# ls elementary mathematical equality essentially a self-identity statement?

In a lot of places equality is defined that for two expressions A=B, A=B means that A and B have the same value (A=B). This relationship seems strange as we are slightly abusing use/mention, if '=' denotes a relationship between expressions then in '1+1=2' '1+1' and '2' are denoting themselves.

Doing this seems a little bit strange considering we generally use '2' and '1+1' to denote values themselves.

If we treat equality as a relationship a mathematical object has with itself, it is not that useful, however the variety of the names and what they inform us of is, for example x+1=2 is equivalent to 2=2 but the former is more interesting.

In this way, is the elementary mathematical use of the equals sign essentially just an identity statement? (Ignoring the intricacies of the idea that different objects that we equate aren't always mathematically the same due to their construction, as I usually get linked this article and it's not really what I'm asking).

• "1+1" and "2" are two expressions denoting the same number but they are not the same numeral (i.e. name for number). May 8, 2023 at 10:39
• Compare with "Napoleon is the First Emperor of France" where "is" can be read as "is equal to". May 8, 2023 at 10:39
• @Stef - agreed. But I've written " When we write a statement, it must be clear if we use words to speak of physical objects or of linguistic expression." When in everyday speaking we write "Rose has four letters" also children understand that we are speaking of the name of the flower. The same with mathematics: if we write the expression "1+1=2" every mathematician understand that we are computing with numbers (use) and not working about syntax (mention). May 8, 2023 at 11:59
• ChatGPT? Is that you? May 8, 2023 at 13:20

There are actually lots of different equality/equivalence relations one can define (in the following simply R to avoid any associations you might have with the symbol =). They all have in common:

• reflexivity (xRx for every element x)
• symmetry (xRy => yRx for every element x,y)
• transitivity (xRy, yRz => xRz)

= is overloaded in mathematics, sometimes it denotes identity, but not always. When two expressions are compared, a+b = c+d actually means equality between the numbers the expression evaluates to.

Equality of a program or an expression might range from: evaluates to the same thing, computes the same function, computes the same function AND does so with the same complexity (for various measures of complexity).

An equality between logical sentences means, or might mean, the worlds that make them true are identical, not the sentence itself.