- Hi, everybody, Jade here.
Today's video is about one of the most famous paradoxes
in mathematical history.
It's called Russell's paradox named after Bertrand Russell,
super cool dude, very influential philosopher/
mathematician/ logistician/ social activist/
Nobel prize winner, so yeah, we'll get to know all about him
but this is probably one of my favorite paradoxes
and it's really one of those ones
that just makes you really question everything, you know,
question everything you know.
So yeah, I'm super excited for this video.
Just before we get into it, I just wanna say that
this is probably the most genuinest video I've ever done.
There's a lot of subtleties and there's a lot of philosophy.
So if you find that you're getting a bit lost
or you need to like rewind
or watch the video again, that's totally normal
and I'd actually be pretty surprised
if you've got everything the first time.
So yeah, let's get right into it.
So first, I'm going to introduce you
to a somewhat simpler version of the paradox,
just to soften some of the concepts
that we're going to talk about later on.
So, this one is called the Barber paradox
and it goes like this.
There is a town with one barber and his task is to shave
all and only those who do not shave themselves.
Now, the question is, does the barber shave himself?
Think about it, if the barber does not shave himself,
then he should and if he does shave himself,
then he shouldn't.
So?
So, keep this little riddle at the back of your mind
throughout the rest of the video
but now, let's get right into the math.
So Russell's paradox is ultimately about
the foundation of mathematics,
but some of you might be unfamiliar with that concept,
what is a foundation of mathematics?
Well, let's draw an analogy with the other sciences.
The sciences are kind of like a tree.
Biology is the study of living organisms,
which are the result of millions and millions
of chemical interactions.
So, one could conclude that all branches of biology
stem from chemistry or that chemistry is in some way,
foundational to biology.
Likewise, chemical reactions are ultimately
physical interactions.
So, one could conclude that all branches of chemistry
stem from physics.
If we keep going down in this way,
we'll eventually come to the elementary particles
that make up our universe, electrons and quarks.
The laws that govern these particles
can be considered the foundations of science
in that, the rest of the sciences stem from them.
If we were to discover something completely new about them,
it would somewhat affect our understanding
of the other sciences,
even if only at the theoretical level.
In the 19th century, each branch of math
was pretty disconnected from the other branches
and there was no unifying trunk.
Mathematicians at the time wanted to unify
the branches of mathematics
and therefore, a foundational theory was needed.
But what made it tricky is that,
the approach to figure out the foundations of math
is very different to that of the other sciences.
We learn about the other sciences
by observing the outside world,
whether it be watching cells under a microscope,
measuring orbits through a telescope
or smashing particles together at insanely high speeds.
But we can't really learn more about mathematics
by observing the outside world.
Math seems to be something that just is.
We can imagine a universe where the fundamental
building blocks are different or where time goes backwards,
or where everything was made of anti-matter,
but we can't really imagine a universe
where two plus two equals five.
Math is different in that it seems to be true by definition
and even though it's very good at describing our universe,
it doesn't seem to be about our universe.
Why?
That is the question at the heart of this paradox.
Now, it's here that we steer away
from the harder sciences and head more toward
the study of what reality really is, philosophy.
We come to the first character in our story.
The question, why is math the way it is,
captured one of the greatest philosophical minds
of all time, Plato.
Plato thought that mathematical objects,
numbers, shapes and the relationships between them
were objective truths, that is,
objects that existed independently of us and our world.
He thought they existed in their own world,
which he called the World of Forms.
But one of Plato's pupils, Aristotle, had his own views.
Unlike Plato, he thought that numbers themselves
weren't objects but properties of objects.
For example, if there were four cows in a field,
it wasn't that cows were an object
and four was also an object,
but the number four was a property
of the collection of cows.
He also thought they didn't exist
in their own World of Forms, independent of our world
but they described features of our world
and so, belonged to our world.
Another colorful philosopher, Immanuel Kant, soon came along
and he didn't like the view
that mathematical objects were objective truths
and believed that to understand the basic principles
of math, you needed some kind of intuition.
That yes, math described our world,
but we also brought it to the world from our experience.
Now, this is where our quiet hero of the story comes along,
Gottlob Frege.
Yes, interestingly, Bertrand Russell is not the main player
in Russell's paradox but a German mathematician named Frege.
The story of Frege is somewhat tragic
as he was largely ignored during his life
and 20 years of his life's work was unjustly
washed down the drain.
But here, he will be remembered as a God among us.
Frege disagreed with Aristotle's view
that numbers were properties of objects
because of this argument.
If numbers are properties of object,
then one number should belong to any object
and it shouldn't be influenced by matter of opinion
but then, imagine a pair of shoes,
is it one pair of shoes or is it two shoes?
Depending on how we conceptualize an object,
the number belonging to it changes.
Again, imagine a deck of cards,
depending on how we conceptualize it,
it could be one deck of cards or 52 cards.
So Frege concluded that numbers don't apply to objects,
but to concepts.
He also disagreed with Kant that to understand math,
you needed intuition and experience,
stating that the laws of arithmetic can be known
from the basis of reason alone.
"I sought to make it plausible that arithmetic is a branch
of logic and need not borrow any ground of proof
whatever, from either experience or intuition."
Frege's main goal was to reduce mathematics to logic
and show that logic is in fact, the foundation of math.
This idea is known as logicism.
Now, the notion of logic that Frege had in mind
is pretty similar to how we would use it
in everyday language, but just to make it super clear,
logic is a tool for reasoning
about how different statements affect each other
through nothing more than deduction and inference.
For example, if we take it as a fact
that old dogs have a good sense of smell
and Tifa is a dog, deductive reasoning will tell you
that Tifa has a good sense of smell.
I don't know if this is the best example though
because she's pretty old,
I don't know if she can smell much anymore
but you get the idea.
Frege started his quest of reducing math to logic
by coming up with a definition of what number was.
Now, this might seem like a somewhat trivial question
to some of you, what is a number,
but remember at the heart of Frege's paradox
was the idea that you don't need any kind of intuition
or experience to understand mathematics
and when you think about it, have you ever actually tried
to describe what a number is without using the word number
or words derived from the word number?
I've spent some time trying as I was writing this video
and I found that I always ended up using the word number
again, or a word that was directly related to number,
like amount or quantity.
Give it a go right now and let me know
what you come up with in the comments.
Frege defined numbers by using this idea
of concepts and extensions.
A concept is pretty much any idea you can think of,
the color red, shape, Bill and Bob, Shakespeare's plays,
hipsters, elephants with no ears, literally any idea
and an extension is the set of all things
that fall under that concept.
For the concept the color red,
its extension would be the set of all red things
past, present, and future.
The concept Bill and Bob would be made of the extension
of the Bill and Bob you were referring to.
If we have a concept like square circles,
which doesn't make sense,
their extension would simply be the empty set,
a set with nothing in it.
Numbers, Frege stated, are extensions of concepts.
For example, the number four is the extension of the concept
of all things made up of a collection of this many objects.
The number seven is the extension of the concept
of all things made up of a collection of this many objects.
Frefe built his foundation of mathematics
from the axiom that all concepts we can possibly think of
have a corresponding extension and therefore,
there are as many extensions as there are concepts.
He named this the General Comprehension Principle
and it sounds reasonable, right?
Can you think of a concept that doesn't have
a corresponding set to go with it?
I don't think I could, but Bertrand Russell did.
Frege was just about to have his work printed
when he received a letter from Bertrand Russell
in June, 1901 which read along the lines of,
"Dear colleague, I find myself in complete agreement
with you in all essentials.
In regard to many particular questions,
I find in your work discussion, distinctions
and definitions that one seeks in vain
in the works of other logisticians.
There's just one point
where I have encountered a difficulty.
Consider the set of all sets
that are not members of themselves.
Is that set a member of itself?"
This simple question shattered Frege's foundation
and caused him a mental breakdown so severe,
he ended up in hospital and later, caused him to write,
"My efforts to throw light on the question
surrounding the word number seems to have ended
in complete failure."
Let's break it down.
First, a word about sets.
Sets are a collection of things.
You can even have sets within sets.
Take, for example, the set of all sets of bird species.
There exists a set of all penguins, the set of all seagulls,
the set of all pigeons
and the set of all other bird species
and they themselves would make up the set
of all sets of bird species.
Most sets are not members of themselves,
that is, they don't fall under themselves.
For example, the set of all teapots is not a teapot,
the set of all turtles is not a turtle
but what about the set of all things that are not turtles?
Because it's a set and not a turtle,
it is a member of the set
of all things that are not turtles.
Therefore, the set of all things that are not turtles
is a member of itself.
A lot of sets are members of themselves
and there's nothing too extraordinary about that.
But the question that Russell asked was,
is the set of all sets that are not members of themselves
a member of itself?
Think about it, if it is a member of itself, then it is not
and if it's not a member of itself, it is.
Sound familiar?
I'll say it again, if it is a member of itself,
then it's not and if it's not a member of itself, it is.
Now the reason that this was so catastrophic
is because the last thing you want
in a neat systematized axiomatized description
of anything is a contradiction.
Imagine if one of the foundational laws of physics
predicted that gravity both always pulls and always pushes.
It would be a pretty useless law.
When your system is able to derive two opposite theorems,
this puts into question the entire theory.
Numerous attempts were made to correct this hiccup,
like Russell's Theory of Types,
which tried to put sets into some kind of hierarchy
but many thought that this fix
was too artificial and ad hoc.
Frege eventually felt forced to abandon
many of his views about logic and math.
Even so, as Russell points out,
Frege met the news of the paradox with remarkable fortitude.
"As I think about acts of integrity and grace,
I realize that there is nothing in my knowledge
to compare with Frege's dedication to truth.
His entire life's work was on the verge of completion.
Much of his work had been ignored
to the benefit of men infinitely less capable.
His second volume was about to be published
and upon finding that his fundamental assumption
was an error, he responded with intellectual pleasure,
clearly submerging any feelings of personal disappointment.
It was almost superhuman and a telling indication
of that of which men are capable if their dedication
is to creative work and knowledge
instead of cruder efforts to dominate and be known."
Apparently, he didn't know about the breakdown.
Anyway, after a series of fortunate events,
the foundations of mathematics is now known
to be a system called Zermelo-Fraenkel Set Theory,
which has a lot of Frege's original ideas
incorporated into it.
But although this is currently the most widely
accepted theory, the search still isn't over
with candidates like category theory
and homotopy type theory becoming serious contenders
for that much desired position
at the bottom of the math tree.
But that is for another video.
(upbeat electronic music)