Another trick question!

I saw this posted on a forum. I've been racking my brain trying to prove this wrong but can't. Is space really infinitely divisible? Is there any smart answer to this one to prove it wong? :yuck:

A runner wants to run a 100 meters - in a finite time. But to reach the 100-meter mark, the runner must first reach the 50-meter mark, and to reach that, the runner must first run 25 meters. But to do that, he or she must first run 12.5 meters.

Since space is infinitely divisible, we can repeat these 'requirements' forever. Thus the runner has to reach an infinite number of 'midpoints' in a finite time. This is impossible, so the runner can never reach his goal. In general, anyone who wants to move from one point to another must meet these requirements, and so motion is impossible, and what we perceive as motion is merely an illusion.

These things are so interesting! I just wish I could figure them out :grumpy:

Thanks :)

-jen

arildno
Homework Helper
Gold Member
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Think of a related problem:
Suppose you've got a rod of finite length.
Now, if space is infinitely divisible, you can divide the rod into infinitely many parts.
Does it therefore follow that, because you have an infinite number of parts, the length of the rod is infinite?
Of course not!
Dividing the rod into a finite number of parts doesn't change its the rod's total length at all!
Why should it be different if you divide it into infinitely many?

Your original example is basically a "temporal rod" that you in a clever way divide into infinitely many time intervals.
Just because you have infinitely many time intervals doesn't mean that the sum of these is infinite.

I sorta understand what you're getting at... :shy:

Let me see if I got this right... what you're saying is that 100 meters is a 100 meters no matter how infinitely you divide it. And no matter how much you divide the time line (the time it takes to move across that distance), it doesn't change the sum of the time it takes to run the distance? ( I think I just made that more complicated )

thanks :)
-jen

arildno
Homework Helper
Gold Member
Dearly Missed
The point is:
Just because you can prove you've got an infinite number of time intervals to run through, doesn't meant that the sum of these (the total time) is infinite.

In your case, the first time interval is the time needed to run 50 meters,
the second interval the time needed to run 25 meters and so on.

oh ok! Thank you

Man I wish they could teach this good at school!!! You guys rock!

I just hope I'm not being a pest with these silly questions. :shy:
-jen

Gokul43201
Staff Emeritus
Gold Member

arildno is right. If you make the sum of 1/2 + 1/4 + 1/8 + 1/16 + ... you get 1 not infinite.

BobG
Homework Helper
Jen,

You've got the problem backwards.

Before you can run 100 m, you have to run 50 m. Before you run 50 m, you have to run 25 m. Before you run 25 m, you have to run 12.5 m ...... before you run 5 nanometers, you have to run 2.5 nanometers, and ..... well, you get the picture. Eventually you realize it's impossible to ever get started.

Which is cool . You get to sleep in! :rofl:

That's not Zeno's Paradox, it's ONE OF Zeno's paradoxes . Aristotle's response to that one in particular was that there are two senses in which something can be infinite: in divisibility, or in extent. In a finite length of time someone can come into contact with things infinitely divisible and so a finite length of time can be taken to cover a finite length. Zeno's thing is usually understood to mean that to go a finite length one must cover an infinite number of points and so must get to the end of something that has no end.

Zeno's 2nd paradox is all about Achilles & the Tortoise. The Tortoise can't be overtaken by Achilles because Achilles must first arrive at the point where the Tortoise was when they started, so the Tortoise will always be ahead.

The 3rd one is all about an arrow flying through the air. Zeno said that time is made up of 'instants', so when something moves it's really infinitely many 'instants' back-to-back. So an arrow flying through the air is standing still, since at any instant it occupies a definite position in space. So it can't be in motion.

^^ there are some more for StupidHead

ps- re: the 3rd one when you see a flag waving in the air is it the flag waving, or is it the wind waving? Or both, or neither? Maybe it's just your mind that's waving. hehe

Math crap

excuse me if this accidentally posts more than once, my browser's been acting strangely with the forum. anyways,

It is correct to say that if something can be divisible infinitely many times, it is not necessarily infinite in itself. So you view this trip as such - our runner travels 50 meters, then 25 meters, 12.5, etc. etc. Well, since the numbers (distances) to add are becoming smaller and smaller, the total sum converges to a certain point, rather than continues to infinity. Let's see it mathematically.

So we first go 50 meters, because that is the midpoint. Then we must go to the next midpoint. Thus, the ratio of the terms to add (distances to travel, sequentially) have a common ratio of 1/2. That is, 50+25 (which is 1/2*50) + 12.5 (which is 1/2*25)... This is a convergent geometric series. Geometric because each term has a common ratio for which to find the next term (1/2), and convergent because the absolute value of the ratio is less than one (1/2). The abs(ratio) being less than one means that each progressive term is smaller than the previous.

The formula for solving a convergent geometric series is

(First term)/(1-Ratio)

So our first term is 50, with our ratio being 1/2.

50/(1-1/2)=100

This should be Algebra II material I believe. Now you can do fun stuff, like lets say we have a square inscribed within a circle. A second circle is inscribed within the square, a second square is inscribed within the smaller circle, and more squares and circle to infinity. What is the sum of all the circles minus the sum of all the squares?

Well whatever, I hope a mathematical explanation helps.