Tuesday, February 26, 2008

Karl Pilkington on art

Karl Pilkington, as I might have mentioned before, is something of a hero of mine. Here, he visits the Hayward Gallery, and chats to some artists. His interview with one David Shrigley, in particular, is priceless. Shrigley tries to argue at one stage that an artist doesn't need to be able to draw well, in the same way that a guitarist in a band doesn't need to be able to play the guitar well.

Now, I'm no expert on music, but I reckon that Brian May, Jimi Hendrix, and the woolly-capped fellow from U2, can probably all play the guitar quite well, and that they had to learn their craft first, and then build the art upon the craft. A concept which seems to find little favour in modern art...

Gauging What's Real

I'm currently reading Gauging What's Real, by the philosopher of physics Richard Healey. This is an excellent treatise on the interpretation of classical and quantized gauge field theories. These theories are the prevailing theoretical paradigm employed in the standard model of particle physics. In fact, Healey's book forms the perfect counterpoint to my own, for whilst the balance of exposition in my own book is about 75% mathematical structures, 25% philosophy, the balance is somewhat reversed in Healey's text.

One of the criticisms I would make is that Healey's account of the mathematics of gauge field theory fails to employ the far simpler approach pioneered by Andrzej Derdzinski. The essence of Derdzinski's approach is that an interaction can be represented by an interaction bundle δ, a free particle field can be represented by a free-particle bundle σ, and the interacting particle field can be typically represented by the tensor product bundle δ ⊗ σ.

Nevetheless, Healey's book is much recommended.

Thursday, February 21, 2008

Monday's sunset

The major enviro-aesthetic event of the week was surely Monday's spectacular Wagnerian sunset. During the day, the sky was bereft of cloud and haze, and, from a West Berkshire perspective, aircraft contrails carved deep white furrows across a field of pure azure overhead. After the Sun slid beneath the rim of the horizon, a forboding golden glow bathed the entire Western hemisphere of the clear cold sky, silhouetting the stark dendritic outlines of the Wintering trees.

Saturday, February 16, 2008

Is it abnormal to be normal?

I think it's true to say that most of us, when we are young, dream of growing up to become statisticians. Sadly, very few of us chase that dream, and instead we grow up to become, say, premiership footballers or Grand Prix drivers. Nevertheless, there are a select few who persist, and who consequently spend their working lives looking at random variables, probability density functions, and cumulative distribution functions. Still, my beating heart.

Needless to say, statistics doesn't really do it for me. The problem with statistics, as a branch of mathematics, is that its value derives largely from its practical utility, not the intrinsic interest of its mathematical structures. Whilst statistics has vital application in all branches of science, finance and commerce, it is precisely this universality and utility which divest it of intrinsic interest. Statistics performs a service for other sciences, rather than being of value in its own right.

Nevertheless, any subject can be interesting once you tunnel sufficiently deeply, so consider the following well-known statistical scientific truism:

If a number of independent random variables combine in an additive fashion, then the collective result is a normal distribution. (i.e., a bell-shaped, Gaussian distribution).

If a number of independent random variables combine in a multiplicative fashion, then the collective result is a lognormal distribution. (i.e., a distribution whose logarithm is a normal distribution).

Now, I'm not sure whether this common wisdom is really correct. As far as I can make out, the first clause in a simplified statement of the central limit theorem. This asserts that the sum of a collection of n independent and identically distributed random variables will converge to a normal distribution as n tends to infinity. The central limit theorem has one particularly important implication for measurement science and the estimation of measurement error:

Suppose that the variable to be measured has an arbitrary distribution, and suppose that one measures the value of the variable by taking a collection of sample measurements, each sample consisting of n measurements; if one calculates the mean value from each sample, then the distribution of sample means will converge to a normal distribution, centred upon the true value of the measured variable, as the size of the sample, n, tends to infinity. Hence, whatever the distribution of the variable being measured, whether it is normal or not, the collection of sample means will have a normal distribution. This is crucial, because it enables one to estimate the 95% or 99% confidence interval, (the uncertainty or measurement error), in a measurement estimate, using the simple formulae or tables of values associated with the normal distribution.

So far, so good. But in general, will a sum of independent random variables give a normal distribution? The central limit theorem doesn't entail that it will, for the central limit theorem requires the contributing random variables to be identically distributed. So what happens when a sum of independent random variables with different distributions is taken?

The second assertion, that a lognormal distribution results from random variables combining in a multiplicative fashion, also needs to be tightly qualified. The assertion is largely based upon the following property of the logarithmic function:

log (A x B) = log A + log B

Thus, given a collection of independent random variables with identical distributions, their product will possess a distribution well-approximated by the logarithm of a normal distribution (applying the central limit theorem again). However, what if the collection of variables are not identically distributed?

Moreover, it has been noted that if the number of steps in a multiplicative process is itself subject to a statistical distribution, then the result will not necessarily be a lognormal distribution. For example, if the number of steps in a multiplicative process is subject to a geometric distribution (a discrete version of the exponential distribution), then whilst the body of the distribution will be lognormal, the tails will exhibit power law behaviour. That's interesting.

Saturday, February 09, 2008

Sheila law

It is time to face up to the fact that there are large communities of young women in this country who respect not the conventional mores and manners of English society, but are beholden instead to a set of principles called Sheila law. Named after the colloquial term for an Australian woman, Sheila law attributes highest-value to a life of foul-mouthed, heavy-drinking, promiscuous behaviour. Those who subscribe to Sheila law regard the whole world as governed by Sheila law, and reject other standards of behaviour.

Sheila law is especially insidious because it is intolerant to all forms of criticism. Any expression of reproach directed towards those who live under Sheila law is met with aggression, hysteria and abuse.

One moral code for everybody would be something of a danger, and it is time for a constructive accommodation with some aspects of Sheila law. It seems inevitable that at least some form of Sheila law will be incorporated permanently into English mores and morality, and officially recognising this will improve relations with young modern females.

We need to look at Sheila law with a clear eye and not imagine, either, that we know exactly what we mean by Sheila law and just associate it with Australian women, or whatever. I do not think we should instantly spring to the conclusion that the whole of that world of modern female morality and practice is somehow monstrously incompatible with human dignity and respect, just because it doesn't immediately fit with how we understand it.

Monday, February 04, 2008

Bus lanes

Any call to abolish bus lanes is, of course, somewhat passe, but there is still an ongoing need to question the logic behind devoting an entire carriageway of the public highway to an apunctual, inefficiently engineered, pollution-generating form of public transport.

In those places where two lanes previously existed, the introduction of a bus lane approximately halves the capacity of the road. This approximately doubles the amount of congestion and, perforce, doubles the amount of pollution.

Those who attempt to justify the imposition of bus lanes, such as Transport for London, claim that bus lanes 'encourage' people to travel by bus. In fact it is claimed, without any reference to supporting documentation, that bus usage in London is up by 40% in the last 5 years. However, even if this were true, and even if this were attributable to the imposition of bus lanes, it doesn't follow that congestion will decrease as a consequence. If the capacity of a road has been halved, then approximately half of all road-users will need to travel by bus to restore the level of congestion to even its pre-existing level. And there simply aren't a sufficient number of buses for that to happen.

So the net result, then, is more congestion, more pollution, more frustration, and fractionally shorter bus journeys. Does this represent a net gain? I'm not sure that it does.