What is a point?

Date: 8/26/96 at 21:32:45
From: Anonymous
Subject: What is a point?

Define a point, please.

Date: 8/27/96 at 22:15:6
From: Doctor Robert
Subject: Re: What is a point?

The word "point" is often left undefined in geometry texts. It is 
pretty easy for us to conceptualize a point, but it is quite difficult 
to define exactly. I would say that a point is an entity that has 
only one characteristic. That characterstic is its position. But 
then, you see, I haven't defined position, have I? Anyway, a point 
has no size, color, smell, feel, etc. It has only its position. 

A point is often represented by a smudge of chalk on a blackboard, 
or a smudge of pencil lead on a piece of paper. But, of course, 
those are only representations. A point can really only exist 
in our minds. 

I hope you're not more confused than when I started. 

-Doctor Robert,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 8/27/96 at 22:15:6
From: Doctor Ken
Subject: Re: What is a point?

Hi! The question you asked is actually quite a subtle one, and in a 
certain sense it makes a difference what kind of math you're doing. 
If you're doing Geometry, then no amount of physical torture will get 
me to define a point; it's an undefined term, and it's undefined on 
purpose. 

So this is really about math in general: what is an undefined term? 
Well, as you may know, math is about making definitions and proving 
theorems. For example, I can define a circle to be a set of points 
equidistant from a given point, and I can prove a theorem about that 
circle, such as "the perpendicular bisector of a chord of a circle 
will pass through the center of the circle." But in order to make 
these definitions and prove these theorems, I need to start from 
_something_.

That's the role of the UNDEFINED TERMS and AXIOMS in math. When I 
defined the circle as a set of points, I used objects I already had 
(points) to define a new kind of object (the circle). So every time 
we define a new object we have to have some old object to base it on. 

If you think about the structure of math as a tree, there has to be 
something at the bottom of the tree, some objects that aren't defined. 
A point is one of these objects. It is undefined. It is just an 
object. In geometry, people usually think of points, lines, and 
planes as undefined objects (also known as undefined terms).

So what are axioms? Well, they're statements that we don't have to 
prove (much as undefined terms are objects we don't have to define). 
A traditional example of an axiom in geometry is the statement "given 
any two points, there is one and only one line that passes through 
them." This statement is just accepted as true, so that we have a 
starting point, something we can use to prove theorems. It also does 
something else: it tells us something about points and lines. Any 
concepts we have in our heads about points and lines MUST satisfy this 
axiom. If we're thinking of points as bottles of beer and lines as 
telephone poles, then we have a problem, because I can show you a 
couple of bottles of beer that don't have a telephone pole connecting 
them.

So that's how it goes in geometry. In other parts of math, for 
instance when you're using the coordinate axes, a point may not be an 
undefined term at all - it can just be a list of numbers, such as 
saying "consider the point (3,2)." For that, check out the answer 
from Dr. Chuck, which follows.

-Doctor Ken,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 8/27/96 at 22:15:6
From: Doctor Chuck
Subject: Re: What is a point?

A "point" is an infinitely small entity at a specific location on a 
number line, plane, 3-D space, etc. When we talk about points, we are 
referring to one specific location.

For example, along a number line the number "2" exists at just one 
point. I said that points are infinitely small because the point at 
'2' is different from the point at '2.000000001'. Here's a picture of 
a number line:

                                   The point 2
                                        |
                                       \/
-infinity <==...---(-1)-----0-----1-----2-----3--... ==> infinity



Okay, so this makes sense: if you want to distinguish one place 
along a number line, you "point" at it. You label that place 
with the corresponding number, and refer to it with that number. 

Now, how do you distinguish a location in 2-dimensional space (i.e. 
a sheet of paper)? Imagine that we have two number lines, one 
horizontal and the other vertical. We are "pointing" at a place "p":

              ...
               |
              2
               |
               |
               |
               |
               |
              1            p
               |
               |
               |
               |
               |
...---(-1)-----0-----1-----2-----3--...
               |
               |
               |
              ...

How do we describe where the point 'p' is? We can't just say p is 
at 2 because we don't know which number line that refers to - is it 
at 2 along the horizontal number line, or 1 along the vertical number 
line? 

To describe where 'p' is, you must talk about where it is both 
horizontally AND vertically. So, you can say:

   'p is at 2 horizontally, and 1 vertically'. 

However, this is a mouthful to say. Because describing points in 2 
dimensions is really useful, people have defined some conventions to 
make life easier. They call the horizontal number line the 'x-axis', 
and the vertical number line the 'y-axis'. The convention for talking 
about 2-dimension points is to write: ( position along x-axis , 
position along y-axis ).

Therefore,

   'p is at (2, 1)'

2-dimensional points can be described by any pair of numbers. 
For example, (4,5)  (6.23432, 3.14...)  and (-12, 4)  are all points.

Sometimes people want to describe a point in three dimensions. 
To do this, they need to use a triplet of numbers like  (1, 2, -5)  
- do you see why? 

I hope this helps.

-Doctor Chuck,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 8/2/97 at 12:54:51
From: James R Sheldon
Subject: Re: What is a point

I was reading your answers to this person's question, and I remembered
seeing some stuff about points in a Geometry book:

From The University of Chicago School Mathematics Project Geometry:

4 descriptions of a point:

   A point is a dot.
   A point is an exact location.
   A point is an ordered pair of numbers.
   A point is a node of a network.

The dot one is used often in computers, the location and ordered pair 
are used in Geometry usually, and the node is used to solve problems 
like the Konigsberg Bridge Problem.

I found the networks the most interesting since I had never seen those
before.