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

• danne89
In summary, Newton solved Zeno's Paradox by redefining "time" as the area swept around a center of force.
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.

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

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.

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.

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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?

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

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

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.

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

I don't see any difference between saying

"It's at rest for zero time"

and

"It's never at rest"

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?

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)

danne89 said:
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.

Ok. But indeed how did he fix it?

danne89 said:
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.

danne89 said:
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.

## 1. What is Zeno's Paradox of the Arrow?

Zeno's Paradox of the Arrow is a philosophical thought experiment that questions the concept of motion. It argues that an arrow in flight is at rest at every instant of its motion, and therefore, it cannot move.

## 2. Who was Newton and what was his solution to the paradox?

Isaac Newton was an English mathematician, physicist, and astronomer. His solution to Zeno's Paradox of the Arrow was his first law of motion, also known as the law of inertia. This law states that an object will remain at rest or in motion in a straight line at a constant speed unless acted upon by an external force.

## 3. How does Newton's solution address the paradox?

Newton's solution addresses the paradox by acknowledging that an object in motion will continue in motion unless acted upon by an external force. In the case of the arrow, it is the force of the bow that propels it forward, overcoming the resistance of air and other external forces. Therefore, the arrow is not at rest at any point during its motion.

## 4. Did Newton's solution completely resolve the paradox?

No, Newton's solution did not completely resolve the paradox. While it provided a scientific explanation for motion, it did not address the philosophical question of whether an object can truly be in motion at any given moment.

## 5. Are there any other proposed solutions to Zeno's Paradox of the Arrow?

Yes, there are other proposed solutions to the paradox, such as the concept of infinite divisibility. This argues that an object, such as an arrow, can be divided into an infinite number of points, and therefore, it is always in motion between those points. However, this solution has also been met with criticism and remains a topic of debate among philosophers.

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