Is the meaning (semantics) of an arithmetic expression its value? Eg, is "$2$" the meaning of "$1+1$"? I think that in some math book I read long ago it said that the meaning -- semantics -- of an arithmetic expression is the value of the expression. For example, the meaning of this expression:
1 + 1

is 2.
Does that seem right to you? Have you seen a math book which says this?
 A: Short answer : number $2$ is the meaning of the expression "$1+1$" if by meaning is understood  denotation.
So, precisely, $2$ ( number $2$ itself, an extralinguistic object  ) is the denotation of $"1+1"$ ( an expression).

Meaning has two sides or aspects : (1) intension ( or " Sinn / sense" as says Frege) and (2) extension or " Bedeutung/ denotation" ( Frege again).
As to expressions ( contrasted with sentences) intension or sense is the conceptual aspect of meaning.
The intension of "$2+2$" is " the number that is the sum of $2$ and of $2$ ", or, more technically, "the image of the ordered pair $(2,2)$ under the function addition".
The extension or denotation of the expression " $2+2$" is the object this expression refers to, namely the number $4$.
One may know the sense of an expression without knowing its denotation. Example, I know the sense ( intension) of $" 3.567 \times 5.974231"$ , but I'd have to use a calculator ( or to do a calculation myself) in order to know what this expression denotes, that is, what is its value.
Two expressions can have different senses, but the same denotation. For example : " $log (100)$ " and " $8^{\frac {1} {3}}$".
In the case of sentences ( not expressions) , the sense is the (objective) thought they express, that is , the proposition they convey; and the denotation is their truth value ( namely: True , or False).
Henri Poincaré is supposed to have asserted that " mathematics is the art of giving the same name to things that are different [ " l'art de donner le même nom à des choses différentes" ]  . It seems to me that the inverse is also true :  mathematics is the art of giving different names to things that are identical; I mean, the art of using names that have different senses, but the same denotation.
Not grasping this fact plays a crucial role, it seems to me, in the problem some people encounter with mathematics. Do math teachers pay enough attention to it?
As to the quotation : https://www.apmep.fr/IMG/pdf/503_Villani-Metz.pdf
