Definition of "in between" in geometry

Question

Hi!

I'm having a discussion with a stubborn person.

The problem:

Given is a rectangular triangle. A and C are points of the hypotenuse and B is the point with a perpendicular angle. Nothing fancy...

He insists to call B "between" A and C. He used Pythagoras to support his claim.

He said: "You can use (C) as a point to determine the distance between (B) and (A). Is this a true or false statement?"

Of course that is ridiculous.

First I tried to explain collinearity. A point is "between" two other points if collinearity is satisfied.

Second I tried to explain the convex hull to show that the area enclosed consists of every possible point which can be seen as "in between" two objects.

Then he came up with an example he said: "The nose is between the ears". I'm not kidding!

My response:

We live in three spatial dimensions. And in 3D the nose is not between the ears.

Appearance: On a photo, a 2D representation, the nose appears to be between the ears.

But appearance is not the same as the real thing. You have lost relevant information!

Similar questions would be: "Is my left ear between my right ear and the nose?" or "Is my right ear between my left ear and the nose?"

By your definition, each of the three objects could be seen as between the other two objects.

This is not possible. Your definition is wrong because it does not correspond to reality.

The correct answer is: Part of the brain is between all three objects.

Similar: When you look at the moon and hold your thumb in front of your eye, your thumb appears to be as big as the moon. Will you then walk around and tell everybody that your thumb is as big as the moon?

How to deal with such a person?

Is there a better way to describe the problem?

Was I right to use the above definitions?

Is there a standard definition to go with?

Answer 1

"Between" is an ambiguous term with no rigorous mathematical definition. You don't say 2 is "between" 1 and 5 in mathematics, you say that 2 is in the interval [1,5]. As far as your collinearity claim, you still don't say that the point is "between" those two points, you say that the point is on the line that passes through the other 2 points. Both of you are using ambiguous meanings of the word "between" which are technically correct and technically wrong at the same time.

TLDR: There is no correct definition of "between" in mathematics, you're not wrong to use your definitions but you're also not right, and the "problem" has no correct answer due to ambiguity.