idea-mongering

•21 January 2018 • Leave a Comment

Even though I am well established in my career, to the point where I am now more often asked when I’ll retire rather than when I started, I still find myself part of the eternal conversation, what do you want to do when you grow up?  Often, this comes up when I’m talking with students, but it has also been known to arise as part of the perambulating conversations we have during the beer-at-the-end-of-the-day sessions that happen from time to time.

And I know what I’ve always want to be.  I want to be an ideamonger.  A forger and fashioner of ideas.  What I find interesting is that looking back, it’s clear to me that I haven’t always known that this is what I want to be, but reflecting on my path to this point in my life provides some clear illumination that I’ve been moving towards this nonetheless.

Mathematics, my main professional interest, is very much a forge and battleground of ideas.  We explore the abstract, seeking to glean what we can from the ideas and concepts before us, sometimes exploring their consequences until we are forced to come up for air, and sometimes exploring their practical consequences.

Mathematics has allowed me to have a career spanning research and education, so not only can I explore ideas for their own sake and the sake of their consequences, but I’m also able to transmit those ideas through my teaching.  And it’s more than transmitting the ideas themselves; it’s also teaching the process of exploring ideas, challenging them, forcing them to reveal themselves.  This is something that I’m finding more and more interesting, somedays I have to admit more than the exploration of the ideas themselves.  And it also explains the occasional lecture I give on a topic unrelated to mathematics.

And the writing I do, when I do writing, is also an exploration of ideas.  I recently made the mistake of going through my GIANT FILE OF STORY IDEAS, as vast a list as it is, and I was able to see some general grouping of some of these ideas into coherent areas of exploration.  For me, I’ll admit that this makes the act of writing a bit more difficult, as I find myself distracted by the ideas more than the individual stories, but that’s just one more thing I’ll need to work through.

But I’m also finding that things are coming together in interesting ways.   A long time ago, I read Jokester by Asimov and it’s one of those stories that always stayed with me.  In part it’s shaped how I think about mathematics, in that part of the process of discovery and proof is asking the right question, since the right question will point us in the right direction.

But it’s more than that.  Education is changing and a large part of what’s driving that change is that facts are no longer expensive to store and transmit.  Rather, it’s the process by which we interrogate facts that is becoming the interesting thing, and so we need to be better at asking interesting questions.

the shining city on the hill and the great white whale

•13 January 2018 • Leave a Comment

I suspect that my memory is a bit suspect on this, which given the time that’s passed wouldn’t surprise me, but I have the memory from an AP English class in high school that the shining city on the hill is a reference from Pilgrim’s Progress by John Bunyan, an allegory with the subtlety of a 2×4 upside the head. The shining city on the hill was the goal towards which the hero Pilgrim was journeying along the Straight and Narrow Path, through the Slough of Despond and the other stops along his journey.

The shining city on the hill has become a not uncommon reference, one that I’ve used on more than one occasion, as the end point that we might not reach in our efforts. But while we might not reach the shining city, we need to have a goal towards which to work. Towards which to strive. Towards which to direct our efforts, rather than bouncing around in some administrative Brownian motion, bouncing randomly from one thing to another, which can happen in large and complicated organizations.

The great white whale is of course Moby Dick. I remember taking a literature class as a university student from a Professor who felt that Moby Dick was one of the great American novels and indeed one of the great novels in absolute terms. I’ve read Moby Dick a few times now; I try and reread it every few years, and it is one that I enjoy more each time I read it. Ahab’s obsession towards finding and killing Moby Dick still resonates, being quoted by no less an illuminary than Khan Noonien Singh

So why have I brought together the shining city on the hill with the great white whale? I’ve been reflecting a lot recently on the objectives towards which large institutions direct their effort and their energy. And I think there’s a central question that we always need to address. Are we moving towards the shining city on the hall, moving towards the heavenly destination in which all things will be better, or are we at sea, bound by our irrational obsession to find and kill the great white whale that in the end we cannot do more than lose our ship to.

Change is a complicated thing, and the decisions we need to make in that process of change are complicated decisions, often and necessarily based on partial information. We rarely have the luxury of making decisions based on perfect or complete information and we do the best we can. But I do think we can and must continue to reflect on the goals of the institution and to keep evaluating that basic decision, is our goal the shining city on the hill or is our goal the great white whale.

the old reading project and the new

•8 December 2017 • Leave a Comment

On 1 January 2017, I set myself a challenge. I would read the Sir Richard Burton translation of the Tales of 1001 Arabian Nights. My original plan was to read one Night per day, and finish in (roughly) 3 years. But I got slightly more ambitious, cranked my pace up to four Nights per day sometime during the spring, and I finished the tale of Ma’aruf and Dayla, the last of the Tales, yesterday during a morning coffee break.

It’s been an interesting journey. I’ve had a physical copy of Sir Richard’s translation on my shelf for 20 years or so, 3 hardback volumes with very small type and very thin pages. And it’s interesting that reading it on the Kindle made it a much easier read. I carry my machine with me all the time, but I would not have been able to easily carry the bricks of the physical volumes. And sometimes, my best reading time was lunchtime or coffee, not the morning when I first woke to greet the day and not the evening, the night, when I was tired and sleep was regretfully more attractive than even reading.

I smiled when I came across Sinbad and the 7 voyages of a man never happy to be sitting at home. I remember Jinn and Ifrit freed from sealed stoppered bottles and rings. I remember caves of treasures but SPOILER ALERT it wasn’t Aladdin or Ala Al’din who found the cave and the Jinn to satisfy all his desires. Rather it was Ma’aruf who found the cave of treasure and the Jinn sealed into the ring while plowing a field.

There were long fascinating stories in the middle, like the many Tales of the battles of Gharib, and all in all, I’m happy I made the effort. The question is, what reading comes next.

I have always viewed the buying of book, the owning of books and the reading of books to be related but separate pleasures, and I have a lot of unread books on the shelves. Next though, perhaps I’ll tackle the source of one of my favourite quotes in these troubled times: Long is the way and hard that leads from darkness into light, which is a paraphrase but I think a reasonable paraphrase for most situations.

the power of Hofstadter’s principle

•27 November 2017 • 4 Comments

Those who know me, know that I am a fan of Hofstadter’s principle, also know as Hofstadter’s law, which states that the task at hand will always take longer than you think, even when you take into account Hofstadter’s principle.

Part of the power of Hofstadter’s principle is its self referential nature.  Part of its power is the way it feeds on itself, and I want to take a moment to reflect on that.  If we were to phrase the principle as saying, the task at hand will always take twice as long as you think, even if you take into account Hofstadter’s principle, then twice becomes four times, four times becomes eight times, and very very soon, every task takes infinite time.

Now, I have projects that seem to be taking infinite time to complete, but that is merely a subjective illusion that comes from how I feel the passage of time.  We as humans have a very poor relationship with infinity, but that’s a story for another time.

But every iteration taking a bit longer than the previous iteration, this is something we can hold in our heads.

One reason I like Hofstadter’s principle is that it can be so easily recast for the situation in which we find ourselves.  The task at hand will always be harder than you think, even when you take into account Hofstadter’s principle.  And the task at hand will always be more complicated than you think, even when you take into Hofstadter’s principle.

In this, Hofstadter’s principle is akin to Parkinson’s Law, which (loosely) states that work will expand to fill the available time.  I think the two, Hofstadter’s principle and Parkinson’s law, march hand in hand, and together they have a significant impact on the way in which we work.  The way in which I work.

And so I find myself engaged in constant combat, against the self referential nature of Hofstadter’s principle, and against the expansionist principle of Parkinson’s law.  My daily hope is that awareness of Hofstadter and Parkinson will be part of my toolkit in, if not defeating them, at least moderating their effects.

the truth I find in an old story

•20 November 2017 • 1 Comment

In Zen Flesh, Zen Bones, story 14 of 101 Zen Stories, Paul Rips relates the story of an old monk and a young monk going down a road during a storm. They come across a well dressed young woman unable to cross a muddy intersection. The old monk picks up the young woman and carries her across to the dry road on the other side of the intersection. The monks continue down the road until the young monk, troubled by what had just happened, reminds the old monk that monks in their order do not touch women and how could he break the rules so blithely. The old monk then says, ‘I left the girl there. Are you still carrying her?’

This is one of my favourite (very) short stories and one that I come back to time and again. Because I know that I have a tendency to carry things beyond when they can reasonably be carried. For instance, I am a maker of lists, and some of the items on my lists have acquired tenure, they’ve been on my lists so long. They have acquired almost an untouchability, saying to me, how dare you try and remove me from your list. I have been here so long that I have become an indelible, unremoveable item on your list.

This comes back to the previous entry in these musings of mine, of how to get stuck into the large things. The comment made by Jacey Bedford, friend and colleague, is the old adage that the only way to eat the elephant is one bite at a time. (And apologies to elephants, who have been in the news recently for much less worthy reasons.)

It’s an interesting thing, this comfort we develop sometimes in carrying the same thing for a long time. For such a long time indeed that the act of carrying itself becomes almost a comfort. That the thing we are carrying becomes a comfort blanket, a favourite stuffed bear, and this can make it hard to set it down on the side of the road and walk away.

But this is a common thread in Zen, that we are sometimes, often, the source of our own discomfort, because of our own tendency to carry the source of our discomfort with us. For me, I am beginning to think that one among the many things I carry is this comfort that arises from procrastination. That the fact that something remains on my list is somehow a good thing, a comfortable thing.

And so now, I bid adieu and go back to the list, the new things and the old things, and we will address one of each this evening. And perhaps, if I am strong and able, I will carry on with so attacking my list, the new things and the old things. It is an impossible dream that someday the list is empty, but at this point my sincere wish is that I can look on my list, and all of the things staring back at me are new untenured things.

time leaks

•18 November 2017 • 3 Comments

In poker terms, a leak is an aspect of one’s play that leads to less than optimal results in the game.  Always playing one’s favorite hand is a classic example of a leak, or playing certain hands in exactly the same way.

This idea of the leak is a much broader idea, somewhere between the poker leak and the more literal definition of leak, and one that I spend a lot of time thinking about.  This is the time leak.

I sometimes get to the end of the day, I look back and I wonder, where did the day go.  Where did my time go today.  During the work week, I spend a lot of time in meetings, and this is just part of the role I currently have.  Some meetings are reasonable or valuable uses of time, though it must be said that this is not true of all meetings.

This I suppose is a variant of Parkinson’s Law where meetings expand to fill the scheduled time.   I am partially responsible for this, as some of the meetings are mine, though not entirely.

But other days, and these other days often includes weekends, I don’t have the excuse of meetings.  I don’t have the time I spend teaching, which is often the high point of my working day.

I wake up in the mornings and I have a clear list of what I want to accomplish over the course of the day.  It is always an ambitious list, and I sometimes make good progress against my list.  But not always.

But there is something I have come to recognize how much I struggle with.  There are some large projects which have been on the list for too long, and I am not good at making day by day progress on these large projects.

And this is now my task, to segment these large projects and to develop the habit of clearing the daily list.

Habits are hard to form. They require cultivation and curation, and it is very easy to begin a project like this tomorrow.  There is an old quote, I’m not sure of the original source, that tomorrow is the greatest labor saving device known to humankind.  It’s not true, but it is always easy to believe it today.

So I need to become something of a plumber, seeking out and sealing the time leaks in my own internal plumbing.  We’ll see how it goes.

persuading the world that mathematics can be fun 2: cohomology and local triviality

•5 November 2017 • 1 Comment

I’m feeling particularly brave today, and with this level of bravery comes a degree of foolhardiness that strikes me as reckless.  But I have set myself a quest, and part of that quest is to bring mathematical ideas out into the light.

So let’s do this.   Consider an impossible object.  The particular I want to consider is the Penrose triangle or impossible tribar, thought the same sort of discussion may well hold for some of the other objects on the list.

If we restrict our attention to just one of the corners of the Penrose triangle, we see nothing out of the ordinary.  Just the corner of a triangular object that might have been built out of pieces of wood or steel.  Not much to see here.

But it is when we look at the Penrose triangle as a whole, not one small piece at a time, that things go terribly, horribly wrong.  It is not possible to build a Penrose triangle in normal space, regardless of the materials used.

Lots of people find the Penrose triangle and other similar impossible objects interesting for all sorts of reasons.  For me, there is one particular aspect that I find interesting, one among many, and this is that the Penrose triangle illustrates a phenomenon that mathematicians find fascinating.

This is the phenomenon of getting something interesting globally, about an object as a whole, even when it’s not all that interesting locally, in its small pieces.  The how of getting this something interesting is where the mathemagic happens and so I don’t want to go into detail here.

But I feel that the general idea of taking two objects that are made up of the same looking small pieces, but being able to distinguish one from the other, is I think one of the great ideas in mathematics.

And it’s a remarkable slippery and subtle idea.  To convince you of this, let me end with an example.

Take a line, an ordinary straight line.  If we look at very small pieces, say intervals of length ε (a symbol that mathematicians use to refer to the very very very small), then we can build up the line by overlapping lots (and by lots, here we need to mean infinitely many) of these small intervals.

If on the other hand we look at a circle of radius 1 000 000 000, we can also build the circle by overlapping lots (though by lots, here we mean finitely many even though the number needed will be huge) of these small intervals.

The line and the circle are very different, and yet we can build them out of the same small pieces, by using enough of these small pieces.

The line and the circle are very different; we can keep going around and around and around a circle, something we can’t do for a line.  We keep going the same direction on a circle and we keep passing the same point, which we can’t do on a line.

So, different objects, built from the same small pieces.  Or impossible objects, built from possible small pieces.  Both reflections of the same basic idea, that even when things looks simple on the small scale, they get interesting when viewed on the large scale.