I am sure our discussion shall be fruitful.

I am equally optimistic.

I am going to derail our discussion for a bit, please forgive me. I am a much better student than teacher. I am too busy learning to pause long enough to explain my own ideas to others. Perhaps this is why I am poor at doing so. Again, forgive me. As I explain the following, it will not be smooth, and it may seem as if I am talking down to my audience. I am just struggling to explain something difficult in the simplest way possible.

There are some common problems and mistakes made when people conceptualize the finite and infinite in segments. Those segments could be time or length, for simplicity, I will stick with length.

All of the following regards the line segment from Zero to One, inclusive:

0----------1

What I am about to explain goes against the grain to say the least. My calculus instructor and I hammered this out over a decade ago. I was lucky to have her for three years of Calculus, and spent enough time in her office, that by the second year, I had finally convinced her that my account of the line segment is correct, and the one her profession espouses is wrong. I understand that most mathematicians disagree with what I think, but don't make the fallacy of ad veracundium. Mathematicians are wrong, and I am right. (said without hubris or ego, just an understanding of both sides)

Problem Number 1:

People make the mistake of equating a limitless supply of units of precision with an infinite number of each unit of precision.
__What I mean:__
The unit segment is of length -1-. Let us create a standard unit of length and call it a Dash. You will see that in my line segment, I have created it by using 10 Dashes. A Dash is of length -.1- There are 10 Dashes in the line segment.

Now, let us create a smaller unit of measurement, and call it a Dish. Let us say that there are 10 Dishes in a Dash. There are 100 Dishes in the line segment.

Let us create another standard unit of measurement called a Dosh. There are 10 Doshes in every Dish. There are 1,000 Doshes in the line segment.

As you can guess, we can do this for an

**Eternity**. We can keep creating new D*shes, and for each one, there will be more of them in the line segment than the previous D*sh. Now, let us look at the problem again:

People make the mistake of equating a limitless supply of units of precision with an infinite number of each unit of precision.
Just because we can keep creating new D*shes, each with one more decimal place of precision, does not mean that any of them have an infinite supply in the line segment. For each D*sh, I can tell you how many of them are in the line segment. Imagine that we test this out:

You are in charge of making up the D*shes, I have a calculator with keeps multiplying my previous answer by -10-. Each day you and I wake up, you create a new D*sh by dividing yesterday's by -10-, I then use my calculator to multiply yesterday's number by 10, and tell you how many of your new D*shes there are in the line segment.

Each day, your D*sh will get closer to zero, and my answer will get closer to infinity, but we will never, ever, ever, ever get there. Neither of us. You are creating a fraction, whose denominator keeps increasing by a factor of ten, and I am creating a new fraction, whose numerator keeps increasing by a factor of ten, which will never get us to zero or an infinity. Every day, you will be wrong to say there is an infinite number of segments, and every day, I will be correct to say that there is a finite number of segments (and point to my calculator for proof). You will never be right, and I will never be wrong.

What confuses most people is that we can do this for an ETERNITY, which they confuse for INFINITY.

__Problem Number 2:__
**People think that they can start counting in Dashes, and when they run out, move to Dishes, and again to Doshes.**
__What I mean:__
It is generally thought that there is an infinite number of points in the line segment. The way our brain fools us into thinking this also relates to the next problem that I will bring up, but let me point out this half of the problem first. When we think of small segments of our line segment of length -1-, we tend to pick a small unit, and imagine that we can just create a smaller unit (as we just went over in the last problem). What we are doing, without usually knowing it, is counting in one D*sh, and when we run out of room, moving to a new D*sh, and counting some more. Here's what happens:

We count in Dishes (.01), and find that there are 100 of them in the segment. But WAIT, I can think of a smaller unit, a Dosh (.001), so we haven't really run out of room, I just count these smaller things and find that more of them fit in that last bit of room before we run out of space. And when they run out, we create another D*sh of size .0001, and so on. And our brains fool us into thinking that since we can create a new D*sh, there are an infinite number of Dishes!

This problem is closely related to the previous one, but it manifests itself differently when we start thinking about line segments. The important thing to learn here is that you must pick a unit of precision and stick to it. You can switch to a new D*sh just because you ran out of space. This is why I often speak of God brewing a cup of coffee. I don't want people to start talking about Grinding Beans as a unit of measurement. In this way, and by naming -.001- a Dosh, I demonstrate something very, very important. When you add a decimal place, those two things are both Numbers, but that is all they have in common. They are as different as Grinding Beans and Placing Filters. You can't keep switching from one to the other and think that you are proving anything about the prior one.

__Problem Number 3:__
**Designations of POINTS on a line is not the same as measuring the LENGTH of a line.**
__What I mean:__
Points on a line are designations of location, not length. They are a mystical thing of size Zero. Whether or not there is an infinite number of them in the line segment has no bearing on the length of the line segment. This is the toughest hurdle to overcome, and I don't expect anyone to make it in a day.

When someone points to one of these points, they are NOT actually placing their finger ON THE LINE. They aren't even pointing AT THE LINE. What they are doing is DIVIDING the line into two parts. Remember, the Point has no length, so it can not be a "part" of anything. It is a place of division. And we can, once the segment has been divided, talk about the lengths of these two segments. One will have a length from this division to the Zero, and the other from the division to the One. Both segments will be of finite length.

Now, the next problem made is that you can now divide the line segment a tad closer to the One, and further from the Zero. We are moving the Point to the right a little bit. How far to the right do you want to move it? Pick any length, and you have created a unit measurement again, a D*sh. The segment to the left has grown by a D*sh, and the segment to the right has decreased by a D*sh. You can move a smaller amount, if you like, and divide somewhere else, you have created another D*sh.

Again, the fact that we can do this for an ETERNITY, does not mean that the line segment has an INFINITE number of units of length.

Understanding all of this resolves many mathematical paradoxes, which is how I know it to be correct.

Peace,

swivel