Are the assertions "$2 + 2$ equals $4$" and "$2 +2$ is $4$" identical

Question

Are the assertions "$2 + 2$ equals $4$" and "$2 +2$ is $4$" identical? Or is this a linguistic, psychological or murky philosophical thing rather than a mathematical thing

Answer 1

I think that we should first ask "In what language are these assertions made?". The assertion does not belong to the mathematical language, where it should be written "2+2=4" or more basically, using the language of the Peano axioms: "$s(s(0))+s(s(0))=s(s(s(s(0))))$". If it's in the English language, then one must ask if "equals" and "is" mean the same thing all the time or only when they are used between two numbers or numeric expression.

Answer 2

Yes, from mathematical point of view (and in the context of the ordinary addition), this two particular assertions are identical (in the sense that they present the same information).

However, in different mathematical expressions, the word "is" can have different meanings. In other words: sometimes "is" doesn't mean "equals" (like in "2+2 is even").

So, as in everyday language:

• "is" cannot be always interchangeable with "equals";
• what makes clear the meaning of "is" is the context.

Answer 3

Clearly, 2+2 is not identical to 4. 2+2 is the sum of two twos but 4 is a single number. What we mean by either way of saying it, is that 2+2 is quantitatively equivalent to 4. And I wish we would use the identity symbol and reserve the equals sign for conditional quantitative equivalence.

This also applies to 0.9999... is quantitatively equivalent to 1. They are obviously not identical. Mathematics should not rule out common sense.