Math is Power?

Date: 03/13/2003 at 09:11:20
From: Carl  
Subject: It is said that "math is power". Why?

It is said that "math is power".  The way I see it, it is no more 
"powerful" than reading and writing, in fact I would think reading is
more powerful. If you are not able to read, how could you do math?
 
I returned to college last year to complete my education after a
thirty-year hiatus. I dropped out after three years of school, 
as school just didn't hold my interest anymore. Now after thirty 
years I've returned to school to finish what I'd started. I am doing
research on why "math is power", and frankly I am having a difficult
time answering the question. I have many ideas but none that seem to
make sense to me.  I have been in management for over twenty-four
years and the only math I have used is basic computation: addition,
subtraction, percentage, etc. Can you offer any insights?

From my days in high school I have struggled with math. When I
entered the business world I always had a fear of math and how it
would/could impede my advancement. Since I received my first promotion
(1978) math has been of little use other than basic addition,
subtraction, etc. It apears to me that math has a relationship to
almost everything we do and everything we use: computers, radio,
television, paper media, etc.  But the question still eludes me: 
Why is math power?

Date: 03/13/2003 at 11:58:52
From: Doctor Ian
Subject: Re: It is said that "math is power". Why?

Hi Carl, 

Thanks for writing to Ask Dr. Math. 

> It is said that "math is power". 

I thought the more common saying was 'knowledge is power'.  

Personally, I believe that power is power, although other things are
often sources of power.  But to say that a source of power _is_ power
is sort of like saying that a lake _is_ a fish:

  Is Geometry a Language?
  http://mathforum.org/library/drmath/view/55427.html 

So I guess the first thing I'd say to you is, don't worry too much
about people who make broad claims of the form '_____ is _____'. 

>The way I see it, it is no more 
>"powerful" than reading and writing, in fact I would think reading is
>more powerful. If you are not able to read, how could you do math?

Just to play devil's advocate for a moment, I can imagine an
illiterate shepherd in a primitive culture who understands that if he
has a sack containing one pebble for each of the sheep he's supposed
to be watching, then he can, by matching up pebbles with the sheep he
sees, 'compute' the number of sheep who are missing at any given time.
 
That's clearly mathematics (it is, in fact, a model of subtraction),
and it's clearly useful, and it doesn't involve reading.  And, within
the context of his culture, his ability to keep track of his sheep
might be much more valuable to him (i.e., might confer upon him much
more power) than being able to read.

Of course, you and I live in a very different culture!  Mostly I'm
just trying to make a point about how difficult it is to make
categorical statements of any kind, about anything.  

>I returned to college last year to complete my education after a
>thirty-year hiatus. 

Congratulations!  And good luck with your studies. 

>I dropped out after three years of school, as school just
>didn't hold my interest anymore. Now after thirty years I've 
>returned to school to finish what I'd started. I am doing 
>research on why "math is power", and frankly I am having 
>a difficult time answering the question. 

That's not surprising.  The point of the URL I cited earlier, 

  Is Geometry a Language?
  http://mathforum.org/library/drmath/view/55427.html 

is that you'd probably make your task considerably easier (and the
result more useful) if you spent some time up front refining your
question so that you know exactly what it is that you're trying to
figure out.   

>I have many ideas but none that seem to make sense to me, I have
>been in management for over twenty-four years and the only math 
>I have used is basic computation: addition, subtraction, percentage, 
>etc. Can you offer any insights?

Sure!  That's what I'm here for, to offer insights.  :^D

Here's one insight:  In some places in the United States, people do
something called 'noodling', which is a kind of fishing, specifically
for catfish.  To noodle, you go to the kinds of places where a catfish
might live, and you wiggle your fingers as bait.  What you want is for
a catfish to try to eat your hand, so you can grab his jaw and wrestle
him out of the water.    

There are several problems with this!  For one thing, catfish get
pretty big, so sometimes the catfish pulls you under the water to
drown before you can pull him out of the water to suffocate.  Also, it
turns out that snapping turtles and poisonous water snakes often live
in the same kinds of places as catfish.  There are other problems,
too, but you get the point.  

Now, imagine a man who has spent all his life noodling for catfish,
not because he thinks of it as a sport, but because he doesn't know
any other way of fishing.  He might tell you that he's been getting
along fine with just his fingers, and his perception is accurate.  But
his perception is also limited!  He doesn't know that there are people
who stand on the shore, safe and dry, casting baited hooks into the
water and pulling fish out.  And if he doesn't know about something,
he can't conceive of missing it.  

I don't know who first came up with the idea, but many people have
divided knowledge in to four areas:

  Things you know,                Things you don't know,
  and know that you know.         and know that you don't know.

  Things you know,                Things you don't know,
  but don't know that you know.   but don't know that you don't know. 

For any given person, 'mathematics' would normally be split up among
these four areas.  For example, I would say that for a 'normal' person
in our society, the split would include: 

  Addition, subtraction,          Algebra, geometry, 
  multiplication, division,       trigonometry, calculus, 
  percentages.                    statistics. 

  Various problem solving         Much more powerful problem-
  techniques (e.g., how to        solving techniques.  Also, 
  deal with simple story          what things like algebra and
  problems).                      calculus are good for, and
                                  why the people who use them
                                  daily care so much about them.

Do you see the point I'm trying to make?  Just because you didn't know
about something and didn't use it doesn't mean that it wasn't there to
be used, and that you wouldn't have been much better off knowing about
it so you could use it.                             

A natural question to ask at this point might be:  Why haven't you
known about all this stuff for all this time?  As you noted yourself,
you left school because it didn't hold your interest.  And I suspect
that one reason it didn't hold your interest was that your teachers
never explained to you (in a way that you found satisfactory) why you
_should_ find it interesting.  We get questions of the form "Why
should I care about ____?" all the time, e.g., 

  Why learn to factor?
    http://mathforum.org/library/drmath/view/60957.html 

It's not easy to answer these kinds of questions in a single email
exchange (and I'm certainly not going to claim that the answer I just
cited is perfect), but my own experience suggests to me that students
would be a lot more open to math if we placed less emphasis on
particular techniques, and more emphasis on the what it means to
approach a problem mathematically, e.g.,

  Factoring vs. an equation
  http://mathforum.org/library/drmath/view/61333.html   

  How many pencils?
  http://mathforum.org/library/drmath/view/56828.html 
  
  Why factor?
  http://mathforum.org/library/drmath/view/53277.html 
  
I say this as someone who actually got an undergraduate degree in
mathematics without really understanding any of the things I'm telling
you today.  My first job out of college was working as a software
engineer at NASA, and the thing I treasure most about that job was the
chance to work with some real mathematicians, who gave me a glimpse of
how the world looks through a mathematical lens.  

One of the things that constantly knocked me out was how I'd be
sitting around discussing a software problem with one of these guys,
and he'd suddenly say something like "Well, there's a theorem that
says blah, blah, blah", and it would be some seemingly random thing he
picked up in a course in graduate school, which he hadn't thought
about for 20 years, but which turned my hard problem into an easy one
just by changing the way I looked at the problem.  

One of the biggest lessons I learned from these discussions was that
it's worth learning as much as you can about anything you can, because
you can't predict in advance when you're going to find a connection
between two seemingly unrelated concepts.  Sometimes that's just fun,
and all you get out of it the pleasure ("Cool!") of having an insight
into something.  But sometimes it changes something you thought was
impossible into something easy, or changes a task that you thought was
going to take a year into a task that will take two hours - or better
still, a task that you don't even have to do, because someone else has
already done it for you.  

Of course, my experience is mostly with small projects, involving a
few people over the course of a few years.  But consider getting the
same kind of leverage on projects involving many more people, over
much longer periods of time.  

For example, a friend once told me about an image processing project
he was working on as part of a large defense contract, which involved
using radar from high-flying planes to spot ships on the ocean.  It's
a tough problem, because you have lots of noise from waves - it's kind
of like trying to spot a needle on a piece of crinkled tinfoil.  One
day, he was discussing it with someone, who asked him a couple of
simple questions:  "Radar is light, isn't it?  And water polarizes
light, doesn't it?"  And those two questions changed the nature of the
problem!  If you figure out how the water is polarizing your radar
waves, you can use a filter to eliminate the polarized part of what
comes back, and what's left - if there is, indeed, a ship where you're
looking - is like a picture of a car sitting in the middle of an empty
parking lot.  

In other words, by finding a nice, mathematical way to describe what
the background noise _was_, they were able to make that noise - and
thus the problem itself - disappear.  

Conversely, sometimes mathematics can tell you that something you
thought would be straightforward is actually impossible, so you should
stop wasting time on your current approach and look for a different one. 

To the extent that people _do_ say that 'mathematics is power', I
think this is the sort of thing they mean by it.  

>From my days in high school I have struggled with math. When I
>entered the business world I always had the fear of math and how it
>would/could impede my advancement. Since I received my first 
>promotion (1978) math has been of little use other than basic 
>addition, subtraction, etc. 

The noodler from our earlier example would say that he's had little
use for anything other than his fingers when catching fish.  What
would you say to him now?  

When I got to NASA, the only programming languages I knew were FORTRAN
and Pascal, and all the scientists we worked with used FORTRAN, so I
wrote all my programs using that.  After about five years, I started
getting interested in artificial intelligence, which introduced me to
a different language, Lisp.  And suddenly programs that would take
days to write in FORTRAN were taking minutes to write in Lisp.  Huge
parts of what I had always considered to be 'normal' parts of
programming tasks (like making sure that you have enough memory
dynamically allocated or statically declared) simply disappeared, not
because the nature of the tasks was different, but because Lisp
provides very flexible data structures that you have to build by hand
in other languages.  

Now, on the one hand, I could say that during those first five years,
Lisp was of little use to me.  Or, I could say that during those first
five years, I had no idea how useful Lisp would have been to me.  

Lisp was in the lower right-hand corner of my 'know/don't know'
diagram, and it could easily have stayed there forever.   If it had
stayed there, I could happily have argued that knowing FORTRAN was
sufficient for doing anything that I wanted to do with a computer. 
Would that perception have been accurate?  

>It apears to me that math has a relationship to
>almost everything we do and averything we use: computers, radio,
>television, paper media, etc. The question still eludes me: Why is
>math power?

Here are two other ways to think about the extent to which math is (a
source of) power.  One involves what you can do if you're familiar
with math:

  What is mathematical modeling?
  http://mathforum.org/library/drmath/view/61551.html 

The other involves what other people can do to you if you're not:

  Understanding graphs
  http://mathforum.org/library/drmath/view/61632.html 

  Is algebra useful in the real world? 
  http://mathforum.org/library/drmath/view/61611.html 
  
I think both of these ways of looking at math are relevant to your
question. 

I hope this helps!  Feel free to write back if you'd like to talk more
about this, or anything else.  

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