Order

A set $S$ is said to possess an order (or, if necessary to disambiguate, a total order) if, for every pair of distinct elements $x,y\in S$, either $x$ is before $y$, or $x$ is after $y$, with respect to some order-relation.

Besides what Objective Mathematics means by "order," there are two other significant uses of the word. One refers to a command (an order) given to other people, as in "he ordered a pizza." In the other usage, a set of things is said to possess an order if there is some pattern to which the things adhere; that pattern may or may not involve an order-relation. Both of these usages are related to the mathematical usage of the term; it's not a coincidence that they are all described by the same word. But getting into that would take us too far afield. I do not mean either of those things by "order."

Examples

Library books are ordered according to the Dewey Decimal System.

A single-file line of people could be ordered by height, with the shortest person in the front, and the tallest person in the back.

Natural numbers have a default ordering (by construction), where n+1 comes after n.

The marks on a clock are ordered clockwise.

To learn math, one must proceed in a certain order: some things must be learned before other things.

The alphabet is ordered. Little kids learn to remember this ordering by singing the alphabet song.

In feudal Europe, there was an order of succession (who inherits the throne after whom) which was largely agreed upon. The king's eldest son inherited the throne after the king died. If the king's eldest son was also dead, then his eldest son would inherit the throne. If the king's eldest son was dead and had no sons, then the king's second-eldest son would inherit the throne. Etc etc.

Non-examples

The books on my bookshelf are not ordered: they are in random positions with respect to one another.

Most single-file lines of people are not ordered by height.

The law of causality states that things act in accordance with their nature. One formulation of this law is that the universe has order.

If a riot happens, the police are supposed to come in and restore order to the streets.

In the Harry Potter series by J. K. Rowling, there is an organization called The Order of the Phoenix.

Before and after

The concepts of "before" and "after," in their primary usage, are identifications of properties which are irreducible primaries, and thus they admit only ostensive definitions. Nevertheless, I will give a circular "definition" to indicate what is meant by them. The primary usage of these concepts, and the way that they are first arrived at by a conceptual consciousness, is as measurements of relative time-ordering. That is, we directly perceive that some events occur before other events, and that some events occur after other events. Then, by a process of measurement omission, we create the corresponding concepts.

Besides the primary usage, there are many derivative usages of "before" and "after." Consider the following example, wherein paragraphs are sorted from least abstract to most abstract.

1. In most contexts, the number 1 said to be before the number 2, because, when counting, the sound "one" is uttered before the sound "two." Here, the latter usage of "before" is primary; the former is derivative. Likewise, in most contexts, the letter X is said to be before the letter Y, because "X" is always pronounced before "Y" in the alphabet song.
2. Having established an ordering of the letters of the alphabet, we can extend it to an ordering of the English language, by using the rules of alphabetical ordering. Alphabetical ordering tells us, for any two distinct words, which one is before and which is after.
3. We can use alphabetical ordering to give an ordering to many other things. For example, a set of people (say, on an attendance sheet in a school) could be ordered with respect to the alphabetical ordering of their names. In such a situation, Bob may be said to be before Cindy.

What is happening here is that we are using "before" and "after" to define an order-relation on things like the alphabet, the English language, and a set of people.

Order-relation

An order-relation is a binary relation satisfying reflexivity, antisymmetry, and transitivity (properties which are defined below). The reason for this concept is that there are many relations which are like "before" and "after" in some important respect, but mean something else. The concept subsuming such relations is that of an order-relation.

Examples

"To the right of" and "to the left of"

"On top of" and "underneath"

"Higher pitch than" and "lower pitch than"

"North of" and "south of"

"Later in the alphabet than" and "earlier in the alphabet than"

The traditional concept

Standard mathematics says that, by definition, an order-relation on a set $S$ is a function $R:S\times S\rightarrow \{0,1\}$ such that, for any $x,y,z\in S$, it satisfies

1. Reflexivity: $R(x,x)=1$
2. Antisymmetry: If $R(x,y)=1$ and $R(y,x)=1$, then $x=y$.
3. Transitivity: If $R(x,y)=1$ and $R(y,z)=1$, then $R(x,z)=1$.

Although I accept that any order-relation will indeed satisfy properties (1) through (3), I do not like using them in the definition of an order-relation. I don't feel like it perfectly captures whatever essence it is that examples of order-relations share in common. For the time being, however, I don't have a better definition.

Partial order

A set is said to possess a partial order if some, but perhaps not every, pair of elements are comparable using some given order-relation.

Examples

The leaves of a tree (a real, physical tree; not a mathematical tree) possess a partial order with respect to their height above the ground. For some ordered pairs of leaves, the first is higher than the second. For others, the first is lower than the second. For yet others, it cannot be said that the first leaf is higher than the second, nor can it be said that the first leaf is lower than the second; they are at approximately the same height.

A basket of blueberries could be partially ordered by their perceptual size. Perceptually, we can see that some blueberries are larger than others, and some blueberries are smaller than others. But without introducing an explicit measure of volume, there are many incomparable pairs of blueberries, where we cannot tell which one is the smaller, and which one is the larger.