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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?

ngc1300
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    Does $4 \implies 2+2?$ and does $3+2 \implies 4 + 1$? And just what sort of statement is $5$. I walk into your living room and say "Five" and you say "what about five" and I say "five". The thing if $2+2$ is a value. ANd $4$ is a value. And as values they are one and the same. A number of value with no verbs or context isn't really a staement of any sort. We can't just say "Horses" or "Hoofed animals of transport" and "Horses $\implies$ Hoof animals of transport" doesn't really make sense as we aren't saying anything about them. – fleablood Mar 28 '18 at 23:31
  • Probably a bunch of other duplicates too... but yeah, four should be enough. – Asaf Karagila Mar 28 '18 at 23:34
  • $2+2$ and $4$ are "names" and the conncetives, like $\to, \lnot, \land$ "connect statements. – Mauro ALLEGRANZA Mar 29 '18 at 06:18

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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).

coffeemath
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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.

vadim123
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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.

Bram28
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  • I've been thinking a lot lately about how ubiquitous identification of an expression with its value is unfortunate for later areas of math where it is valuable to make this distinction. If I were raising a child to be a mathematician, I would maintain this distinction throughout their math education (assuming I had full control of it). But as the vast majority of children are not going to become mathematicians, this distinction does not seem particularly helpful to them. – Derek Elkins left SE Mar 28 '18 at 23:55
  • @DerekElkins I would recommend against that. They try it in languages like perl and end up having to put $ arbitrarily in front of value expressions. Just look at any language where they have "value of ($)" rather than "the lvalue of" (& in C, or just the object itself in C#) or "the expression of" ($\ulcorner \urcorner$ in godel numbering) and the horrible tedium of the prior languages is apparent. – DanielV Mar 29 '18 at 00:02
  • @Bram28 To second this sentiment, since $\vdash$ doesn't actually require any propositional interpretation, it is simply a statement of mechanical reachability, it would make plenty of sense to define some rules of inference to lead to $2+2 \vdash 4$. And $\to$ is usually defined in a way to mimic the behavior of $\vdash$. – DanielV Mar 29 '18 at 00:06
  • @DanielV Except you don't have to put the $ arbitrarily in front of value expressions. Instead having the $ or not means different things which is my point. But I agree that it can easily be done in a way that is tedious, though I'm confident it can be done with less tedium than the way Perl does it. Still it is likely to create some tedious pedantry in a way that won't pay off for most, which is why I don't consider it particularly helpful for most. It's unfortunate that the identification is a thing that needs to be unlearned later – Derek Elkins left SE Mar 29 '18 at 00:22