# Is '=' or 'equals to' a relationship between Mathematical objects.

The Wikipedia definiton of equality gives it as a 'relationship between two expressions'

This confuses me as when we define mathematical expressions like $$2+2=4$$ it makes no sense to say that '=' or 'equals to' relates the two expressions as it would mean that '$$2+2$$' and '$$4$$' are representing themselves as expressions as opposed to denoting or naming the objects.

I think in mathematical contexts when we use expressions it is always to unambiguously name an object, for example: '$$2∈N$$' is meaningless if '$$2$$' represents itself as an expression.

I understand there is a relation between the expressions whose value are the same but is defining this relationship as 'equality' and saying it is denoted by '=' correct? It seems that '=' should denote a relationship an object has with itself. If the expressions act as names, then placing the symbol between the names means we discuss the objects, not their names.

Is there an explanation of whether 'equality' is at object or expression level?

Because if we treat '$$2+2$$' as denoting a number just as we treat '$$4$$', then all of a sudden $$2+2=4$$ being a statement about expressions seems strange to me.

It seems if we see as '$$2+2$$' and '$$4$$' as denoting mathematical objects, can we then treat equality as a self-identity statement?

• Beware: use/mention errors like that are commonplace in mathematics, so much so that most mathematicians don't even notice them. You might find more satisfying answers in philosophy of language. Nov 27, 2022 at 20:23
• Mathematics, for the most part, has a syntax and semantics (in the mathematical, formal sense of these words). So where $2+2$ is a syntactic expression, it is interpreted as an object in the mathematical universe on the side of semantics; the same for $4$. The statement $2+2=4$ is stating that the semantic interpretation of these two expressions is the same object. Mathematicians are trained so hard to make these translations from the first week of university, that it becomes natural to most, and unless you study further foundations of mathematics, you won't even know where the problems lie. Nov 27, 2022 at 21:23
• Don't expect Wikipedia to resolve subtle technical distinctions. It's better used as the start of such an investigation. Nov 27, 2022 at 21:36
• @Confused: Yes. Just like "The OP of this question" and "user 1124513 on this website" refer to the same user. Nov 27, 2022 at 21:38

The $$=$$ sign is overloaded so we need to be careful in which sense we're using it. The first is as a definition when we give a name to some other thing. A familiar example would be $$y=mx+b$$ which is us defining $$y$$ to be $$mx+b$$. Sometimes the notation $$:=$$ will be used for definitions like this and I prefer it.
The second is as a statement. I might make the claim the $$2+2=4$$ and want to evaluate the truth of the statement as true or false. I could also make the statement that $$2+2=5$$ and want to see if that's true or false as well. This is often used in programming languages to made decisions about branching or looping and they will typically have some notational implementation to distinguish the case when it's being used as a definition and when it's being used as a logical statement such as writing $$2+2==4$$ instead.
Finally, and most importantly, is as an equivalence relation. To understand exactly what they are we first nee to know what a relation is but thankfully it's pretty simple. A relation $$R$$ on a set $$A$$ is a subset $$R \subset A \times A$$. We say $$a \in A$$ is related to $$b \in A$$ when $$(a,b) \in R$$. This is a very general construction and can be used to create important relations like orderings. In a practical sense evaluating a statement about relationships like $$2+2=4$$ can sometimes be done by seeing if the element $$(2+2,4)$$ is in $$R$$. Lets look at what makes equivalence relations special.
An equivalence relation has three properties, which are reflexivity, symmetry and transitivity. Reflexivity means that $$(a,a) \in R$$ for all $$A \in A$$, which just means $$a=a$$. Symmetry means if $$(a,b) \in R$$ then $$(b,a) \in R$$ or if $$a=b$$ then $$b=a$$. Finally is transitivity. This means if $$(a,b) \in R$$ and $$(b,c) \in R$$ then $$(a,c) \in R$$. Alternatively if $$a=b$$ and $$b=c$$ then $$a=c$$.
This relation has another structure which is given by the fundamental theorem of equivalence relations, which is as a partition of a set. A partition of a set $$A$$ is for some indexing set $$I$$ we for sets $$A_i \subset A$$ for $$i \in I$$ $$\cup_{i \in I}A_i = A$$ and $$A_i \cap A_k = \emptyset$$ for all $$i,k \in I$$ with $$i \neq k$$. Basically you split the set up into bins labeled $$A_i$$. I don't think it's an understatement to say this is among the most important theorem in modern mathematics so it's worth knowing. It has applications everywhere.
• If we view equality as an equivalence relation, it should suggest that '=' is a relation on an object with itself, namely the object on the left and right of the equals sign form a pair that is an element of $=$, the set of pairs of the form $(a,a)$, For any object $x$ it has a relation with itself, that it forms a pair $(x,x)$ that is an element of $=$? Nov 28, 2022 at 11:37
• @Confused you've got the right idea. Sometimes relations are written with infix notation so instead of $(x,x) \in R$ you might write $xRx$. Relations don't have to be from the same set and you can have a relation between $A$ and $B$ which is just a subset of $A \times B$ that doesn't require the set to be the same but equivalence relations, and most important relations, are typically on $A \times A$, the keyword being "on" to identify this case. Nov 28, 2022 at 12:05