Friday, July 20, 2007

Mathematics: something really ugly yet beautiful.

Is that the case with maths? Is it really this horrible looking thing, which as sometimes happens, appears beautiful to certain people but horrible to others?

'Mathematics is logically imperfect', so what is it that we find beautiful and great about it? Did the perfection or ultimate truth of it attract you? How would you feel, upon discovering that it's not really that perfect after all? At a younger age I used to like the fact that an answer was either right or wrong in maths. If it was right then you were left feeling content, and if it was wrong you struggled with the question to get the right answer- which provided a feeling of more great than just contentment, upon arriving at the right answer.

Compared to other subjects, like English and History, I was always left frustrated in them whenever I got a big red cross. Namely because I didn't know what the right answer was. The teacher used to argue that the 'beauty' of English is that there is no right answer, but I wasn't taken in by these comments. (Especially on certain exercises). My battles with the comprehension exercises and I think, the lack of tools in my English back pack, always had me struggling. Hence, the immense dislike this 13 year old bean felt for English.

It was ultimately this 'challenge' to get the one right answer, using whatever means possible, that kept me attracted to maths like a magnet. It was the desire to overcome the difficulties that lay across my path. It was the feeling of having conquered a nation by defeating the question. And the more common feeling to myself- that of having scored a goal or taken a wicket. The adrenaline rush and excitement of the victory is always immense and leaves you wanting more. Maybe that is why I end up liking the areas of maths (analysis) which cause me the greatest distress? The largest amount of joy is associated with these areas- if I'm ever to overcome my difficulties.

It is now as I'm doing some reading, that I realise that everyone has a different motivation to studying maths.

The one word that I can unfortunately come up with, to describe why I want to continue my study of mathematics, is that 'maths is cool.' If I was to strain my 'ickle'* brain a little more, I'd say that the diverse nature of maths is alluring and the satisfying pleasure of sometimes following a proof and other times constructing one myself has me hungry for more. Yes, I was ignorant of maths until I started university (as I continually say), and only now am I getting a proper birds eye view of everything. But I think my motivation is still the same. I still want to study exciting concepts and be able to understand them. To overcome difficulties.

What am I getting at? Well, I'm currently reading a book and it has a mention of Russel's Paradox. I read this during the year sometime but didn't think much of it, but today I read about Godel's theorems of undecidability. Are these the questions that everyone asks upon reading the two theorems? The theorems in question are,

First Theorem
'If axiomatic set theory is consistent, there exists conjectures which can neither be proved or disproved.'

Second Theorem
There is no constructive procedure which will prove axiomatic theory to be consistent.

I've foolishly not read ahead and decided to post this, but although we assume certain axioms when we prove, we can never be sure of them being true. As I continue my mathematical 'bull' (1st post!) doesn't that make Maths like Physics or the sciences, in where experimental deductions are used as results? Or is my arrow a long way off. Don't worry I actually like these two theorems since, like I've said, sometimes a scratch on an otherwise perfect car can appear beautiful. (As long as it's not my car :D). That example stinks, but e.g the mss building. It looks old and 'ugly' yet there is something beautiful about it. (apart from the fact I had an office there). As these thoughts swirl about, I'm reminded of the following picture:

How perfect is maths then? Or is it only the logicians how face this problem? As Russel said upon Godel's discovery,

'I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere...'

Once again, I don't really know what I'm thinking but I'm trying to splutter something out which doesn't make sense. (as usual).

What I may be getting at is the fact that when I was talking to the teachers who didn't teach maths, I said something to the effect that in maths, once you have a proof of something you can't contest it, because it is based on logical premises that are all true. But are Godel's theorems saying that step 1 of the proof may or may not be true? I know in maths we tend to 'assume' a lot of things, and say that if this is the case then something else is true etc, but should we be assuming things that haven't been proved. Tell me when to shut up, but I'm actually quite happy about 'this chink' so to speak. It's not making me look at maths in another light, but rather I'm glad that Russell didn't find the certainty in Mathematics the way he wanted.

Am I going against the 'code of a mathematician' by saying these things? I know certain axioms are obvious, like the addition of real numbers is commutative, but .... I think it's best that I carry on reading. :D This makes maths all the more interesting. Oh, and this post does not make any sense!

I've been having a week full of quotes, so why stop now:

"Wir müssen wissen,
Wir Wirden wissen.

We must know,
We will know."

That was inscribed on David Hilbert's tombstone and it sounded neat.

Internet problems have delayed this post, but is any one else getting the Harry Potter book in a few hours? Don't be alarmed if you see Voldemort around ... I'll be away for the weekend, so have a Potty one! My post on my final week at the school and overall thoughts will have to wait- don't look too happy!!


Anonymous said...

beans said...

I take it you don't like this blog. :p

Nice animation indeed- it was needed to shake me out of my \sout{annoying} 'OK' weekend.

'Rain rain go away; come back another day.' Is how I'd describe half of the proceedings.