May Almighty illuminate our intellect and inspire us towards the righteous graph...

At the moment I'm reading The Origin of Wealth by Eric D. Beinhocker because Amazon kept bugging me with it every time I bought Taleb.

And those Amazon algorithms know their stuff: Beinhocker is like Taleb only more polite, less bombastic, and generally more interesting.

TOoW leaves out most of the epistemological stuff and concentrates on wealth. So far Beinhocker has gone over why neoclassical economics is nonsense (it treats the economy as a closed, non-dynamic system, which it isn't, and treats people as perfectly rational, which they aren't).

Beinhocker is working his way towards describing complex adaptive systems of the Murray Gell-Mann variety...

In the meantime he also writes about non-linear equations and deterministic chaos: like this equation here:

B_t+1 = r * B_t * ( 1 + B_t)

Where B is the value of something at time t, and r is some other number.

When r is set to 1:


Now if you set r to 2:




Now if you set r to 3.3:


Now if you set r to 4:


Which is, apparently, chaotic.

I had always thought that in mathematical terms chaos meant "randomness", but in fact the two are very separate ideas.

A system is chaotic if it:

1. Is sensitive to initial conditions,
2. Is topologically mixed, and
3. Has dense periodic orbits.

Now I understand the first of those points, but not the second or third.

More reading to do methinks...

Maths and Melancholy

For a while now I've been meaning to write a very long post about maths, science, religion, and education.

The fact that I never finished the essay is testament to the fact that education is an immensely complex issue, and invites ignorant and uneducated opinion.

Take this appallingly titled article by Simon Jenkins: Maths? I breakfasted on quadratic equations, but it was a waste of time. Right, OK. Where to begin?

This all seems to be in response to something called The Reform Report, which has been compiled from a think tank called Reform.

Anyway let's look at what Jenkins says:

"In the age of computers, maths beyond simple and applied arithmetic is needed only by specialists. Ramming it down pupils' throats in case they may one day need it is like making us all know how to recalibrate a carburettor on the offchance that we might become racing drivers. Maths is a "skill to a purpose", and we would should ponder the purpose before overselling the skill."

Riiiight. So a journalist thinks that in the age of computers complex maths is needed only by specialists.

If economic prosperity is still considered a Good Thing, then surely preparing students for high-paying and rewarding roles in finance, economics, engineering, business, and science by promoting maths is a positive step. Anyway, let us ponder:

"When Kenneth Baker invented the national curriculum in 1987, it never occurred to him to question its content. Science and maths lobbied hard and captured the core, alongside only English. Not just history and geography, but economics, health, psychology, citizenship, politics and law - with far better claims to vocational utility - were elbowed aside."

All of those subjects have a strong claim to vocational utility. But there is a distinction between vocational utility and simple utility.

Learning psychology is fair enough: but without knowledge of statistics how are you to interpret pschological studies? Learning economics is good: but a central part of economic modelling relies on a knowledge of mathematics.

Maths is a subject that ensures all doors into future careers are kept open. Liberal arts still offer enormous choice but you are still locked out of some career paths.

Anyway as Ben Goldacre points out, there is some questionable use of maths in the Reform report itself.

I feel much more comfortable with the third way: no more conflict between arts and science and engineering, just an understanding that well-rounded people should be versed in as many subjects as possible.

Whither Carnot Efficiency?

An interesting story about a new method of generating solar power from the inventor of the super-soaker. The article claims efficiencies of 60 % are possible.

As some of the comments point out this idea might not be feasible. The Carnot efficiency of a heat engine is given by:

efficiency = 1-(T COLD/T HOT)

(with absolute temperatures used)

The article suggests temperatures as high as 600 degrees centigrade. So (assuming T COLD is room temperature):

1 - ((273+25)/(600+273)) = 0.66.

Giving a theoretical efficiency of 66%.

Of course it's possible there is some error in my understanding of the article and/or theory.

However the endoreversible process is a slightly more accurate method of measuring the efficiency of a heat engine (at least according to the Wikipedia article), which is given by:

efficiency = 1 - (T COLD/T HOT)^0.5.

So:

efficiency = 1 - ((273+25)/(600+273))^0.5 = 0.41

Giving a theoretical efficiency of 41%, rather less than as advertised.