#!/bin/env slides
Paradoxes
Bruce Hill
Warning
🚨 This is a no-meta-jokes talk. 🚨
This talk is not a paradox.
What Is a Paradox?
A paradox occurs when two things you know to be true contradict each other.
A paradox means that one or both of the things you know to be true are actually false.
Paradoxes are a way to fix your understanding of the world.
Paradoxes aren't something to marvel at, like how big the universe is, or how weird Quantum Mechanics is.
Paradoxes are a you problem, not a universe problem
Obligatory Etymology Slide
Greek "paradoxos": "contrary to expectation, incredible," from para- "contrary to" (see para- (1)) + doxa "opinion,"
Baby's First Paradox
The Liar's Paradox
This sentence is false.
(Originally "All Cretans are liars", spoken by a Cretan)
I think this paradox is pretty lame. If anything, the two contradictory premises are:
- All grammatically correct sentences are either true or false
- The sentence
This sentence is false.is grammatically correct.
I think it's pretty obvious that (1) is false. Lots of sentences just don't have coherent "truthy" interpretations. The sentence is just incoherent.
Zeno's Paradoxes
Achilles and the Tortoise
Achilles is a fast runner and he's racing against a tortoise. Since he's faster, he gives the tortoise a small head start.
In order to run past the tortoise, Achilles first has to catch up to where the tortoise is now.
In the time it takes Achilles to catch up to where the tortoise was, the tortoise has moved ahead by a bit.
Repeat ad infinitum. Achilles will never pass the tortoise because he's constantly stuck catching up.
Achilles and the Tortoise
False Premise: it takes infinitely long to do an infinite number of things.
Zeno didn't know about calculus. You can have an infinite sum that adds up to a finite quantity.
See also: Zeno's Arrow, where an arrow has to get halfway from its current position to its target infinitely many times.
Easy Mode: Over
Okay, those are some of the simple paradoxes. I don't think anyone here is confused by them. We live in a post-Zeno world.
From here on out, shit's gonna get real.
Envelope Paradox
There are two envelopes with money in them. One envelope has twice as much money as the other. You are allowed to pick one envelope to take.
If you have a chance to change your mind, and pick the other envelope, should you?
If your envelope holds $X, you have:
- 50% chance to double your money and get $2X
- 50% chance to halve your money and get $0.5X
E(value) = 0.5 * 2X + 0.5 * 0.5X = 1.25XSo, it seems like you would make more money by switching!
Would you like to switch again? Infinite money!
Envelope Paradox Resolution
The mathematical formulation is bad. Another way to frame the calculations is:
There is $X in one envelope and $2X in the other. If you switch envelopes:
- There is a 50% chance you had the big envelope and lose -$X by switching
- There is a 50% chance you had the small envelope and gain +$X by switching
E(gain) = 0.5 * -X + 0.5 * +X = 0
Raven Paradox
Consider the hypothesis:
All ravens are black.
If this is true, then it must be true that:
If something is not black, it's not a raven.
If you see a black raven, this should bolster your confidence in the hypothesis, right?
If you see something yellow and it's not a raven, that should also bolster your confidence in the hypothesis. Every non-black, non-raven you see makes you more confident in the hypothesis.
"Look, a blueberry! I guess all ravens really are black, huh?"
This is obviously absurd.
Raven Paradox Resolution
The paradox all along was the idea that this is absurd.
Actually, you should increase your confidence when you see non-black non-ravens. It's just that you've already seen a ton of non-black non-ravens.
Imagine you visit a foreign country and you have a theory
that every word for an animal ends in -mo. A cat is a
blorpmo, a dog is a squeemo.
You flip to a random page in a translation dictionary and
see a word that ends in -snoz. You guess that it therefore
must not be an animal, and sure enough, it means "bottle".
You do this a few times and every random word that doesn't
end in -mo is, sure enough, not an animal. Your theory
seems more plausible.
In the raven case, it's just that we've already seen too many non-black non-ravens. At this point, each additional one we see only nudges the probability a teeny tiny bit.
The Friendship Paradox
This is a fact from graph theory:
On average, each person's friends have more friends than that person does.
Our intuition is that: "Some of my friends are more X than me, some are less" and on balance, that averages out.
It would be true if you were talking about being good looking or being tall or skill at pinball machines, but not "having friends"!
What's going on?
The Friendship Paradox Resolution
The issue is sampling bias.
Assume there are:
- 10 people with 1 friends
- 10 people with 15 friends
- 10 people with 29 friends The average person has 15 friends
However, everyone has at least 10 friends who are friends with everyone, a few people who are friends with half of people, and most people aren't friends with any loners.
If you have an average number of friends, you're probably friends with a lot of people who are more social than you and not very many people who have no friends.
The Twin Paradox
Facts from Einstein's theory of Special Relativity:
- Time passes more slowly as you approach the speed of light
- Velocity is relative and no perspective is privileged
Imagine you have two twins on Earth.
One twin (Sigourney) goes on a round trip journey to Beta Hydri, traveling 24.38 light years at 99.9999% the speed of light.
The other twin (Eartha) stays home.
From Eartha's perspective, Sigourney is traveling at near the speed of light, and must, therefore, be aging more slowly. After about 48 years, Sigourney will return home, having aged only 24 days.
However, from Sigourney's perspective, it looks like Eartha and the rest of the planet Earth was whizzing away at 99.9999% the speed of light. As distances contract at high speed as well, it looks like Beta Hydri is only 12 light days away, so after reaching the destination 12 days later, Sigourney turns the ship around and makes the 12 light day journey back to Earth. From his perspective, Eartha must have aged only about 49 minutes because she was traveling at relativistic speeds.
Who is right?
The Twin Paradox Resolution
It turns out that the acceleration matters. Sigourney is turned into jelly by the relativistic acceleration on her trip, and that's a big difference that is absolute and not purely in the eye of the beholder. Sigourney doesn't think that Eartha is undergoing intense acceleration.
When Sigourney reaches the turnaround point in her journey, any signals sent from Eartha would catch up to him "all at once", and as she decelerates, she sees message after message coming from Eartha all in rapid succession. Sigourney knows that Eartha has aged years as soon as Sigourney slows down.
It's almost as if the acceleration is what makes the universe realize that Eartha is going to be older than Sigourney when they reunite.
If Sigourney never accelerated, the paradox would be unresolved.
Simultaneity
What if Sigourney wasn't Eartha's twin, but was actually born "at the same time" as Eartha, thirty years ago on a spaceship mid-flight on its way past Earth to Beta Hydri?
What does it mean for Eartha and Sigourney to be the same age? It means they were born at the same time.
What is "the same time"?
Turns out that's a whole can of worms and the universe does not care what you think.
Takeaways
Paradoxes help us understand when we're wrong:
- When our intuitions are at odds with the universe
- When we screwed up our math
- When we aren't actually making sense
You can't cause a paradox in a time-travel movie, you can only uncover when the screenwriter is contradicting their own worldbuilding rules.