Topics Mathematics

Question What does it mean in mathematics for two things to be equal, or for two things to have the same "identity"? For example, because anything divided by zero is "undefined", can we say that 1/0 = 2/0? What about the relational database concept of "null" which is supposed to stand for "unknown"? In relational algebra, they say NULL is not equal to NULL, but doesn't that violate the law of identity that everything is equal to itself?

Accepted:
July 11, 2007

Comments

I will begin by acknowledging that neither the philosophy of mathematics nor the metaphysics of identity are my specialties. But if you'll take what I say with a grain of salt, perhaps I might make a helpful observation nonetheless. Keep in mind that "being equal" mathematically is not exactly the same thing as "being identical," mathematically or otherwise. "Being mathematically equal" means, one might say, having the same mathematical value, in the sense of amounting to the same thing. Identity on the other hand means being the same thing, in the sense of having all the same properties. This difference can get confusing because commonly in symbolic logic the equal sign, "=", is used to express identity. But, if you follow me, then "2+2" equals "4" but is not identical to "4." Why? Well, while they both have the same mathematical value, they don't both have the same properties. "2+2", for example, is a formula composed of two Arabic numbers and a symbol for the mathematical relation of addition. One might also say "2+2" expresses a discursive process, the process of adding two different numbers. "4" on the other hand is a single number, the number between 3 and 5, and a number made by three straight lines with no curves; it does not represent a discursive process. I suppose it's possible, too, to know what 4 is without knowing that 2+2 equals 4--just as it's possible to know what 1369 is without knowing that it's the square of 37.

I'm afraid I can't help you with the question about relational algegra except to say that perhaps you mean the law that everything is identical to itself rather than equal to itself.

I think it is important to distinguish here between the meanings of expressions and the things that those expressions denote. Peter is right that the expressions "2+2" and "4" are different expressions, and they are not synonymous. But they both denote the same thing, namely the number 4. Now, in the equation "2+2=4", is the equal sign being used to express a relationship between meanings of expressions, or does it express a relationship between what is denoted by those expressions? (In other words, is this an intensional context or an extensional context?) I would say it expresses a relationship between what is denoted by the expressions, and the relationship is identity: what is denoted by the two sides of the equation is one and the same thing, namely the number 4. So I disagree with Peter's conclusion about what "=" means in mathematics. I would say "=" in mathematics means "is identical with".

I would say that the situation here is very much like the situation in the sentence "The morning star is the evening star." The expressions "the morning star" and "the evening star" are different expressions, and they mean different things. But they both refer to Venus, and the "is" in the sentence means "is identical to".

Now, what about the example "1/0 = 2/0"? In most contexts, I don't think any mathematician would write this. To say that "1/0" is undefined means that it doesn't denote anything, so it makes no sense to use it as if it did denote something. In particular, it makes no sense to say that it is equal to something. (I said "in most contexts", because I can imagine a mathematician might choose to enlarge the number system by adding a new element called "infinity", and he might choose to extend the definition of "/" by defining the values of "1/0" and "2/0" to both be infinity. In that case "1/0 = 2/0" would be true, but the expressions "1/0" and "2/0" would no longer be undefined.)