SIMON PAMPENA: The point about points is that our idea of
points, like when we go like this, we go,
here's my point, 3.5.
I'm going to move it over here.
Here's 3.5, right?
That's wrong.
That's not 3.5.
If that was really 3.5, we wouldn't be
able to see a thing.
A point is not a small tiny circle.
That's just a representation.
And what it does is it gives us the wrong sense of how
things are.
Like if you remove a point, OK, and if it was, in your
mind, of a ruler, you should be able to hold it up to the
sun and see a little bit of sunlight shine through.
But that's wrong.
It's impossible.
Because if that was the case, we'd be removing a little bit
of length, and then we wouldn't have one.
So this is pushing our notions of what maths is, which
is why I love it.
I love this stuff.
BRADY HARAN: But your whole premise, your discussion here
is how do we measure numbers.
SIMON PAMPENA: Yeah.
BRADY HARAN: Who wants to measure numbers?
Who said numbers were measurable in the first place?
Why is this a problem for you?
SIMON PAMPENA: Why is this a problem?
BRADY HARAN: You don't measure numbers.
Numbers are--
SIMON PAMPENA: Why this is a problem?
BRADY HARAN: Numbers are what you do the measuring with.
SIMON PAMPENA: Why is this a problem?
Because we are using stuff that we haven't proved.
We're using naive notions of measure.
That's why that's a problem.
We're making assumptions.
This almost broke maths, where I'm going with this.
It almost broke maths.
And that sounds like I'm being really full-on, but it's
absolutely true.
And I know I'm going to be able to show you the exact
reason why it almost broke maths.
But what I'm talking about here, I can tell you right now
what this led to.
This led to the computer, what I'm talking about.
The hair's standing up in the back of my neck when I talk
about this.
That's how exciting this is.
It's so exciting that people started going, oh, I know how
to measure.
You take one.
You take the final.
You take the initial.
Sure.
That's all right.
Well, what happens when you remove points?
What happens when you remove an infinite number of points?
Let's just think about it like this.
Let's go from 0 to 12.
So the length of that is 12.
So 12 take 0 is 12.
That's the length.
So then I'm going to wrap it around a clock face.
Whoops.
That's not a very good clock.
It's more like an egg.
So there's 12 there, 9, OK?
And the reason why I'm going to show you this is because
now I'm going to mark out every fraction.
So it's almost like every single way that you could
divide up time in a way that would be sensible.
I'm actually going to put holes, tiny little holes all
around the clock face.
Now, this is where I'm hopefully going to break your
brain, because now I've removed every single fraction
on the segment of the number line.
And the fact that I've made it 12 doesn't matter.
What's the length of this line now?
Remember, the real number line contains numbers other than
fractions because most of the real number line are
transcendental numbers.
What's kind of crazy is that in this case, though, between
any two fractions, so between any two fractions, so let's
just say here between 0 and 1, you have removed 1/2, yeah?
But you've also removed 1/4, and 1/8, and 1/16, and a 32th.
Rah, rah, rah.
It keeps going do, do, do, do, do, do, all the way down.
So between any two fractions is another fraction, and
between any two fractions is another fraction.
So in maths, they call it dense.
It's dense with numbers.
So it should make you feel a little bit uneasy that you
might be making a mistake.
What we can play with is how many points there are.
So what you can do is you can count out all the numbers.
This is a fraction here.
So you could go-- it's this on this, whatever it is.
So that's 2 on 1.
This is 2 on 2.
This is 2 on 3.
This is 2 on 4, et cetera, et cetera.
So each point represents a fraction.
And you've kind of covered every single possible fraction
that there is, right?
And what we're trying to do is we're trying to show that
this, all the fractions, are countable or listable.
So this is the really great thing.
And what you do is you draw this snake.
So you draw the snake like so.
And you draw it like that.
And you're basically going, I can create a
list of all the fractions.
If we know that we can list all these points, let's make a
list of them.
So let's talk about all the fractions.
Let's call it F, all the fractions between 0 and 12.
So 1/2 will be in there.
2/3 will be in there.
4/3 will be in there, whatever.
If that's listable, let's measure the
length of all of them.
So the length, what's the length of 1/2?
Well, that's 0.
What's the length of 2/3?
Well, that's 0.
What's the length of 4/3?
Well, that's 0.
Boo, boo, boo, boo, boo, boo.
So I've got an infinite sum of 0.
So 0 added to itself infinitely many times?
0.
So it's because of the number, in a sense, the number, even
though it's infinite, and infinite really isn't a
number, but the fact that the type of infinite it is--
a countable infinite, and it's listable--
means that if we remove a countable number of points, a
listable number of points, we haven't changed the length
because the length of all those points have to be 0.
BRADY HARAN: If I remove your feelings--
SIMON PAMPENA: Yeah?
BRADY HARAN: --your size won't change.
Just like if I remove the numbers, the infinitely small,
abstract numbers from a line, the length of the
line doesn't change.
Numbers don't have volume or size, because they're just an
abstract thing.
They're a describing word.
Are we starting to come towards philosophy here?
SIMON PAMPENA: No.
I'm just a bit weirded out by you removing my feelings.
Without my feelings, I can't do maths.
Because it's all very emotional.
See, this is the thing, is that you're putting a nice
analogy on something when this is already an analogy.
The problem is that humans--
and mathematicians--
we are really analogizing.
Even the maths that you see, in a sense, it's an analogy.
Just like how Roman numbers was like-- they
thought they had it.
They thought, yeah.
We know numbers.
They're never going to change.
We're in charge.
They're never going to change.
And they've changed.
And it's because there's been a better way
to represent numbers.
And I think what I love about this is that we draw a line.
Well, first of all, you can't draw a line.
It's impossible.
It's just a very, very thin rectangle.
And you draw points.
That's ridiculous.
You can't draw points.
But there you go.
We draw points.
And then suddenly, we go, oh, I know what to do there.
Because I know how to use a ruler.
You do final, take initial, fantastic.
Suddenly something make sense in maths.
[BUZZER NOISE]
It doesn't make sense.
Suddenly, you're confronted with these conundrums of like,
what happens when I take away a point?
Well, nothing, because there is no next point, like there
is no hole per se.
And so, when you try thinking about it, you actually have to
start questioning every single assumption.
And this is where this led into an
analysis of set theory.
What assumptions were mathematicians
making about maths?
Very heavy questions.
What mathematicians were able to do--
should I go here?
OK, I'll go there.
What mathematicians were able to do, were able to create
sets of numbers that you couldn't measure.
And it was linked to these kinds of ideas.
So you're actually picking all-- the guy who did it was a
guy called Giuseppe Vitali.
And he came up with this idea of consider the solid segment
between 0 and 1 and consider all the fractions.
Then he considered every single copy of that in 0 to 1.
And what that means is it's very complicated.
It's called a partition.
But basically, he found all these different copies of it.
Then he just took one point from every single copy.
And that set you could not measure.
There was no way you could get a sensible answer.
Getting 0 is a sensible answer.
This gave no sensible answer.
It gave a contradiction.
You look at it one way, you get one answer.
You look at it the other way, you get a different answer.
That's a problem.
Because I thought I knew how to measure.
I thought I knew how to use a ruler.
I thought I knew maths.
And this guy realized that there were these sets that you
couldn't measure, and that caused massive problems.
Mathematicians were very, very upset.
The thing that I realized when I studied this is that it made
me question what I knew about numbers.
So my life I see as a timeline of discovery of numbers.
Why not?
I discovered numbers the way, watching "Sesame Street" or
whatever you do, and you get a sense of, oh, yeah, one, two
fractions, you start getting your head around these things.
But suddenly, when you come to the point where you think
about how numbers are related and points and decimal
expansions, all that sort of stuff, you start
going, wow, 3 is weird.
I thought I knew 3, but 3 is very strange.
The abstraction of these numbers are very strange.
And what it really drove me to is to question everything.
You have to question everything in maths.
There's so much that we bring in from the outside world,
which just gives you the wrong idea.
We thought we knew measure because
measurement was about numbers.
So a ruler, you could go, there's a ruler.
There a real ruler.
There's a mathematical ruler.
It looks like a ruler.
Surely I can't go wrong with this.
But what you can do, the sorts of sets that you can make, the
removal of points, you're really doing things there's no
way you could do in reality, no way at all
you could do in reality.
So this is the stuff.
And then exploring these things,
that's what really gets--
I mean, people actually started to question reasoning
out of this, how we reason.
And it actually did lead to the computer.
It actually led--
Alan Turing was the guy who was trying to reformulate some
of the questions that came out of this.
He started reformulating Kurt Godel's work.
And out of that came the Turing machine, which is, for
all intents and purposes, the first computer.
It might not be exactly.
But anyway, he's the guy who came up with it.
And now we're just trying to think.
Now we're trying to go, how do you think?
What are the assumptions we make?
So yeah, so for me, this was the crowning glory of my
mathematical education, was really getting to the bottom
of like if I was a musician, it was like, what is music?
What is it?
Why does it make you feel a certain way?
BRADY HARAN: But it sounds like your conclusion is, we
don't know.
SIMON PAMPENA: What don't we know?
BRADY HARAN: What numbers are.
SIMON PAMPENA: I would love that because I I'm assuming
it's going to be completely incomprehensible to me.