"unfinished"

From Witley Court, the ruins of a Jacobean palace, complete with an impressive fountain and a stunning Baroque chapel.

1

Water and light

One... two – the bells announce the hour. Still nothing. Hell, it’s hot here. I’m not sure now if I’m standing in the best place – yeah, I’m facing the front of the fountain, but more crucially, I’m also facing the sun, and judging by the glaring look it’s giving me, it doesn’t like being stared at. A few more agonizing minutes, and the cupids on either side of the fountain, having decided they’ve kept us waiting long enough, lazily shoot some water from their bows.

The long moments in which nothing happens were probably intended to increase the tension to an unbearable level; I stifle a yawn and contemplate my over-heavy mountain boots. One by one, the fountains surrounding the central sculpture shoot into the air. They don’t change direction, the figures don’t move, and yet it all suddenly springs to life. The graceful arcs of water emphasize the curves of the centerpiece, they glitter and dance in the sun. Their delicate watercolor draws my attention to different parts of the scene. In the center there, with a mighty, but still only metaphysical sweep, Perseus swings his sword and pierces the sea monster’s jaw. Its teeth, like those from a T-rex fossil, look surprisingly sharp – but its eyes have already been dulled by Death’s scythe. It’s very artistic, the way the creature’s watered down blood squirts into the air. All the tension of the story is concentrated in that fountain, and it shoots higher and higher, a man-made cloud. It’s not quite the fountain in Geneva, but it’s got the same monumentality; and it’s tall enough for me to doubt whether even the largest sea monster could have as much blood as all that.

“I won’t be stared at if I hide well,” the sun reflects as it looks down into the clear fountain pool. And it wraps its face in a cloudy veil, and I know I have chosen the best place to stand. Droplets of water fall on my face and arms. The battle has been won; I persisted in watching the spectacle despite all obstacles and was duly rewarded by the retreat of my sunny enemy.

2

Problems with perspective

On the spectrum of audio guide makers, the ones from Witley Court should occupy the opposite end to “practical” people, somewhere towards the “poetic” section. But of course they wouldn’t know what I mean, as they don’t understand the meaning of the words “opposite end” – when they used this expression to denote the position of the next point on the tour, it didn’t mean, as some might have supposed while standing with their back against a wall, the part of the room in front of them, but a different room altogether, located not in front of them, but behind.

This wasn’t the only time on this trip we had doubts about geometry. In the Baroque chapel, we debated whether the painting on the ceiling was “upside-down” or not. With the character’s feet closer to us and the sky away from us, it did look uncomfortable, especially when compared with a second wall-painting, orientated the other way around. But then it’s only natural to draw the sky further away than the ground, isn’t it? The matter was further complicated by the fact that the artist realized that the painting was going to be hung on the ceiling, so he foreshortened the figures a bit to suggest that they are actually above us. Above us and at the same time having their own private ground and sky? Makes my head spin.

But neither this painterly drawback nor the broken CD player in the corner emitting cho-cho-cho-choral music of a slightly more mi-mi-mi-minimalistic nature than intended by the composer prevented us from appreciating the place.

The amazement we feel at seeing a thing is unfortunately a function of how common it is (this function is probably different for everyone and depends on many other variables; but on average it’s an inverse proportionality). In Poland I couldn’t stand Baroque architecture – argh, the overflowing gold, the kitsch angels... – while Gothic churches were a source of constant awe. But this year I’ve come to feel that flying buttresses are the same everywhere, and though gems such as Winchester or Yorkminster still squeeze a gasp out of me, your average Gothic or Neogothic minsters will only warrant a polite nod. Baroque churches of the type found on the Continent, on the other hand, are almost unheard of here, and so the chapel in Witley Court caused the intended widened eyes and increased heartbeat. There were no chubby angels here, and the gold leaf adorning a relaxingly white wall was overdone with much more style than is found in a typical Polish Baroque church.

No witty ending for this article.

Cantor's Seductive Sets

This article has absolutely nothing to do with travelling, it's something I wrote for a competition for a math article (actually for a "maths" article, as it was a British contest). The results have just been out, and no, I haven't won it. I actually didn't stand a chance, because there was a word limit, and I accidentally sent them the version with a couple hundred extra words. Arghhhh...

PS Sorry 'bout the varying sizes of the pictures; I still haven't quite got the hang of inserting pictures here.

A Beautiful Concept: Cantor’s Seductive Sets

“There are just as many even numbers as there are natural ones.”

“What…? But how…?” you might be thinking…

See for yourself: let’s pair ’em up:

Generally, every number n in the set of natural numbers is paired with 2n in the set of even numbers, and so in the land of numbers there are no bachelors. As this land can be quite conservative, bigamy isn’t allowed, and so there must indeed be as many even numbers as natural ones.

But how can a set have as many elements as a subset that isn’t that set? There can’t be as many women as people if there exists at least one man…

Welcome to the world of infinity, where these strange things do indeed happen! Your passport is an open mind, and the most important law goes like this:

If and only if there exists a one-to-one correspondence between two sets, both sets have the same number of elements (so-called cardinality).

Working under this law, we soon discover that there are just as many rational numbers (fractions), as integers (natural plus negative numbers), as natural numbers, as even numbers. This infinity, the infinity of the natural numbers, is called ‘countable infinity’. Is it the only one?

Nope. In fact, there are more points on any line segment, no matter how small, than there are fractions in the vast expanse of Numberland.

And, as if this weren’t enough, there’s an infinite (at least countably infinite, that is) number of infinities – every set of subsets of a set has more elements (a larger cardinality) than that given set. I’ll try to outline the proof for your benefit and amusement.

Imagine you are in a room with infinitely many children. Perhaps uncountably many yelling, screaming brats... You give each kid a piece of paper (so much for saving trees...) and ask them to make a list of all their friends present in the room. Could it happen that each possible list of children was written down by one of the kids?

Given the sheet of paper, each child is faced with a problem. Is he his own friend? Consider all the kids with a low self-esteem, who decided they don’t like themselves. Make a list of them. Could this list have been written by any of the children? Suppose a child had indeed made this list. Is this child, then, her own friend? If she is, she should be on her list – but her list names only the children who aren’t their own friends. Suppose then that she’s not her own friend. Then we should include her on the list of kids that aren’t their own friends – but that’s the list of her friends, and she isn’t her own friend!

Clearly, the list of children that aren’t their own friends can’t be matched up with any child, so not every possible list can be written down by some child.

The conclusion? The set of all subsets of a given set always has a larger cardinality than the set itself. (Don’t quite see it? Disregarding all children’s rights, throw the yelling and screaming brats, the whole infinity of them, into a nearby dump, replace them with elements of any set, and look at what’s left of our story.) The second conclusion? Low self esteem doesn’t pay off...

***

“Do any of you love set theory?” asked my maths teacher once upon a calculus lesson. Undaunted by the jeering silence that followed, he continued. “Because, you know, you’re at the age in which young people are often under the negative influence of that theory.”

Setting aside my teacher’s charming cluelessness about the level of teenagers’ vulnerability to beautiful mathematical concepts, what could he have meant by that foreboding “negative influence”?

At the time Georg Cantor (though I owe the topic of this article to him, I still haven’t introduced him, so... everyone, meet Cantor, the brilliant 19^th century German mathematician, father to all the different infinities!) created set theory, it was truly revolutionary. Up till then, the only acceptable type of infinity was a potential one – numbers could be as large as one wanted, but they never actually reached infinity. Cantor changed all this – he wrote about infinity as if it was as much of an existent being as any of our little number friends.

But how can infinity exist? When we talk about 0’s and 25’s, we can pretend that we are just talking about abstract properties of ‘the real world’. But when we start talking about the infinity of real numbers, where in the universe can we find anything that even resembles it? This is one of the reasons many of Cantor’s contemporaries rejected his theory. Another objection they had was Cantor’s use of the so-called reductio ad absurdum proof. This is the sort of proof the ‘brat story’ is – we assumed that the children could have made all these lists, and then showed that this assumption leads to a contradiction, so its negation must be true.

Kurt Gödel, a brilliant 20^th century logician, once proved that God exists. His proof was strictly logical, starting with a carefully blended mixture of axioms, and ending in the elegant filet mignon of the conclusion; the same type of proof used in mathematics. Our belief in points, numbers and infinity is no more or less justified than that in God. Mathematics proves that certain numbers and points exist – but what that existence means no one has much of an idea. At best, mathematics seems to be an impossibly nerdy religion.

Cantor’s set theory, even when clad in the unladylike rags of infinitely many brats, is too elegant to be thrown into the dump. Perhaps my maths teacher knew more about young men’s nature than he let on... After all, it’s not all that difficult to capture a young man’s heart, and set theory is the perfect seductress. She might not be true, but she sure is beautiful.

An essay on public transport and television

Maybe the place you live in isn't much of a travel topic, but there you are, you can't write about the same things all the time. And I'll be describing some of my first impressions of Bristol, from the time when I was still a tourist here.

It’s been nigh on nine months since we moved to Bristol... There’s a month and a half to go, but the time’s as good as any to do a little summing up.

August 2008. The smiliest stewardesses I’m ever likely to meet – already I feel we’re in the West. Then there’s their accent, a reminder that, try as I might, I won’t be able to understand everything people are going to say to me. Well, that premonition sure wasn’t an overstatement.

Next was – ah, yes, the hotel TV. I’ve no idea how it happened, but we ended up watching CBeebees – BBC’s toddler program. It was nice to see Postman Pat alive and well after all these years (and considerably more British than I had remembered him from the Polish version); but the real hit was Brum – a show dedicated to – you guessed it – a cutely stupid car. “Brum, brum, brum, brum, brum!” my brother and I chanted the main theme and rolled to the ground laughing.

I distance myself from the childish extravagances of my 18-year-old self. I am now, after all, 19, and long past such bad taste. And so I’ve never watched CBeebees since.

The fact remains that British telly (as they call it here) is a universe better than Polish TV and at least half a world ahead of American television. Even their commercials are funnier (except the ones that boast “Prepare your funeral with us and we’ll give you a free pen!”). Britain is possibly the world’s only country in which you can spend a whole day watching national television and not consider your time wasted.

Continuing on the track of unimportant technological innovations – hurrah, Bristol has double-decker buses! Before I got used to left-hand traffic, sitting in the front seat of the top floor had the hang-on-for-your-life quality of a rollercoaster. Then, as the views out of the huge front window lost their shiny new appeal, I began to notice that the front seat is rather uncomfortable (you can’t stretch your legs out in it), and so I moved my strategic seating position to deeper reaches of bus space. But I’ve always remained on the top floor – partly out of habit, partly because the idealist in me believes I’ll someday lift my head up from my book and look out the window, and partly because the lower floor is taken up largely by old ladies who can’t climb stairs.

During one of those first days we visited Bristol cathedral. I never bothered to look around too carefully, there was an evensong concert I wanted to listen to; besides, I could always come back later. After all, I was going to live in the same city. Guess what, nine months later, that coming back still hasn’t happened, even though I pass the cathedral every day on my way back from school.

It’s not much of an exaggeration to say that the only times you ever notice the place you live in are the days you move in and move out. (Presumably for those who spend their whole lives in one house that would be the day they’re born and the day they die?)

Nonetheless, writing this text has got me into a state akin to that of someone seeing their home for the last time. I almost wish we didn’t have to leave Bristol in July... Then again, as I’ll probably be heading for Oxford next, maybe I don’t. Bring on the ochre-colored arches of Oxford’s Gothic cathedrals, the long-benched tables in its medieval halls, the evil-faced grotesques adorning its buildings, so that I may fail to notice them for a few years, and open my eyes only to say goodbye.