Introduction to the Mathematical Paradox
Consider these two examples:
“This statement is false,” & “I always tell lies.” These two statements cannot be true and false simultaneously. The reader is left in a conundrum.
- If the statement is true, then it’s wrong because a statement is false.
- If the statement is false, then that the statement must be true since the statement is false.
Thus a paradox results in a puzzling problem that cannot be resolved. With paradoxes, logic is sometimes defied. Other times it involves circular reasoning. And occasionally, the paradox results from an invalid argument. Upon more thorough analysis, it’s often discovered that definitions need to be clarified and rules refined.
In the grandfather paradox, a time traveler emerges fifty years earlier to see what life was like in a previous generation. However, when he explores his family history, he kills his grandfather before his mother and father were born. Thus he cannot exist. So traveling back in time causes an enigma!
Let’s remember the dialogue between Marty and Professor in the movie Back to the Future.
Marty: You mean I’m going to see where I live? Am I going to see myself as an older man?
Doc: No, no, no, Marty. That could result in a [gasps] Great Scott! Jennifer could conceivably encounter her future self! The consequences of that could be disastrous!
Marty: Doc, what do you mean?
Doc: I foresee two possibilities.
1: coming face to face with herself 30 years older would put her into shock, and she’d pass out. Or,
2: the encounter could create a time paradox, the result of which could cause a chain reaction that would unravel the very fabric of the spacetime continuum and destroy the entire universe! Granted, that’s a worst-case scenario. The destruction might be very localized, limited to merely our galaxy.
Marty: Well, that’s a relief.
Here’s another one.
Bobby has had five birthdays. Each time he has a birthday, his family throws a huge party. On his fifth birthday, when Bobby is 21, he graduates from college. Is that possible? If his birthday is on February 29, yes!
A form of Russell’s paradox is given in the story of a librarian who discovers a problem. The librarian has a set of catalogs of various books in the library. Some of the catalogs list themselves in the book. Some of them do not. To simplify this problem, the librarian makes two more catalogs;
• The catalogs that list themselves. And,
• The catalogs that don’t list themselves.
The question the librarian must resolve is this: Should the catalog which lists the other catalogs that do not list themselves be listed in itself? If it is listed, then by definition, it should not be listed, but if it is not listed, then by definition, it should be. This paradox explains that the set of all sets that do not contain themselves leads to a contradiction.
Bertrand Russel is the father of set theory. He struggled throughout his life to resolving problems related to sets.
Some paradoxes are both true and false. For example, what if you walked into a room and stood in the doorway one foot inside the room and one foot outside. You are at the same time both inside and outside of the room. Then I asked you this: Are you in the room? You have no answer because you are inside and you are not.
In a similar vein of thinking, math students constantly tell that they believe math is dead. Because the subject has been the same for centuries. Mathematics is dynamic and ever-changing. So is our world static or dynamic? Can we change our patterns of behavior? Are we pre-programmed to live a scripted life? Some individuals believe that we have a prescribed destiny and that destiny determines their fate. Others believe that they can dream it and achieve it. For me, both are correct.
Deepak Chopra wrote an article that appeared on April 19, 2011, Huffington Post, the paradox of scientific proof and public opinion. It’s an interesting article because he talks about how massive amounts of data desensitize us. The wrong conclusions people conjure and how public opinion sometimes trumps scientific fact. He presents the spurious conclusions people make and how sometimes science leads people down that path. He uses the link between cell phone use and cancer which most people believe are highly correlated. However, recent studies with huge datasets show that cancer is not significantly increased by cell phone use. So, which is more believable; scientific evidence or public opinion?
The word paradox originated in the 1500s from the word’ paradox. ‘Para’ meaning contrary to. And ‘doxon’ meaning opinion. In our current lexicon, the word paradox is commonly used to refer to irony or a contradiction. Paradoxes typically come from philosophy, mathematics, or English. However, it can be found in the creativity of the mind’s imagination.
For example, MC Escher’s artwork is fascinated and leads those who view his art down a path of incredulity. Consider his work called drawing hands or his drawing called reptiles. If you look at Asher’s images closely, you will question what’s truly happening, and we’ll see remarkable perspectives that defy one another. Even abstractions leave the viewer to new depths of thinking.
Paradoxes make us think, wonder, and delve into our curiosity. Critical thinking is at the heart of the questions we ask about life. Paradoxes can bring out our passion and challenge us to discover what has yet to be created.
I leave you with a paradox to ponder: Nothing is impossible. Thus if it is impossible, then it’s wrong because nothing is impossible.
Galileo once said:
“Mathematics is the language with which God has written the universe.”
Thus if we are to use this language, we must be cognizant of the questions that arise when using paradoxical arguments or confusing statements.