5 Mind-Bending Paradoxes Explained

When a statement seems contradictory, a situation impossible, or a scene so puzzling we just can’t wrap our minds around it, we call it a paradox. There are all sorts of paradoxes, including unsolved problems in biology, confusing concepts in relativity, mathematics, logic, and much more. Today we’re going to go over some of the most popular paradoxes out there, and break down exactly what makes them so perplexing.

The Bootstrap Paradox

The first one we’ll dive into is called the bootstrap paradox, and it is one of the many mind-boggling issues that arise when we think about time travel to the past. Long story short, an item can travel backward in time and become its own cause for existence in the future. Sounds confusing, so let’s break it down.

Imagine you walk into your local library, looking for an interesting story to read. After scanning the shelves for a bit, you pull out The Lord of the Rings by J.R.R. Tolkien. Lost in his fantasy world, you decide that this is the best book you’ve ever read, and you now have a burning desire to personally thank Tolkien for his works.

Unfortunately, Tolkien died in 1973, so you head home and do what any determined fan would: you build a time machine in your garage and travel back to the 1920s, where you find Tolkien in his 30s. You spend several hours gushing about your favorite characters, moments, and battles from his book, and ask him to autograph it. But Tolkien is flustered, and has no clue what you’re talking about, because he won’t publish the book for another 15 years and hasn’t even started writing it yet.

Finally, you decide it’s time to go home. But as you’re stepping into your time machine and warping back to the future, you realize that you accidentally left your book in Tolkien’s house, leaving it in possession of its author.

What happens here is that Tolkien is inspired by your words of praise, and finding the copy of the book you left behind, decides to claim it as his own and publishes it as The Lord of the Rings. It becomes a historic novel, eventually ending up in your local library, where you pick it up, fall in love with it, and head back in time to give it to Tolkien.

On the surface, this might seem fine, but it begs the question: Where did the book come from in the first place? It can’t have originally come from Tolkien, because he got the idea from you when you traveled back and gave it to him, and it didn’t come from you, because you got the idea from Tolkien when you found his book in the library.

And this is the paradox: the book exists in a time loop, with no beginning. Cause-and-effect has been broken because time travel has allowed the effect to become its own cause.

If it still doesn’t make sense to you, here’s a couple more examples.

An episode of Doctor Who describes a situation in which a man goes back in time with some of Beethoven’s music, but upon finding that Beethoven doesn’t exist, decides to become Beethoven and publishes the music himself. Not only does this leave the origin of the music up in the air, but even the name Beethoven seemingly has no origin.

And finally, in the 1980 movie Somewhere in Time, the character played by Christopher Reeve is given a pocket watch by an elderly lady. Later in the movie, he travels back in time and meets this lady in her youth, where he gifts her the pocket watch. So according to the woman, the man traveled back in time and gave her the watch, which is why she gave it back to him later in life. But according to the man, the woman gave him the watch first, which is why he went back in time to give it to her. Following the trail of the watch’s origin, you end up stuck in a loop with no beginning.

Hilbert’s Hotel

Our next paradox comes from the world of mathematics, and highlights the inability of the human mind to easily comprehend infinity.

Hilbert’s Experiment of the Grand Hotel is a thought experiment that imagines a gargantuan hotel. It’s so big that it has not just a few thousand rooms, not even a few million, but an infinite number of rooms. Stranger still, every room is occupied by a single person, meaning if you randomly select any number, that room is taken, all the way from room number 7, to room 400,000, to room number six octillion. No matter how high you go, they are all occupied.

But you really want to stay in this hotel, so you come up with an idea to make some space for yourself. Every occupant simply needs to move up one room, meaning the person in room 1 moves into room 2, the person in room 2 moves into room 3, and so on, all the way up the chain. This leaves room 1 vacant, and you can now take it for yourself. This method works for any number of finite guests.

Bringing in 10 new guests simply means that everyone needs to move up 10 rooms; bring in 50, and everyone needs to move up 50 rooms. No matter the number, you can always find room.

It’s a strange idea that because infinity does not behave like the numbers we encounter in our daily lives, even when it’s full, you can still make space in it. But it gets even weirder.

Not only can you fit any number of finite guests into the hotel, but even an infinite amount. If an infinite amount of people shows up looking for a room, all the manager has to do is move the guest from room 1 into room 2, room 2 to room 4, room 3 to room 6, 4 to 8, 5 to 10, and so on. Doubling each room number always ends in an even number, leaving every single odd-numbered room vacant, and the infinite number of guests can settle in for the night in rooms 1, 3, 5, 7, etcetera.

But this is where we’re going to stop with the math behind Hilbert’s Grand Hotel, because mathematicians have gone wild from here, coming up with solutions to housing an infinite number of cars each carrying an infinite number of people. Now imagine the hotel is on the beach, and an infinite number of ferries arrive each carrying an infinite number of cars each carrying an infinite number of people. The math gets pretty crazy, but you don’t need to understand the math to get what this paradox is about.

The reason Hilbert’s Hotel is a paradox is because it leads to a result that is counter-intuitive, yet certainly and provably true. Our instinct tells us that when a hotel is full, it’s full, and that’s the end of the story, but when working with infinity, things are never quite so simple.

The Ship of Theseus

Our next paradox is a classic, first recorded by Greek historian Plutarch nearly 2000 years ago in his biography Life of Theseus.

The story goes as follows. A ship carrying the legendary hero Theseus sailed from Athens to Crete, and because Theseus was such an important figure, the ship was preserved for many generations. And over the years, as old planks began to crack and rot, they were replaced with new ones. Likewise, the aging sails were eventually replaced with newer fabric, as well as the oars, the rudders, and everything else.

After all these years of renovations, it was realized that not a single piece remained from the original ship—every last bit had been replaced with something newer. So then here’s the question: Is this still the ship of Theseus, or is it an entirely new ship?

If you take the side that it is an entirely new ship, it only leads to more questions. It’s unclear when this transition happened, because presumably at some point in the renovations it was still the original ship, and only after a certain number of parts had been replaced did it become something completely new. But how many parts? Surely not just one, but where do you draw the line? Maybe once you pass 50%, maybe 75%—there’s no definitive answer.

But here’s where it gets even trickier. What if someone had secretly been saving all the discarded parts from the original ship, all the old planks, decaying sails, and splitting masts, and once they had every single piece, reconstructed the old ship with the original parts. Sitting side by side with the one that had been renovated over the years, they both lay claim to being the actual Ship of Theseus. One ship claims this through continuity, and the other through originality.

This is the paradox—a paradox of identity.

Whichever ship you believe to be the actual Ship of Theseus largely comes down to your own opinion. And several so-called solutions have been thought up over the years, but none of them can quite nail it down.

You might have already been familiar with this paradox not just through the ship example, but from others around the world. Old tales from several European nations speak of knives who have had their blades and handles replaced, and an ancient Buddhist text describes a man whose body parts were removed by demons and replaced with those from a corpse, leaving him questioning his true identity.

These concepts might sound abstract and a bit detached from reality, so let’s make it personal. Throughout your life, your cells are dividing and dying off; in fact, every day, more than 50 billion cells die in your body, constantly replaced in a fine-tuned biological process. Because some cells live longer than others, it’s estimated that ALL of our body’s cells are replaced every 7-10 years. So at what point do you consider your body one that’s entirely new?

The Lottery Paradox

Imagine there’s a big lottery in town, with a single guaranteed winner, and you just can’t resist buying a ticket. Not because you’re addicted to gambling—you can quit whenever you’d like—but because you can’t stop thinking about becoming that sole winner and buying that super yacht you’ve always wanted.

After heading home, you turn on the TV to see the devastating news that exactly one million lottery tickets have been sold. You stare at the one in your hand, and realizing that you essentially have no chance of winning, you throw it away. After all, there’s a million other tickets; it is very reasonable to assume that yours is losing, hence why gambling is a losing game in the first place. After talking to your friend, you explain your logic to him, and he also throws his ticket out, realizing that he too has no real chance of winning, and the ticket was nothing more than a waste of money.

This is perfectly logical thought, as 999,999 out of a million is well beyond the threshold to consider something a certainty. Therefore, it’s reasonable to assume that ticket number 1 is losing, and it’s also reasonable to assume that ticket number 2 is losing. This logic follows all the way through to the one millionth ticket, and applying the same logic across the board, it has now become reasonable to assume that not a single ticket will win. But this can’t be, because you know that one of them will win.

And this is where the paradox arises, in these contradictory yet seemingly true statements. You have one rational line of thought telling you that you are completely justified by your natural intuition in saying that every single ticket will lose, while another rational line of thought tells you that one of them must be winning.

To some, this doesn’t seem like a paradox; after all, it’s just a matter of probability. But to philosophers, this isn’t a purely mathematical paradox like Hilbert’s Hotel was—this is a paradox of rational human thought. At what point do we draw the line between being justified in something being a certainty, while it is still technically possible? And how can this rational logic lead to two completely true, yet contradictory statements?

This puzzle was first put forward in the 1960s, and its underlying logic has been grappled with ever since, with many solutions having been put forward, such as restricting the logic to find the difference between something being highly probable and something being absolutely certain. To cognitive scientists, it seems to be a problem of information aggregation, or how we combine results from different lines of thought.

Regardless, much like the other paradoxes, this is a level of logic that doesn’t affect one’s day-to-day life, and even figuring out solutions to it does not influence anyone’s decision making when it comes to gambling. And this is considered by some to be the secondary Lottery Paradox: what compels people to engage in an event when the odds are so astronomically stacked against them that they know they have next to no chance of winning?

The Grandfather Paradox

For our last paradox, we’ll dive into one that you’ve most certainly heard of before. The grandfather paradox—perhaps the most popular time travel problem ever thought up.

Let’s start with the basics. You build a time machine and travel back many decades, encountering your own grandfather while he was still in his youth. For whatever reason, you end up taking his life, meaning that he doesn’t ever meet your grandmother, and never has kids with her, leading to you never being born.

If you are never born, then nobody goes back in time to eliminate your grandfather, meaning he continues his life as normal, meets your grandmother, and the two raise a girl who ends up giving birth to you. Now that you exist, you can go back in time and kill your grandfather, starting the whole cycle over again.

It’s an utterly confusing mess, and has left everyone scratching their heads for years. Its relative simplicity yet far-reaching consequences are part of why it has become so popular, and there is quite a bit to unpack here.

One solution to the paradox arises from quantum mechanics, specifically the many-worlds interpretation. This theory suggests that when an event can happen one of two ways—say, you either kill your grandfather or you don’t—both options actually happen, branching off into multiple universes or timelines. This solves the issue of you not being born, because the universe in which you kill your grandfather is not the one in which you were eventually created.

By the way, this theory also seems to fix the Bootstrap paradox mentioned at the beginning of the video, because it means that the object’s time loop actually begins in a separate universe. But this idea leaves so much to be explained with these alleged alternate universes that there’s really just more questions than answers.

Another solution comes from what is known as the Novikov self-consistency principle, a conjecture put forward by a Russian physicist proposing that the universe outright prevents any actions that would create contradictions or paradoxes. If this is true, it could mean that the laws of the universe do not allow time travel to the past in any form, or it may mean that all we can do is observe the past without interfering. The most concerning implication of this, though, is that it may indeed allow us to travel to the past, but anything we do only serves to preserve the timeline as it was, meaning that, all along, our past actions only contributed to the current state of the world. Put simply, you can indeed travel backwards, but nothing you do matters as you cannot change the future.

If this is true, it brings into question the notion of free will, because it appears that our actions are already predetermined to preserve the timeline. And so, perhaps the simplest explanation is the most boring one: that time travel to the past simply isn’t possible, and we’ll never need to deal with the potential consequences of such an invention.

Key Takeaways

• The bootstrap paradox involves an item traveling back in time to become its own cause.
• Hilbert’s Hotel paradox illustrates the counterintuitive nature of infinity in mathematics.
• The Ship of Theseus paradox questions identity when all parts of an object are replaced.
• The Lottery Paradox highlights the contradiction between rational thought and probability.
• The Grandfather Paradox explores the logical inconsistencies of time travel to the past.

Sources

• Original Side Projects video: 5 Mind-Bending Paradoxes Explained

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