Does it make sense to relate something to itself

Question

We have reflexive relations in mathematics like equality that maps an element x with itself such that xRx is always true.

These are called relations however, xRx really can be seen as a self-identity statement then something that relates x with something else, Perhaps my naieve view is that a relation is between two objects. I understand that there may be a relation between expressions but the equivalence relations (equality for example) are between objects.

Is it simply mathematical language, or is it acceptable to relate something with itself?

Answer 1

Yes, it’s fine; in fact probably essential if you want to make sense of eg. notions of sameness or proximity. After all if the thing isn’t allowed to be similar to itself then similarity seems to break down.

Answer 2

You are onto something. Yes, it don't make sense to relate a thing to itself. Its like tautology. Not wrong but not useful either.

Ofcourse a thing is similar to itself, even more, its same to itself. What do you get from it? That things don't change on their own. A one year old is well aware of that.

Now, its useful to compare a thing to itself in past or in future. People say "you have grown up" or "you are getting weak". They are comparing you from today to you from past. Two different things so the comparison is useful.

In mathematics when its said "x for all x such that..." they are not comparing a thing to itself. They are comparing a thing to an idea of the thing, an object to the die its casted from.

Its like having a table. The columns define how data in it should be. For example we decide to put a person's name, father name and date of birth in a table; 3 columns. Now each row is the object, it should fit in the table. The table is the die. Object is whats casted. We cannot for example store 4 data points about a person in the table, it has to be 3 data points. Data casted become a row in the table.

Its like having variables. The mathematics line says that pick anything, lets call it x, as long as its fitted in the die i.e. have such and such properties like this another x thats already casted, it has such and such other properties as well or the function give such and such output.

Answer 3

You are asking why does an expression exists, and if it has some meaning, without actually considering the goal of its existence. The question goes in the opposite sense: what symbol can be used to express an entity having a specific type of relationship with itself?

Self-references have multiple senses:

• A producer that is able to consume its own content;
• A consequence that is caused by itself;
• An output that goes connected to an input on itself, etc.

For example, when you think of yourself, you are self-referencing you: you subject (observer) are looking at yourself as an object (observed).

Another example, the first UML diagram in the page: a Person in its manager role can manage itself.

In the mathematical concept 1 can be divided by 1 check the role of both members: it is clearly different.

Answer 4

A good question. I believe we can derive one example of a reflexive relation viz. equality from the transitive and symmetric relational properties of equality

1. x = y
2. y = x (symmetry of equality)
3. x = x (transitivity of equality, from 1, 2) [reflexivity of equality]

The question is, are all transitive and symmetric relations also reflexive?