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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?

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    $\begingroup$ maybe it refers to semantics in logic en.m.wikipedia.org/wiki/Semantics_of_logic $\endgroup$
    – Luis Alexandher
    Commented Jan 8, 2022 at 23:48
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    $\begingroup$ It seems wrong, but, without the precise language of the book, it is hard to tell if your missed a nuance in the text or if the text was sloppy. $\endgroup$
    – Thomas Andrews
    Commented Jan 8, 2022 at 23:50
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    $\begingroup$ All the math books I have, do math. If one of them veered off in this strange direction, I'd put it down, and never pick it up again. $\endgroup$
    – Gerry Myerson
    Commented Jan 9, 2022 at 1:15
  • $\begingroup$ Wouldn't this be better placed at philosophy.stackexchange.com ? $\endgroup$
    – user700480
    Commented Jan 11, 2022 at 9:58
  • $\begingroup$ Gottlob Frege pointed out that this view is wrong, since if it were correct the claim “$2+2=4$” would be vacuous. Frege articulated a distinction between two different kinds of “meaning”: the sense of an expression and its referent: “$2+2$” and “$4$” have the same referent but different senses. $\endgroup$
    – MJD
    Commented Jan 26, 2022 at 21:42

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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

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