critics and supporters) misunderstand his points, and alot of the criticism I saw boiled down to some nonsense like “higher education is a joke” or “this is your brain on Common Core.” Some of Carr’s points in the thread are very true and worth thinking about. But I do think Carr over-generalizes some of the math he’s talking about. To be fair, I’m guilty of over-generalizing the math in pretty much every single piece I’ve ever written about math, so I am far from innocent. Still this is my take on that whole 2+2=5 debacle.
Carr’s Thread
Let’s start by going over what Kareem Carr himself says in the Twitter thread:
Carr begins by saying that a statement like 2+2=4 is a kind of generalization; the equation is associated with something in reality, but it’s not actually part of reality. This is true: concepts like gravity and protons are tangible things in the world, but to riff on a Keith Devlin quote, you will never see a living, breathing “2+2=4" strolling down the sidewalk. An equation is a kind of abstract shorthand for some relation between numbers.
The next tweet is where things get shaky. Carr gives the example of people who say “I put a rooster and a hen together and… later there were three of them (1+1=3).” He then says that people who say stuff like this are making a “tremendously deep point,” forcing us to realize that numbers are abstractions.
I almost agree with him here, but it’s not the number that’s abstracted in this example, it’s the function of addition. The numbers are not the abstracted part. The number 1 in this example clearly refers to “one hen” or “one rooster,” and the number 3 clearly refers to “three chickens.”
What is addition, really? Addition is an abstraction of counting. When I say 2+2, I am saying “count 2 and then count another 2.” In Carr’s example, you are definitely not counting the chickens. You are using the notation + as a function of chicken mating. If we take addition to mean counting (the normal definition of addition), then “1 rooster + 1 hen = 3 chickens” makes no sense, because counting 1 rooster and 1 hen leaves you with 2 chickens total. But this equation kind of makes sense if you instead take + to mean “the act of mating” instead of “the act of counting.”
The Arithmetic of Chicken Mating
When someone says “I put a rooster and a hen together and… later there were three of them,” and then equate that to 1+1=3, what they are really showing is that we don’t have an operator for poultry fornication (operators are symbols like +,*,÷). They’re not making a profound point about addition, they’re just showing a gap in the supply of operators we currently have.
So let’s fix that, and make an operator for chicken mating: the symbol ⨁. The symbol ⨁ means “these two animals successfully mate.” We can build arithmetic around this symbol.
If I have “1” chicken, I need to know the gender of that 1 chicken. Of course, two male chickens cannot mate with each other, so we might decide to put exponents on our numbers to denote gender, like so:
- 1ᴹ = 1 male chicken
- 1ᶠ = 1 female chicken
- 1ᵁ = 1 chicken (unknown gender)
Assume that when chickens mate, they only have 1 chick. Now we can actually use our new chicken arithmetic. For example, 1ᴹ⨁1ᶠ=3ᵁ (3ᵁ because we do not know the gender of the newborn chicklet)
Does 2+2=5 in chicken arithmetic? Maybe, but these could also be true:
- 2ᴹ⨁2ᶠ=6ᵁ
- 2ᴹ⨁2ᴹ=4ᴹ
Let’s clean up our notation and state that 2ᵁ=2 and in general nᵁ=n for some natural number n. You will notice that:
4 ≤ 2⨁2 ≤ 6
We can generalize the above equation to get what I call the Fundamental Theorem of Chicken Mating Arithmetic:
2n ≤ n⨁n ≤ 3n (for any integer n)
Of course, chicken arithmetic is something I made up five minutes ago. But the point is, we can use the conventions of math to describe chicken mating much more accurately than addition can. Here’s what I think is actually going on when people say “1+1=3” with an example like this:
- They put two chickens together and the chickens have a chick. They want a mathematical way to express this event.
- They know that 1 (something) 1 = 3 where “something” is a function. But there’s no well-known term for the function of mating.
- Putting two chickens together to mate is kind of sort of close to addition, so they use addition notation since it’s close enough.
People who say 1+1=3 in this scenario aren’t really using addition, but they’re reaching for the closest thing available to a function of mating, and the closest thing happens to be addition.
What Carr Gets Right
The thesis of Carr’s Twitter thread is not “2+2=5,” despite what clickbaity articles will lead you to believe. Carr’s real thesis, in my view, comes midway through the thread:
He’s absolutely right. We have to be careful lining up math to reality. Just because a number has some property, that does not mean the property also applies everywhere that number shows up.
One property of the number 55 (or any number) is that it is equal to itself: 55=55. But if you and I both took an algebra test, and we both scored 55 out of 100 points, does that mean our knowledge of algebra is equal to each other, since 55=55? No, not at all. Our knowledge could be vastly different. You might know a lot about solving equations but struggle with the quadratic formula, and I might have the opposite problem. Just because our scores are equal does not mean our knowledge is equal. We may use the number 55 to measure something in the world, but that does not mean properties of the number 55 will also apply to the thing we’re measuring, in this case algebra skill.
Mathematical psychologist Jean-Claude Falmagne realized this as well, and fleshed this idea out into something called learning space theory. He also founded the learning software company ALEKS, which shows students’ knowledge not with numerical scores, but with a pie chart detailing a student’s exact knowledge state, the topics they know and don’t know.
We have to remember that Carr is a legit data scientist. In data science, we compress the world down to a series of metrics and numbers. This is super useful, but a lot of nuance can get lost. For example, I could throw together a graph of book sales for a bookstore in a given week:
But does this mean that Sunday and Monday were equally good days for the store, since the sales were the same? Not necessarily. Maybe on Monday we overperformed, selling way more books than we normally do on Mondays.
We run statistics on a business to get clarity on those fuzzy, fundamental questions. What kinds of books get sold? What are the strengths of the bookstore? Why does someone choose to buy a book from us? Statistics helps us get closer to the answers, by converting everything to numbers, but it comes at the cost of abstraction. This doesn’t make statistics bad, it’s just something to be cognizant of.
Is it True? Is 2+2=5?
So is 2+2=5? It could be true, I guess… but in a trivial way. Here’s what I mean. Is the phrase “Pigs can fly” true? Well, sure, if we redefine “pigs” as “a kind of watercraft that looks like this” and we redefine “fly” as “floating on a riverbank.” In that case, yes, pigs can absolutely fly. But surely there’s a problem here, because if we can redefine words to mean whatever we want, then words are useless.
A word is a category. Often, we start with a broad word, a category of things (animal). Then we take that word and make a more specific word for a more specific sub-category (dog) and we keep going, making more words for even more specific things (beagle). The same thing happens in math. We take a concept (number) and we make it more specific (prime number) and we keep making it more specific as needed (Mersenne primes).
It’s totally true that symbols can mean different things in different contexts. For example, there’s a branch of math called group theory, which studies how certain operations work in certain sets. “Operations” means actions like addition, multiplication, etc.
In group theory, the notation * is used for any operation. We are used to seeing * as “multiplication,” but in group theory, using * to mean “addition” or another operation is entirely normal.
If our operation * means “addition” and our set is integers, then an equation like 2*3=5 makes perfect sense in group theory, even though this would not make sense if we treated * as normal multiplication.
But there’s a difference between using * to mean “addition” in group theory and using 1+1=3 when describing chickens. Mathematicians reading a group theory textbook understand that they are using * for something other than the “normal” meaning. People who say 1+1=3 don’t seem to realize that they’re using something completely different than addition.
Communication requires us to agree on what symbols mean in context. In France, a price could be written as “2,56” whereas in the U.S. it would be “2.56”, but if I’m in France, everyone around me knows that the comma denotes a decimal. I could walk up to the cashier and claim that the price is actually $256 because my way of thinking is different, but I’ll probably just get thrown out of the café. It feels like a stretch to say that I’m making a deep point by claiming the price is $256 — it feels more right to say that I misunderstood the situation.
I like Kareem Carr’s Twitter, and it does kinda suck that Carr’s most interesting points were lost in the uproar of “2+2=5". Carr is absolutely right that numbers are abstractions. He is right when he says “math is logical but it is not always neat and nice.” We do need to remember that properties of numbers don’t always apply to real-world scenarios. . But if you say 2+2=5, it’s much more likely that you simply misunderstood addition.
