Fractal Revelation

Lots of stuff written about how to understand the Bible, especially the Old Testament, is written quite badly. It (the stuff, not the OT) is unclear and uninteresting and often repeats the same obvious points. I think it is because they don't tend to realise that there are a lot of good analogies between the way that the Bible works or the way that preaching works and the way that a lot of the rest of life works. [Dale Ralph Davis seems to be a major exception to this rule because he applies everything so well and keeps it relevant.]

This is an attempt to explain what sometimes gets called "multiple horizons" (a bad title) in a way someone like me would understand easily...

Back when I was a maths geek, I used to like patterns called fractals. Here's a picture of one:

What is special about fractals is that you can zoom in on them as much as you like, and they still look the same. With that fractal, if you picked any of the little circle things and zoomed in on it, you'd get exactly the same pattern as you do with the big picture.

You get the same sort of idea in lots of other fields besides maths - sometimes in a piece of music, one little bit of it shows you something of the pattern for the whole. Or in a book, one scene or action represents the pattern of a much larger section.

Well, it's pretty much the same in the Bible. Often we get little episodes that have the same pattern as much bigger ones. For example, 1 Samuel 2, God has mercy on Hannah, who is sad because she can't have children. Hannah sees that as a pattern of how God has mercy on Israel, which is barren, by giving them a king, which is what Hannah is part of. But we can now see that that itself is part of a much bigger pattern - God having mercy on a spiritually barren world by giving us Jesus. It's the same pattern at lots of different levels, like a fractal.

Recent Economics

My knowledge of statistics is decent; my knowledge of economics is not. But even I (after a little reading) can tell that the sub-prime loans crisis was due to people forgetting that probabilities need to be independent for simple maths to work on them.

Here's an example:

Let's say the probability of Man Utd winning the Premiership is 1/3, the probability of me getting over 60% on my patristics exam is 1/2 and the probability of Arsenal finishing outside the top two of the Premiership is 1/4.

Now how I do in my exams is pretty much independent of what goes on in football, so I can say that the chance of Man Utd winning the Premiership and of me getting over 60% on my patristics exam is 1/3 x 1/2 = 1/6.

But the chance of Man Utd winning and Arsenal finishing outside the top two aren't independent. If one happens, the other is more likely to happen. So the probability of that isn't just 1/3 x 1/4 = 1/12. It's going to be more than that.

That's roughly what happened with the sub-prime mortgages. They took lots of events which actually were connected to each other, in this case the chances of poor people in America being unable to pay their mortgages, but just multiplied the probabilities when assessing the risks. If someone had done that in a GCSE maths class I was teaching, I'd have told them off. But when they're doing it with the global economy, they probably deserve to be sacked and sued until they can't get any kind of mortgage, especially because the people their incompetence hurts most will be the poor. It pretty much always is.

Here's a great rant:

As a result of America's mortgage crisis, we have learnt that, to play in banking's premier league, you need much more than a degree from Harvard Business School, the morals of an alley cat and an unbridled lust for riches. These attributes help, but the sine qua non of a seat at the top table is a willingness to suspend disbelief until junk loans to trailer-dwelling welfare claimants can be diced, sliced, spiced and resold as triple-A securities. This takes some doing, as the business - quite clearly - makes no sense.

It is like a restaurateur opening up cans of dog-meat, sprinkling it with herbs, presenting the mix as steak tartare, and charging £25 a portion for something that will poison most of his customers. The difference is, passing off chopped donkey for Aberdeen Angus would be illegal; the repackaging of toxic sub-prime mortgages as high-quality investments was not.

Game Theory and Confessionalisation

In one of my exams yesterday, I had prepared for (well, kinda) and expected a question on Calvin because there had been one on every single past paper. But there wasn't, so I ended up blagging about confessionalisation from the point of view of game theory. This is a tidied up and shortened version of what I said...

Confessionalisation was a big feature of the Reformation, after the start. In 1528 (for example), there were lots and lots of different groups all across Europe, and all believing different things. You could go to London or Amsterdam or Munich or Venice and find lots of people who believed lots of different things. By 1650, that wasn't possible in the same way any more. The number of beliefs held had greatly decreased, and each region was much more homogenous.

I argued that it was an inevitable consequence of the view that religion could be legislated, and that different states could have different beliefs - the policy known as cuius regio, eius religio (each region, its own religion). If we assume (as a simplification) that there is only one issue, and it can be represented by your position on a line, then we have the following picture.

Points A and B represent the religious positions of two rulers, who are not on friendly terms. Ruler A will therefore suspect anyone to the right of him of being a sympathiser with B - they all appear to him to be in the same direction, so he will persecute everyone to his right. Ruler B, likewise, will persecute everyone to his left. This will lead to a polarisation of the centre - if they remain in the centre, they will be attacked by both the right and the left; the pressure on them greatly reduces whichever direction they travel in. Therefore the centre will largely disappear.

The second stage is that of consolidation. In this, the fairly large number of early positions consolidate over a long period of time into fewer positions. This is largely due to the influences of education, particularly of ministers and rulers, and the production of literature. If one fairly homogenous group produces large quantities of literature which become available in other areas, or if they have training facilities with such a good reputation that they attract people not just from within the group but from similar groups, they are likely to win "converts" to their cause, and to grow at the expense of those other groups. And the larger they get, the more significant this effect is. This is largely what happened with Calvinism. Due largely to better publishing and writing, and better training facilities (as well as better systematic theology and God blessing them), Calvinism grew rapidly at the expense of most of the other Protestant groupings, to the extent that the only major Protestant groups left by 1650 were Calvinists, Lutherans and Anglicans.

Of course, I simplify hugely. But it seems to work fairly well for a first order model. Apparently, though, historians aren't overly fond of this sort of thing, and I haven't heard of many people applying even very simple game theory like this to history... Never mind, it was a fun sort of idea, and I'm still a physicist in my methodology.

Donkeys

While I was in Israel, one of the locals said something interesting. He said that many of the paths in use were created by donkeys, and had often been there for thousands of years. The reason he gave for this was that donkeys allegedly always take the easiest route from A to B, even over difficult terrain. That got me thinking.

It should be reasonably easy to set up a situation with a donkey and a mathematically well-defined surface, and to see whether it does indeed take something very close to the minimum energy route from A to B, and what sort of distance this holds over. Does it work as far as the eye can see? How does it cope with variable terrain types - e.g. mud, solid grass, rocks? Does it follow the sort of route that would be predicted by a proper mathematical minimum energy route, or does it follow a local energy minimization, or even just a picking a vaguely easy route for the next 20m that leads in the right sort of direction?

I'm sure there's a PhD project in there somewhere, for a mathematician who likes donkeys...

Is Hebrew Mathematical?

I'm learning Hebrew at the moment. And one thing that people say to me quite a bit is that I should appreciate Hebew, because it is a very mathematical language. And to be honest, I've never really understood that, until today.

I used to be quite good at maths, well - until my second year at Cambridge. Maths was great, because all I needed was to understand a few simple facts, and then everything else was obvious. But Hebrew didn't seem to be like that at all....

As far as I can tell, Hebrew has a very large number of odd and silly rules, but if you follow those rules, it tends to work.

Today I finally made the conceptual leap. Hebrew is like maths, as encountered by people who aren't very good at it.

Miscellany

Maths

Here's an interesting maths fallacy (and thanks to DH for pointing me to the page with it on).

Sci-Fi Stuff

I also had a nifty scientific idea this morning. I was watching Firefly (TV sci-fi series, they only ever made one series of it) yesterday. One of the odd details is that all the planets and moons are meant to be in the same stellar system, but they all look remarkably like Earth. Odd that.

Anyhow - I was wondering how one might go about getting enough light and heat from the Sun to make them warm enough for habitation, and I figured that putting a lens at the Lagrange Point could well do it. Then I realised that such a lens would need to be pretty huge - nearly planet-sized even, and so would effectively require you to demolish a planet / moon to make it.

Then I thought of an interesting way round - use a Fresnel Lens (like those magnifying sheets you can get). You'd still need a big lens, but it'd be essentially 2D rather than 3D, so would need far far far less material and at least be possible.

And finally...

It's my birthday! Woo!

Pascal's Wager

This is an old argument thought up by the French physicist, mathematcian and philosopher Blaise Pascal. I'd be interested to see people's thoughts on it.

It's not meant to be an argument that says Christianity is true - Pascal helped invent probability theory in maths; it's meant to be an argument about which way is the better bet, Christianity or atheism.

The Argument

The argument basically goes like this...

Lets assume there are only two possibilities for belief - Christianity and Atheism and that one and only one of them is correct. For the sake of argument, lets say that the probability of the Christians being correct is p, so the probability of the Atheists being correct is 1-p.Each belief has a group of people who follow it - the Christians and the Atheists.

If Atheism is correct, then both groups will end up exactly the same - pushing up the daisies. If Christianity is correct, then the Christians will end up very much better off than the Atheists. Lets call the amount the Christians will be better off than the Atheists in this scenario H.

So the Christians stand chance p of gaining H - their expected gain is pH.

The Atheists, by contrast, have no expected gain. Whether they are right or not, they do not end up better off than the Christians. So, said Pascal, if you're betting, it's much better to bet on being a Christian than an Atheist.

Criticisms

The usual criticisms I hear of Pascal's Wager go something along these lines:

1. Probability theory doesn't work on this.

2. Pascal only gets the results he does because he has massively oversimplified the situation. There are far more possibilities than just Christianity and Atheism. And surely leading a good life has to count for something, even if I don't believe in God.

3. It doesn't work, because Atheists get to make the most of their life now, whereas Christians are busy worrying about the next one.

My Analysis

I don't understand 1. Why shouldn't probability theory work on this?

2. really doesn't work either. Yes, you could introduce other ideas - Islam, Buddhism, etc. But it wouldn't change the fact that Christians end up better off than Atheists on average unless you start introducing silly religions that treat Atheists better than Christians in the afterlife, and give them a higher probability than Christianity. I don't see any of those religions around, and I don't think I've ever heard any arguments for their validity... Of course, you might decide that Islam (for example) comes out as an even better bet, but that doesn't change the fact that Christianity comes out better than Atheism

The third objection is better. Being a Christian does cost stuff in this life (but also, I think, makes this life a lot more livable). Lets say that being a Christian in this life costs an extra amount c compared to being an Atheist.

In that case, Christians would gain pH and lose c, so their expected gain would be pH - c. Atheists would still have zero gain.

So in order for Atheism to be a better bet than Christianity, c > pH. In other words, the cost of being a Christian would have to be bigger than the probability of Christianity being true multiplied by the gain Christians get if they are right in the afterlife.

In other words, to be an Atheist rather than a Christian, you are either very poor at betting or sure of at least one of the following options:

• that it would cost an immense amount to be a Christian, so great that even a decent chance of an eternity of paradise would not make up for it
• the chance of Christianity being true is vanishingly small (to which I would say that if so many otherwise sensible people believe it, it's not going to be that unlikely)
• the promised reward in Christianity is not that great.

So what do people think? Which is it?

the golden ratio, φ

The Da Vinci Code makes quite a bit of the so-called Golden Ratio, φ, which is mathematically equal to (√5+1)/2. [In the DVC, it is described as being 1.618, which is accurate to 4 significant figures, but not exactly.]

It comes up all over the place in nature, but not as often as claimed in The Da Vinci Code. Wikipedia's article on φ is pretty helpful on clearing up some of the fallacies about it. It doesn't mention the "sacred feminine" at all, and I've never come across references to φ in that context outside the Da Vinci Code.

But why should it matter?

The book of nature is written in the language of mathematics.
Galileo Galilei

As Galileo said, mathematics is a very profound and useful tool for describing the world we live in. One of the simple tools used in mathematics is the Quadratic Equation, an equation which looks like this:

ax² + bx + c = 0

where a, b and c are any number at all. The simplest non-trivial quadratic equations have a, b and c all as either 1 or -1. This gives us 8 possible equations.

x² + x + 1 = 0 (equation 1)
x² + x – 1 = 0 (equation 2)
x² – x + 1 = 0 (equation 3)
x² – x – 1 = 0 (equation 4)
-x² + x + 1 = 0 (equation 5)
-x² + x – 1 = 0 (equation 6)
-x² – x + 1 = 0 (equation 7)
-x² – x – 1 = 0 (equation 8)

Of these 8 equations, only four have real solutions (equations 2, 4, 5, 7). The solutions to those four equations are:

equations 2 and 7: -φ or φ – 1 (which is also 1/φ) equations 4 and 5: φ or 1-φ (which is also -1/φ)

Because φ is so heavily involved in the solutions to those equations, that means it has a few useful properties:

• If you square it, you add 1
• If you divide 1 by it, you take 1 off
• If you cut a line into two parts, one bigger than the other by a factor of φ, then the ratio of the total to the longer section will also be φ.

Those properties of the number mean that it does come up in nature and in geometry a fair bit, but not as much as claimed in the DVC.

Nothing mystical, just maths.