Is It Really Possible to Draw a Graph?

Date: 02/01/2007 at 22:23:24
From: Zak
Subject: What is the best way to draw a graph

While doing pre-calculus, a common thing for our teacher to ask of us 
was to "draw a graph".  For example, she might ask us to draw f(x)=x.
(Continue reading, it gets more interesting!)

To start off, by definition this graph is a "collection of points,"
and points have no dimension (depth/length/height).  By drawing the
graph with a pencil, you are giving the graph a "3D" framework, 
because the "thinness" (I can't think of a better word) would be
infinitely small, too small to see.

The dimensions of a graph are too small to draw, and therefore only a
"representative graph" can be drawn.  My teacher looked at me 
strangely.  I was wondering if you could help answer this question.

Date: 02/02/2007 at 08:36:30
From: Doctor Peterson
Subject: Re: What is the best way to draw a graph

Hi, Zak.

You're right: When we draw a graph, we aren't really drawing that set
of points; we are making a representative picture of the graph.  In
the same way, if I asked you to draw your house, what you put on paper
would not be the house itself, just a picture of the house, showing
approximately what the house looks like.  That doesn't mean that you
can't draw your house!  You just have to know what we mean when we say
that.

In geometry, when we draw a circle or line, it is never a real circle
or line, because those have no thickness, are not made of atoms, and
(in the case of the line) extend forever.  Our paper is not really a
plane.  All these things are just representations or models of
abstract concepts in our minds: ideal lines, ideal circles, ideal
graphs.  Sometimes we confuse the real with the ideal, and imagine
that what we are drawing is the actual circle or graph; it's important
to call attention to the difference, as you are doing.

Bottom line: You CAN draw a graph--but it's only a picture, not the 
real thing.

Does that clarify things?


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