Hi, I'm Toby and welcome to the joys of mathematics. If you're an old viewer then welcome back
and if you're new,
there's a chance you ended up here after watching a now viral video of mine that I uploaded three years ago called
VeggieTales predicts modern internet humor.
The original video didn't get many views at all until about a week ago where it now has over a million.
The video was copyright claimed, so I don't want to show it again, but here's a bit of a live action replay for you:
After a discussion about how in the future
entertainment will be randomly generated, Larry tells Bob that jokes are funny when they are
unexpected. Bob asks, "whatever happened to it's funny because it's true?"
Larry says 2+2=4 is true, but not funny.
They then ask a robot who tells them that the answer to two plus two is
w e e d e a t e r, but that's beside the point.
We know that two plus two equals four.
That's true, or is it? When proving something in mathematics
we have to start with a series of statements that can be taken as true without argument.
These are called our axioms and
depending on which set of these statements we choose will determine how complicated it is to prove something,
even if what we're trying to prove, such as 2+2=4
seems trivial and obvious, the axioms you start with can be the difference between
fitting this into a video for YouTube or taking 300 pages in a text book full of sub theorems and
exercises to even get to this point.
Today, in this video, we are going to lay out the piano axioms and go from there to try and prove that 2+2
really does equal 4. It's going to get quite fundamental because we can't take anything for granted.
We can't even take numbers as a concept for granted.
We have to define what numbers are before we can start adding them to each other, and likewise
we need to define what addition even means.
Before we go making any wild claims, We first want to understand what does equality really mean.
What does it mean to equal 4? We need to set up some rules to follow so that we
understand what this actually means to us.
These rules might seem to you too simple and obvious to even bother writing down, but we really want to be precise
here. We want to know exactly what equals does, what it means. We don't want to have any room for
contradictions. We're going to define what numbers are soon,
but in the meantime I've represented numbers of a certain type with letters. Our first rule is that X is equal to X.
You know one variable is equal to itself. So for every natural number, which is what we'll go on to talk about, an
equality means that it is equal to itself. Our second rule if X is equal to Y then
Y is equal to X. It doesn't matter if you're on the left or the right side.
This equal sign, it doesn't matter. Whatever is on one side is equal to what's on the other. In math,
we would call this symmetric. It's a mirror image of itself.
Our third rule is that if X is equal to Y, and Y is equal to Z,
then following our logic here, X is going to be equal to Z. In math we would call this
transitive. Our fourth rule is that if A belongs to some group, some collection,
(We don't really need to know what it is) and A is equal to B, then
this equality means that B also belongs to that same group. A and B are said to be
"closed" under equality and anything they are equal to will also be part of the same group.
These are the laws that we need to understand what it means to equal something.
And now at least we can use an equal in our equation and know that we defined it from some first principles.
So now that we have equality out of the way, we can start to think about what it means to be a number and,
again, numbers are something you might think you're familiar with, but we really want to be precise.
We're going to use the piano axioms,
which I mentioned earlier, and for the purpose of writing on this chalkboard here,
I have replaced a word you might have heard of, which is the phrase
"natural number" with a little flower, a little natural flower there.
The natural numbers are just, you know
the numbers that you would count things with (0, 1, 2, 3, 4, 5 and onwards) and yes
I am including 0 in this list. That's a bit of a personal choice.
I could have chosen to just start at 1 but in fact our first axiom is that 0 is a natural number.
So number 2, our second rule, is that for every natural number
"n" there is this "S of n" [or S(n)], which is said to be the
successor of n, and it is also a natural number. You might think that you are familiar with the idea of a
successor or the concept of next, maybe through Ariana Grande's hit song
"Thank You, Next". Maybe that will help you intuitively, but, to be honest, you don't really need to know
that this means next. You don't need to know what next is
Because we're defining it through all these rules.
Don't forget what we're doing here is defining the natural numbers themselves.
So you don't need to bring any preconceived ideas to this. We can just start from here
so if we have this successor of n, the successor to every natural number, and at this point
we know that 0 is a natural number,
then we know there's going to be some
S of 0 which is also a number and natural number that we can go on to use later on.
We could give a label to this successor of 0, we could give it our very familiar label of
1 let's look at another one of our rules, number 3: for natural numbers m and
n, m is equal to n if and only if the successor of m is
equal to the successor of n. This gives us a little bit more of a constraint on what a successor can be.
You definitely can't have two
numbers that are equal to each other if their successors aren't also equal to each other, so you can't have
essentially double ups where the successes are equal, but the numbers themselves weren't equal, so that helps us define
recursively what our numbers are going to be. Our fourth rule here
is that for every natural number "n", the successor of n can't be 0 and
That's alright. It kind of, amongst other things, rules out any of the negative numbers being natural numbers
because, through this way, if we were to have, you know, our next successor a successor of
1,
We could call that 2.
You know that we're not going to run into any problems because we're going to keep going that the
successor of 2 is equal to 3, and
this would have only really been a problem if the successor of
-1 was equal to 0, which we can't have so
no success is equal to 0, and our fifth rule is, I haven't written it out in full
but I've called it induction.
It means that if we keep defining our natural numbers in this way, say we're going to have
the successor of the successor of 1, also known as the
Successor of 2, we could call that
3 (very familiar names I'm assigning to them) then we can just keep going like that and
by induction this is going to eventually give us our entire set of natural numbers.
There's no room here for you know a couple of random extra guys to come in
somewhere later down the chain who only relate to each other in terms of that
They are each other's successor and they don't relate to anyone else who's come before them. They don't stand on the shoulders of giants
they're just in this own little loop of their own. Well my induction we've got no room
we've got no time for any of that
This is the way that we are going to define our natural numbers and this is going to give us all of them.
Alright, well now we know what numbers are, we've given labels to them as well.
And we've only got one thing left to do and that is to define addition.
The reason that we need to define addition is because addition really could do anything.
It could be any function that map's one number to another.
Addition to us seems so simple because it's a really good way as we know it to describe
real-world examples. If we have one ball and we put it in a bag with another ball then we have two balls.
And in that case 1+1 would be equal to 2 and we would be very familiar with that idea
But we could have had some other
definition of addition. Maybe an addition plus sign with a squiggle around it, which would add
1 and 1 to give us some other answer. If
this addition was defined to be just combining two balls of slime,
you put one ball of slime and another ball of slime into a bag. Well, what will you get at the end?
You'll probably still have
one big ball of slime.
So, in this way,
you could have, you know, 1+1=1 it all depends on your definition of this function here on your addition.
So we want to be clear about what we mean by our addition symbol and it's going to be defined recursively
using these two rules:
Number one: if you have A and you add
Zero to it, you're going to still have A
Our second rule is that if you have A, you add the successor of B,
then what you're going to have is the successor of A+B.
This makes sense
if you think of some examples from how you're familiar with the addition, say if
B was two you'd have the successor of B being 3,
plus maybe A is 1 so that would give you 4, and then you'd have the same thing here, 1 plus
2, the successor of 1 plus 2 would be 4 as well.
So, hope that these two rules of addition make sense to you, because now we're going to use them to try and figure out
our elusive equation:
2+2
Now this is going to be equal to 2 plus...
well, what was 2 from before? we defined 2 to be
the successor of
1 and
Using this second rule here about successes, this would be equal to
the successor of
2
plus 1
Now let's think about what one was from before.
What did we define that to be? We defined 1 to be the successor of 0 so we can rewrite this as the
successor of 2 plus the successor of 0.
Running out of room a little bit on the board here.
Now we can employ rule 2 again, and sort of rewrite our brackets in a way so
this is going to be equal to
the successor of
the successor of
2 plus 0. All right.
Now we can employ rule 1 from before: if you add 0 to something it remains
unchanged. We're adding a 0 to 2 in here
so this is all just going to be 2. So we've got the successor of the successor of
2.
Getting very close now. the successor of 2, we defined before when we were doing our axioms, as
Being the label 3. So now we have the successor of
3, and the successor of 3 is nothing but
4 itself
so there we go by using our definition of addition and also our way to
construct the natural numbers and thinking about what a number actually is in terms of successes,
We've managed to show that 2 plus 2 is indeed
equal to 4. If we want to think of a little moral to the story we can think about how our choice of axioms is
determined how easy it would be for us to prove something later on. We really want to have that
Broad and detailed base and foundation of knowledge to work from so that we can go on to find out
Interesting things that we know to be true.
Doing it this way means that we are relying less on,
you know just believing other people and
Believing in things and having trust, and instead we know this to be true because we've seen it
ourselves from the first principles. We built this foundation
ourselves and now we can stand upon it to be confident and
What we're doing now and to be confident in going on to use some of these ideas. If you have a shaky
foundation of truths that you know to be true without argument then it's going to be hard to
believe anything further on, and you might be more at risk of believing things that actually turn out to be false
so if you are able to
maybe try to think more
often about how to go back and boil things down to a set of statements that you can take as true without
argument, and maybe those statements are so fundamental
they seem trivial to write down, but at least then you will know what they are and
you'll know what's really important to you. Now
it is time to thank this episode's sponsor Brilliant.org. Across the bottom of your screen, we'll be scrolling a few key areas of
mathematics and science which are related to what we've done here today. These are all also courses on Brilliant.org.
Brilliant is a site where you can take courses and quizzes on many interesting areas of
mathematics and science and it's a great way to further your mathematical foundations.
If you go to Brilliant.org/tibees
you can sign up for free
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So thank you to Brilliant and thank you for watching.
