How Can a Line Have Length?

Date: 04/14/98 at 22:00:50
From: Magnus Collin
Subject: The impossibility of a line having Length

We have a line, OR a line segment, OR a ray (it really doesn't
matter). If this particular linear entity (let us say that it is a 
line segment, for the sake of hypothetical simplicity) consists only 
of infinite points, all with zero volume, mass, diamater, cross-
sectional area, etc., how can it have length?  Why are we able to take 
a particular line segment and give it a length of 5?  If a substance 
can only be composed of volumeless points, how can it have volume in 
and of itself?  

In other words, in Geometry class, the teacher might draw up a line on 
the blackboard and put a tiny dot on it and call that miniature mark 
"Point B." The mark of the chalk on the line is visible, and we tend 
to see B as just a tiny little point on the line, and we think, "Oh, 
there are infinite points right next to each other which make up the 
line."  In reality, the points are 100% without substance; how can 
they come together and make up one bigger continuous substance 
(composed uniquely of volumeless points) which has the physical 
property of length? 

It would be like trying to add a bunch of zeros together, even an 
infinite number, and hoping eventually to arrive at 5. Even if you 
have infinite points, if each of them has a length of exactly zero, 
they can never result in a new form that has length. 

Thank you very much for your time, and I would really appreciate any 
comments or fallacy which you might find in my logic.

Date: 04/15/98 at 12:19:43
From: Doctor Daniel
Subject: Re: The impossibility of a line having Length

Hi there,

You're basically asking this question:

How is it that Euclidean points (which have zero length, area, volume) 
can unite to form Euclidean line segments (with length), planar faces 
(with area) and solids (with volume)?

It's a fair question, and frankly, one that philosophers have 
struggled with a lot. Here are a couple of quite similar ways it's 
usually answered:

1) It's essentially defined away by concepts like calculus (in its 
more advanced flavor of measure theory). There, the cardinal concept 
really is that the line from 0 to 1 has length 1, and a point is 
simply the interval between two lines that just touch. (Think about 
the space from 0 to 1 on the number line and the space from 1 to 2.  
The "interval" of their intersection is the point 1.) So it's not such 
a big deal (in this manner of thinking) that their intersection has 
length 0; even though the line is the union of the points, it is the 
line that starts out with length 1, which causes the point (interval 
with just 1 point) to have length 0, not the point of length 0 which 
somehow unites to form something of length 1.  

2) Here's a somewhat older way of dealing with the problem, which 
dates to the 19th century. Think of a point as just being the limit of 
the process "Take an interval and cut it in half." Obviously, the 
point has length 0, but it's also clear that all of the intervals 
along the way don't. It's just that, as the limit of this process, we 
can't say that the point has length, say, .0000000000001, since I can 
show you an interval that's even shorter that includes the point.  

Both of these answers date to the last 100 years. Prior to that, 
though, the question has a tremendously long history. Maybe you've 
heard Zeno (a Greek philosopher)'s version, which is essentially:
"Why is it that 1 + 1/2 + 1/4 + 1/8 + ... = 2?"  I do think there's a 
great value in these answers, though, in that they typify what 
mathematics has taught us in the last 100 years.  

Good luck!

-Doctor Daniel,  The Math Forum