Euclid's Axioms for Geometry.
Everyone knows Euclid's Axioms for Geometry – yes, even you! That's because Eucild's Axioms are what you do when you go to primary school and you learn about shapes. Even if you didn't learn anything in primary school, you still know Euclid's axioms, that's how easy they are. It's a shame that the “Lines and shapes” chapter in your primary school maths book isn't called “Euclid's Axioms for Geometry and their immediate consequences” because doing geometry with axioms is really cool.
To understand axiomatic maths you have to pretend you know nothing about the world. In particular, you know nothing about Geometry What are points? What are lines? Can you draw a line between two points? You don't know. Now Euclid is going to tell you five statements about geometry – these statements are the infamous axioms. The axioms are incredibly obvious to anyone who's been alive for more than 30 seconds, but they are all you need to prove pretty much anything you would ever want to know about geometry. Here they are:
Two points can have a line drawn between them
You can draw any line as long as you like
Circles exist
Every right angle is the same
Parallel lines do not meet *
There is so much you can prove just from these axioms - from really obvious facts (triangles have three sides and three corners) to less obvious ones (Pythagoras' Theorem, the angles in a triangle add up to 360 degrees). You can even prove facts about algebra. You can prove that any number can be uniquely split up into primes using Euclid's axioms. I have no idea how you prove this, but Wikipedia tells me that you can so that's enough proof for me.
Euclid wrote his book of axioms and proved everything I've told you about in 300BC. That's about 2300 years ago. You do hear a lot about how smart the Greeks were, but seriously: they were smart. Maths before the Greeks was all: “I have two apple in one hand and three in the other, I wonder how many apples I have?”. Axioms are really really important in modern maths. And proofs are really really really important - maths is proofs.
*This axiom is interesting - you can prove most things without it and actually for a long time people thought you didn't need it at all. The maths of General Relativity uses a special kind of geometry where parallel lines can meet, it's trippy.
The Axiom of Choice
Mathematicians since Euclid have generally been pretty impressed by his axioms. In the early 20th Century they were so impressed they decided to copy him and invent their own collection of axioms – but this time they didn't just want to describe geometry they wanted to describe ALL OF MATHS. Maths is pretty big and pretty complicated and we know a lot about it, so boiling it down to a few simple statements is kind of a big deal.
To do this they “invented” sets. What's a set I hear you ask? Well, anything is a set! {1,2,3,4} is a set, it contains the numbers 1, 2, 3 and 4. But 2 on it's own is also a set, but it's a pretty boring set because it doesn't contain anything. Every number up to infinity is a set, and the collection of numbers up to infinity is a set (we write the set of all numbers like this: {1, 2, 3, …} with the ellipses just meaning “and so on”). {a,b,c} is a set and so is {@,£,$,%} and, best of all, YOU are a set! (you're my favourite set). So if we know about sets we know about anything. Okay, we know the maths of anything. Even though all cancers form a set I can't tell you how to cure cancer using set theory :(
So all the mathematicians worked really hard and worked out all the axioms, but there's one problem... we call that problem the axiom of choice. We need this axiom to prove lots of really important, really obvious things. If we don't have the axiom of choice we can't prove that multiplying two non-empty sets together gives another non-empty set. Whole chunks of maths need the axiom of choice to get anywhere. But the axiom also proves lots of things that just aren't true. For example if the axiom of choice is true then we can take a sphere, split it into pieces and put it back together so that we get two spheres of the same size... (Hey guys, you can now get this joke, yay!)
So no one is very sure if The Axiom of Choice is true or not. It's terrible! The moral of the story is never to believe anyone who tells you maths is all logical and consistent, because it's not.
The Russell Set
Remember how I just said that everything is a set? Well, that was the original definition but it turns out to be a terrible definition that makes no sense. Don't believe me? I'll show you:
Lets think about a set that contains all sets that do not contain themselves. Pretty confusing set, but there is no reason why it shouldn't be a set. Let's call this set the Russell* set.
Now, is the set Russell a member of the Russell Set?
If it is then
it doesn't contain it's self
and so it's not.
But if it's not then
it's a member of the Russell set
and so it does contain it's self.
As I said, it's pretty confusing. There is lots of clever ways of redefining sets so that we don't run into these problems but it's further proof that logic and mathematics is actually a bit crazy!
*Named after after Bernard Russell, who pointed out this whole problem. You've probably heard of his - he had his fingers in many pies. And he was like... really good at all those pies.)
An infinity of infinities.
Okay remember that set we talked about – the set of all numbers from 1 to infinity, or {1, 2, 3, …}. We call that set the set of all natural numbers. And this set is infinitely big right? And you can't get any set bigger than this? Right? Wrong! We actually have an infinity of sets bigger than this. It's crazy, even crazier than I can adequately explain.
The set of all numbers and all their negatives {…, -3, -2, -1, 0, 1, 2, 3, 4, …} might look like its bigger, but actually its the same size as the set of natural numbers. The set of all fractions is also the same size – even though there is an infinity of fractions between any two fractions! But the set all decimal numbers – numbers like 4.12392701 – is a “bigger infinity”. It's hard to explain, and it doesn't seem to make any sense. The man who proved this ended up dying in an asylum with his proofs widely mocked. But it's okay because we all believe him now!
Knowing about sets bigger than infinity is really useful for explaining the axiom of choice. Basically, what the axiom of choice says is that if we have a really big infinity of infinitely big sets we can always chose an element from each of the sets.
5. Gödel’s Incompleteness Theorem
Okay, now for my final depressing reason that maths doesn't actually make sense. You know how I told you how great axioms are, well...
In 1931 when he was only 25 Kurt Gödel proved that any systems of axioms complicated enough to prove anything interesting would be incomplete: there will be statements that are true that we cannot prove.
Doesn't it blow your mind that someone can prove something like that?
And the Incompleteness Theorem matters – there is things that we do not know and cannot know. We will never be able to prove that there exists a set bigger than the natural numbers {1,2,3,4...} but smaller than the set of all decimal numbers. Wow.
Did I mention that Kurt Gödel also went crazy and thought everyone was poisoning him?
So for the sake of my sanity (and yours if you've made it this far) I'm going to stop at 5. After all maths is nuts and maybe 10 = 5 ;)
Love you peeps, I'm off to work on my actual maths project!
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