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.
Archive for the ‘Overview’ Category
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.
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.
Let’s begin with a short practical experiment. Pick up a pen, or whatever similar sized object is handy, hold it a short distance above the ground, and drop it. The result — that the pen falls to the ground — is not a surprising one. The point of the experiment was not to note the result, however, but rather to note our lack of surprise at it. We expect the pen to fall to the ground; our expectation is based not on knowledge of the future however, but on abstraction from past experience. Chambers Dictionary defines “abstract”, the verb, to mean “to generalize about something from particular instances”, and it is precisely via this action that we come to expect the pen to fall to the ground. By synthesis of many previous instances of objects falling when we drop them, we have generalized the rule that things will always fall when we drop them*. We make this abstraction so instinctively, and take it so completely for granted, that it is worth dwelling on it for a moment so we can see how remarkable it actually is.
The Narrow Road draws its title from Oku no Hosomichi (The Narrow Road to the Interior), the famous travel diary of Matsuo Basho as he journeyed into northern Japan. My aim is to follow a similar wandering journey, but instead travelling into the abstract highlands of pure mathematics, pausing to admire the beauty and sights along the way, much as Basho did. That means we have a long way to travel: from the basics of abstract or pure mathematics, through topology, manifolds, group theory and abstract algebra, category theory, and more. There may well be some detours along the way as well. It is going to take a long time to get to where we are going, but along the way we’ll see plenty of things that make the trip worthwhile. Indeed, as is so often the case, the journey means more than the destination.