So let me start by asking you a question,
you're the Numberphile guy: what is a
number?
(Brady: A number is an abstract way to represent)
(a quantity.)
That's a very good answer. - (I just thought of that!)
It's a very good answer.
I spent I think most of my PhD years
thinking
about the answer to that, and that's more
or less what I had. - (You should- you should)
(have given me a call!)
(I would have told you!)
You're a hard man to reach.
(Okay, good answer)
(Okay. Are we right? What is a number?)
Right, so
as you said it's just an abstract thing
that represents quantity. Now
if you ask somebody they might tell you,
you know, by giving examples. So what is a
number? Well 0 is a number, 1 is a number,
42 is certainly a number okay?
What about half? Oh okay half is a number.
Minus 3 over 8 is a number;
wait what about the square root of 2? Oh
that's also a number.
What about pi?
You know, what about e? All of these
things are numbers. What about i? What
about the square root of -1 right? Oh
that's imaginary, that's not a real
number. So i is also a number right? It's
just not one that makes sense to you
sort of as a human; but in electric
engineering they use i to talk about
electricity flow and stuff like this so
i is also a number. You can use other
things as numbers
right,
but that still doesn't really answer
mathematically speaking
what's a number?
Now
I'm gonna give it away, I don't have an
answer to that, there is no answer to
that.
But what can a number be?
So
for example a number, if you want to have
a
strict foundation of mathematics; you
want to say I'm going to start with, you
know,
one thing and build math on that. This is
going to be a bit controversial, but the
the de facto standard nowadays is Set
Theory with the Zermelo-Fraenkel axioms - or
ZFC - with the action of choice. We're not
going to go into 'oh that's the the only
one'. It's not the only one, there's a lot
of different ones,
but
ZFC is one of the main foundations of
modern mathematics. First of all what is
a set? A set is a collection of
mathematical objects right? This could be
numbers, or other sets and so on. In set
theory everything is a set.
So if you think about numbers you're
thinking about sets.
Now that sounds odd for a lot of people
because you're not used to this, but if
you think about computers then a letter
is a number. A picture file is also a
number, so all these things are
represented using other things, it's just
a matter of you know coding stuff into a
system. So a lot of times you will see
something that says- so let's say that
the number zero is the empty set. So this
is the set that has no elements, nothing
in it, think about an empty bag. Let's say
that we already have defined the number
n, so the next one is just going to be
whatever n was defined and we add to it
n itself.
So
1 is going to be just zero, which is just
the empty set, and 2 is going to be
zero and 1 - right, because we had zero
and we add 1 itself,
which is
empty set
and this.
So what would 3 be? Well {0, 1, 2} and this
just goes on. So in effect a number is
just
0,
all the way to n minus 1. - (So why is 1)
(the empty set? I thought zero was the)
(empty set?) - So this is the set whose
element is the empty set, so it's a bag
containing a bag containing nothing.
(Okay. So all numbers are just bags)
(of nothing? It's just how many nothings)
(have you got?) - Kind of.
But now we get tricky. So this is just
the natural numbers, so 0,1, whatever -
what's half? What's minus 1, right? All of
these things are not represented here.
So now you can start, you know,
let's call it going a bit crazy with
this. And you can say, okay, I'm going to
define
a collection of natural numbers and say,
this one represents you know -1. And
another collection that represents -2
and so on and so on, and then you can
take those and make- you know, use this to
represent the fractions.
And then you can use
all kind of
constructions to, you know, go from there
and get the real number. So for example,
after I defined all the natural numbers,
let's call this N, how do I get to Z,
which is the integers? So this is also
-1, -2, -3 and so on. Just
for clarity, 0 is a natural number, this
is also a very you know contentious
point
so;
it is.
All right, let's just clarify that.
(I'll put your email address down there.)
How do we get from here to here? Well for
example, so you want to get from the
natural numbers to the integers. So one
way to think about this is to sort of
think about pairs of numbers. So I have a
pair, let's call them x and y,
and
another pair, let's call this x prime and
y prime; and I want to think that they
represent the same thing
if x plus y prime is x prime plus y. So
for example let's put in
0
and 1, this would be x and y, and
this would be
3 and 4. Let's see if I was right:
that's zero plus 4 equals
3 plus 1. So this is 4;
so I was right, everything works. These
two things are kind of the same, so let
me you know put this
approximate, not exactly equal right?
And you can use this now to say, okay,
let's look at all of these pairs of
numbers
that have this kind of thing
and let's call all of these -1. So
-1 is now really just all the pairs of
numbers (x,y)
such that
(0,1)
is the same as
(x,y). And now I can extend the addition
from the natural numbers to this. So -2
would be
(x,y) such that (0,2) is the same as (x,y)
under this weak equality. That, you know,
gives you a way to define -1 and so
on. But you can also redefine 0, so 0 is
now
(x,y)
where (0,0)
equals
this (x,y);
which if you think about this just means
that x equals y. So that's zero but hang
on, zero is the empty set.
And now I'm saying it's not the empty
set - it's this collection of pairs.
And here's the important thing, it's just
coding. You just have to decide how you
code your numbers right. So you say, okay
I'm starting from this, I'm building the
system up
and then at the end- so from this you can
continue and define the rational numbers,
the real numbers, the complex numbers,
whatever. And you can say, let me end
this at the complex numbers, right, so I
went from this to
the rational numbers - so
half, -3/8 whatever - and from here to
the real numbers with pi,
and the square root of 2, and from here
to the complex numbers with i
and whatever. So I can say, okay, let me
define all of these number systems and
each of those has sort of a canonical
way that the previous one mapped into
that. So now I have a canonical way of
putting the natural numbers into the
complex numbers.
So the empty set is now, you know, moved
here to this
particular set and then to that
particular set, whatever I end up here,
moved to this particular set, move to
that particular set. Now I can decide
that the natural numbers are not this
thing;
actually whatever I ended up with as the
copy
in this one.
So
what's a number
from a mathematical, let's say, objective
reality?
It's whatever you want it to be.
(Like a really popular question that)
(everyone always asks on the internet is:)
(do numbers exist or were numbers)
(invented? Or are they- do they just exist)
(in the world? Presumably you think)
(numbers are very much)
(a human invention?)
Well
philosophically speaking I'm very
agnostic, I don't want to have any
concrete set of beliefs because beliefs
lead you to
being sure that you're right and, you
know, you can't really know.
The point here is that
numbers exist
within a context
and
if you're thinking about set theory as
your context
then numbers are whatever you decide to
mean-
whatever you decide that numbers mean in
this context, whatever coding you want.
You want the standard coding for the
natural numbers,
do you want to go into the complex
numbers?
You can use different sets; you can use
you know the empty set to say, I want the
empty set to be 5 and I want zero to
be some other set.
It's fine,
right? It's all about coding and the the
important thing - and this is where you
know
ZFC as a set theory has a lot of power -
is that it just shows that whatever
coding you did,
if it satisfies the basic properties
that you think of for the natural
numbers,
it doesn't matter. You can always
translate this to that and vice versa. So
it's like if you have something written
in ASCII and something written in
unicode,
you can just move between those two,
you know, let's say in English without
all the emoji symbols right.
So you can just move between those two
encodings and that's fine.
(Could mathematics- could)
(you know, modern thought have exist- could)
(exist without numbers? Like were numbers)
(necessary? Or was it just one of the)
(paths we could have taken to reach where)
(we are now?)
So it's a good and interesting question;
I don't know.
But they're so natural. But here's
something that people forget: sets are
also as natural. You know, people like to
say, oh you know numbers started when you
know like caveman was hunting a mammoth
and was like, okay there are two mammoths
and two of us so it's the same
quantity. But to understand that there
are two mammoths you need to understand
the collection of mammoths, and that's
the basic of set theory right? So the
basic is not just numbers, it's numbers
and sets;
or arguably numbers and types or numbers
in some sort of, you know, collections.
And it's just so natural, you know I
don't see how we could have gone without
coming up with numbers.
....transcendental numbers. And so a lot of
the number categories refer to in or out
of these different sets. So rational
numbers are everything inside the blue
line, irrational numbers are everything
outside the blue line.