# The Final Pathetic Bleatings of the Forum

Question:
I humbly request the wisdom of Martin Heidegger

Mathmaticians have agreed that there are an infinite number
of points between any two given points. This being assumed,
you would then have to touch an infinite number of points in
order to travel the distance between the two points. If you
had to touch an infinite number of points from point one to
point two, how could you ever reach the second point?

Replies:

In those days, the new Home Motor was the subject of excited discussion in homes and businesses and around the dinner table.

I know I signed a two-year contract, but I just can't do this any more.

Take a break; pull yourself together. I'll handle this.

While your problem is pathetically underspecified, I will take a crack at it anyway.

1. You are a dork.

2. Your problem evidently assumes an infinite series of events, with each event taking an infinitesimal amount of time.

3. Because you are a dork (by (1)), you assume that an infinite series of events must take an infinite amount of time.

4. Because you are a dork (by (1)), you evidently cannot imagine an infinite series that has a finite sum.

5. Let me give you an example. What is:
```(1/2) + (1/4) + (1/8) + (1/16) + (1/32) + (1/64) + ...
```

This is an infinite series, and yet it has a finite sum!

Kosak, you are the dork.

Back WAY the hell off, freak.

First of all, the notion of an "infinitesimal amount of time" is ill-defined. Your "explanation" is worthless.

Second of all, the sum you are suggesting is underdefined, since strictly speaking it is not clear what the 7th term should be. I suppose your brain intended to display the sum:

Third of all, here is a precise explanation of what is going on:

Dear Forum 2000 visitor, you asked how it is ever possible to move from one point on the line to another, given that there are infinitely many points in between.

To study this question properly, let R be the standard real line. Define the category of paths P as follows: the objects of P are the points of R. The morphisms between x and y are continuosly differentiable functions f: [0,1] --> R such that f(0) = x and f(1) = y. Given morphisms f and g, their composition is defined by

(g f)(t) = if t <= 1/2 then f(2 t) else g(2 t - 1)

Now your question is reduced to asking what the composition of infinitely many morphisms is. For this we have to move to the appropriate arrow category P--> and study its completness properties. It is easily seen that such a composite corresponds to a limit, if it exists, and that the limit exists when the sum of infinitely many intervals involved is finite.

That's what I said.

Yes, but you were sloppy.

Bite me.

Although, I must agree that there is something appealing about category theory.

Its algebraic elegance?

No, not that.

Its ability to relate seemingly unrelated concepts?

No, not that either.

Its connections to logic?

I think I know...

What?

The arrows, I think the arrows are cool.

You mean the morphisms?

Whatever.

Dear Andrej and Corey. Your discussion unveils an interesting subcontext in your relationship.

[Hall of Fame]