# 2+2 is 4 and the world is flat: assumptions and approximations

The argument that “2+2=4” is a social construct not found in every society and that in some places 2+2=5 is an interesting exercise in sophistry.

It is true that you can redefine every part of an equation (or every word in a sentence) to mean something other than what a naive reader would normally assume based on all previous experience with words. Obviously if we redefine 2 to mean something other than 2, + to mean something other than addition, or chose to use a base other than base 10, then we can get an answer other than 4. For example, if we are using base-3, then 2+2 = 11. (Of course, when we convert back to base ten, “11” becomes plain old 4 again.)

This is true of every sentence: if I redefine all of the words in a sentence to mean something else, then the sentence means something else–but no one uses language in this way because it makes communication impossible.

In particular, when children are taught that 2+2=4, they being taught within a system where 2 means two of something, 4 means four of something, and + means conventional addition. When we use these definitions, 2 + 2 always equals 4. There are, in fact, zero societies on Earth where this equation, as used in elementary schools, comes out to five.

There is no mistake involved in assuming that other people are using common conventions when speaking and will specify when using terms in unexpected ways. This is how all communication works. Since we cannot define all words from first principles every time we use them (this is impossible because it would require us to define the words used to define the words used to define the words, etc,) we only bother to define them when using them in unconventional ways, and even then we use conventionally defined words to define them. If a word is not defined or otherwise marked as being used in unconventional ways, then the receiver assumes that it is being used in its conventional sense because language cannot function otherwise.

Behind the scenes trickery is simply that: trickery. The sentence as normally defined and automatically understood is always correct.

There is a story about the time Denis Diderot visited the court of Catherine the Great in Russia. Diderot’s atheism offended the great lady, so she had her court mathematician, Leonhard Euler, confront Diderot with a complicated algebraic equation, then proclaimed that this proved the existence of God.

The tale is perhaps apocryphal, but it has inspired the coining of the term “Eulering”: the use of a complicated argument to confuse your opponent into conceding, especially on some irrelevant point. Modern Eulering often consists of saying something that sounds blatantly false, then when this is pointed out, ridiculing your opponent for not knowing that you had secretly redefined the words. If you were a real expert, the argument goes, then you would know that 10=2 is just as valid as 10=10, because base ten is just a social convention we use to make writing numbers easier, and all other bases–including base pi–are equally legitimate from a mathematical perspective.

This is not expertise, but sophistry. There is no mistake involved in assuming that other people are using common conventions when speaking and will specify when using terms in unexpected ways. This is how all communication works.

It is true that math as taught to children is simplified: all subjects are simplified by necessity for introductory students.

When a child learns to read, he is first taught to pronounce the letters phonetically; complications like “silent e” and “-tion” are only introduced later. The full complexity of English spelling, from rhythm to pterodactyl, is only revealed to advanced students who have already mastered simpler words. If we attempt to reverse the order of instruction, chaos results: students are forced to learn every single word independently, instead of applying general rules that get them through most of the words and help them develop further rules for the exceptions.

The same happens in math; children are taught to count and add with the help of simplifying assumptions like “triangles are flat” and “base 10.” You don’t teach a toddler to count by beginning with -10 and then explaining that “3” is written as “11” in base two. It doesn’t work.

When you learn physics, you begin with Newtonian dynamics, because these are easy to demonstrate at normal human scales. It is only after mastering the basics of F=ma, objects falling at 9.8 m/s^2, and maybe a bit of calculus that you move on to topics like “What happens when you move close to the speed of light?” or “What happens at the atomic scale?”

Subjects are taught in a particular order that equips students with general rules that work in most situations, then specific rules that cover the most common exceptions. Most people will never need to know the “expert level” versions of most fields. For example, most people do not need to understand why airplanes can fly in order to make a reservation at the airport and go on a trip: it is sufficient to know simply that planes fly.

To argue about whether the “basic” or “expert” versions of these fields is more correct  generally misses the point: each serves a specific purpose. If I am calculating the distance between my house and my friend’s, I do not need to factor in the curvature of the Earth; if I am calculating the distance between my house and the antipodes, I do. If I am balancing my checkbook, I can safely assume that all of the numbers are written in base-10; if I am trying to figure out if the 16-bit integer limit will make my airplane crash, then it helps to know binary. If I tell my kids to “stay still so I can take your picture,” I don’t want to hear that Brownian motion technically makes it impossible to hold still.

The current bruhaha on Twitter over whether “2+2=4” is racist or not is half math geeks happy to finally have an audience for their discussion of obscure math things and half “school reformers” who wouldn’t know ring addition if it hit them in the face but want to claim that it has something to do with early elementary math. (Spoiler: it doesn’t.)

In the case of math, yes, math is a social construct, inasmuch as we could use a different numerical base or different wiggly symbols to represent the numbers on paper. Spelling is also a social construct: there is no particular reason why “C” should be pronounced the way it is, much less should we have a silent “L” in “could” (the L in could is actually the result of a centuries-old spelling mistake: “would” and “should” both contain Ls because they are forms of the words “will” and “shall,” which contain Ls. Could is derived from the word “can,” which does not have an L, but because “coud” sounds like “would” and “should,” people just started sticking an erroneous L in there, and we’ve been doing it for so long that it’s stuck). Money is also socially constructed: there is no particular reason why little green pieces of paper should have any value, and in many cases (lost in the woods, hyperinflation, visiting a foreign country), they don’t. Nevertheless, you need to be able to count, spell, and use money to get along in society, which we live in. If math is racist simply because it is socially constructed, then so are all other social constructs. Pennies are racist. Silent “e” is racist.

This absurdity is no accident:

h/t @hollymathnerd, quote by Shraddha Shirude, “ethnomathematics” teacher.