2+2 = not what you think
The factoid 2+2=4
is often used as an example of a claim that is unequivocally and objectively true, but what I like most about it is that it’s precisely the opposite: not necessarily quite true.
Before explaining why, I would like to point out that the objective is not to explore mathematical claims, it’s to raise consciousness about uncertainty. Even things you are 100% certain of, might not be necessarily true. Everything should be doubted.
What is 2+2
? Clearly it’s an addition, which is an arithmetic operation, which is one of the first things we learn about mathematics, so there’s no mystery here. Except philosophers Alfred North and Bertrand Russell spent hundreds of pages in Principia Mathematica to arrive at the conclusion that 1+1=2
. We are not going to deconstruct arithmetic here, merely point out that we are dealing with arithmetic, and that’s not as straightforward as it might initially appear.
You may think you know what a+b
is, but quite likely you are making the assumption that a
is a real number, which is not necessarily true: it could be a vector. Turns out that a
could be many things: a movement in space (rotation) or a movement of a Rubik’s cube. All of these things are explored in abstract algebra, but the short story is that addition doesn’t just apply to numbers.
OK, but 2
is a number, even if we are not sure of which kind of set we are talking about (reals, rationals, or integers), it’s still a number, and it cannot be confused with any other kind of mathematical object.
Not quite.
If I ask you what’s the result of 22+2
, you are most likely going to answer 24
, but if I ask you what’s 22:00+2:00
, you are likely not going to answer 24:00
(which no clock ever shows), but 00:00
. 2
is a number, yes, but it doesn’t necessarily represent the set of real numbers, the set could be much more limited 0..23
, as is the case when we are talking about the hours of a day. So 22+2
could be 0
.
This is modular arithmetic. We don’t have to go into some esoteric algebraic structures, because most humans are already familiar with one non-standard algebra which is used in clocks. And although most programmers would consider modulo to be an operation, mathematicians consider it part of something much deeper.
In modulo 24
we have 24
integers (0..23
), but in modulo 4
we have only 4
: {0,1,2,3}
. So 2+2
can be 0
, which is the same as 3+1
.
Technically we are talking about integers modulo 4, denoted as ℤ/4ℤ
, which is a set of 4 members, but the first member is not 0
, it’s actually {…,-4,0,4,…}
, and 1
is actually {…,-3,1,5,…}
, and so on. So it’s a set of sets.
And to denote that we are operating under integers modulo 4 arithmetic we say: 2+2 ≡ 0 (mod 4)
. If you don’t believe me that it is indeed true, you can check in WolframAlpha: 2+2 = 0 (mod 4).
So there you have it: 2+2
is not necessarily 4
. Most people assume we are dealing with the standard arithmetic, but that’s the problem with a lot of claims: they make assumptions. Some of you may argue that if we are dealing a non-standard arithmetic that should be explicitly stated, and that may be true, but mathematics isn’t the point of this article, it’s to raise doubt about our own certainty. Another claim in the same vein is that lines with a common perpendicular (e.g. parallel) never intersect, this is again making an assumption: Eucledian geometry. The truth is we shouldn’t be so confident of our “unequivocal” facts.
Why insist on 100% certainty? We should always consider the possibility that we are making unwarranted assumptions.