Vsauce! Kevin here. And you have a dilemma.

I have two envelopes and you can only choose one. There’s door number one. And there’s

door number two. Uhh.. Oh.There’s actually three envelopes here. Uhh great. Now we no

longer have a dilemma. Here’s why. Di comes from the Greek for “twice” and

lemma means “premise”. So a di-lemma involves two premises from which you have to choose. Adding a third envelope means this choice

isn’t technically a dilemma — but it does setup a very famous paradox. Wait. Let’s

dissect the word paradox like we just did with dilemma to find out exactly what a paradox

is. Okay, Para comes from the latin “distinct

from” and dox comes from doxa, meaning “our opinion.” “Paradox” translates literally as ‘distinct

from our opinion.’ So there ya go. Now. Distinct from our opinion? That didn’t really

help at all. I thought a paradox was like an unsolvable brain teaser? So how do three

envelopes setup a paradox? What is

a paradox? In 1961, Logician and philosopher Willard

Van Orman Quine outlined the three categories of paradoxes and I have them each hidden inside

these three envelopes. One represents the kind of paradox that you’re most familiar

with. Those that defy logic like the impossible waterfall from this video’s intro. The other

two are… what? Well. Let’s crack one of ‘em open and

find out. Falsidical. This is why Achilles can never catch a tortoise. We’ll use this bootleg Rambo to represent

Achilles and a Ninja Turtle PEZ will be our tortoise. If the tortoise gets a 100 meter

head start, then Achilles starts running, by the time he gets to the 100-meter mark,

the tortoise will have moved another meter. It takes Achilles some more time to get to

that 101-meter mark and in that time, the tortoise has moved forward even further. Achilles will always be catching up to the

place the tortoise was as the tortoise inches forward. The gap gets smaller, but the tortoise

is always slightly ahead. According to Greek philosopher Zeno of Elea, who dreamed up this

paradox 2,500 years ago, the fastest runner in the world can never overtake a tortoise

in a race because you can infinitely divide the distance between them as the tortoise

advances. But that’s ridiculous. We know it’s not

true. Even with a head start I could outrun a tortoise. And I’m no Achilles. So how

can this be a paradox? Zeno knew Achilles could catch up to the tortoise

in real life, but he couldn’t prove it mathematically. He thought there would be an infinite number

of new points for the tortoise to reach that Achilles had to reach… because he didn’t

know that an infinite series of numbers could add up to a finite value — no one knew that

for another 2,000 years. What we now call a convergent series. ½ + ¼ + ⅛ + 1/16

+ 1/32 goes on forever, but it eventually adds up to 1. And at that 1 is where, mathematically,

Achilles finally reaches the tortoise. We knew that Achilles could catch up to the

tortoise, but it took inventing calculus for us to prove why. Which is why this paradox

that confounded great minds for thousands of years is falsidical. Described by Quine

like this: “A falsidical paradox packs a surprise,

but it is seen as a false alarm when we solve the underlying fallacy.” Okay, that’s one paradox envelope downand-

two to go. And behind envelope number two we have: Veridical. For this, we need a game show. Okay I’m gonna replace the two envelopes

we’ve already opened with some prizes. How about we put a million dollars in one of them

and the globglogabgalab in the other. It’s a good enough prize as any. The third envelope

still contains the term for the final type of paradox. Which we’ll get to later. Alright, I’ll shuffle these up. So you don’t

know which is which. Now you’ve got three envelopes. X, Y and Z. Pick the correct one

and you win the grand prize. After you make your selection, let’s say

envelope X, the game show host reveals what’s inside one of the two remaining envelopes.

It’s the glob. Now there are only two envelopes left: the one that you chose and the remaining

mystery envelope. He gives you the option to switch your envelope. Should you do it?

Does it even matter? I mean, your odds of winning at this point are clearly 50/50, right? No. You should always switch. And here’s why.

The odds of winning with your first chosen envelope are 1 in 3. So you have a 33.33%

repeating chance of being right and a 66.66% repeating chance of being wrong. When the

game show host revealed the glob it didn’t suddenly improve your odds to 50/50. The proof

is in the options. After first choosing an envelope, the thing revealed by the host will

never be the money because well that would ruin the tension of the game show. So if your

initial 1 out of 3 pick wasn’t the money and the money is Y, then the host will reveal

Z. If you chose wrong and the money is Z, then the host reveals Y. If you luckily chose

the money the first time, then the host can reveal either Z or Y. It doesn’t matter. No

matter what you’re still stuck in that initial 33% chance that you chose right the very first

time. But if you switch, regardless of the prize revealed, you now leap into the 66%

zone. You’ve doubled your chances of getting the money. To put it another way, when you’re asked

if you want to switch, you’re actually being given a dilemma: Do you want to keep your

single envelope, or do you want both of the other two? It just so happens that you already

know what’s inside one of them. But since the one revealed will never contain the money,

the chances that the other unopened envelope has the money are twice as high as the first

one that you chose. The ‘Monty Hall Problem’ blew up after

a 1990 Parade magazine columnist advocated switching doors in this same scenario from

the game show “Let’s Make a Deal.” When she told readers they should always switch

to improve their odds of winning, nearly 1,000 people with PhDs wrote in to tell her that

she was wrong. She wasn’t wrong. They were. So the Monty Hall Paradox, like the Potato

Paradox we recently covered, is an example of one that is a Veridical Paradox — one

that initially seems wrong, but is proven to be true. Quine said: “A veridical paradox packs a

surprise, but the surprise quickly dissipates itself as we ponder the proof.” Okay. There are paradoxes that seem absurd

but have a perfectly good explanation, and ones that seem false and actually are false

because of an underlying fallacy… even if it takes a major advance in math to prove

it. This last envelope contains the kind we all think of when we all think of paradoxes. Antinomy. The grandfather paradox where you go back

in time to kill your grandfather when he was a child but that means your father was never

born so you weren’t born so how could you go back in time to kill your grandfather?

It’s ridiculous. MinutePhysics proposed a solution to this but these types of paradoxes

are not true or false. Actually, they can’t be true and they can’t be false. As Quine

put it, they create a “crisis in thought.” I am lying. If I’m lying when I say that, then I must

actually be telling the truth. But how can I be telling the truth if I’m lying? The

Liar’s Paradox is an example of Antinomy, which literally means ‘against laws’ and

highlights a serious logical incompatibility. Quine said. Quine said this tape thing was

a good idea in theory but in practice not so much. Quine said: “An antinomy packs a surprise

that can be accommodated by nothing less than a repudiation of part of our conceptual heritage.” Here’s the thing. Antinomies are paradoxes

to us ALL. Falsidical and veridical paradoxes are only paradoxes to those who don’t know

the ‘solution’, but they still have value. Every time we resolve a scenario that runs

counter to our or someone else’s initial expectations, every time we learn the how and why and share

that information…. we’re refining and clarifying knowledge. Which makes all three types of

paradoxes excellent tools for reasoning. Whether or not something is paradoxical to

an individual depends on the accuracy of THEIR expectations. Today, modern mathematics has

given us the ability to show that Zeno’s paradoxes are falsidical. But they were pure

antinomy, unresolved to EVERYONE, for millennia. Quine himself said, “One man’s antinomy

is another man’s falsidical paradox, give or take a couple of thousand years.” Who knows which antinomies of today will be

solved in the future? Right now we struggle with the paradox of the Faint Young Sun: our

current knowledge of stars says that billions of years ago, our sun wasn’t hot enough

to keep the Earth from being a ball of ice. But our geological evidence shows an ancient

Earth with liquid oceans and budding life when everything should’ve been frozen. How could the Earth have liquid water without

a sun hot enough to melt ice? It’s antinomy until we fully comprehend the situation. Maybe

our current understanding of the sun isn’t perfect. Or maybe our knowledge of early Earth

is missing some pieces. A paradox is a problem where the solution

is, or is made to seem, impossible. Sometimes they’re purposely designed for fun because

our minds like puzzles. Sometimes we just stumble on a gap between what we know and

how we talk about what we know, and what is actually true. When we solve an impossible

antinomy, it becomes falsidical or veridical. Someone who knows the answer can see what

the problem was all along: we tricked ourselves… by knowing too little or by asking the wrong

question. In one way or another, all paradoxes come from people. By challenging us to find the flaw or fill

the gap in our knowledge, paradoxes help us define and push our intellectual boundaries.

There’s always more for us to know. Whether we know it or not. And as always – thanks for watching. Hey! If you want to play the Monty Hall Game

yourself you can do that right now over at Brilliant. But the best part about it and

why I’m happy to work with them is that Brilliant helps you learn and refine your

own knowledge. So after you work through the initial problem you can take it to the next

level with variants that make sure you really understand what’s happening. So to support

Vsauce2 and your brain go to brilliant.org/vsauce2/ and sign up for free. The first 500 people

that click the link will get 20% off the annual Premium subscription. Which is an excellent

deal. For everyone.