Is the Number Line Both Continuous and Porous?

Hi Holly,

This is an interesting question. Where exactly are the gaps and the
porosity that you see? Where's the elbow room?

The "densely populated" is the word exactly for the reasons you say:
no matter how close together two numbers might be, there's still TONS
more numbers packed into the tiny space between them!

So the line is truly full of numbers.

You say there are just as many gaps as numbers, but I don't see the
gaps. I think it's really hard to point at one. No matter where you
point, there are still numbers in there! So that's why it's called
"densely populated".

If you want gaps, look at just the integers--there's a big gap between
0 and 1 with no numbers in it, for example.

But when you have the rational numbers, those gaps start to evaporate,
and when you go to the reals, there's even more numbers packed in to
those spaces. Both the rationals and reals would be described as "dense".

I don't see the gaps you refer to, so please help me out--where are
they? I think I'm just not understanding the way you see the number
line. I'd love to talk about this more, so please explain it to me!

I don't think this is a case where you need lots of equations.
Talking about what the words mean, and explaining what you see when
you look at a number line should be just fine.

Enjoy,

Dear Dr. Schwa:

I am both elated and humbled that you answered me and are willing to
discuss the idea of a gappy number line further.

I have said that the idea of DENSE and GAPPY are in conflict.
However, I do not see INFINITE GAPS - INFINITE NUMBERS as in conflict.
I am conflicted about referring to the number line as DENSELY
POPULATED and CONTINUOUS and having no mention of GAPPY. Referring
to the number line as a "densely populated, continuous line with a
infinite number of numbers and infinite number of gaps" seems more
reasonable to me.

As to seeing the gaps, even with an infinite number of numbers
between, for example, the numbers 1 and 2, we must believe that each
of these numbers (all infinity of them) is DISCRETE, yes? Each is
separate, and can be no other, no matter how tightly packed against
its infinite brethren. Didn't some math person (Dedekind or some
such) make some sort of "cut" where he cut it cleanly, *between* two
of the brethren, not taking a piece of either? He cut them AT THE GAP
that exists between each number, is my thought.

............... <---- If I could push these closer, they would
appear to be a solid line. Yet, there is a gap between each. If each
point in a line could be named with its number, each would have its
accompanying gap on either side. There must be a gap or they
would "blend" into some monstrous single number.

Well, that is the limit of my thoughts. Thank you for listening.

Sincerely,

Holly

Hey, I think we all here find your type of question interesting.
Perhaps it's a bit more of a philosophy question than a math question
but a lot of the greatest mathematicians did a lot of philosophy, too.

Your description is reasonable--it's just not what I see when I look
at the line! I see numbers, and any "gaps" are filled in by more
numbers, and any "gaps" between those are filled in by yet more
numbers. It's just numbers everywhere!

This is what a lot of 18th century mathematics was all about. Each
number is itself, yes, and can be no other. I agree. But the
"separateness" is the part I'd argue about.

There's a technical definition, from topology, of what it would mean
for the number line to be "discrete", and indeed you can IMPOSE a
discrete view on the number line if you wanted to. So I'm not saying
you're wrong! In the discrete topology, a number's "neighborhood" is
by definition just the number itself. So it lives alone, separate, no
matter how tightly packed it may be.

At the same time, the NATURAL way to look at it is in terms of
distance. A number's "neighborhood" consists of all the other numbers
within some small distance of it, in the natural topology. And then
no matter how small a neighborhood you look at, there are still
infinitely many other numbers in the neighborhood! So this is a way
of looking at the number line in which there are no gaps, just densely
packed numbers all the way down.

Indeed, it's called the "natural topology", which shows quite a big
philosophical bias, as opposed to the "discrete topology", which
is looked at by the mathematicians as being more of a curiosity than
a really useful structure to impose.

In the natural topology, there's not any real notion of separateness.
Essentially, if you take any number x, and say "there's a gap next to
it", you then need to answer how BIG that gap is--and of course its
size must be 0. That's where thinking of things in terms of distance
gives you a view of the number line in which there are no gaps.

And the same with separateness: there's a sense, pretty obvious to
both of us I think, in which 1 is separate from 2. And indeed, 2
makes the end of the gap that 1 begins. Now, answer a similar
question in the real numbers: if there is a gap after 1, what number
ends that gap? Or, if 1 is discrete, how far is it to the nearest
neighbor?

Dedekind cuts are very interesting little structures. They're
actually used to CONSTRUCT the real numbers from the rationals.

All the stuff I said above, about gaps, and neighborhoods, and in my
previous message about having infinitely many numbers in each "gap"
between two numbers, applies whether you're talking about fractions
(rationals) or decimals (reals).

What Dedekind did was to say that by splitting the rationals into two
groups, one group being all the numbers less than something, and the
other being all the numbers greater than something, you could
nonetheless identify gaps in the rationals.

For instance, you could look at the split of all the numbers that when
squared are less than 2, and those that when squared are more than 2.
You'd find that there's a split at some number near 1.4 ... near 1.41
... near 1.414 ... and so on, but there'd be no exact fraction located
AT the split.

However, what Dedekind also showed was that the real numbers
(decimals) are precisely what you need to fill in those gaps. After
adding them to the system, there are no more gaps in the spaces
created by the Dedekind cuts.

>............... <---- If I could push these closer, they would
>appear to be a solid line. Yet, there is a gap between each. If each
>point in a line could be named with its number, each would have its
>accompanying gap on either side. There must be a gap or they
>would "blend" into some monstrous single number.

This is the crux of the argument, indeed, and exactly what it took
about a century of mathematicians and philosophers to work out.

The interesting thing is that there really is a difference between
real numbers (decimals) which are a sort of limit as that space gets
arbitrarily close to 0, and what actually happens when the space IS 0
and the points are all on top of each other.

VERY LONG decimal numbers that eventually stop are still expressible
as fractions; INFINITE decimals may not be.

Similarly, your dots, when squished together, will turn out to cover
only the fractions; there'll still be all the nonrepeating infinite
decimals uncovered (at least, as I envision the process).

I'm trying to avoid getting mathematical as much as possible, but as
I see it, it goes something like this:

Take all the integers 0, 1, 2, 3, 4, ... and let's look at 9378 as one
example. But the whole list of integers are your infinite list of . .
. . . . . spaced out at intervals of 1 unit.

Now divide them all by 10 to squish them together. Now you have
0, 0.1, 0.2, 0.3, ..., 937.8, ...

And keep dividing by 10 to squish.

If you name any terminating decimal, say 0.9378, you can see when
it'll get hit. And it'll keep getting hit forever after that!

After 4 squishes, 9378 lands on 0.9378.
After 5 squishes, 93780 lands on 0.9378.
After 6 squishes, 937800 lands on 0.9378.

So it's definitely covered! But no INFINITE decimal ever gets covered
in your scheme.

So as you keep squishing, there really ARE gaps left, in the Dedekind
sense. Your . . . . . . ., as it squishes, only covers finite
(terminating) decimals. But Dedekind cuts show how to construct the
real numbers out of this to fill in the gaps.

In summary, then:

There are two types of gaps that I see. One is the big finite-sized
gaps like the ones in the integers, the gap between 0 and 1. The
other is the gap like you see in a Dedekind cut, where you can group
the fractions into two sets, one below something and one above
something, and then you find that there is no fraction equal to the
something! There must be gaps in the fractions, which can be filled
in by these somethings (like the square root of 2, or pi, for
example). When you fill them in, then you have the real numbers (and
in fact Dedekind cuts are often used, around junior year for math
majors in college, as an explanation of what the real numbers represent).

Then, once the real numbers are filled in, I don't see gaps any more,
and I've tried to help you understand why not--but maybe no persuasion
will work. It's always allowed to have your own view of things!
However, mathematicians find the definition of neighborhood so useful
that they are unlikely to adopt the alternative (discrete) definition
where every number is a neighborhood by itself. When working with the
real numbers, it's almost always more fruitful to say "a neighborhood
is any interval centered on your number" and though neighborhoods can
be as narrow as you like, they still contain infinitely many numbers.

Feel free to keep writing back, this is fun.

Enjoy,