A common type-error on the OTIS application

Power Overwhelming 2023-01-17

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 pre-emptively 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

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

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 ${S}$ ?