Reflexivity: How can something be related to itself?

Question

Background: I'm a philosophy student. I'm comfortable with math, but don't have much of a background in it. One of the topics I'm writing about (I-relation in theories of identity) closely mirrors concepts in math. One of those is reflexivity.

My attempt to answer the question: I read the Reflexivity article on Wikipedia, but I'm still foggy on the idea. I get that 1=1 is reflexive, and 1<2 is not. I understand that 1=1 is relating one to one, but it seems so redundant that I can't imagine it being often used in math - but I believe that it is - so I must be missing something. Also, I read that the 'divides' relationship (2 divides 4) is reflexive. I don't see how that is reflexive.

Question

• How can something be related to itself?
• Why is reflexivity a useful concept?
• How is the divides relationship reflexive?

An abstract explanation and concrete example would be helpful.

Thank you.

-Hal

Answer 1

Reflexivity is a natural condition that some relations satisfy. Let $S$ be the set of people on Earth. Define a relation on $S$ as follows: say that $a \sim b$ if person a and person b have the same age. Then clearly $\sim$ is reflexive, because everyone has the same age as themselves. So in this sense a person is "related to their self" because of the way we defined our relation.

Here's a relation that is not reflexive. Say that $a \sim b$ if person $a$ is older than person $b$. Then clearly $a \sim a$ is not true, as a person cannot be older than their self.

Basically, things can be related to themselves or not depending on what you mean by related.

Answer 2

First, let's start with a simple example. Consider the relation of "equality" on, say, natural numbers. Obviously, two natural numbers $n_1$ and $n_2$ are equal if they are the same number, and as usual, this is written as $n_1 = n_2$. I suppose you will agree that, e.g., $1 = 1$, which means that 1 is related to itself by equality.

More abstractly, a mathematical relation is a fairly general object. One way of seeing it is to consider a relation $R$ as a kind of machine that takes two inputs $x$ and $y$ and outputs "yes" or "no" (this is essentially an algorithmic view, but in practice, it's equivalent to the definition on Wikipedia). We say that $x$ and $y$ are related if the machine outputs "yes"; what "related" means depends on the machine $R$.

As for reflexivity, this is a property of relations: A relation is called reflexive if $x$ is related to itself. Examples include equality (everything is equal to itself), less-than-or-equal (we have $n \le n$ for all natural numbers $n$) and so on. As Dylan Yott mentioned, there are non-reflexive relations like less-than (because $1 \not< 1$) or "not equal".

Divisibility is reflexive since $n$ always divides $n$ -- namely, $n = 1 \cdot n$.

Answer 3

In mathematics the term “relation” is defined for mathematical purposes. One could have named it differently. You must never compare a mathematical notion a word can have to the non-mathematical notions it may also have. You mustn't see any relationship between what's mathematically is called “relation” with other notions related to “relation.” (Pun intended.)

To give an example: for Nietzsche there is no longer any “absolute value,” whereas mathematicians hardly can live without one.

Answer 4

The notion of relation is a very general concept, applicable in the most different mathematical and off-mathematical contexts.

In particular, a binary relation on a set $A$ or between two different sets $C$ and $D$, encodes the following intuitive notion: Among all conceivable ordered pairs $(a,b)$ with $a$ and $b$ taken from $A$ (including pairs of the form $(a,a)$) some are "distinguished", "good", "interesting", or the like. These "good" pairs are collected into a "bag", which is nothing else but a set $R$ of pairs. With the intended interpretation in mind such a set of pairs is then called a relation: If the pair $(a,b)$ is "good" then $a$ and $b$ "stand in the relation $R$".

It is helpful to replace writing $(a,b)\in R$ by a notation that supports the intended interpretation. E.g., one would write $a<b$ when $A$ is the real numbers and the "good" pairs are the pairs where the first number is smaller than the second.

Now an example for reflexivity: Let $C$ be the set of all circles in the plane, and consider a pair $(c_1,c_2)$ of circles as "good", when $c_1$ and $c_2$ have at least one common point. This relation is indeed reflexive, because any circle has common points with (a copy of) itself.

Answer 5

1. A name is needed for this property whether it is intuitive or not, since many relations happen to satisfy it, the need to prove or use the property often comes up, and there is a similarity between these use cases that makes it worth abstracting as a concept of its own.

2. It helps to have a uniform treatment of different cases ("consider any line parallel to $L$"), rather than constantly making annoying exceptions ("let $L'$ be any line parallel to $L$, or equal to $L$"). Or "two lines are parallel if and only if they have the same slope" versus "...same slope and different height". If you had defined the relation of parallelism to exclude the case of self-parallel lines, you would find that all these sentences that explicity include it as an additional case are really talking about a different equivalence relation, parallel-or-equal, and introducing a name or notation for the new relation simplifies so many statements that it is easier to use the more uniform thing as the definition of parallel. [This is similar to allowing an 'empty set' and applying definitions like 'number of functions from set $A$ to set $B$' to that case whenever they make sense.]

3. You might be proving that $x \sim y$ for a whole set of $y$ at once, without knowing whether the set contains $x$ or not. It would not make sense to introduce artificial distinctions about hypothetical possibilities just to match a language preference that relations be between different objects.

4. A more basic difficulty, that actually happens in mathematics, is that sometimes you can prove all kinds of relations between a specific pair of things $a$ and $b$, without knowing that they are equal. Maybe you will eventually identify enough similarities to prove $a = b$, but that should not invalidate the earlier proofs that $aRb$ .

5. For some relations it is paradoxical to exclude self-relatedness. "Near", "as tall as", "as fat as", etc.

6. In many situations you want to be able to substitute $b$ for $a$ in a formula or statement (preserving its validity) when they satisfy a particular relation. It disrupts this to not be able to use $a$ more than once.

7. Generally the tendency to be inclusive rather than exclusive about trivial cases (such as "motions" of a space including the one thats keep every point in place, empty sets, digraphs without edges) is forced by the need to have the special cases handled automatically and uniformly by the language when dealing with more and more complicated situations.