Archive for the ‘Journey’ Category

A Brief Tangent

October 24, 2007

I would like to get temporarily sidetracked. The straight road is often not always the best road, at least not when the journey matters at least as much as the destination. Basho often took sidetracks, and they often provided some of the highlights of his journeys. Wandering off track gives you the opportunity to learn a little more about the country that you are travelling through; that’s precisely the sort of sidetrack we’re about to take.

We’ve spent some time contemplating and discussing the intricacies of the infinite. We started off with a very natural abstraction, and quickly got lead into a mire of technicality and complexity. With a little work we came through unscathed and ended up with a new appreciation for the unfolding beauty and complexity that we were lead to. All of that, however, was a matter of “head in the clouds” contemplation as to what it truly meant to be infinite, and what the continuum actually was. Now that we have that understanding it is time to wander off the path and explore the countryside that such understanding begins to open up for us.

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Parallel Paths

September 11, 2007

Some time ago we set out on two diverging roads. One road sent us exploring the infinite, while the other started out looking at discrete patterns, and how they may be abstracted. As we moved along the roads, however, we found the first hints of cross fertilization: our results about the infinite allowed us to propose pattern algebras for infinite patterns — the result being that the algebra was precisely that of numbers. From here on in the two separate paths begin to align, running quite separate but parallel courses. The cross fertilization of ideas will continue, and we will begin to see increasing similarities between these two very separate worlds of abstraction. The paths ahead will soon become interesting indeed.

Grouping Symmetries

August 5, 2007

On his journey north, Basho stopped at Matsushima and was spellbound by its beauty. The town itself was small, but the bay is studded with some 260 tiny islands. The white stone of the islands has been eaten away by the sea, leaving a multitude of endlessly different shapes, pillars and arches, all crowned with pine trees. Each island is different and unique, and each, with its sculpted white cliffs tufted with pine, is beautiful. It is, however, the whole bay, the combined diversity, that ultimately makes Matsushima one of the three great scenic locations in Japan. So far on our journey we have been admiring the beauty of the islands; the subtlety and intricacy of the different algebras that arise from different patterns, different symmetries. It is time to step back and begin to admire the whole — and in so doing gain a deeper and richer perspective on all the sights of our journey so far.

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A Transfinite Landscape

July 2, 2007

Problems that involve infinity have a tendency to read a little like Zen koans. Take, for example, this problem: Suppose we have three bins (labelled “bin A”, “bin B” and “bin C”) and an infinite number of tennis balls. We start by numbering the tennis balls 1,2,3,… and so on, and put them all in bin C. Then we take the two lowest numbered balls in bin C (that’s ball 1, and ball 2 to start) and put them in bin A, and then move the lowest numbered ball in bin A from bin A to bin B (that would be ball 1 in the first round). We repeat this process, moving two balls from bin C to bin A, and one ball from bin A to bin B, an infinite number of times. The question is, how many balls are in bin A and how many balls are in bin B when we’re done? Think carefully!

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Permutations and Applications

May 27, 2007

Numbers are remarkably tricky. We tend not to notice because we live in a world that is immersed in a sea of numbers. We see and deal with numbers all the time, to the point where most basic manipulations seem simple and obvious. It was not always this way of course. In times past anything much beyond counting on fingers was the domain of the educated few. If I ask you what half of 60 is, you’ll tell me 30 straight away; if I ask you to stop and think about how you know that to be true you’ll have to think a little more, and start to realise that there is a significant amount of learning there; learning that you now take for granted. Almost everyone uses numbers regularly every day in our current society, be it through money, weights and measures, times of day, or in the course of their work. Through this constant exposure and use we’ve come to instinctively manipulate numbers without having to even think about it anymore (in much the same way that you no longer have to sound out words letter by letter to read). That means that when we meet a new abstraction, like the symmetries discussed in Shifting Patterns, it seems comparatively complex and unnatural. In reality the algebra of symmetries is in many ways just as natural as the algebra of numbers, we just lack experience. Thus, the only way forward is to look at more examples, and see how they might apply to the world around us.

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Paradoxes of the Continuum, Part II

May 1, 2007

Mathematical arguments can be very persuasive. They lead inexorably toward their conclusion; barring any mistakes in the argument, to argue is to argue with the foundations of logic itself. That is why it is particularly disconcerting when a mathematical argument leads you down an unexpected path and leaves you face to face with a bewildering conclusion. Naturally you run back and retrace the path, looking, often in vain, for the wrong turn where things went off track. People often don’t deal well with challenges to their world-view. When a winding mountain path leads around a corner to present a view of a new and strange landscape, you realise that the world may be much larger, and much stranger, than you had ever imagined. When faced with such a realisation, some people flee in horror and pretend that such a place doesn’t exist; the true challenge is to accept it, and try to understand the vast new world. It is time for us to round a corner and glimpse new and strange landscapes; I invite you to follow me down, in the coming entries, and explore the strange hidden valley.

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Shifting Patterns

April 24, 2007

akaaka to
hi wa tsurenaku mo
aki no kaze

How hot the sun glows,
Pretending not to notice
An autumn wind blows!*

  — Matsuo Basho

What is a haiku? Or, more specifically, what makes a particular composition a haiku, as opposed to one of the many other poetic forms? The defining feature most people will be familiar with is the 5-7-5 syllable structure. Within that basic structure, of course, the possibilities are almost endless, and this is what makes haiku so tantalizing to write: you can shift the words and syllables around to craft your message, and as long as you retain the classic 5-7-5 syllable structure you can still call your work a haiku**.

This is not an isolated trait. We constantly define, and categorise, and classify, according to patterns. We determine a basic pattern, an underlying structure, and then classify anything consistent with that structure accordingly. This is our natural talent for abstraction at work again, seeking underlying patterns and structure, and mentally grouping together everything that possesses that structure. It is the means by which we partition and cope with the chaotic diversity of the world. And yet, despite our natural talent for this, it wasn’t until the last couple of centuries that we had any treatment for this sort of abstraction comparable to our use of numbers to formalise quantity.

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Paradoxes of the Continuum, Part I

March 27, 2007

Infinity is a slippery concept. Most people tend to find their metaphorical gaze just slides off it, leaving it as something that can only ever be glimpsed, blurry and unfocused, out of the corner of their eye. The problem is that, for the most part, infinity is defined negatively; that is, rather than saying what infinity is, we say what it is not. This, in turn, is due to the nature of the abstraction that leads to the concept of infinity in the first place.

The ideas of succession and repetition are fairly fundamental, and are apparent in nature in myriad ways. For example, the cycle of day and night repeats, leading to a succession of different days. Every such series of successive events is, in our experience, bounded — it only extends so far; up to the present moment. Of course such a series of events can extend back to our earliest memories. Via the collective memory of a society, passed down through written or oral records, it can even extend back to well before we were born. Thus, looking back into the past, we come to be aware of series of successive events of vastly varying, though always bounded, length. We can then, at least by suitable juxtaposition of a negation, form the concept of a sequence of succession that does not have a bound. And thus arises the concept of infinity. Is the concept coherent? Does succession without bound make any sense? With this conception of infinity it is hard to say, for we have only really said it is a thing without a bound. We have said what property infinity does not have, but we have said little about what properties it does have.

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A Fork in the Road

March 13, 2007

Alice came to a fork in the road. “Which road do I take?” she asked.
“Where do you want to go?” responded the Cheshire cat.
“I don’t know,” Alice answered.
“Then,” said the cat, “it doesn’t matter.”

— Lewis Carroll, Alice’s Adventures in Wonderland

In the later years of his life, after his journey to the interior, Basho lived in a small abandoned thatched hut near lake Biwa that he described as being “at the crossroads of unreality”*. Now, still early in our journey, we have come to our own crossroads of unreality. We are caught between dichotomies of unreal, abstract, objects. One road leads to consideration of finite collections, and properties of composition (the algebraic properties 1 through 5 from the previous entry); the other road leads to the continuum and questions of ordering and inter-relationship (properties 7 through 10 from the previous entry). The first road will lead to a new fundamental abstraction from finite collections, different from, and yet as important as, the abstraction that we call numbers; this way lies group theory and the language of symmetry that has come to underlie so much of modern mathematics and physics. The second road will lead to deep questions about the nature of reality, and, brushing past calculus along the way, lead to a new and minimalist interpretation of a continuous space through the concept of topology.

Which road do we take? As the cat said to Alice, It doesn’t matter. We are at the crossroads of unreality, and the usual rules need not apply. Which road do we take? Both.

* From the translation of Genjûan no fu by Donald Keene, in Anthology of Japanese Literature.

A Fraction of Algebra

March 5, 2007

As a mathematician there is a story I hear a lot. It tends to come up whenever I tell someone what I do for the first time, and they admit that they don’t really like, or aren’t very good at, mathematics. In almost every case, if I bother to ask (and these days I usually do), I find that the person, once upon a time, was good at and liked mathematics, but somewhere along the way they had a bad teacher, or struck a subject they couldn’t grasp at first, and fell a bit behind. From that point on their experiences of mathematics is a tale of woe: because mathematics piles layer upon layer, if you fall behind then you find yourself in a never ending game of catch-up, chasing a horizon that you never seem to reach; that can be very dispiriting and depressing. In the previous entries we have dealt with subjects (abstraction in general, and the abstraction of numbers) that most people have a natural intuitive grasp of, even if the details, once exposed, prove to be more complex than most people give them credit for. It is time to start looking at subjects that often prove to be early stumbling blocks for some people: fractions and algebra.

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