Wednesday, January 30, 2008

Meditation session

The Buddhist group wasn't nearly as scary as I thought they might be. Their leader was - there's no other word for it - ditzy, and talked well past the limit of the audience's attention span, but no attempts were made to brainwash the participants (unless they've just purged all memory of it from my mind...)

The meditation technique was extremely simple, just a slight elaboration on "breathe slowly and pretend stuff doesn't exist". Currently I'm trying to work weightlifting into my daily schedule, but, once I've made space for that, I'll definitely start trying to fit a little meditation in.

The tea-and-biscuits session afterwards was interesting. Having done my background reading, I was sensitised enough to pick up on a few slightly worrying concepts, but nothing worse than the average Christian group spouts on a regular basis. My concerns were comfortably defused by the comparative openness of the person I was chatting to (the group leader/teacher). By contrast with my last attempt to study meditation, she didn't try to conceal the religious beliefs colouring her thoughts.

I don't find the Buddhist cosmology any more realistic than those of other religions, but I don't find this particular tradition any more scary than most of the other religions out there. As such, I'll keep going to the meditation group for a bit longer.

Incidentally, I had one of those perfect Kodak moments today. During lunch with my friendly neighbourhood evangelists, one made a comment about the importance of scripture. That gave me the ideal opening to reach into my bag and pull out the Koran, the Bhagavadgita and a collection of Buddhist scripture (which I happened to have on me - no really!), with a comment along the lines of "which scripture did you have in mind?".
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Tuesday, January 29, 2008

Curse these cults!

I'm feeling a bit irritated. Yeah, I know that's not exactly a rarity, but at least the reason is interesting.

A Buddhist centre down the road from me is running a meditation class. To me, this sounded great - I've always wanted to learn meditation - and I've been gleefully anticipating the taster session for weeks.

This morning, though, I bothered to look up the group that's doing the lessons. It turns out that the New Kadampa Tradition is to Buddhism what ISKCON (aka Hare Krishna) is to Hinduism. It's a semi-cultish movement based loosely on the original tradition, nothing more.

Why do these groups seem to be so popular? Why do the Hare Krishnas have the manpower to canvass the high street? What's with the sudden growth of these rather imitative little cults?

My conjecture is that the situation is analogous to the phenomenon of invasive species. These traditions typically have their roots in rather minor subtraditions of age-old practice, and grow up fighting their corner against equally sophisticated, often quite similar competitors.

But then the tradition gets exported from its native habitat, into the midst of a population that's never been exposed to it before. This population has no basis for comparing the new tradition with its rivals, and no culture of criticism to provide resistance to its spread. Like bulfrogs in Australia, the tradition grows and spreads. For some reason, there's normally a personality-cult aspect to this phenomenon, although that could just be selection bias on my part.

I like this way of thinking about the formation of religions. As with language, the analogy to biology is not precise - species have no feature corresponding to patois or to combined traditions. But it's easy to visualise, which helps me to spot further examples.

And, of course, there's one very obvious example. A strain of Judaism that was inspired by a charismatic personality, transmitted to a large population of non-Jews, and as a result quickly morphed into a new religion that had little in common with its ancestor.

What would have happened if Srila Prabhupada had been killed off when his movement was just becoming popular, murdered by a government intent on removing the disturbance he caused? What would the backlash have looked like, as his followers fought to spread their truth before it could be obliterated? What would the world have looked like three centuries later, after the Hare Krishnas had become the dominant religious force? What about five centuries later, when "heretical" works had started to be purged? What about two millennia later, when all the newspapers and books that would have placed ISKCON's beliefs in context had long ago rotted away?

What stories would be told of his life? Would we think of him as a Messiah?
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Saturday, January 26, 2008

More ways not to communicate

So I'm in Oxford to meet up with a few friends from work, who are for some explicable reason meeting up a couple of towns over from our actual office. Should be a good evening - drinks at a pub, followed by a nice Lebanese restaurant.

Unfortunately, I get to Oxford before realising I was supposed to look up directions for the pub. Oops. I ask around, but no-one round the train station has ever heard of it.

Brainwave! I've got my mobile with me. I only have the one phone number, it being a new phone, but surely that should be good enough?

"We're sorry, but the person you called is not available right now. Please leave a message after the tone..." Dammit.

Ah, but hold on a sec, this new phone has 3G internet, doesn't it? The invite included the pub's website, so I can look up directions!

"Error: can't connect to network. Have a nice day..." Bugger.

As I write this, I'm sitting in a Coffee Republic, having walked halfway across Oxford trying to find somewhere with wifi. And I've just discovered that the pub is actually two blocks over from where I'm sitting.

Can we please just go back to the days of maps?
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Friday, January 18, 2008

A (stereo)typical conversation

Evangelical: "You know what really convinces me that the Bible is true? Its coherency. So many books, so many stories, and there's no contradictions anywhere."

Me, reaching for a copy of the Bible: "OK, what about this one between ***flickflickflick*** Matthew 2:14 and ***flickflick*** Luke 2:22. One states that the family ran straight off to Egypt and stayed there until Herod had copped it. The other states that, immediately after the "time of purification" (40 days, as per Leviticus), the family went to Jerusalem. FYI, Jerusalem and Egypt are on opposite sides of Bethlehem. How is this not a contradiction?

Evangelist: "Huh, I never noticed that. Did you get that off one of your atheist websites?"

Me: "No, I got it about five minutes into a personal attempt to see how the various Gospel stories line up. It's not exactly a subtle error."

Evangelist: "Mind if I get back to you on this?"

Moral of the story: It's very easy to be sure you're right about something, simply because you haven't solicited enough critical scrutiny. I have no grounds to be smug here - I know I've done the same thing in the past. All I can say is that I'm aware of the problem and do my best to counter it.
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Wednesday, January 02, 2008

Abstract algebra

In our last post, our intrepidly heroic mathematicians were hunting around for a new tool with which to attack some classic conundrums of geometry. The tool they eventually settled on is known as abstract algebra. This post will attempt to summarise it very briefly.

Compared to the 100 pages of lecture notes I'm working from, anyway.

Abstract algebra

The purpose of abstract algebra is to study and categorise various kinds of mathematical structure that crop up with depressing regularity. What do I mean by "structure"? Well, let's take a look at a really simple example: a group.

A group is formed of two components. Firstly, you have a big bag of thingies. Any thingies, it really doesn't matter that much. We'll refer to the bag as X, just to save my having to use the word "thingy" too much.

Secondly, you have a rule that combines two thingies to give a third. We'll refer to the bag as #. The rule must have certain "nice" properties:

1) Closure - the product of two thingies is a thingy. Formally, if x and y are in X then so is x#y.
2) Associativity - if you apply the rule twice to merge three thingies into one, it mustn't matter which two order you do the merging in. Formally, (x#y)#z = x#(y#z).
3) Identity - there is a thingy that can be ignored when applying the rule. Formally, there exists a single element e of X (the identity element) such that x#e = e#x = x for all x in X.
4) Inverse - for any thingy, there must be another thingy such that, when they're merged, you get the identity element. Formally, for any x in X, there exists an element y such that x#y = e.

Got that? No? Well that's not a problem, because I'm going to give you a concrete example.

The set will be as follows: {..., -3, -2, -1, 0, 1, 2, 3, ...}
The rule will be as follows: +

So 1+1=2, 2+6=8, etc. Looking familiar now?

The group that we've just described is the set of integers (whole numbers) under addition. It obeys all the four rules:

1) Add two integers and you get a third integer
2) (a+b)+c = a+(b+c)
3) a+0 = 0+a = a
4) If x is an integer then -x is also an integer

There are lots and lots of other examples:

A) The set of integers under multiplication
B) The set of reflections and rotations of a shape under composition (aka the dihedral group)
C) The set of hours on a 12-hour clock under addition (aka the cyclic group of order 12)

Example A is pretty straightforward - proof that it's a group is left to the interested reader.

Example B is more interesting. Consider, for example, the dihedral group of a triangle. Imagine the triangle is a piece cut out of a sheet of paper - you can flip it over or rotate it around in any way, as long as it still fits in the hole. Why not get yourself a real sheet of paper and try it?

Start off with one point of the triangle sticking directly upwards. You'll quickly find that the valid operations are as follows:

a) Keep it as it was
b) Rotate it 120° left
c) Rotate it 120° right
d) Flip it in the top/bottom line of symmetry
e) Flip it in the top-left/bottom-right line of symmetry
f) Flip it in the top-right/bottom-left line of symmetry

So our set is {a,b,c,d,e,f}. Our group rule is composition, which means doing two of these operations one after another. So for example, b#c = a, and d#e = b. Try it!

Is this a group? It quite clearly has closure, identity and inverses. It turns out to also have associativity, although this is less obvious.

One interesting point to notice is that the operations {a,b,c} form a group in their own right, even without including d, e and f. This subgroup consists of all the ways that a triangle can be rotated. It is called the cyclic group of order 3.

Example C, the clock, is fairly basic - most of the time it behaves just like the integers under addition. Just remember that 4 hours past 10:00 is actually 2:00, and you'll have understood most of what's going on here.

This group also has subgroups. For example, consider what happens if you're only allowed to move the clock on in four-hour intervals. The times you can reach are 0:00, 4:00 and 8:00 (I'm writing 12:00 as 0:00 here - either way, it's the identity element).

This subgroup is in fact the cyclic group of order 3 - so the dihedral group of order 3 and the cyclic group of order 12 both share it as a subgroup, despite having very different origins and properties.


There's one last group that I'm going to describe, because it will be particularly relevant later on. That is the "permutation group".

A permutation is just a rearrangement or shuffle. For example, we could reorder the list "1 2 3 4 5" to "2 5 3 4 1". Call this permutation p.

How does p behave? Well it replaces each 1 with a 2, each 2 with a 5, each 3 with a 3, each 4 with a 4, and each 5 with a 1. Mathematicians have a very concise way of describing this, called cycle notation. A cycle is just a list of elements [a,b,c,...,z], so that the permutation takes a to b, b to c, c to d, z to a, and so on.

In our example permutation p, there are two very obvious cycles. p takes 3 to 3, so [3] is a cycle of p. Similarly, [4] is a cycle of p. p also takes 1 to 2, 2 to 5 and 5 to 1, so [125] is the final cycle. All our knowledge about p can be conveyed by writing it as [125][3][4].

It's easy to note that these permutations form a group - if you do one permutation followed by another, the result is a third permutation. All permutations can be undone, and there is an identity permutation [1][2][3][4][5]. Associativity is slightly less easy, but not hard, to show.

Permutation groups are a particularly important example of finite groups - groups with only a finite number of members. For example, cyclic groups and dihedral groups are finite, but the integers are not finite. The group of all permutations of n objects is called the symmetric group of order n, written as Sn. For example, the group of all possible shuffles of a set of cards is S52.

These symmetric groups are important because it can be proven that any finite group is a subgroup of some symmetric group. This will become relevant later.

This series of posts has gotten far longer than I intended. I'll take a short break before the next post.
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Galois Theory in brief*

* Disclaimer: may not actually be brief.

Galois Theory is a really elegant area of mathematics. It relates two different topics - abstract algebra and geometry - to produce results that are greater than the sum of the parts.


Galois Theory has possibly the most romantic history of any mathematics ever. It was originated by Evariste Galois, a student in France during the 19th century. He was a firebrand of a revolutionary, and kept getting kicked out of universities as a result. Eventually, it seems, someone in government decided that he was too much trouble to have around.

So they set up a honeypot Before Evariste knew where he was, he had been trapped into a pistol duel over a woman (whose identity we don't know). His opponent was one of the greatest duelists in Paris. Galois knew he didn't stand a chance.

The last night of his life, he spent scribbling down diverse thoughts about mathematics. In the margins, he scrawled comments like "I don't have the time! I don't have the time!". The following morning, he went out to fight, and was shot down and left to die. His notes were passed on to a maths professor.

After languishing in various inboxes, the papers reached someone who could see them for the genius they were, and Galois became one of the great heroes of mathematics. It turned out that he'd cracked a set of problems that had been around since the ancient Greeks.


Geometry is the study of shapes - triangles, hexagons, spheres, etc. This subject naturally gives rise to a number of interesting questions, one of which involves the concept of constructibility. A shape is constructible if it can be drawn using only an unmarked ruler ("straightedge") and a compass.

For example, the triangle can be constructed by the following procedure:

Et voila, one beautiful equilateral triangle. So a three-sided shape is constructible.

A four-sided shape turns out to be very easy. It turns out to be possible to bisect any angle*, so just take a straight line (technically an angle of 180°) and bisect it. That gives you two lines at 90° to each other - a right angle. Draw two more perpendicular lines and you've got yourself a square.

Well, a rectangle actually. But the idea's the important thing :)

So you can draw a triangle and you can draw a square. Can you draw a pentagon? The answer turns out to be "yes", but the procedure is so intricate and icky that I'm not even going to try to explain it. By this time, mathematicians were coming to realise that a new approach was needed.

That's where abstract algebra came in. To be continued...

* Procedure left as an exercise for the interested reader. Or google it. What do I care?
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I really like the skeptical atheist community

They raise moral questions that I actually have to think about before discussing.

If anyone's interested then, before clicking that link, I would strongly recommend reciting the Skeptic's Morality Mantra seven* times:

"Ick" is not a valid argument
"Ick" is not a valid argument
"Ick" is not a valid argument
"Ick" is not a valid argument
"Ick" is not a valid argument
"Ick" is not a valid argument
"Ick" is not a valid argument

* Because the regular septagon is the smallest regular polygon that you can't construct with a ruler and compass. Galois Theory kicks ass.
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