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I know the 2+2=4 discourse is no longer raging, but I want to add a point that I hadn’t seen made: *of course* mathematics is cultural, and on the whole that’s a *good thing*. Culture is how humans interact with ideas. What we know as “mathematics” we know only through culture.
To illustrate: our 2.75-year-old daughter has become a huge fan of the spectacular CBeebies (BBC) series @Numberblocks. She has watched all 90 episodes, some dozens of times. In this show, numbers become characters who can combine and rearrange to solve puzzles & tell stories.
One is curious and likes making friends. She’s good at it, too: you can always get a new number by adding one!
Two likes being friends, because it takes two to be friends. Unlike most Numberblocks, he wears shoes, because shoes come in pairs, which is because humans have two feet, which is because humans and many other animals have bilateral symmetry. This is biology, not yet culture.
Three is an entertainer. She likes to juggle, because we tend to think of “juggling” as requiring at least three items, so that one is in the air at all times. This is definitely cultural.
A trend that emerges from here on is that most prime numbers’ personalities are defined by cultural signifiers, because their primary mathematical attribute is, well, being prime.
Four is very proud of being a square! He likes square things, and rectangles, and he’s nervous when round things appear. This is a case where the Numberblock’s personality traits mostly arise from a mathematical property.
Five, like Three, is an entertainer, but for a different reason: we often represent stars as having five points, and a “star” is someone who excels at performing. This is absolutely cultural. Five also has a hand with five fingers, which goes back to biology, like Two.
A cube has six faces, and the cubes that play the most recognizable role in our culture are dice. So Six has pips on each block and likes playing games. (She also likes to rap; I haven’t quite figured out why yet.)
Seven is a very lucky character, obviously. Well, it’s obvious if you’re steeped in a culture that attributes luck to 7. His blocks are all the colors of the rainbow! (Well, a rainbow actually has infinitely many colors, but as we know from Roy G. Biv, only seven of them count.)
To skip ahead, Thirteen is very unlucky. He’s always tripping or slipping or dropping things. In the case of this character, whenever someone says his name “his three falls off” (because he’s made of 10 and 3—do you see the Three in the character design?).
Backing up a bit, Ten is two fives, so she has *two* hands and *two* stars on her eyes (mathematical properties meet cultural signs). When she stretches to her full height, she can become a rocket! Why? Because “countdowns” traditionally start at 10.
Skipping ahead again, Fifteen is a spy. Why is that? Because she’s “secretly” 1+2+3+4+5. Fifteen is not the first triangular number to appear, but it’s perhaps the most distinctive mathematical property she has, so that “secret” property leads to espionage—and the “Step Squad.”
I’ll describe one more Numberblocks character, perhaps my favorite: Eight. Or, as he prefers to be called, Octoblock! Octoblock has eight arms (or are they legs?), like an octopus. And because “eight” also has associations with spiders, he’s a superhero like Spider-Man!
This thread so far has been a recommendation for you to watch @Numberblocks, if you haven’t yet. (It’s available on @Netflix and @YouTube.) But back to the original point: these characters are brought to life by their personalities, which are *culturally* informed.
And, I would argue, the cultural elements are *improving* our daughter’s number sense. She interacts with the Numberblocks not just as representatives of equivalence classes of sets that have the same cardinality, but as fully-fledged characters. She *recognizes* them both ways.
We had a household argument the other day as to whether 7+1 makes 6 or 8. That may seem trivial, but let me remind you that our daughter is not yet three. I believe she *remembered* (imperfectly) a story involving Seven and One and Six, as well as the underlying number fact.
In her mind, the possible answers did not include 5 or 10, because that’s not how the characters she knows interact. And I know she also understands the number statement, because when we put 7 Lego blocks together with another 1, she could count them as 8, and accept the answer.
My point is, the symbols (and words) we use to express 2+2=4 are just as cultural as the Numberblocks’ personalities. We all learned mathematics through culturally-dependent signs, and for some those signs have become a barrier because they do not have access to the same culture.
It’s not that the concept of “two” depends on our cultural understanding, but the way we abstract it from particulars does, and the way we combine numbers through addition (“a symmetric binary operation with identity element 0”) and express equality of abstract numbers does.
The people who tend to criticize the existence of conversations about the cultural-dependence of mathematics tend to be the same people who disparage serious efforts to deepen students’ mathematical understanding (“why not just teach the standard algorithms?”), e.g. Common Core.
