What Is A Paradox?

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.

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