About the Paradox of Achilles and the Tortoise
Given a problem to be solved, we may use a wrong procedure, or we may provide an answer to the wrong question. Neither is helpful in problem solving.
But with paradoxes, we can be wrong in even more ways.
When presented with the paradox of Achilles and the tortoise, some people immediately spring into mental action with the assumption that a paradox is a math challenge. It is not. It is a logic challenge.
With paradoxes, we can be wrong in several ways:
1.
Some people do not know, or understand, what a paradox is. They may say something to the effect of “That doesn’t make sense”.
A paradox has two components: a description of a problem, and a procedure proposed to solve it.
The paradox creator then shows how by applying the proposed procedure to the problem, we obtain a paradoxical result.
The paradoxical result defies logic. This is because the result we obtain, using the proposed procedure, is different from the result we obtain by using other more common (and possibly more intuitive) procedures to solve that same problem.
If we obtain two different results, depending on which procedure is used, then one procedure must be wrong. If we are sure that a common procedure solves the problem, then the proposed procedure must be wrong. The challenge is: Why?
Other scientists have shown that a paradox arises when a problem has not been completely specified, and more than one solution to the problem seems to be intuitive, or rational.
2.
Some people do not know, or understand, what solving a paradox means. They will provide an answer to the wrong question. They will tell you that Achilles will reach the tortoise.
Some may even give you a simple mathematical solution, such as: “We can calculate when Achilles will reach the tortoise by dividing the distance between them by the difference in velocity between them”. This is correct, as a solution of the problem, but it is not the solution of the paradox.
The solution of a paradox is the answer to the question: “How does the paradox formulation misrepresent reality or logic?”. That is, assuming that the problem is correctly described, we need to show why the proposed procedure for the solution of the problem, is conceptually wrong.
Solving a paradox, invalidates the procedure proposed by the author of the paradox, and leaves us with only valid procedures for solving the original problem.
We do not need to solve the problem.
We do not even need to provide an alternative procedure to solve the problem!
3.
Some people do not recognize that Zeno’s proposed procedure for the paradox of Achilles and the tortoise, is wrong.
Well, do not feel bad. This includes most people in the last two millennia. Some people may intuitively recognize that the procedure proposed by Zeno is wrong, but no one, up to now, had a definite explanation of why it is wrong.
I want to believe that some people may have thought of the answer, but made no effort to write it down. Yes, some people have better things to do than solving paradoxes.
4.
Some people try to use Zeno’s procedure, but instead of applying the procedure rigorously, which would never end, confirming Zeno’s paradoxical result, they jump to the expected conclusion (Achilles reaches the tortoise) by applying one of the mathematical tools that helps solving similar, but real problems (e.g., functions with a discontinuity point).
These people will say: “I do not see anything wrong with this paradox”, or: “This is not a paradox. I can show you that Achilles will reach the tortoise, using an approximation method, or using a limit function”. Then they will write a series such as 1 + 0.5 + 0.25 … etc. and start their demonstration.
Why then are these people not proposing a simpler, more common solution to solve the problem, such as the one mentioned in point 2. above?
They want to follow Zeno’s procedure, and use the mathematical tools that are used in similar classes of problems. However, as suggested by those mathematical tools, at one point, very close to the infinitesimal, they are making a conceptual jump. They stop using Zeno’s never ending procedure and declare “close enough for all practical purposes”. I.e., we are so close to x, so we can stop using Zeno’s procedure, and assume x.
Hmmm, being very, very close to catching the train, is not catching the train.
Zeno was an intelligent philosopher and many philosophers and intelligent people since then have struggled with this paradox. We cannot dismiss Zeno’s procedure on the basis that now we have some tool that allows us to use a number that practically solves… the problem and not the paradox.
As mentioned, a paradox is not a math challenge, but a logic one.
For the solution to the paradox of Achilles and the tortoise (Why Zeno’s procedure is wrong), please click on: https://bit.ly/2IM76rF
