All our relations laid bare.

Update: Thank you to Diandra for the kind words about this post!

The Internet is abuzz recently with a depressing bit of news about the state of math education in the United States: the vast majority of adults in the US do not understand what the equals sign means. For a particularly good take on this, please see the excellent article on Cocktail Party Physics. As for my part, I'd like to take this as an opportunity to expand on a point that the author, Diandra Leslie-Pelecky, made in the Cocktail Party post. Specifically, I want to elaborate on the use of the equals sign to indicate a relation.

Fundamental to mathematics is the idea of a relation, which is a formal way of stating that two objects are related in some specific way. For instance, the object "2 + 3" is related to the object "5" by the equality. This notion, however, can hide that something very important has occurred. We have taken a conceptual process, addition, and restated it in terms of a statement about static relations. No matter what I do, I cannot break the relation "2 + 3 = 5." By contrast, if I constrain myself to thinking about the addition process, then it is harder to separate that statement about Platonic ideals from the perhaps imperfect implementation of the addition process. The equality relation, then, tells us about what is.

To take a tangent for a moment, mathematics can be thought of as the process of identifying and abstracting patterns. The concept of addition, for instance, is an abstract way of discussing and modeling a very common pattern in the natural world. We need not specify whether we are adding apples or planets; the pattern is the same. Thus, taking the relational view is a natural step in this methodology of abstraction, as relations tell us about the patterns that we can identify in other patterns. We recognize that the pattern "three objects" is indistinguishable from the pattern "one object and two objects added together," and so we say that these two patterns are equal to each other. In doing so, we make no statement about which pattern precedes which in a process, meaning that we can represent the "three objects" pattern as the "1 + 2 objects" pattern should we find the latter more convenient.

Of course, processes exist, and so mathematics would be much less useful were it not able to describe them. One may well point out, for instance, that the concept of a function has a very clean intuitive description as a mathematical formalization of a process. That is, the expression "f(x)" means "take x and do f to it." Notice, however, that we ultimately rely on the idea of a relation to make sense of functions. We say things like "y = f(x)," meaning that the result of f acting upon x is related to y such that the two objects are indistinguishable. We can thus remove any notion of dynamics from our description, focusing on the pattern that our process introduces. One can even go as far as to think of a function f as a kind of relation between other objects, so that "x f y" means that the objects x and y are related by the action of f.

This shift to relational thinking is very powerful, and underlies not only much of mathematics, but also much of our language. When I say "I am Chris," there is no naming process implied, but only the statement that the concepts "I" and "Chris" are related by the verb "to be." That is, in announcing my name, I am relating the concepts of self and name. Recognizing this has been a boon to the Semantic Web, and is used to express concepts in terms of abstract relations, such as is done in Notation3. The entire concept of the World Wide Web comes back to the concept of a very special relation known as the hyperlink.

Understanding the notion of a relation can be seen to be critical to understanding the means by which we understand and model the world around us. A key part of being human is our capacity to identity patterns and relationships between objects around us, and it is precisely this capacity that we bring to bear in mathematics.

PS: This is why ":=" is often preferred over "=" for indicating "set equal to" by mathematicians.