If numbers are infinite in both directions, physical contact is impossible.

[tdahlin]

This is a paradox that has been bothering me since I was taking algebra in high school. Let's say that I want to represent the distance between to objects. Given that numbers are infinite in both directions, by which I mean that there is no limit to how large, or small a number can be, there should be no limit to how close, or how far to objects can be from each other. Let's say I were to drop a a ball from a distance of one foot above my desk, the ball will move towards my desk, and the number representing the distance between the ball and the desk will become increasingly smaller until it comes into direct physical contact with the desk, at which point the ball will be a distance of 0 inches from the desk. however this should be mathematically impossible. The number representing the distance between the desk and the ball should just keep getting smaller and smaller, never actually reaching zero. Mathematically, no matter how close the ball gets, it can still get closer without touching. even if you were to represent the distance as a decimal followed by 100 googul zeros with a 1 at the end, you can still add one more zero, or another 100 googul for that matter. clearly however, there is a point at which a physical distance may not become any smaller, and the two objects must touch. In fewer words, there are infinite numbers between any two numbers, no matter how close or far apart those two numbers are, thus in the one foot between the ball and the desk, there should be infinite points in space, and in order for the ball to reach the desk it must pass through every point in space between the two objects. how can the ball pass through every point if there are infinite points?

 

[Vorde]

http://en.wikipedia.org/wiki/Zeno's_paradoxes

 

[xAxis]

... Let's say I were to drop a a ball from a distance of one foot above my desk, the ball will move towards my desk, and the number representing the distance between the ball and the desk will become increasingly smaller until it comes into direct physical contact with the desk, at which point the ball will be a distance of 0 inches from the desk. however this should be mathematically impossible. ?

Why? You sad yourself when it comes to direct physical contact with the desk the distance will be zero.
@Vorde: This has nothing to do with Zeno's paradox

 

[xAxis]

... how can the ball pass through every point if there are infinite points?

Points don't have a lenght. The length of those infinite points is still finit. So your "paradox" reduces to "how can ball pass some well defined length :)

 

[Vorde]

This has everything to do with Zeno's Paradoxes, and it comes from an improper understanding of limits largely. The general idea of 'you shouldn't ever be able to get anywhere' because you can always make the distance gap smaller than it is is a central notion of Zeno's Paradoxes.

I like to say that the solution is that there is nothing requiring one to take smaller and smaller steps, so you can just get there easily.

 

[Drakkith]

however this should be mathematically impossible. The number representing the distance between the desk and the ball should just keep getting smaller and smaller, never actually reaching zero. Mathematically, no matter how close the ball gets, it can still get closer without touching.

Whether it CAN get closer without touching it is irrelevant. What matters is what it is actually doing. It is moving with a certain velocity that mathematically says that it WILL move X distance in Y time. You can divide these in half to your hearts content, but the reality is that the ball WILL reach your desk in a finite amount of time.

 

[mfb]

This is a paradox that has been bothering me since I was taking algebra in high school. Let's say that I want to represent the distance between to objects. Given that numbers are infinite in both directions, by which I mean that there is no limit to how large, or small a number can be, there should be no limit to how close, or how far to objects can be from each other.

"Distance" is defined as absolute value - and this is bounded by 0. It has no upper bound, but it has a lower bound.

 

[sophiecentaur]

It depends on what you mean by "touching". There is a big difference between the mathematical and physical approaches to this question. There is no time that two physical objects can be said to be touching in the sense that two ideal mathematical shapes could be touching (i.e. outer edge of one object is in exactly the same position as the outer edge of the other). Real objects are 'fuzzy'.
When two atoms (which have finite size but are not really like billiard balls) come close together, there will be a separation between their 'centres' when there repulsive forces start to get high. They start to 'push against each other'. That could be when you say they are touching - but nothing is actually touching anything. The internal electric fields and forces start to increase at a high rate as you try to decrease the separation.
Zeno's paradox (which no one takes seriously, in any case) cannot be brought in here.

You may eventually do a Maths Analysis course (could have another name, these days) in which all the strict definitions of limits, inequalities and infinitessimals etc. etc. are dealt with rigorously. Once you get that far, I think you will find that the verbal 'blather' of a lot of the Philosohpers will be seen to be pointless and you won't worry so much.

 

[Ibix]

Disturbingly, if Wikipedia's is to be believed, people do still take this seriously.

To me, the resolution of the apparent paradox is simple to sketch, although formal limit theory is needed for rigour. Basically, people seem to get hung up on the idea of there being an infinite number of distance steps to travel, and fail to remember to slice up the travel time the same.

For the ball moving at constant speed, covering a distance d in a time t, it's fairly easy to see where the trick in the paradox is. Split d in two: then the ball travels a distance d/2 in a time t/2, twice. Split it again: now the ball travels d/4 in time t/4. Again: d/8 in t/8. Again, and again. Always, the total distance is d and the total time is t. This is clear for any finite number of splits. Is it true for an infinite number? That is where you need formal limit theory - but it should be obvious that for there to be a problem, something special and unexpected has to happen. In other words, the trick is that Zeno's Paradox relies on magic happening when infintesimals arise.

Since calculus is spectacularly successful and based entirely on an understanding of the behaviour of infintesimal quantities, that isn't really a defensible position, to my mind.

 

[sophiecentaur]

There's a simple two word reason for what we find and that's "Convergent Series". There's no reason why the language of Maths can't be just as valid as the language of words.

 

[phinds]

Whether it CAN get closer without touching it is irrelevant. What matters is what it is actually doing. It is moving with a certain velocity that mathematically says that it WILL move X distance in Y time. You can divide these in half to your hearts content, but the reality is that the ball WILL reach your desk in a finite amount of time.

Ah, there you go being logical again. You're no fun at all.

 

[Drakkith]

Ah, there you go being logical again. You're no fun at all.

You know me, Mr. Killjoy.

 

[cragar]

maybe we should do this at the edge of a black hole and ask how long it takes for someone watching at infinity. just kidding

 

[HallsofIvy]

maybe we should do this at the edge of a black hole and ask how long it takes for someone watching at infinity. just kidding

You go first!