While scrolling through my twitter feed I came across a series of threads all essentially asking the same thing: what is the meaning of the phrase 2+2=5?

If you’ve read Orwell’s 1984, you’ll know this as the dogmatic statement used by the Party to brainwash Smith into mental submission via doublethink. If you haven’t read 1984… I don’t know what you’re doing with your life go read it.
The main take-away, though, is that this phrase has very strong political undertones of anti-intellectualism and governments trying to politicize truth and deny reality to “control the masses”.

Nazi theory, indeed, specifically denies that such a thing as “the truth” exists. There is, for instance, no such thing as “Science”. There is only “German Science”, “Jewish Science”, etc. The implied objective of this line of thought is a nightmare world in which the Leader, or some ruling clique, controls not only the future, but the past. If the Leader says of such and such an event, “It never happened” — well, it never happened. If he says that “two and two are five” — well, two and two are five.

George Orwell, “Looking Back on the Spanish War”

So this is going to be a mess to tackle, as you may already be able to tell. Let’s dive into it.

These are the types of tweets and posts that had people concerned. It started around summer but even now there are conversations and threads continuing.

The main premise of these kinds of posts is that “2+2=4” is a truism only in “white” culture, since those symbols and mathematical concepts were standardized by white societies. For example, we use base 10 but Babylonian societies used base 60 for their number system.

I think there’s a miscommunication taking place here on both sides. On the one hand, the guy who said “math is not universal” probably didn’t intend to say that the way he did. What I THINK he meant was “the mathematical symbols and constructs we use aren’t necessary universal”. And to that, I agree. Of course for someone working modulo 3, the expression “2+2 = 1” might as well be true.

The key distinction here is that there are different ways to represent the same abstract concept. Whether you represent 4 as “10” in base 3 or as “-1” in modulo 5, we are all talking about the same thing here: the 4th smallest positive integer.

But what of the argument that “2+2 = 5”?

I think the argument here enters a realm of philosophy more than pure mathematics at this point, since we have to consider how we construct mathematical knowledge in the first place. There are essentially two schools of thought: mathematical platonism and mathematical formalism.

Mathematical Platonism is the idea that the concepts of “2” and “4” and “addition” exist beyond the realm of human though. That is, if there were no humans or intelligent life there to write it down or imagine it, the concept of “2” and “4” would still exist. This goes not just for basic numbers and operations but even for the more advanced things, like perfect circles, straight lines, a mathematical limit and so on. This is probably the view that most people hold about what mathematics is.

In this scenario, we can say without a shadow of doubt that “2+2=4” and it always will be. Because the integers 2 and 4 are said to exist in a place outside of our brains.

Mathematical formalism, on the other hand, is essentially the idea that math exists only insofar as we dream it up. Whereas mathematical Platonism says that the concept of the derivative always existed in some abstract space and humans merely ‘discovered’ it, formalism tells us that the differentiation did not exist in our universe before Newton and Leibniz thought it up.

One of the interesting ideas brought up by the conflict between these two schools of thought is whether unsolved problems are “True” in some sense. Most of modern math is built from 9 axioms known as the ZFC. Every theorem is practically built FROM these base axioms using pure logic. However, we know some problems, such as the Continuum Hypothesis, are undecidable from these axioms, meaning it cannot be proved or disproved from those 9 statements alone. Now, some Platonists might argue that there exists an abstract universe of sets where the hypothesis is determinately true or false, and our axioms simply don’t reflect that well enough. A formalist would say that the hypothesis is simply undecidable and remains that way, and we can either add an axiom to make it true or false, but we hold that power.

The question of whether “2+2 = 4”, then, is really up to us to decide. If you are a mathematical Platonist, the statement is simply true. Given the existence of “2”, “4” and the addition operation in some higher space, obviously 2+2 is equal to 4. However, if you are more of a formalist, you might be of the opinion that “it depends”. Depends on the axioms we use and the system of arithmetic we are working in. Sometimes it may equal 4, sometimes it may equal 5, other times it may equal 0.