A common type-error on the OTIS application

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}?

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