What exactly is an equation?

Solution 1:

One approach: An equation is a predicate, $P(x)$, of the form $s(x) = t(x)$ where $x$ is a free variable (or vector of free variables) and $s(x), t(x)$ are terms -- expressions which evaluate to elements of the universe (e.g. real numbers if you are doing mathematics over the reals) when values are substituted for variables. This means that $P(x)$ is something which evaluates to either $true$ or $false$ when $x$ is replaced by members of the universe. In the 1-variable case it can be thought of as a function of the form

$$P(x): U \mapsto \{true,false\}$$

where $U$ is the domain of discourse.

To solve an equation is to determine $P^{-1}(true)$, the set of all values which make the predicate true.

Solution 2:

Short answer touching on foundations and notation.

Mathematicians use equations to tell their readers that two expressions (the things on either side of the $=$ sign) are actually two (different) names for the same underlying object. But there are contexts in which that simple meaning can be lost or forgotten.

In elementary school kids (and teachers) are uncomfortable writing $$ 3 = 1 + 2 $$ because they want to think of the $+$ and $=$ as operations, analogous to the buttons on a calculator, so want to read them only from left to right.

In beginning algebra the equation $$ 3x + 2 = 8 $$ is meant to be "solved". That is, you are to find the values of the variable $x$ that make the two sides of that equation represent the same number, namely $8$. The rules that say you can "do the same thing to both sides" essentially preserve the fact that the two sides continue to name the same object.

If all the $x$'s are on the same side of the equation you can (but need not) think of this as asking for the preimages of the other side under the given map.

Later on when you encounter $$ f(x) = x^2 $$ you can be confused because there's nothing to "solve". This equation tells us that we will use "$f$" to name the squaring function.

When you encounter a "functional equation" like $$ f(x+y) = f(x) + f(y) $$ you may try to solve it: find all the "values" for the function $f$ that make the equation true (for every $x$ and $y$). In this case (assuming continuity or some other weak regularity) the answer is $$ f(x) = cx \quad \text{for some constant } c $$ but you can't get to that conclusion by transforming

the equation to simpler equation(s) and end up with an equation that can be solved by inspection.

In all these cases the equality tells you two things are really the same. What you do with that information depends on the context.

Solution 3:

The universe of equations does not have uniform semantics.

Equations can be tautologies. $$ 1=1, x+x=2x, \tan x = \frac{\sin x}{\cos x}, \frac{\mathrm{d}}{\mathrm{d}x}\mathrm{e}^x = \mathrm{e}^x, \dots$$

Equations can be explicit definitions. $$ f(x) = f(f(x-1)), \\\frac{\mathrm{d}}{\mathrm{d}x}g(x) = \lim_{h \rightarrow 0} \frac{g(x+h) - g(x)}{h}, \dots $$

Equations can be implicit specifications. This seems to be the only kind of equations you are discussing. $$ 8x+7 = 15, x^5+x-1 = 0, 0 = \int_{\Omega}f \cdot g^* \,\mathrm{d}\mu, \dots $$

These semantics can be nested. $$ f''(x) = - f(x), \mathrm{Tor}_n^R(A,B) = (L_nT)(A), \dots $$

Frequently, these equations are (implicitly) augmented by specifying the sets from which the variables may be drawn. For instance, \begin{align*} g &\in \{\mathbb{R} \rightarrow \mathbb{R}\}, \\ U &\subset \mathrm{dom}(g), \\ m(U) &> 0, \\ H_0(U; \mathbb{Z}) &= \mathbb{Z}, \\ x &\in \mathrm{int}(U), \\ h &\in \mathbb{R}, \text{[footnote]} \\ x+h &\in \mathrm{dom}(g), \\ \frac{\mathrm{d}}{\mathrm{d}x}g(x) &= \lim_{h \rightarrow 0} \frac{g(x+h) - g(x)}{h} \end{align*} Changing these specifications may alter what the final equation denotes, if it denotes anything.

[footnote]: There's notational abuse here. "$h$" only appears as a dummy variable in the limit -- it is expanded to infinite sequences. We really want to say that every element of such a sequence is real and when you add $x$ to any of them, you still land in the domain of $g$. This is (in some sense) really a defect of our usual notation for limits that we do not get to specify the set from which the sequences are drawn. Some would write these two conditions with the "$h \rightarrow 0$", but this doesn't change that these constraints are really to be applied to the various $h_i$.