Morality in Mathematics

Suppose for whatever reason you wanted to visit me. I could tell you to go to $40.4432^\circ N, 79.9428^\circ W$ by turning your steering wheel $117^\circ$ counterclockwise in 8 seconds. Or I could tell you to go to Carnegie Mellon by taking a left turn at the next traffic light.

Both these sets of instructions technically convey the same information. But not all information is made equal. Telling you to turn left is much more helpful, because it tells you the point of what you are doing.

Mathematics is much the same. Often, I will describe a mathematical argument as being “morally (in)correct”. My friends think this is a strange thing to say. Surely mathematical arguments are either correct or incorrect. What on earth does “morally” mean?

Mathematical arguments can be seen as a series of directions. We start from a certain point (a collection of assumptions), and we arrive at a destination (a conclusion). And it very much can be the case that one set of directions is far clearer than another.

Let me give you a fairly advanced example. Consider Sylow’s Theorem. It is perfectly possible to prove it without using the words “orbit” or “stabilizer” once. But when you do so, you will inevitably end up saying “turn $117^\circ$ counterclockwise” instead of “turn left” a bunch of times. Here, these left turns are just the theory of group actions (which include orbits and stabilizers).

Definitions and theorems are much the same. You should not just treat them as lists of causes and lists of effects.

For instance, I could tell you that the implicit function theorem says that given real/complex Banach spaces $X$, $Y$, and $Z$…

But your eyes would probably glaze over that. At the very least, mine did the first time I was learning about it. Why should you care?

But now let me tell you the following. Consider a function like $f(x, y) = x^2 + 2y^2 - 2xy + 5x + 7y - 20$.

Given some value for $x$, we can find a value for $y$ such that the equation $f(x, y) = 0$. It turns out that we can do even better; there is some function $F$ taking values of $x$ into values of $y$ such that this equation is satisfied (more precisely, $f(x, F(x)) = 0$). And because $f$ behaves very nicely, so does the function $F$. In fact, if $f$ is $k$ times continuously differentiable, then so is $F$.

Now we have a reason to care about the Implicit Function Theorem. Equations like $f(x, y) = 0$ can be solved, and the function that carves out the solutions is just as nice as $f$ itself.

Hygiene

This is why we care about mathematical hygiene. Picture an unnecessary proof by contradiction that goes like this:

Assume for the sake of contradiction that $X$ is false. Insert direct proof that $X$ is true. Contradiction.

This isn’t bad because it makes the argument longer; it barely makes a difference to the length. Rather, it obscures the point of the argument. The point of this argument is not that we may derive a contradiction by assuming “not $X$”; the point lies in the direct proof of $X$. But using a proof by contradiction makes it seem otherwise.

Orthogonality

Being morally correct and technically correct are separate things. Of course, they are correlated: if you are egregiously wrong, then you are likely not morally correct either, and a good portion of correct arguments are morally correct.

Though it may be rare, you can be morally correct without being technically correct. Usually it is some technical issue that prevents you from being fully correct. Take the existence of the algebraic closure of a field for instance.

Definition (Field Extension). We say a field $K$ is an extension of $F$ if $F \subseteq K$.

Definition (Algebraic Extension). Suppose $K$ is a field extension of $F$. We say it is algebraic if every element in $K$ is a root of some polynomial in $F$.

For example, $\mathbb C$ is an algebraic extension of $\mathbb R$ because every complex number is the root of a real polynomial.

Definition (Algebraic Closure). An algebraic extension $K$ over a field $F$ is an algebraic closure if $K$ is algebraically closed.

Notice that a field is algebraically closed if and only if it has no (non-trivial) field extensions.

Here is the theorem we are trying to prove: every field $F$ has an algebraic closure. The proof idea is simple: consider the collection of all algebraic extensions of $F$ endowed with the order $\subseteq$. Then Zorn’s Lemma produces a maximal element $\overline{F}$, which is algebraically closed because it has no non-trivial field extensions.

This argument isn’t technically correct: the collection of algebraic extensions of $F$ is not actually a set. But you and I ought to agree this is morally correct. Telling you this will make it much easier to understand the actual proof; in fact, reading this and the proof will be much faster than just reading the proof.