There has been a really weird… phenomenon, I think mostly on Twitter, where some people referred to 2+2=4 as a basic, incontrovertible fact, and some other people have been invoking higher mathematics to say that sometimes 2+2=5. Since I’ve got a master’s degree in math, I thought I might as well explain what’s going on, and when 2+2=5.

To get an obvious point out of the way so as to avoid misunderstanding, in normal conversation, when people talk about 2+2=4, they are clearly referring to the natural numbers under one of the the ordinary definitions of addition (such as under the peano axioms, where addition is combined succession). Given that set and that operation, 2+2 equals 4, always, no exceptions.

Where higher level mathematics comes in is that a mathematician is free to consider any set whatsoever, and to define any operations on it that he wants. Most of these sets and operations won’t be interesting, but they will be valid. Let me give you an example of such a set with such an operation, so you can see what I mean.

Let us define the set F, such that F={2,5} (that is, F has two elements, one called “2” and the other called “5”). Let us define an operation on F×F→F (the cross product of F, mapping to F), called +. It is defined such that + of any two elements in F is the element “5” in F. Thus, 2+2=5. Under this definition, it is also the case that 2+5=5 and 5+5=5.

OK, so we now have a set F and an operation + such that in F under + as we’ve defined it, 2+2=5. Whoohoo. This is obviously a completely uninteresting set and operation. There’s nothing to say about it. There’s no reason one would ever want to actually define it, other than as a joke. (There is a certain similarity, here, to the classic set of outcomes to a coin flip, “heads I win, tails you lose”, if that is of any interest.)

But this is it. It’s not some great truth of higher mathematics, it’s a trivial side-effect of what is doable in higher mathematics which has generalized the more common mathematics.

I should note that there are very interesting sets and operations to define. For example, given a prime number m, the set of non-negative integers less than m, with addition defined to be addition modulo m, is quite interesting. (For those not familiar, modular arithmetic is basically doing arithmetic and if the result is larger than the modulus, you keep subtracting the modulus until it isn’t. Thus (2+6) % 7 = 1, since 2+6=8, but 8 is bigger than 7, so we subtract 7, and 8-7=1. You can also define modulus in terms of remainder after division.) These are quite interesting because the operation we’ve defined as + is closed, so every number has an additive inverse. It would take too long to explain why that’s interesting, but the short short version is that it’s the basis of many cryptographic algorithms, including some of the most widely used public key cryptographic algorithms like RSA which underly most encrypted web traffic.

Thus I hope it’s clear that it is not the case that higher mathematics is all uninteresting. It is only that the abstractions which are used in higher mathematics have boring side-effects. This is no different from the oddity of “Buffalo” meaning (1) an animal, (2) a city in NY and (3) to confuse someone allowing one to claim that “Buffalo buffalo buffalo buffalo buffalo” (meaning that buffalo from the city of Buffalo confuse buffalo from the city of Buffalo) is a perfectly valid English sentence. It is, but this is, at most, an amusing accident. It doesn’t mean that English is stupid, nor that people who study language are stupid. It’s just that complex things occasionally have odd quirks. What is true of the English language is also true of higher mathematics.

And none of this means that the people who say, “well, actually, sometimes 2+2=5…” aren’t intentionally missing the point.

In closing, let’s never forget the point in this XKCD comic: