The Narrow Road

The good news is that I am now gainfully employed in a full time job doing mathematics research. The bad news is that I have a lot less spare time to write new entries here. While the long term goal of eventually developing enough mathematics to motivate category theory and topos theory is firmly in place, the rate at which we get there will be drastically reduced. In the new year I am afraid that you can expect fewer entries, each of which will likely be shorter. Hopefully by reducing the quantity of work per entry and I can still keep them somewhat frequent (I am aiming for about 1 per month), and continue to advance toward the goal of motivating and explaining some of the truly interesting and inspiring ideas in modern mathematics.

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 of 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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This is a brief apology to let readers know that I am still working on the site. I have been otherwise occupied for several weeks, but the larger issue currently is that I am struggling to write the next entry “A Brief Tangent”. The reason for this is that I am battling to find the right way to introduce and discuss the ideas I wish to cover, and this has proved harder than I initially expected. Many words have been written, and not a small number erased again, and the entry is progressing slowly. I hope to be able to finish it in the next few weeks and get it posted. Thank you for your patience.

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.

I am now the very happy husband of Tuula Kristiina Talvila. The ceremony was held at Rockcliffe Park Pavilion in Ottawa on the 24th of August. I could not hope to be married to more wonderful woman, and am looking forward to a lifetime together with her. Hopefully, understandably, new entries may be somewhat delayed.

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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On Friday the 13th of July I successfully defended my thesis, entitled “Pro-finite Lie rings and p-adic Lie algebras“. From here I have only the minor hurdles of getting the Faculty of Graduate Studies to sign off on formatting, and organising copying and binding, to successfully complete my degree requirements. I will therefore be graduating in October, having moved to Ottawa at the end of this month, and gotten married to my fiancée Tuula on the 24th of August. For those who are curious, a copy of the my thesis can now be found in the PDFs section on the lower right; I can’t promise as to how understandable it will be for a general audience, but you may be interested to know what a mathematics Ph.D. thesis looks like.

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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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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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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