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5 Paradoxes to Make You Question Reality

Welcome to this blog post, where I’ll be discussing 5 paradoxes to (hopefully) blow your mind. So, get thinking outside the box, and let’s jump in…

1. Buridan’s Bridge

Statement

This is a paradox described by Jean Buridan, a philosopher in the Middle Ages, in his book Sophismata.

It is described as follows:

Plato is guarding a bridge, as Socrates approaches, wanting to cross.

Plato tells Socrates that if the first thing he replies is the truth, he will be allowed to cross.

If it is false, he will throw Socrates into the water.

Socrates then replies with “you will throw me into the water.”

Plato guarding the bridge from Socrates.

Discussion

Clearly, the paradox arises since, if Plato throws Socrates into the water, then what Socrates said would have been the truth, and hence Plato should have let him cross the bridge.

However, if Plato lets Socrates cross, then he will not be thrown into the water, and hence his statement would have been false, meaning he should have been thrown into the water!

Many people, including Buridan himself, have tried to solve this paradox.

Buridan says that Socrate’s response is technically neither true nor correct, since it is in the future tense, and hence its truth cannot be determined.

He also states that because Plato’s promise was given without proper thought, he does not have to fulfil the promise.

Plato should have ensured that his promise could not cause any contradiction.

Another interesting decision Socrates could have taken would have been to stay silent; is silence truth or untruth?

Finally, a funny thing Plato could have done; let Socrates cross the bridge, but then throw him into the water once he has crossed.

This way, Socrates has told the truth, and was allowed to cross.

For more information on this one, check out its Wikipedia page here.

2. The Potato Paradox

While you might not technically consider this a paradox, it produces a result which is very counter-intuitive.

Statement

This comes from ‘The Universal Book of Mathematics’:

I bring home 100 kg of potatoes which consist of 99% water. I leave them out overnight to dry so that they consist of 98% water.

What is their new weight? Surprisingly, it is actually 50 kg!

Potatoes
Jacket, Dauphinoise, Croquette… So many reasons to love potatoes

Answer

Why is this so?

Well, initially, in 100 kg of potatoes, there are 99 kg of water, and 1 kg of solids.

When the potatoes have been left to dry, the amount of solids in them cannot change, so there is still 1 kg of solids.

However, this is now 2% of the total mass since the potatoes are now 98% water, and 2% solids.

So, if 1 kg of solids is 2% of the mass, then the total mass must be 50 kg!

I’ll be honest, when I first saw this, I would have assumed the weight had only change by a kg or two, so you’re not alone if you thought so too.

For more information on this one, check out its Wikipedia page here.

3. The Twin Paradox

You’ve probably all heard of this one at some point, but I wonder if you’ve ever understood why, in fact, there is no paradox.

Statement

The paradox reads as follows:

Alice and Bob are twins. Alice goes on a journey to a star 30 light years away. She travels so fast that, to her, the trip only takes one year.

Meanwhile, her twin, Bob, remains on Earth.

When Alice returns, Bob has aged 60 years, and she only one year.

However, from Alice’s point of view, wasn’t it actually Bob who travelled 30 light years and back? Shouldn’t she have aged more?

Clock
I was looking for a fancy clock photo and saw this; unfortunately I think it might be from Harry Potter…

Resolution

This apparent contradiction is easily solved by taking care to look at each twin’s inertial frames.

Bob only has one inertial frame, his rest frame at Earth.

Alice, however, has two; her rest frame travelling towards the star at some speed +u relative to Bob.

And another, her rest frame travelling home at -u relative to Bob.

Now, there are three events which happen in this situation: Alice leaves Earth, Alice reaches the star, and Alice returns home.

Note that these things to not happen simultaneously in Alice’s and Bob’s rest frames.

By applying the Lorentz transformations to each of these event times, you can easily show that the total time elapsed for Bob, will in fact be longer than that for Alice.

The crucial thing to understand here is that the problem is not symmetric.

Bob always remains stationary in one rest frame, but Alice is not always stationary in only one frame; she has two different rest frames for each stage of her journey.

For more information on this one, check out its Wikipedia page here.

4. The Monty Hall Problem

This is another one you’ve probably heard before, and while it’s technically not paradoxical, it’s still interesting. It was initially posed by Steve Selvin.

Statement

The problem is set up as follows:

Imagine you’re on a game show, and you have to choose one of three doors to win the prize behind that door.

Behind two of the doors is a goat, and behind the other is a car.

You choose a door, and the host, who knows what’s behind each door, opens a different door to reveal a goat.

If given the choice, should you switch your door for the other, unopened one?

Monty Hall Problem
Which door is the other goat hiding behind? I reckon the car’s behind door one.

Answer

You might naively think that it would make no difference if you switched or not; the probability of the car being behind your door would now be 1 in 2 since it is either a car or a goat.

You wouldn’t be alone, in fact many academics believed this line of reasoning, and refused to accept that their answer was wrong.

So, what is the answer then?

As you may have guessed, it is actually beneficial to switch doors. Why?

Here is a rather simple, intuitive explanation courtesy of Matthew Carlton. It’s not rigorous by any means, but it does help you to understand.

If you initially picked a goat, with a 2/3 chance, then you will always win by switching, since the other goat will have been revealed.

Alternatively, if you initially picked the car, with a 1/3 chance, you will not win by switching.

This means that switching will win 2/3 of the time!

Another way to think about it is: whatever door you’ve chosen, there will always be a goat in at least one of the other doors.

So, the host might as well be offering you the chance to switch your single door to both of the other doors before revealing anything, which I’m sure you’d rather do.

monty hall problem
Clearly, the probability of the car being in doors 2 or 3 is 2/3.
monty hall problem
Revealing the goat doesn’t change the probability of the car being behind doors 2 or 3; we knew there’d be a goat in at least one of the doors!

Vos Savant put it quite well in her article when she said,

“Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?”

I’d recommend looking at that article here. The amount of hate she received from people, adamant that she was wrong, is quite alarming.

For more information on this one, check out its Wikipedia page here.

5.The Ladder Paradox

What an inventive name, you can see why it’s claimed the top spot on this list.

So, what is it? Well, it’s actually quite similar to number 3 in that it concerns Einstein’s theory of Special Relativity.

Statement

Imagine a ladder is travelling close to the speed of light towards a garage which has a length shorter than the ladder’s rest length.

At rest, the ladder will not fit inside the garage, but at this speed, the ladder will be Lorentz contracted in the garage’s rest frame.

The ladder may now be short enough so that, for a time, it fits completely inside the garage. We could even shut both of the doors to show that the ladder does in fact fit inside.

Now, consider the reverse situation in the ladder’s rest frame. Imagine now that the ladder is stationary, and the garage is rushing towards it.

In the rest frame of the ladder, it is now the garage which will be Lorentz contracted. This means that the ladder is now too long to fit inside the garage.

We can no longer shut both doors with the ladder fully contained inside the garage. We have a contradiction!

Resolution

Okay, maybe we don’t actually since this one can be resolved. In fact, there’s no problem in the first place. Why?

It stems from the relativity of simultaneity: events which appear to be simultaneous in one frame needn’t be simultaneous in another.

In the garage frame, the back and front doors are closed simultaneously to demonstrate that the ladder is in the garage.

Ladder Paradox
In the ladder frame, the ladder is fully enclosed in the garage and both doors are shut simultaneously.

However, in the ladder frame, the back and front doors are not closed at the same time. In fact the back door is closed first, just as the front of the ladder meets the end of the garage.

Then, the front door will close afterwards, just as the back of the ladder meets the front of the garage.

Ladder Paradox
In the garage frame, the ladder is never fully inside. The back door is shut first, then a little bit later, the front door is closed

So there you have it, no contradiction!

If you’re having trouble imagining this, just remember, saying the ladder fits inside a garage is equivalent to saying the ends of the ladder are both inside at the same time.

When you make a statement like that at high speeds, you’ve got to be careful. Time isn’t absolute!

For more information on this one, check out its Wikipedia page here.

Thanks For Reading

Thanks for reading this blog post, I hope you enjoyed it.

If you liked this sort of thing, let me know and I can write some similar stuff.

And, as always, if you have any questions about anything I’ve said, comment below or contact me here.

Take a look around my other posts, I’m sure you’ll find something interesting!

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