*[This is a transcript of the video embedded below.]*

2 plus 2 equals 5 is the paradigmatic example of an obvious falsehood, a falsehood that everyone knows is wrong. Because 2 plus 2 is equal to 1. Right? By the end of this video, you will know what I am talking about.

George Orwell is known to have used two plus two equals five in his novel Nineteen Eighty-Four as an example of a blatantly false statement that can still be believed.

The same example was used as early as 1789 by the French priest and writer Emmanuel Sieyès in his essay “What is the third estate”. At that time, the third estate – the “bourgeoisie” – made up most of the population in France, but was not allowed to vote. Sieyes wrote

So you can see that using two plus two is five as an example of an obvious falsehood has a heavy legacy. And if you say otherwise, it can understandably upset some people. For example, mathematician Kareem Carr was recently cheered on Twitter for pointing out that 2 + 2 isn’t always four.

He was accused of “waking up” for allegedly excusing wrong math as being okay. Even he was surprised at how upset some people were, because his point of view is of course absolutely correct. 2 + 2 is not always four. And by that I don’t just mean that the symbol “4” could be replaced by the symbol “5”. You can of course do that, but that’s not the point. The point is that two plus two is a symbolic representation of the properties of elements of a group. And the result depends on what the 2’s refer to and how the mathematical operation “+” is defined.

Strictly speaking, without these definitions, 2 + 2 can be pretty much anything. Hence the joke that you shouldn’t let mathematicians scrap your restaurant bill because they haven’t agreed on how to define addition.

To see why it is important to know what to add and how, let’s look back for a moment to see where the “normal” law of addition comes from. If I have two apples and add two apples, that’s four apples. Law? Law.

Okay, but how about that. If I have a glass of water at a temperature of 20 degrees and pour it together with another glass of water at 20 degrees, then the water together has a temperature of 40 degrees. Um. No, definitely not.

If both glasses contain the same amount of water, the final temperature is half the sum of the temperatures, i.e. still 20 degrees, which makes much more sense. Temperatures do not add up according to the rule of two plus two equals four. And why is that

This is because temperature is a measure of the average energy of particles and average values are not added up like apples do. The average height of women in the United States is 5 feet 4 inches and that of men is 5 feet 9 inches, but that doesn’t mean the average American is 11 feet 1. You have to know what you are adding to know how to add it.

Another example. Suppose you turn on a flashlight. The light moves at the speed of light. And as you know, the speed of light is the same for all observers. We learned that from Albert Einstein. Yeah, the guy again. Let’s say I turn on the flashlight while you’re running at me at, say, ten kilometers an hour. At what speed does the light come on * you? That is the speed of light plus ten kilometers per hour. Law? Eh, no. Because that would be faster than the speed of light. What’s happening?

What is going on is that the speeds don’t add up like apples either. They only do this approximately if all the speeds involved are much smaller than the speed of light. Strictly speaking, however, they have to be added according to this formula.

Here u and v are the two speeds you want to add together and w is the result. C is the speed of light. You can see immediately that if one of the speeds, say u, is also the speed of light, then the resulting speed remains the speed of light.

So if you add something to the speed of light, the speed of light doesn’t change. If you come running towards me, the light from my flashlight will still come at you at the speed of light.

Indeed, if you add the speed of light to the speed of light because you might want to know the speed at which two rays of light are approaching head-on, you get c plus c equal to c. In units of the speed of light, 1 + 1 is 1 according to Einstein.

These are a few examples from physics for quantities that only have different laws of addition. Here’s another one from math. Suppose you want to add two numbers that are members of a finite group to keep things simple, let’s say one with only three members. We can give these elements the numbers zero, one and two.

We can then define an addition rule for this group, which I write as a plus sign with a circle around it to make it clear that it is not the usual addition. This new addition rule works like this. Take the usual sum of two numbers, divide the result by three, and take the remainder.

So for example 1 + 2 = 3, divide by three, the remainder is 0. This law of addition is defined in such a way that it keeps us in the group. And with this law of addition, you have 1 plus 2 equals 0. By the same rule, 2 plus 2 equals one.

I know this looks strange, but it’s perfectly normal math, and it’s not even very advanced math, it’s just not generally taught in school. This remainder after division is usually called the module. So this law of addition can be written as the plus with the circle equal to the normal plus mod 3. A set of numbers with this law of addition is called a cyclic group.

This works not only with 4, but with any whole number. For example, if you take the number 12, it just means that if you add numbers to anything greater than 12, you will start over from zero. This is basically how clocks work, 8 + 7 = 3, add another 12 and that comes back to 3. We’re pretty used to that.

Clocks are a nice visual example of adding numbers to a cyclic group, but the measurement of time itself is not an example of cyclic addition. Because the “real” physical time does not run in circles, of course. It’s just that on a simple clock we may not have an indication of the time change from morning to evening or to the next day.

In summary, when you add numbers you need to know what you are adding and use the correct law of addition to describe what interests you. If you take two whole numbers and use the standard law of addition, then yes, two plus two equals four. But there are many other things that these numbers could stand for, and many other laws of addition, and depending on your definition, two plus two can be two or one or five, or really anything. That didn’t “wake up”, that’s math.