Top 10 Pieces of Mathematical Awesomeness

1. 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:

1. Two points can have a line drawn between them

2. You can draw any line as long as you like

3. Circles exist

4. Every right angle is the same

5. 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.

2. The Axiom of Choice

Mathematicians since Euclid have generally been pretty impressed by his axioms. In the early 20^th 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.

3. 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.)

4. 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!

xxxxxxxxxxxxxxxxxxxxxxxxxxx

challenge post; the concept of lovely

go here, watch and smile :)

This is still one of my favourite videos on YouTube. In fact all of the The Beckoning of Lovely videos are.

So my challenge is: do something lovely. Anything lovely; for yourselves or for others, it doesn't matter. Then come here and write about that lovely thing.

It doesn't have to be extravagant, in fact simple pleasures are often the most satisfying. Practising the concept of lovely can surely only lead to a little bit more happiness in a world that often isn't so accommodating for such a feeling :)

No strict deadline, but before the new year would be pretty nice :)

(Oh, a reminder to Jenny and Mhairi; we're still excitedly awaiting your top ten lists!)

She is a real royal lady, true patron of the arts

Hey girls, it's Friday! (or at least it was when I started writing this, and then didn't finish on time to publish)


I recently joined a group on facebook called "that awkward silence when an Arts student talks about their future prospect" - for the purposes of irony, really. That was before the figures for the arts cuts were public knowledge, so now you could say my membership is of a more bitter irony now.


On visiting that page now, though, God, those people are so annoying. The Arts students are getting all uppity about the snub, whilst non-arts students maintain a sense of smug self-righteousness when it comes to criticising the arts in general. The same mantra; good luck getting employed with that kind of bullshit degree - not even worth the paper its printed on. Blahblahblah arts is stupid.


It's OK, the opinions of ignorant individuals doesn't bother me. I know that the arts industry as it stands in Britain is one the fastest growing, a mass employer, and that we're envied by many countries worldwide for our arts scene. This can be credited to a culture of established accessibility; art is no longer considered as high-brow as it once was, and within education bodies the concept of "creative-thinking" is very much encouraged. It's incredible - to have so much culture absolutely bursting within our tiny isles - it's not only an economical plus, it's a social one too. If we as a society are exposed to arts - and that is arts in all forms, from literature to film making to philosophy and beyond - we learn to express ourselves and communicate ideas in an entirely fulfilling means of conversation. Arts is popular for a reason; its the fibre of life - its everywhere, as it should be. 


What's not OK, is the government's policies on the arts. In recent weeks we've seen a 30% cut of arts funding in England (though only 15% should apply to frontline services) and in Somerset county council the decision has been made to cut funding for the arts by 100% (incidentally, funding for the arts in Somerset only took up 0.004% of the budget - that's how dispensable it was seen from the offset). In Scotland we are facing similar cuts - though the means by which this will be carried out has yet to be decided by the SNP. Essentially, this is going to cause a lot of restrictions on the industry - including those who view it. This holds great potential to see a return to the old, elitist structures of culture - where those who are currently interested in the arts remain there (if they can afford to, financially and in regard to their future prospects, that is), whilst nothing is done to encourage those who are not involved in arts to engage in them. You'll see this through the maintenance of crown jewels such as Opera Houses and the Royal Shakespeare Company, but you can forget any fundings for non-yearly festivals or radical new project ideas. Oh the Conservatives, they just love their hierarchies.


I'm not saying the arts should have priority. I definitely, definitely agree that in face of a smoothly running NHS or a thriving arts scene, the NHS should always have the preference. What I'm saying about this is that the manner in which these cuts are being implicated - so fast and so brutally- is going to have dire consequences to not only economic growth, but is going to undermine the happiness of the nation during times where the nation's spirit is going to be in desperate need of lifting.


What is more frustrating, however, is the fact that the Conservatives' cuts (I say Conservative here because Clegg's backbone has been entirely indistinguishable since he entered the Coaltion) are so blatantly driven by ideology, but the policies of which highlight so many contradictions its sickening. I've always thought this of the Conservative ideology, that it's founded on contradictions that ultimately lead to its failing. What we need to understand about Conservative ideology is that they believe in upholding tradition - the very name denotes this fact, "to conserve". These traditions are preserved through symbols; for example, red phone boxes, the royal family, London cabs - such things maintain a sense of Britishness through their symbolic status to our country. And the reason Conservatives are so obsessed by the concept of the nuclear family? Of course, it's the oldest tradition there is. 


But tell me, because I really want to know, how exactly do you maintain a person's ability to read such symbols, that are inherent to the success of this ideology, when you take away the means by which these symbols are read? If you try to steer people away from the arts, further emphasis must be placed on non-arts subjects in order to encourage potential candidates to favour these areas instead. But symbolism is learned through a creative process - and traditions are abstract in nature; most traditions are merely signfiers of greater concepts. How can you maintain them if you undermine the creative process, if you remove the way in which applying symbolic meaning to life is learned?


Aside from that, symbolism is important to learn not just in respect to the success of this archaic, bullshit ideology, but also to life as it is right now. Take away the arts, then you take away a world-view. Can we really only see the world through facts and figures? I don't know about you guys, but the reason I read books and watch films because of a distinct need to make sense of the world through metaphorical means, through a process that conveys ideas that are often too big to comprehend as a whole - and without those books and films that shaped my childood, that got me through adolescence and continue to offer me an entirely new world-view even now, I can't even think about what a narrow-minded individual I'd be. And that is just two outlets the arts provide. We are surrounded by design; we love to play computer games and watch brilliant TV programmes and buy snazzy, edgy furniture. We as a nation are at the forefront of the arts, we're internationally acclaimed for our talent and - let's not be too modest - our brilliance. And it just makes me so very sad to think that our government is going to dismantle this within five years or more (i really, really hope it's only five though).


I think this is where I draw distinct preferences between the Conservative ideology and the Liberalist ideology; Labour at least strove to provide opportunity for all those willing to take it. The Conservatives, meanwhile, are preoccupied with providing opportunities to those who can afford the expenses of them. On other fronts I'm still making up my mind.


Either way I'm certain; take away the arts from the wider public, then take out the nation from imagination.

it's friday, it's fireworks and remember remember

Remember telling me "don't scream, and let go of the handles, it's way more funner!"? It's funny how that sort of sums up us. Remember snorting rainbow dust? Remember seriously discussing which of each others' body parts we'd prefer to have when we merged as one person? Remember those hours over phones and msn discussing The Story? Remember how we used to be so emo it hurt? Remember our despair that Billie Joe was 40-something and married? Remember oreo-milkshake? Remember getting drunk on cococabana and lying on our backs watching the stars fall from the sky?

Remember the uncool cool area? Remember the spongy grass at the short-cut to the bakeshop, and the pound you lost there that seemed to be absorbed by the ground? Remember when Mr Hayward took us for tea at Hoswick Visitor's Centre? Remember when you asked me to come rowing, but I forgot and you appeared at my door and I was still in my pajamas? I'm still proud of how fast I managed to change into daywear. Remember how scared we were of Mr Lindsay? Remember how he told us his life-story in Japan, to us only and for no real reason other than to talk about his favourite subject (himself)? Remember life is like soy-sauce, love is like rice?

Remember activity Wednesday and Newspaper? I was so shy, and I accidentally drew on your jeans, and you just laughed. Remember town days, and sitting on your knee in the Havly, and getting kicked out of the Havly for being "too loud"? I'm sure you're tiring of such a phrase, but I could listen to you all day at whatever volume you choose. Remember Thursday Period Four? Remember all those discussions about teachers and boys and boy wizards and time-travellers? Remember sitting on a rock by the sea at 1AM drinking lambrini and discussing my boy troubles? Remember then standing up and realising how drunk we were, and then rolling down a hill in the wet grass?

Remember that time when you came up to Shetland and it was glorious? Remember "she's from Boston!"? Remember the Easter dinner? Remember how lemony it was, and how delicious it was, and how Harry didn't like the sweet-potatoes? Remember "do the Ashley face!"? Remember the emotional moneys? Remember me telling you how freaking hot you are as we looked through your myspace photo albums? Remember buying cocktail pitchers from Flints and pulling funny faces for the camera?

It's just the surface. But I don't ever want to forget.