The UHT

There are two things that you need to know about the UHT.

First YOU NEED TO KNOW THAT IT IS FALSE!

Second you need to know what it is.  I’ll explain that right now.

The UHT is the Universal Homomorphism Theorem.  Don’t let the words scare you.  I’ll explain them in a second.  Chances are you or someone you know believes in the UHT.

A homomorphism is any mathematical operation that you can do to a number (we call those functions) with the property that if you do it to the sum of two numbers, you get the same answer as if you do it to the two numbers seperately and then added the results together.

In math-speak, f is a homomorphism if f(x+y)=f(x)+f(y).

The UHT is the belief that all functions are homomorphisms.

Here’s the problem: THERE ARE LOTS AND LOTS OF FUNCTIONS THAT ARE NOT HOMOMORPHISMS!

Here’s one example:

The square function f(x) = x^2

It is not true that (x+y)^2 = x^2 + y^2.  For instance, does (1+1)^2 = 1^2+1^2?  If you believe that the answer is yes, then come to my house and I will give you 1^2 + 1^2 dollars if you will give me (1+1)^2 dollars.

Today a friend told me of a group of students who insisted that \sin(x+y)=\sin(x)+\sin(y).  The students also said that one of their teachers had taught them this (supposed) fact.  The true fact is that \sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y) .  When the teacher was confronted he or she (is reported to have) said “Well, we were teaching them to do it that way (\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y) ), but they weren’t getting it, so we just started teaching them this easier way (\sin(x+y)=\sin(x)+\sin(y)).”

This teacher knew that the UHT is false, but taught it to the students anyway.

Let’s leave the math-speak behind and talk plain English.  The sine function describes waves.  Let’s say you’re standing in the ocean on a sandy beach.  A wave approaches.  Now the water level is at your ankles.  In a couple of seconds it is up to your knees.  What happens next?  You know that in a couple more seconds the water-level will drop back to ankle-level.  But the UHT says that it doesn’t.  The UHT says that the water will keep rising.  A couple seconds after being at your knees it will be at your waist.  Then your chest.  By the end of the day the entire continent will be inundated.  But the wave won’t stop there.  It will keep going.  Rising and rising and rising forever.  That is what the UHT says.  Let’s all be glad that the UHT is false.  Let’s hope that there are not any other teachers that teach it.

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2 Responses to The UHT

  1. Arlynda says:

    Horrible I tell you! Seriously horrible! Remedial math classes abound for a reason, and it isn’t just student ineptitude.

  2. Evelyn says:

    If my students are ever struggling, I will start teaching them “easier” ways to do problems. I’ll start by telling them that every function equals zero. That will simplify things! I will call it the Universal Triviality Theorem.

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