When does 2+2=5?

It would be an interesting question as an entrance exam for a class: to see what students would come up with. What kind of class would it be, though? Mathematics? Logics? Philosophy? Linguistics? Dadaism?

Since I don’t have the authority to offer a class, in whatever subject, let’s keep it an open question, and offer some ideas for answers (all contributions welcome).

- When 3+3=7

Not a bad try, but is it sufficient? Can we derive, from 3+3=7, that 2+2=5?

It could also mean that 2+2=4+4/6=4.666~

Frankly, it could also mean that 2+2=7

- When 1+1=3 and 3+3=7

This makes sense. We have not changed the nature of integers, but the nature of the operation of addition: in itself, the action of adding up two equal integers adds up one more unit to the result. Why not?

Then, does 1+2=4? Does 4+2=7? In this case, adding up any two integers adds up one more unit to the result. Why not?

There could be or not be a particular characteristics to the addition of two equal integers, which would consist in the addition to the result of one more unit. Why or why not?

I could not say, but this begins to question the nature of integers, and if the coexistence of two equal integers (say, 3 and 3) differs in any way from that of two different integers (say, 2 and 4) of hypothetically equal sum.

In short, what are we talking about when we talk of 1, 2, 3? 1, 2, 3 what?

- Never

This is indeed an acceptable answer, yet not one that is likely to get one admitted into the class, unless… it is followed by an analysis of the assumption contained in the word “when”, in our initial question.

We did not ask: “does 2+2 ever equal 5?”, nor: “could 2+2 ever equal 5?”. No, we asked “when”, implying that it sometimes does, and that the person or persons to whom we are asking the question may, indeed should, be able to say something about it.

This is an entirely arbitrary assumption, one whose motivations reveal much about the questioner and its intentions: a discussion thereof being potentially of equal interest, or equally liable to reveal the qualities necessary to pass the test, as any attempt to answer the question itself.

- Sometimes

There we go. Problem solved.

Depends on the nature of integers.

Let’s observe cellular reproduction, where one cell equals 1. With the addition of time t, 1+t=2, 2+t=4, 4+t=8. In this scenario, 2+2 never equals 5, but 2+2=8.

Yet, for accuracy, we had to introduce time t to represent what happens, what makes reproduction an operation. This asks, in return, whether there is or isn’t a temporal component to other operations? When we say or write “2+2”, hasn’t some time lapsed between the first and the second 2? Isn’t the linear space connecting characters on a surface a visual representation of time? The time it takes us to write, think, read what it is that the characters represent?

In which case, what would justify making t explicit in one set of operations, and not in another? Couldn’t we also write 1=2=4=8 in the case of cellular reproduction, and 2+2+t=4 in the case of classic arithmetics? Then, why not consider that by its function, the symbol “+”, which represents the addition but also connects visually, on what tends to be a metaphorically (at least) temporal line, the added numbers, intrisically contains the value of t? A value which could or not be equal to 1, producing or not 2+2=5?

(to be continued)