A paradox is the realization that a simple problem has two apparently contradicting solutions.
Whether intuitively, or using a formula, or using a program, we can easily solve the problem. However, someone challenges us with another method to solve the same problem, but that method leads to a different result.
A paradox is not an irrational statement. Our definition excludes many examples of such statements classified under the term paradox.
For example, there is a whole class of semantic “paradoxes” based on circular thinking, or self-reference, essentially similar to the “Barber paradox”, which do not fit our definition.
Russell’s paradox, the generalized set theory definition of the Barber’s paradox, is not only applicable to set theory. Such self-referencing statements can be made whenever we choose to use logic irrationally in our thinking.
These are statements that contradict themselves. Our mind can conceive a number of irrational thoughts. Such contradictions do not have two explainable solutions. They do not even have one solution.
In their simplest form, they consist of two sentences each simply denying what the other says. For example, the classic Liar’s paradox, or “This sentence is false”, can be of interest as a subject for conversation, but are essentially irrational, self-contradicting statements.
Non Intuitive, Solved, and Unsolved Problems
To be consistent with our definition, if a problem has a solution, but its results are counter-intuitive, it is not a paradox. It’s a logic, mathematical, or scientific curiosity. The results may be surprising, instructive, and interesting, but there is no open challenge. In this case, the reader will normally be satisfied with the solution, after learning the details.
When a problem is not solved, it is usually the object of research. The reader is challenged to find a solution. Some problems have no solution; but if two (or more) contradicting solutions have been proposed for the same problem, and they lead to different results, the reader is challenged to find a reason why the solutions are contradicting, and why a solution should be rejected, as based on erroneous reasoning.
Thus, a classic paradox is a problem that may have several accepted, common methods to reach the result, but someone proposed a method that reaches a different, contradicting result.
An Example: Solving a Problem
Let’s suppose you were taking a test at school, working in a group, and the teacher assigned to the group a problem that admits only one possible result.
If one student in your group proposed a first method to solve the problem which leads to a result, but another student proposed a second method that leads to a different result, which method should the group use to solve the problem and submit the result? Both results cannot be right, thus one method must be wrong. For the sake of this example, let’s suppose that the second method is right.
The way for your group to pass the test is to demonstrate why the first proposed methods is wrong, thus the group could use the second method, find the result and submit it to the teacher.
This is the process we need to use when solving a paradox: We need to show what’s wrong with the proposed method: Why the method is based on flawed reasoning.
The word “method” is synonymous with a program, a procedure or a formula for solving a problem.
As an example, let’s take the famous paradoxes, attributed to Zeno, a Greek philosopher that lived around 450 BC.
Two of his most known paradoxes are: Achilles and the Tortoise, and the Dichotomy problem. These two paradoxes are related.
Zeno proposed methods for solving those problems, that are paradoxical. Those methods seem to indicate that Achilles will never reach a tortoise, or that no-one can go from point A to point B.
If we wrote a recursive program literally following Zeno’s proposed procedure, our program would never end normally, because the condition for the end of the recursive process would never occur. Our program would confirm the paradox!
On the other hand, we could write a program in several other ways, using a sound method, and solve the problem, but those solutions would not solve the paradox.
It doesn’t matter how many other methods we can find that solve the problem.
Solving the Problem is not Solving the Paradox
To solve the paradox we need to think at a meta-level:
Solving the paradox is showing why the proposed method is conceptually wrong, or why the proposed method cannot work to solve the problem, or how the paradox formulation misrepresents reality or logic.
Only then we can be assured that we can discard the wrong method and “pass the test”.
The several arguments proposed by philosophers and mathematicians in the last two millennia did not solve Zeno’s paradoxes. They mostly proposed other methods to work out the correct result (i.e. Achilles reaches the tortoise).
For the solution of Zeno’s paradox of Achilles and the Tortoise please follow the link:
Zeno’s dichotomy paradox can be solved in the same way, as described in the article at the above link.
For the solution to Zeno’s paradox of the Arrow you can follow the link: https://bit.ly/2vj4hIt
Homework assignment: Solve Zeno’s paradox of the Moving Rows.