2+2 -> 4 or 2+2 =4

Question

Why do we write $2+2=4$?. If we start with the expression $2+2$, certainly after we define the rules of arithmetic and numbers, etc., we would get what we mean by the symbol $4$. In other words, if I am not mistaken, $2+2 \implies 4$. However, if given $4$, this does not necessarily imply $2+2$. We could arive at $4$ from many different arithmetic expressions. Should we not be writing $2+2 \implies 4$ instead?

Answer 1

The OP is correct that the process described in $2+2$ yields $4$ as its outcome.

The reason we write $=$, is that $2+2$ is known to result in a number. $4$ is also a number. We write $=$ to denote that the numbers on both sides are the same number.

Answer 2

The symbol $\implies$ is typically used between two statements $P,Q$ as in $P \implies Q,$ to mean "if $P$ then $Q.$" Your use here seems unusual in that neither $2+2$ nor $4$ are statements in the logical sense (i.e. true or false).

Answer 3

I suppose $2 +2 \rightarrow 4$ could indeed be used to indicate some kind of computation, i.e. that the process of adding $2$ and $2$ produces the number $4$. Such a notation would need to be carefully defined, but in the context of algorithms or computations it would make sense.

However, as far as the typical mathematical truth that '$2$ and $2$ is $4$' is concerned, We typically write $2+2=4$: it's a statement that expresses the identity of the result of adding $2$ and $2$ on the one hand, and $4$ on the other. This has useful arithmetical and algebraic applications, as we can replace any occurrence of $2+2$ with $4$, and vice versa. That is, the equality is a two-way street relationship, which is not something suggested by the $2 +2 \rightarrow 4$ notation.

Finally, in logic, $2 +2 \Rightarrow 4$ would mean '$2+2$ implies $4$', which makes no sense, since only statements can imply statements, and neither $2+2$ nor $4$ is a statement.