[Edu-sig] Re: Edu-sig digest, Vol 1 #216 - 1 msg

Kirby Urner pdx4d@teleport.com
Fri, 19 Jan 2001 17:00:50 -0800


At 02:33 PM 01/19/2001 -0500, Morris, Steve wrote:
> > > Whatever happened to kicking back and appreciating the
> > > masters, getting a sense of what high quality mathematics
> > > is like without actually trying to be a master yourself?
> > > That's how we do it most music and art and literature.
> > > For some reason, math is supposed to be different.  Why
>
>There is a little difference. Music can be good or bad. Bad music is still
>music. Math is right or wrong. Wrong math is not math, it is a mistake.
>While math can be beautiful that isn't its purpose. The purpose of math
>isn't personal appreciation. The purpose of math is to get the answer to a
>specific kind of question. If you get the wrong answer you are usually worse
>off than no answer.

We should teach correct, verified math.  Not just "noise" 
(bad or wrong math).

But in many cases the answers have already been obtained
(thousands of years ago perhaps), and the appreciation comes 
from seeing the solutions.  It's like finding out what's in 
the toolbox, what's there to use. If you don't know you have 
a screwdriver, and what it's for, you'll be at a disadvantage.

But that doesn't mean you had to invent the screwdriver. 
Nor does it mean you have an immediate need for one.  Just
nice to know it's there and what it does.

>That being said there is significant value in understanding what can be done
>with math even if you don't learn how to do it. Understanding someone elses
>proof is still valuable even if you couldn't have derived it yourself.
>Understanding the reach of math lets you know when to ask for a master.

Exactly.  And it doesn't have to be the proof that you
understand.  You may just want to use the tool, not prove
that it's correct -- you leave that to others.

This may sound "lazy", but even within the domain of 
mathematics, people leave it to others to verify and 
prove a lot of the time, and just use the results to get
on with it.  

In Mathematica, for example, you have all these "black boxes"
that spit out everything from Bernoulli numbers, to large
probable primes, to pictures of convex hulls given a point
dispersion.  Given the math you're exploring and/or applying
at any given time, it may or may not be relevant to open 
each and every one of those black boxes and understand 
their inner workings (besides, many black boxes have more
black boxes inside them, and so on).

It's similar to using an API, an object library.  Here are
these utilities and here are the commands for invoking them.
You may not have time to read the source code, nor will you
always understand what you read.

I'm not arguing that math should be treated as "black boxes"
by everyone.  The analogy with open source is a good one.
We trust math because it's open source, and lots of good
minds eyeball its codes, looking for lapses, errors and so
on.  But any given user/appreciator of the library isn't
necessarily going to stop and study the source.  And when it
comes to math appreciation, we shouldn't forbid kids to 
access powerful "math objects" just because they're not 
at a level where they can completely comprehend how they
do what they do.

By analogy, we shouldn't forbid kids from using Python 
just because they haven't learned how it works under the
hood in any real detail.

Kirby