A Riddle Or Not + Zeno's Moving Arrow

[Intoscience]

The riddle or not:

A barber lives and works in a small town, he is the only barber in the town. The barber only ever cuts and shaves the hair of all the men who live in the town that never shave or cut their own hair. He never ever cuts and shaves the hair of all the men that do their own.

Zeno's moving arrow paradox:

Zeno suggested that the arrow never really moves because each of the smallest possible moments of its journey = 0 time and 0 distance so concluded that even every moment even to infinity only ever adds up to 0.  This has since been proven a fallacy and as we all know in reality an arrow shot from a bow will have a journey of some distance over time. 

So my question is, was Zeno's concept on the right lines/ With modern understanding of the quantum era, do we not describe space & time in discrete quantities, i.e the Planck scales where each moment of space and time may well be made up of a undetermined quantity though never 0?

In addition, if the arrow (though not physically possible) was to travel at C then from its own perspective would it not indeed never travel anywhere (obviously assuming instant light speed acceleration)? I appreciate this second thought is rather metaphysical and not to be taken too literally.    

[Genady]

On 11/10/2021 at 4:12 AM, Intoscience said:

With modern understanding of the quantum era, do we not describe space & time in discrete quantities ... ?

No, position and time are described by real numbers in QM.

[studiot]

On 11/10/2021 at 8:12 AM, Intoscience said:

Zeno's moving arrow paradox:

Zeno suggested that the arrow never really moves because each of the smallest possible moments of its journey = 0 time and 0 distance so concluded that even every moment even to infinity only ever adds up to 0.  This has since been proven a fallacy and as we all know in reality an arrow shot from a bow will have a journey of some distance over time. 

So my question is, was Zeno's concept on the right lines/ With modern understanding of the quantum era, do we not describe space & time in discrete quantities, i.e the Planck scales where each moment of space and time may well be made up of a undetermined quantity though never 0?

This Zeno paradox is deeper than any of the others and was not properly answered for 150+ years after the others.

The other Zeno paradoxes rely on sequences of integers and their reciprocals.

This one relies on something deeper.
The solution came after it became necessary to integrate many functions that could not be integrated by the Riemann integral, commonly taught in high school today.

As you likely know, the Riemann integral is the sum of lots of small rectangles that make up the area under a curve.
In fact it is the limit as the width of these rectangles ten to zero.

But Zeno's question is what happens when that limit is reached ie the width is zero?

The generalisation the the Riemann integral was introduced by Lebesgue (1875 - 1941) adn this ushered in what today is known as measure theory.

https://en.wikipedia.org/wiki/Henri_Lebesgue

 

The other approach to this issue was also developed in the first half of the 29th century by Paul Dirac and is known as the Dirac Delta function.

 

[MigL]

I wasn't familiar with this version of Zeno's paradox, Studiot.
The only one I'm familiar with involves the arrow travelling successive half-distances.
Thanks for the info.

[studiot]

Since several people liked my comment, (Thank you all) , I will expand a little.

The issue here reaches many parts of theoretical Mathematics, but one part goes to the heart of applications in Science.
This is in probability theory and therefore in statistics and quantum mechanics.

The probability of an event P(E) is defined as the limit of the relative frequency of that event as the number of trials tends to infinity.

For instance consider rolling an n sided die.
As the number of sides increases the number of different possible outcomes increases.
As the number of possible outcomes increases so the probability of any given outcome (ie an event) decreases.

So as n tends to infinity P(E) tends to zero.

So we have the apparent paradox to resolve of how can we have a probability when we know that the die must end up showing one face or another, yet the probability of showing any one face is zero.

In QM we resolve this by taking the probability between x and (x + δx) and taking a limit as δx tends to zero.

In 'shut up and calculate ' mode we don't think about this we just do it and get 'the right answer'

 

[Genady]

Aren't there two different concepts: one is 'probability', the other - 'probability density'?

Edited January 6, 2022 by Genady

[studiot]

52 minutes ago, Genady said:

Aren't there two different concepts: one is 'probability', the other - 'probability density'?

Yes probability and probability density are different.
 

I have already defined probability. Probability is a non negative number less than or equal to 1.

'Probability density' is introduced to overcome the problem of division by zero, as with all densities. Probability density is a function.

Edited January 6, 2022 by studiot