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 for positive real numbers , , . Find the minimum possible value of .
The wrong solution I keep seeing goes like so:
Nonsense solution: By AM-GM, the minimum value of is . Equality occurs if , which means . Since , this gives , so the minimum possible value is .
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 or (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
To call these equations false is misleading. If your friend asked you whether , you would say “no”. But if your friend asked whether equals the 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 , , satisfying some constraint. So this minimum should be an absolute constant, hence independent of , , .
In other words, if and are nonconstant functions, then
- the statement “ with equality when ” does make sense; but
- “ is the minimum of with equality when ” 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 . Or rather, although , it can take any value as close to as you want, say by taking , .
By the way, food for thought: what’s the maximum possible value of ?
1 thought on “A common type-error on the OTIS application”
seems to be the maximum: