Paradoxes are self-contradictions. For example, writing ‘This is a blank page’ on a blank page means that it is no longer blank.
Another example is the Liar’s Paradox, commonly demonstrated by the two statements below:
The sentence below is false.
The sentence above is true.
Both cannot simultaneously be true, making it a paradox. Below are five more examples of paradoxes.
The Grandfather Paradox
Time travel has been a fantasy of mine (and presumably everyone else’s) since I could comprehend the concept. Going back in time to explore the past (trying to ride dinosaurs) and rectify my wrongs would have been a dream come true.
However, as I got older, I learned about the Grandfather’s Paradox. It has been written about in various stories since the early 1900s, and has baffled physicists, dashed dreams, and been the foundation of many sci-fi movies — like Back to The Future — ever since.
The paradox states that if you went back in time, and killed your grandfather before your father was born, your father, and consequently, you wouldn’t have been born, meaning that you couldn’t have travelled back in time and killed your grandpa in the first place.
Although there are some possible solutions to the paradox, they are only conjectures and cannot be proved until time travel is achieved in reality, a seemingly impossible feat. Until then, time travel continues to remain a mystery.
The Black Hole Information Paradox
In the mid-1970s, Stephen Hawking proved, in a groundbreaking discovery, that a black hole slowly radiates, and eventually dissolves, what is now known as Hawking Radiation.
This, however, presented a conundrum for scientists: the discovery implied that information entered a black hole but was subsequently lost forever, contradicting a key principle of quantum physics, which stipulates that the universe’s overall information (such as particle velocity and spin) is preserved.
As expected, this vexed many physicists, and John Preskill went so far as to place a bet with Hawking and Kip Thorne, with others even declaring ‘war’ on Hawking’s solution. Nevertheless, three decades later, Hawking conceded the bet, turning on his previous theory, coming out with a new explanation, stating that information does indeed escape from black holes, but in an unrecognisable state.
Nonetheless, his solution is still debateable, along with other posited solutions, as all lack conclusive proof. Black holes have been widely studied, yet we are far from understanding them, although we are getting closer.
Achilles And The Tortoise
Travelling back through time, in the 5th century BC, Zeno of Elea put forth a paradox, in which the light-footed Achilles is challenged to a race by a slow, crawling tortoise, with the condition that the tortoise is given a head start.
Of course, Achilles agrees, and confident of victory, gives the tortoise a head start of 100 meters. However, Zeno then creates a problem for Achilles. By the time Achilles covers the 100 meter gap between him and the tortoise, the tortoise has created a new gap. Even though it is smaller, by the time Achilles sprinted across that gap, the tortoise has opened a new gap, and this goes on ad infinitum.
This means that Achilles will never be able to cross the tortoise, but wait, something seems way off. Can’t Achilles, a trained, swift warrior outrun a slow, steady tortoise?
Fortunately, this paradox has definite solution. It involves converging sequences, and their sum to infinity. For example, let’s take the following sequence:
½, ¼, ⅛, ¹⁄₁₆, …
Now, we know that this is a geometric sequence, and that its common ratio is ½. Using this information, and the fact that this series is converging (this is when the common ratio is between -1 and 1), we can calculate its sum to infinity with the formula 𝑎/(1−𝑟), where a is the first term of the sequence, and r is the common ratio. In this case, the sum to infinity is 1.
Using this logic, we can conclude that an infinite number of decreasing distances has a finite sum, thus allowing Achilles to outpace the tortoise and maintain his dignity.
The mind in itself is also a paradox; it uses itself to understand itself. We are surrounded by paradoxes, and adopting the ‘paradox mindset’ can enrich our daily lives by opening new doors and providing new perspectives.
To end, I would like to share a paradoxical quote by Socrates, one that encourages us to be humble, but determined in our pursuit of knowledge.
“The only thing I know is that I know nothing.”