SIMON PAMPENA: I want to take a little bit of a departure
from looking at specific numbers and ask the question,
how do you measure numbers?
This was one of the most challenging things I came
across studying maths at uni, was this notion of how to
measure, mathematically measure.
So this is what I'm going to do.
I'm going to show you the real number line.
Yeah, you're familiar with that right?
So think about this.
This is the real number line.
It's there in your head.
So here's my question.
What's the distance between 3 and 4?
What's the measure?
How long is that section?
BRADY HARAN: 1.
SIMON PAMPENA: 1.
How did you get that?
BRADY HARAN: Because I know the number that I need to add
to 3 to get to 4 is 1.
SIMON PAMPENA: OK, that's good.
So you know the number.
So you know the answer, basically, you're saying?
BRADY HARAN: Yeah.
SIMON PAMPENA: OK, well, imagine you
didn't know the answer.
So imagine this is the first time you've ever tried to
measure the distance between 3 and 4.
What would you do?
BRADY HARAN: I'd make a piece of wood the distance
between 0 and 1.
And then I would take it to 3 and 4 and see how many of them
do I have to put on there.
SIMON PAMPENA: OK, that's good.
That's good.
I like that.
It's actually good that you brought in reality.
Because that's going to really bite you.
It's going to really bite you.
But that's cool.
That's cool.
I like the fact you said that.
So basically you said, OK, well, I'm going to look at a
bit of a number line which I know what the length is
because it starts at 0.
And you're going to go and drag it over
until you measure it.
But another way you could do it is you could say, this
number, take that number.
So you're comfortable with that.
You know that that distance there is 4 take 3.
Now this is what's called naive.
And that's not a putdown.
It's naive.
It's a naive way of measuring numbers.
I'm hopefully going to take you down the rabbit hole now.
3.5 is a point right there.
I'm going to put a little kind of, little bit of
choke wire around it.
3.5 is a point.
Do you agree that it exists?
3 and 1/2?
BRADY HARAN: I'm happy with that.
SIMON PAMPENA: You're happy with that, OK.
What's the length of 3 and 1/2?
BRADY HARAN: What's the length of it?
Well, to give you a length, I need a start
point and an end point.
SIMON PAMPENA: No, I'm just talking about the point, just
the point, one point.
So what's the length of a point?
BRADY HARAN: Points don't have length.
SIMON PAMPENA: What's another way of saying that?
3.5 take 3.5 is 0.
We use exactly the same logic that we used here for the
segment of line.
We're going to use the same logic for the point.
Now, this sounds like I'm being
trivial, maybe even annoying.
But I am getting somewhere.
I'm making a point about points.
Because what now is the length of 3 to 4 if we
remove the point 3.5?
What is the length of that line segment?
BRADY HARAN: The length is still 1.
SIMON PAMPENA: Why?
BRADY HARAN: Because the thing you removed took up no space.
SIMON PAMPENA: That's right.
So this idea is that we've got a line segment, 3 to 4, which
is made up of a continuous set of points.
There's only points between 3 and 4.
But if we remove 3.5, we haven't changed it.
We haven't changed it at all.
And part of the reason behind that is because the real
number line has this property that there is no next number.
What's the number after 3.5?
BRADY HARAN: 3.50000000000000--
SIMON PAMPENA: Yeah, exactly.
But you're not even saying that correctly.
Because the way that we describe numbers doesn't
contain the awesomeness of what's going on here.
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.