# Why do all equalities require two different things?

All equalities by definition require two different things. In 1+1=2, we presume that 1=1.

However, although the two different 1's in 1=1 can either refer to a single object or two objects which we consider similar, we demand they have to refer to two different objects in 1+1. We don't count the same object twice.

Although we agree to ignore/ not consider the differences between the two entities in 1+1 to treat them as similar, we don't really stay true to our own conditions. We make use of those differences to be able to tell the two things apart. It's like saying 'the differences don't matter at all — the two things are similar', while also saying 'come on, we all know that they are different things' . We are simultaneously aware and not aware of the differences.

The concept of '2' require similar-different objects, and such can only exist in our subjective perception.

I would love to know your opinion regard this.

Read my argument in detail here: Math is Subjective

• Comments are not for extended discussion; this conversation has been moved to chat. Oct 20, 2021 at 16:08

All equalities by definition require two different things. In 1+1=2, we presume that 1=1.

No, they don't. In particular, equivalence relations are defined to be reflexive, allowing you to equate an object with itself. For this reason, the fact that 1=1 follows from the definition of =. Also, the "equality" relation denoted by "=" is generally defined to relate each element of a set only to itself and not to any other elements, as opposed to other equivalence relations which may be defined less strictly.

However, although the two different 1's in 1=1 can either refer to a single object or two objects which we consider similar, we demand they have to refer to two different objects in 1+1. We don't count the same object twice.

It does not make sense to talk about "two different 1's," because "1" does not refer to a physical object, like an orange, rather it refers to an element of a set, usually the multiplicative identity of a ring. It is a simple exercise to show that the multiplicative identity of a ring is unique, so we can be sure that the symbol "1" refers to the same thing each time it appears. This is important, because it would not really make sense to write mathematical expressions in which we use a certain symbol multiple times, with it meaning different things each time.

• So the "two different 1's" can refer to the same object in 1+1? Your answer doesn't explain why we don't allow ourselves to count the same ring twice, because, a ring wouldn't lose its 'multiplicative identity' after it has been counted once. Oct 20, 2021 at 9:20
• @SameeraBandara Not only can the "two different 1's" refer to the same object, they must refer to the same object, else the symbol is not well defined. I'm not really sure what you mean by "count the same ring twice," since I have say nothing about counting rings. When we write the expression 1+1, we are saying that we have defined a binary operation on, for instance, a ring R (or perhaps some other algebraic structure) +(·,·):R×R→R, and we referring to the output of the operation + for the inputs 1,1. 1+1 is addition, not counting. Oct 20, 2021 at 19:04

This question deals with the difference between two distinct concepts:

• Identity: Two symbols refer to the exact same object
• Class equivalence: Two symbols refer to two different objects of the same class

In other words, if I have a red playground ball, it is 'identical' to itself, but it is 'equivalent' to every other object in the class 'red playground balls'. We use the equals sign '=' to indicate both identity and class equivalency in mathematics (in logic, we use the triple-bar '≡' to represent identity), because it's generally understood in math that we're dealing with abstractions where the difference between identity and equivalency is mostly irrelevant. Thus when we say "1+1=2", we are implicitly saying "one object of class X with one (different) object of class X is equivalent with two objects of class X". It doesn't matter if the two objects on the left side are the same as the two object on the right side — on this level of abstraction the nature of the objects doesn't matters as long as they are all of the same class — but there still has to be two distinct objects on each side, because we can't add an object with/to itself.

The concept of '2' doesn't rely on sense perception: it's easy enough to say that one purple unicorn plus another purple unicorn is two purple unicorns. But it does rely on the concept of class membership, because if there are no classes of things, then there is only one of everything we see (and possibly only one non-dual thing that is the undivided form of everything).

the two different 1's in 1=1 can either refer to a single object or two objects which we consider similar

No, the two 1's in 1 = 1 both refer to exactly the same thing, namely, the numerical value. There is only one numerical value denoted by the symbol '1'.

we demand they have to refer to two different objects in 1+1. We don't count the same object twice.

No, in 1 + 1, the two 1's refer to exactly the same thing, namely, the numerical value.

If we said one bat plus one bat equals two bats, the two occurrences of 'one' would still refer to the same thing, namely, the unique numerical value. The two bats have to be different, though, but this is implied.

Arguments to be logical have to eschew equivocation and fuzziness.