Context, Language, and False Equations

Date: 02/12/2001 at 18:03:06
From: Rheanna
Subject: Algebra

Is there such thing as a "false equation?" My math teacher says no, 
but a friend and I are trying to prove him wrong by taking a survey. 
So far yes and no are tied with 4 each. I need you to break the tie.

Date: 02/13/2001 at 12:55:49
From: Doctor Ian
Subject: Re: Algebra

Hi Rheanna,

Have you ever heard the story of the Emperor's nose? It's a story 
about a Chinese man who decides to find out how long the Emperor's 
nose is by asking a lot of different people... none of whom have ever 
seen it!

The moral of the story is that you can't prove anything by taking a 
survey.

(A modern, real-life version of the story occurred during World War 
II, when a group of about 100 university professors in Nazi Germany 
signed a letter saying that Einstein's Theory of Relativity was 
incorrect. Einstein's response was instructive: "If they were 
correct, one signature would be enough.") 

Having said that, let's consider your question. Is there such a thing 
as a false equation? 

We can certainly write down an equation that isn't true. For example, 
there is nothing to prevent us from writing

     1 + 0 = 0

Leaving aside for the moment the philosophical issues involved in 
deciding what '+' means, this is clearly an equation. And it's 
clearly false. So it's clearly a false equation. 

Ultimately, an equation is just a shorthand version of a sentence. 
And a sentence can certainly be false, can't it? I can write: "The 
current President of the United States was born in 1492." This is a 
sentence, which happens to be false. Using mathematical notation, we 
might write the same thing this way:

     year_of_birth( president( united_states, 2001 ) ) = 1492

This is a false equation, corresponding to a false sentence. 

There is no big problem with having false equations. The only thing 
you need to remember is that a false equation, like a false sentence, 
isn't really useful for anything, except possibly for misleading 
people. And for constructing indirect proofs. 

Well, okay, there _is_ one other thing that you need to remember, 
which is that an equation, like a sentence, _always_ assumes a 
context. And it's really difficult (perhaps impossible) to write an 
equation - or a sentence - that isn't true in _some_ context!

This is easier to see with sentences.  Try to think of a sentence 
that is clearly false - for example: "Frogs have wings."

Certainly toy frogs can have wings, and fictional frogs can have 
wings, and it's not impossible that a mutant frog might have wings.... 
So depending on the context of your sentence - which is to say, how 
you define words like 'frogs' and 'wings' and 'have' - it may be 
false, or it may be true. 

Similarly, an equation like 

     1 + 0 = 0

is false in the context of standard arithmetic, but it's easy to set 
up algebraic systems in which it happens to be true. So you can't 
really say that an equation is false without _completely_ specifying 
the context in which the equation is to be evaluated.  

What this means is that you can assert that '1 + 0 = 0' is false, and 
I can assert that '1 + 0 = 0' is true, and unless we've established 
that we're talking about exactly the same context, we can _both_ be 
correct. 

Which raises the question: Is it possible to _completely_ specify a
context? Read this story by Lewis Carroll before you answer:

   What the Tortoise Said to Achilles
   http://www.lewiscarroll.org/achilles.html   

This is probably more information than you wanted, isn't it?  :^D

Believe it or not, I'm trying to let you in on a secret about math, 
which is that these kinds of conversations about basic notions 
like 'true' and 'false' can be quite a lot of fun... and very valuable 
in learning to defend yourself against people who want to manipulate 
you with subtle forms of lies... and essential for understanding what 
the limits of mathematics are. 

(A fun game to play is to take turns with a friend trying to say 
something that is absolutely true or absolutely false - that is, true 
or false in _any_ possible context - and then trying to find a context 
that makes it false or true.) 

It also turns out that the issue of determining the context in which a
sentence (or an equation) is to be evaluated is one of the things 
that makes it so difficult to get computers to be able to understand 
natural language.  

This is because most of what people communicate to each other using
sentences (including equations) isn't contained in the sentences at 
all, but in their shared understanding of the world.  

Think about the last time you had a conversation like this: 

  "Mom, can I go to the mall?" 
  "Have you finished your homework?" 

On the surface, the question and response have nothing to do with 
each other. A complete explanation of _why_ this question generates 
that response could fill a book. If you heard it at a friend's house, 
you would have no trouble at all understanding it.  But how would a 
computer make sense of it, without already knowing about school, and 
homework, and why kids are always trying to get out of doing their 
homework, and why parents consequently bug them about it, and what 
people do at malls... ?

The central problem of getting computers to understand language, 
then, has much less to do with dictionaries and grammars than with 
getting computers to experience the world in the same way that humans 
experience it, so that they will use similar contexts when evaluating 
the truth or falsity of sentences.

This is a tough problem, a problem that will only be solved by people 
having conversations like the one we're having right now.

I hope this has been helpful, and not too overwhelming. Let me know if
you'd like to talk about it some more, or if you have any other 
questions. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/13/2001 at 19:56:02
From: Rheanna
Subject: Thank You!

Thank you for the answer on if a false equation is real. I had never 
thought about it that way. I really appreciated your work in sending 
that e-mail. It was not too overwhelming, and it was just what I was 
looking for. Thanks again!