Newton's Solution to Zeno's Paradox of the Arrow

[danne89]

How did Newton solve Zeno's Paradox of the Arrow which is stated as follows:
1. When the arrow is in a place just its own size, it’s at rest.

2. At every moment of its flight, the arrow is in a place just its own size.

3. Therefore, at every moment of its flight, the arrow is at rest.

Did he redefine the meaning of an instance? I cannot understand.

 

[matt grime]

No, I think he'd've pointed out that number 1 is not a meanignful statement.

 

[HallsofIvy]

Newton was quite comfortable with "instants". That was essentially what he meant by his "infinitesmals".

What Newton did was give a specific definition of "instantaneous" speed.

In Zeno's time (and up until around Newton and Liebniz {Fermat and DesCartes amoung others had the basic concept}) the only way to calculate "speed" was "average speed"- distance moved divide by the time interval. At a given instant, there is no distance moved and no time interval and so you cannot find (average) speed. That was the point of Zeno's "When the arrow is in a place just its own size, it’s at rest." The calculus allows us to define speed at a given instant.

 

[DaveC426913]

I have to say, statement 1 is not at all intuitive - I wouldn't let it pass as a given.

I can see how Newton put numbers to that.

 

[danne89]

Hmm. I find the first premiss quite intuive. When you fotograph a object you catch its position in an instance (or very smal time interval). When you look at a movie, you have numberless of pictures, which simulates motion. Why isn't it obvious that in an instance, the object is only in ONE place?

 

[HallsofIvy]

Yes, but that's not what "at rest" means.

 

[danne89]

Clearly it's! Do the pictures on your fotos move?

 

[Jimmy Snyder]

All of this confusion reminds me of an old joke:

Police officer: Did you know you were traveling at 90 miles per hour?

Driver: But that's impossible officer, I haven't been driving for an hour.

The arrow is moving all the time. Divide zero by zero all you want, the arrow still moves.

 

[danne89]

If the arrow is moving in an instance, yhen the heck is it resting?

 

[Jimmy Snyder]

I don't see any difference between saying

"It's at rest for zero time"

and

"It's never at rest"

 

[danne89]

So no object is never ever in a state of rest. So if a wait for an infinite amount of time, my foto will start move?

 

[fourier jr]

i don't think it was Newton who fixed zeno's paradox about the arrow, i think it was aristotle. morris kline's "mathematical thought from ancient to modern times" has some good info on the 3 paradoxes & whoever fixed them all (i think it was aristotle)

 

[Jimmy Snyder]

So no object is never ever in a state of rest. So if a wait for an infinite amount of time, my foto will start move?

No MOVING object is ever in a state of rest.

 

[danne89]

Ok. But indeed how did he fix it?

 

[DaveC426913]

Ok. But indeed how did he fix it?

By inventing calculus, he made it possible to calculate speed at a given instant.

Read HallsofIvys response (post#3). That explains it eloquently.

 

[arivero]

Did he redefine the meaning of an instance? I cannot understand.

In the deep end, it is said that he uses Kepler's second law to redefine "time" as the area swept around a center of force. In this way the problem becomes completely geometric, you are only asked by relationships between Keplerian areas and Cartesian coordinates, and time is hidden under the carpet. Zeno paradox still lives there in the area postulate, of course, and ultimately it was found that the angular momenta must be a halfinteger multiple of a basic quantity, thus reviving Zeno problem.