The Solution of the Paradox of Achilles and the Tortoise
The
Solution of the Paradox of Achilles and the Tortoise
by
Giuseppe Gori
In
this article I explain Zeno's paradox of Achilles and the tortoise
first proposed by Zeno almost
twenty five centuries ago.
I show what
it means to solve a paradox, I propose
an
explanation
of
this paradox, and I show
why other explanations
of
this paradox proposed by notable philosophers and mathematicians
during the last twenty three hundred years were unsatisfactory.
The
History
The
paradox of Achilles and the tortoise (one of a set of similar
paradoxes) was first introduced by Zeno, a Greek philosopher that
lived in the South of Italy approximately 490-450 BC.
According
to the procedure proposed by Zeno, Achilles will never reach the
tortoise, as every time Achilles reaches the point where the tortoise
was, the tortoise has moved further ahead.
The
paradox is described, for example, at:
https://en.wikipedia.org/wiki/Zeno%27s_paradoxes
This
is not a difficult problem
to solve, and it is easy to calculate the time it will take for
Achilles to reach the tortoise. However, by following Zeno's
reasoning the problem seems unsolvable. Thus we have a paradox:
two different results for the same problem, depending on which
procedure we use.
The
paradox has remained unexplained until now.
From
the same Wikipedia source we can learn about some of the proposed
explanations:
- Aristotle (384 BC−322 BC) remarked that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small.
Aristotle also distinguished "things infinite in respect of divisibility" (such as a unit of space that can be mentally divided into ever smaller units while remaining spatially the same) from things (or distances) that are infinite in extension ("with respect to their extremities"). - Before 212 BC, Archimedes had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller. Modern calculus achieves the same result, using more rigorous methods. These methods allow the construction of solutions based on the conditions stipulated by Zeno, i.e. the amount of time taken at each step is geometrically decreasing.
- The ideas of Planck length and Planck time in modern physics place a limit on the measurement of time and space, if not on time and space themselves.
- According to Hermann Weyl, the assumption that space is made of finite and discrete units is subject to a further problem, given by the "tile argument" or "distance function problem"
- Infinite processes remained theoretically troublesome in mathematics until the late 19th century.
While mathematics can calculate where and when the moving Achilles will overtake the Tortoise of Zeno's paradox, philosophers such as Brown and Moorcroft claim that mathematics does not address the central point in Zeno's argument, and that solving the mathematical issues does not solve every issue the paradoxes raise. - Debate continues on the question of whether or not Zeno's paradoxes have been resolved. In The History of Mathematics: An Introduction (2010) Burton writes, "Although Zeno's argument confounded his contemporaries, a satisfactory explanation incorporates a now-familiar idea, the notion of a 'convergent infinite series.' ".
- Bertrand Russell offered a "solution" to the paradoxes based on the work of Georg Cantor, but Brown concludes "Given the history of 'final resolutions', from Aristotle onwards, it's probably foolhardy to think we've reached the end.
From:
https://plus.maths.org/content/mathematical-mysteries-zenos-paradoxes
(A mathematics magazine):
- “So Zeno's paradoxes still challenge our understanding of space and time, and these ancient arguments have surprising resonance with some of the most modern concepts in science.”
- “In his Lectures on the History of Philosophy (first delivered in 1805-6) Hegel said that "Zeno’s dialectic of matter has not been refuted to the present day: even now we have not got beyond it, and the matter is left in uncertainty."
- Bertrand Russell, in his “Recent work in the Philosophy of Mathematics", written in l90l, held that “From him [i.e. Zeno] to our own day, the finest intellects of each generation in turn attacked the problems, but achieved, broadly speaking, nothing."
Russell, however, thought that the problems involved in Zeno’s paradoxes, which he identified as "the problems of the infinitesimal, the infinite, and continuity”, have not merely been advanced by the mathematicians of his own age but have been completely solved. - Some philosophers, notable among them Bergson, refused right from the beginning to accept the newly-advanced theories of continuity and infinity and the consequent definition of motion, and endeavored to offer alternative definitions of motion, continuity and infinity. Since the mathematical solution found a number of champions, including no less able an advocate than Russell, and there was no dearth of the defenders of the philosophic tradition, there ensued a battle royal which brought to the fore scores of re-interpretations of the paradoxes and as many 'solutions'.
- No solution, however, was found to be tenable, and soon philosophers despaired of finding a solution that would be acceptable to all.
- In the 1950s, Prof. Ryle, in offering a solution of the Achilles-Tortoise paradox, feared that the fate of his solution would be, like that of his predecessors’ solutions, "demonstrable failure”, and, Prof. Lazerowitz went to the extent of opining that the paradoxes are (valid) theorems of some metaphysical wish-fulfillment language.
- The position as it then obtained, (that is, philosophers and mathematicians inability to overtake Zeno) was ably summed up by Prof. Owen in his masterly paper of 1957-58.
- Many a solution has appeared since then, including one by Shamsi, the author of the artice, but none succeeds in resolving the paradoxes as a whole.”
What
Does Solving a Paradox Mean?
As
mentioned
at the beginning of this article, a paradox
proposes
the existence of two different results
as a solutions
for the same problem.
These results
are
inconsistent
with each other, depending
on which procedure is
used.
Only one can be correct.
As
Brown and Moorcroft
suggest, we
are not looking for a mathematical demonstration that Achilles
reaches the tortoise. Assuming they are both running in the same
direction, we know he will. We can calculate the exact time, given
the distance between the two and the two speeds, using a simple
formula:
t
= distance/(difference in velocity)
Instead,
explaining or invalidating a paradox is to show a fault in the
paradox formulation, or the proposed solving procedure, so that we
can exclude this procedure and demonstrate that there is only one
result for the original problem.
The
solution of a paradox is the answer to the question: “How
does the paradox
formulation
misrepresent reality or logic?” That
is, we need to show why
the proposed method is conceptually wrong.
Solving
a paradox, invalidates the formulation of a problem proposed by the
author of the paradox and leaves us with only
valid procedures for solving
the original problem. You can read: “Solving a Paradox, What
does it Mean?” at: https://bit.ly/2IudnVO
Why
were the previous proposed solutions for the paradox not
satisfactory?
The
fault of most, if not all, the proposed solutions to Zeno's paradox
is the assumption that Zeno's proposed procedure was correct. The
procedure seems to be logical when it is first introduced to us, but
we will see that the procedure proposed by Zeno is deceptive.
The
authors then tried using Zeno's deceptive
procedure to reach the expected correct result for the original
problem.
The
Explanation of Zeno's Paradox:
Zeno's
proposition invites the solver to do a series of steps each time
changing system of reference:
STEP
1: The
starting system of reference:
The point where
Achilles
starts the
race and
the
tortoise is
well
ahead,
STEP
2:
After
a while, we
are
then
asked
to use a
new system of reference:
The point where
Achilles
reached
and
where
the tortoise initially
started,
with
the the
tortoise now
a
bit further ahead,
STEP
3: Then again we are asked to use, recursively, a new system of
reference with the new starting point for Achilles and with the
tortoise still further ahead,
with
every
step
we
are asked to freeze
the process and then
continue by re-creating
and examining
the original problem using
a different system of reference.
After
Zeno's proposed first step, or first change of system of reference,
the problem, as presented in the second step, is exactly the same as
the original, the only change being a difference in "scale".
No progress was achieved in solving the problem.
Changing
system of reference essentially restarts the problem-solving
procedure.
This
realization implies that the problem is never going to reach a
conclusion as the step by step procedure is reiterated.
If
the system of reference is changed at every step, our working
spacetime shrinks with every step, the solution becomes elusive and
the tortoise becomes apparently unreachable.
Zeno
proposes a procedure that never ends, for solving a problem that has
a trivial solution.
A Programming Analogy
Zeno's
proposed procedure is analogous to solving a problem by recursion,
a well known problem solving technique available in modern
programming languages. However, recursion would be the wrong
technique to solve the original problem.
If
the problem was programmed exactly as Zeno suggested, the program
would never end normally, simply because the condition for the end of
the recursion process (Achilles reaches the tortoise) would never
occur. Our computer program would confirm the paradox!
Our Solution of Zeno's Paradox
Our
solution of Zeno's paradox can be
summarized by the following statement:
"Zeno
proposes observing the race only up to a certain point, using a
system of reference, and then he asks us to stop and restart
observing the race using a different system of reference. This
implies that the problem is now equivalent to the original and
necessarily implies that the proposed procedure for solving the
problem will never end."
That’s
it. You cannot change system of reference in the middle of a problem
that uses a system of reference, whether openly stated, or implied.
As
an analogy, you cannot solve a problem involving measurements by
using English Imperial measures at the start of calculations and then
switch to metric measures (without proper conversions) in the middle
of calculations.
In
another similar paradox, the Dichotomy paradox, Zeno
uses the same argument to paradoxically state that we cannot go from
point A to point B, because it would take an infinite number of steps
and we would never arrive. In this case as well, we can show that the
proposed method involves changing the system of reference in the
middle of the proposed procedure. Thus the proposed method for
solving the problem is invalid.
An
example
The
following is not a “solution” of the paradox, but an example
showing the difference it makes, when we solve the problem without
changing the system of reference.
In
this example, the problem is formulated as closely as possible to
Zeno’s formulation.
Zeno
would agree that Achilles makes longer steps than the
tortoise.
Let’s
assume that one Achilles-step is about 20
tortoise-steps long, and let’s also assume that both Achilles and
the tortoise make the same number of steps in the same amount of
time. For example, two steps per second (the exact amount doesn’t
really matter).
If
the tortoise starts the race 20 Achilles-steps ahead of him, then
after 20 steps Achilles reaches where the tortoise was (See diagram
below: Tortoise starting point).
In
the meantime, the tortoise has made 20 of her steps, and she is now
one full Achilles-step ahead of him.
We
have not changed our system of reference. We referred to both
starting points. These did not move relatively to each other. We
could choose any fixed ground point. To please Zeno, let’s
continue by
referring to the tortoise starting
point,
where Achilles currently is.
When
both runners make one more step, step 21, the tortoise will have
moved by one of her steps and she will still be ahead of
Achilles by that one tortoise-step. Achilles is now one Achilles-step
ahead of the tortoise starting point.
Now,
let’s continue, without changing the system
of reference. This is the key point.
We
do not redefine the problem and use the current positions of the
runners as new starting points, as Zeno proposes, but we refer to the
information about the race we have already accumulated in our
knowledge base.
Achilles then completes his 22nd step, and he is two Achilles-steps ahead of the tortoise starting point. The tortoise will have completed her 22nd tortoise-step from her starting point. Hence the tortoise is now behind Achilles by 18 tortoise-steps.
Thus,
if we do not change the system of reference, the paradox does not
appear.
The Dichotomy
paradox
In
another similar paradox, the Dichotomy
paradox,
Zeno uses the same argument to paradoxically state that we cannot go
from point A to point B, because it would take an infinite number of
steps and we would never arrive. In this case as well, we can show
that the proposed method involves changing the system of reference in
the middle of the proposed procedure. Thus the proposed method for
solving the problem is invalid.
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