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.