One
of the most famous Zeno’s paradox is a
paradox about a perpetual race between
Achilles, the best runner in Greece, and
an old lazy tortoise. This paradox has
been solved, discussed, explained, declined
during more then twenty four centuries,
time enough to congratulate it (paradox)
with an immortality, or at least accept
it as a longlived tale.
The
sense of the paradox is:
Achilles, a symbol of quickness must overtake
a tortoise, symbol of slowness. Achilles
runs ten times as quick as the tortoise
and gives her ten metros odds.
Achilles runs ten meters, a tortoise runs
one meter,
Achilles runs one meter, a tortoise runs
one decimeter,
Achilles runs one decimeter, a tortoise
runs one centimeter.
And so on until the infinity. As a result
according to Zeno Achilles can run everlastingly,
but never will be able to run down a tortoise.
Such is the immortal paradox.
Starting from Aristotle until 20 century
there were a lot of attempts to refute
this paradox, but no one refutation seemed
ideal.
Is this paradox really irrefutable? In
reality Achilles overtakes a tortoise
very easily and even does not suspect
that the famous paradox forbids him to
fulfil such an illegal action.
Let
us consider point of view of refuter of
the paradox and paradox defender.
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Part
2. Refuter:
Refutation of an irrefutable paradox
Point
of view of refuter:
The
mystery of the paradox is in its haze
formulation. Zeno in a disguised way gives
us a limitation on the length and time
of the race.
Let us be fair to Zeno. Soon of all he
did not intend to delude descendants with
a help of haze formulation. Perhaps he
did not realize it himself.
To solve this paradox it is necessary
to consider an infinite geometric series
and know that a sum of the infinite series
could be a finite digit. Necessary actuarial
mathematics was introduced by Newton and
Leibniz many centuries later. In the times
of Zeno philosophers guessed that the
sum of infinite series was infinite too.
Zeno offered us to consider a series:
and so on until the
infiniteness.
This sequence is
a positive convergent geometric series.
Scale factor and
common ratio :
A sum S is equal
to:
In our case:
It means Achilles
runs over the tortoise in 100/9
meters from his first position
and as his speed is 10 m/sec it happens
in 10/9 seconds.
With the given terms
every new interval: centimeter, millimeter,
micron and so on increases the length of
the race by less and less value and every
new partial sum is less than its integral
sum of 100/9 meters. It will be so until
time 10/9 seconds expires.
Such way Zeno had given us the length of
the race, limited 100/9 meters and time
of the race 10/9 seconds.
Refuter's
conclusion: Zeno’s
Paradox stops being a paradox if honestly
to add a couple of words to its formulating:
Achilles is not able to overtake a lazy
tortoise if the length of the race is
less than 100/9 meters or the time of
the race is less than 10/9 seconds.
If
Achilles isn't given time to run he will
not move at all.
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Part
3. Paradox Defender:
An internal core of the paradox
Point
of view of refuter: Calculation
of the moment of "meeting" of
Achilles and tortoise is not a subject
of the paradox. What is at the core of
Zeno's paradox is the idea that in order
for Achilles to catch up with the Tortoise,
he must first perform an infinite number
of acts. And, that seems to be impossible.
Really,
it is not obligatory to consider infinite
series to get the answer of 100/9
meters. It is possible to solve
a problem algebraically.
Let x is a distance which
was crept by Tortoise till the moment of
a meeting with Achilles.
Then 10 + x is a distance
which was run by Achilles till the moment
of a meeting.
Speed of Achilles is 10 m/sec,
speed of Tortoise is 1 m/sec.
Time of movement from start up to a meeting
is the same for both runners.
Let us write an equation.
x/1 = (10 + x)/10.
x = 10/9 is the distance
which was crept by Tortoise till the moment
of a meeting.
10 + x = 100/9 is a distance
which was run by Achilles till the moment
of a meeting.
It is
difficult to believe, that Zeno could
not find a solution. It is even more difficult
to imagine, that Zeno never overtook nobody
or did not see, how it is done by others.
Paradox
Defender's conclusion:
Calculus allows
us to calculate the sum of an infinite
numbers of terms, but does not explain
how somebody is capable to finish going
through an infinite number of points,
if he has to go through these points one
by one.
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Part
4. Refuter:
Achilles does not go through an infinite
number of points one by one
Refuter's
argument:
Achilles
does not need to go through an infinite
number of points one by one.
The first section is 10 meters and Achilles
has to run 10  20 steps to get over it.
The second section is 1 meter and Achilles
has to run couple steps to get over it.
The third section is only 1 decimeter.
That's it. Achilles's foot is bigger then
third, fourth, fifth and so on sections
and Achilles covers all these sections
simultaneously with one step. He does
not go through an infinite number of sections
one by one.
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Part
5. Paradox Defender:
Paradox and theory of relativity or can
we find conditions that paradox still
apply?
Paradox
defender 's argument:
Well,
Achilles does not go through an infinite
number of points one by one if he runs
with terrestrial speed. But we may make
him to do it if his speed is commensurable
with a speed of light.
Let
us try to find such conditions, that paradox
still apply. To do it we will have to
consider the famous paradox from the point
of view of theory of relativity.
First of all, we have to deprive Achilles
of the ability to cover some space sections
simultaneously with one step, and to make
him go through an infinite number of sections
one by one.
How to do it?
We have to find a way to decrease Achilles's
size with each new smaller section. Such
way he will have to run each new section
despite the length of this section is
very small.
In
accordance with Einstein’s theory of relativity
the length of any object in a moving system
will appear foreshortened in the direction
of motion. The higher speed the less size.
The length has its maximum in the system
in which the object (Achilles in our case)
is at rest. If L0
is a length of Achilles in this system
then the length L
in the moving system is
,
where c
is speed of light, V
is speed of Achilles.
If speed V
of Achilles is commensurable with a speed
of light then length’s decreasing will
be substantial.
Of course, even the quickest Achilles
is not able to run so quickly. Well, let
place our famous runner in a rocket.
Speed
of Achilles’s rocket is commensurable
with a speed of light.
L
= 0.1L0
when V
= 0.95c
To keep decreasing, Achilles' s rocket
has to fly with acceleration.
L
= 0.1L0
when V
= 0.95c
L
= 0.01L0
when V
= 0.995c
L = 0.001L0
when V
= 0.9995c
L
= 0.0001L0
when V
= 0.99995c
And so on.
What will be with
time?
A clock in a moving system will be seen
to be running slow. The time will always
be shortest as measured in its rest system.
The time measured in the system in which
the clock is at rest is called the "proper
time".
,
where T0
is "proper time" interval.
T
= 4.47T0
when V
= 0.95c
T
= 14.14T0
when V
= 0.995c
T = 44.72T0
when V
= 0.9995c
T
= 141.42T0
when V
= 0.99995c
And so on.
Events
from point of view of detached observer:
Seeming decreasing of length in the direction
of motion and slowing down of time will
be from the point of view of detached
observer.
Destination between Achilles and tortoise
decreases. Achilles appears thinner and
thinner, tortoise is closer and closer,
but now foreshortened rocket is not able
to cover some space sections simultaneously,
and Achilles in his rocket has to fly
through an infinite number of points one
by one. It makes Zeno right.
Events
from point of view of Achilles:
Achilles himself will not feel any changes
inside a rocket, he will not feel that
rocket moves. For him the rocket itself
and tortoise ahead will be "frozen",
and he will see that all surroundings
rush back.
What
happened with Achilles relativistic mass?
It will increase.
The increase of effective relativistic
mass is given by the expression
,
where m0
is "rest mass".
So,
the length of Achilles’s rocket continues
to decrease in the direction of motion.
Relativistic mass continue to increase,
the time is slow down. It is a state of
singularity.
Paradox
Defender: Aim
is achieved. Achilles can not overtake
a tortoise.
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Conclusion
Calculation
of the moment of "meeting" of
Achilles and tortoise is not a subject
of the paradox.
A
real core is a question is an eternal
division of the length of section possible?
In other words is space discrete or continuous?
When
we calculate the moment of "meeting"
of Achilles and tortoise we do not answer
real Zeno's question, we simply disregard
it.
We
can ignore true sense of paradox and neglect
a question about "discrete or continuous
space" while we consider terrestrial
speeds. Even the fastest Achilles is not
an exception.
To
make Zeno right we made Achilles in the
rocket to “jump” into singularity state.
Now Achilles in his rocket has to go through
an infinite number of sections one by
one.
Now the question of "continues or
discrete space" is important. If
space is continuous then rocket with Achilles
will run eternally.
If space is discrete then … what will
happen when Achilles’ rocket achieve the
smallest last Rubicon?
There
is the third possibility. We will talk
about it later.
by
Tetyana Butler
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