# What are (algebraic) equations, really?

This is a much more philosophical question that I'm used to asking, but it's been nagging at me a bit. Let's say I have an algebraic equation $y = x$, $x, y, \in \mathbb{R}$. The best interpretation I have for this is: for any given $x$, you produce a $y$ with the same value. But, philosophically I have some issues:

1) Why am I allowed to choose an $x$ in the first place, and how do I know that the equation will produce a $y$ for every $x$ I choose? It's "obvious" that it does, but it seems like this is a sneaky way of saying something about the operator "=" that is often left out of textbooks.

2) What about more complicated equations that involve algebraic operations, like $y = x^2$? I need to define certain operations, like multiplication, in order for this to work. However, this isn't very satisfying. For example, the real numbers form a field and therefore in order to define multiplication we require $\forall x \ne 0 \exists x^{-1} | x*x^{-1} = 1$. But how do we really know that this exists for all $x \in \mathbb{R}$?

am I just going way too deep into an ultimately meaningless question?

• I would not look at it in a way that an $x$ produces an $y$, but rather as a relation on the set of $\mathbb R^2 \ni (x,y)$. – flawr Jul 13 '16 at 15:05
• So you're saying that all pairs $(x,y)\in \mathbb{R}^2$ that satisfy the relation already exist, independent of my description, and my description is just a useful way to talk about the pairs? – Michael Stachowsky Jul 13 '16 at 15:08
• Generally when you think of e.g. a line or of a plane, you consider a subset of $\mathbb R^n$ that satisfies some conditions, for example an equation $ax+by+c=0$ or $Ax+By+Cz+D=0$. – flawr Jul 13 '16 at 15:11
• You can't tell what color a widget is unless you know what a widget is. You can't tell whether non-zero reals have inverses unless you have a def'n of R and show that something exists that satisfies this def'n. This leads to "foundational " topics, which are covered in texts on set-theory...... And what IS the def'n of R ? – DanielWainfleet Jul 13 '16 at 18:32

The equation $y=x$ is a predicate about the variables $y,x$. It may be true or false. For each possible pair $(x,y)\in\mathbb{R}^2$, some pairs satisfy the equation and some do not. Hence the equation partitions $\mathbb{R}^2$ into those pairs that satisfy the equation, and the rest. It is possible that all pairs satisfy the equation, or that none do.
It is common to draw $\mathbb{R}^2$ as a pair of perpendicular axes, and indicate the satisfying pairs for a given equation as dots. This is called the graph of the equation.
As for your second question, in order to meaningfully talk about $\mathbb{R}$, much less $\mathbb{R}^2$, we need to have a mathematical definition for it. Part of this definition includes the field axioms. Unfortunately, the definition is pretty complicated, and students are taught to use $\mathbb{R}$ before they really understand that definition.