There’s a common error I keep seeing on OTIS applications, so I’m going to document the error here in the hopes that I can preemptively dispel it. To illustrate it more clearly, here is a problem I made up for which the bogus solution also gets the wrong numerical answer:

Problem: Suppose $a^2+b^2+c^2=1$ for positive real numbers $a$, $b$, $c$. Find the minimum possible value of $S = a^2b + b^2c + c^2a$.

The wrong solution I keep seeing goes like so:

Nonsense solution: By AM-GM, the minimum value of $S$ is $S \ge 3\sqrt[3]{a^2b \cdot b^2c \cdot c^2a} = 3abc$. Equality occurs if $a^2b = b^2c = c^2a$, which means $a = b = c$. Since $a^2 + b^2 + c^2 = 1$, this gives $a = b = c = 1\sqrt3$, so the minimum possible value is $1/\sqrt3$.

The issue is that the first line does not make sense. It’s worse than just “false” or “wrong”: it’s a type-error, meaning it cannot even be formulated into a statement which could then be regarded as either true or false.

What do I mean by “type-error”? In mathematics, coherent statements are usually either true or false. Examples of false statements include $\pi = \frac{16}{5}$ or $2+2=5$ (from the Indiana Pi bill and 1984, respectively). However, it’s possible to write statements that are not merely false, but not even “grammatically correct”, such as

• $\pi = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
• $\log(\lvert \mathbf i + 3 \mathbf j) = \cos(\mathbf k)$
• $\det \begin{pmatrix} 5 \\ 11 \end{pmatrix} \neq \sqrt{2}$.

To call these equations false is misleading. If your friend asked you whether $2+2=5$, you would say “no”. But if your friend asked whether $\pi$ equals the $2 \times 2$ identity matrix, the answer is a different kind of “no”; in the words Tom Leinster (section 3.3), the best response is “your question makes no sense”.

In this case, one seeks a minimum of a function in three variables $a$, $b$, $c$ satisfying some constraint. So this minimum should be an absolute constant, hence independent of $a$, $b$, $c$.

In other words, if $f(a,b,c)$ and $g(a,b,c)$ are nonconstant functions, then

• the statement “$f(a,b,c) \ge g(a,b,c)$ with equality when $a=b=c$” does make sense; but
• “$g(a,b,c)$ is the minimum of $f(a,b,c)$ with equality when $a=b=c$” is a type-error. The minimum value of a function is not supposed to depend on the inputs.

And of course, the actual minimum value to this example problem is $0$. Or rather, although $S > 0$, it can take any value as close to $0$ as you want, say by taking $a = \sqrt{0.999998}$, $b = c = 0.001$.

By the way, food for thought: what’s the maximum possible value of $S$?