Saturday, 27 February 2010

More Random Thoughts

I spent most of Thursday journeying to and from Uckfield to attend the East Sussex SACRE meeting on behalf of Hastings Humanists. Since I have a Senior Railcard and a Buspass this was not expensive, just time-consuming. When I returned, and all next day, I had a headache. Whether this was due to bumping about on the bus, or waiting for it in the cold and wet, or some other cause I'm not sure, but at least it seems to have cleared up today. At any rate it stopped me going to the chess club on Friday evening.

While in Uckfield I chanced to go into a Health Food shop and bought a jar of Barley Cup as a possible substitute for drinkng too much Coffee. It doesn't have any distinctive taste that I can detect, just a smooth texture. I did try flavouring it with some Malt Extract, bought at the same shop, but Honey would probably be better. Since I arrived in plenty of time for the meeting I looked around to see what cafes were available and ended up in a Poppins restaurant, which provided a nice lesagna with baked potato and salad.

I'm still working on the knight's tours book. I had hoped to get it finished for my 70 th birthday, but there is still a lot to be done. At present I'm on the chapter dealing with tours on oblong boards. I completely rechecked the tours on the 3x9 board, finding 146 as reported on the KTN website back in year 2000, although there is a minor misprint there, the number of {1,1} tours, with ends a diagonal step apart, is 28 not 29. The next section to check is on the 4xn boards, where I did some work trying to generate recursion relations for the numbers of half-tours, which has never yet been reported on the KTN site.

Saturday, 20 February 2010

Random Thoughts

I played another couple of chess games on Friday evening, at a slow rate without clocks, and won both of them against a player who seemed quite strong, so perhaps I'm getting back into the right frame of mind. One ended in a knight checkmate, the other in a queen against rook superiority. The more rapid play games which we played on previous weeks require one to react much more instinctively, rather than contemplate each move carefully.

Why are there no longer any malt-flavoured cereals being produced? I used to like malted shreddies when they were produced by Rowntrees, but as soon as Nescafe took them over they changed the recipe so that the malt taste was far less. I complained at the time, but got no helpful response. Now they have removed the malt altogether! This seems to be part of their policy of claiming that everything is "whole grain".

My article on "Howard Jacobson and the Temple of Darwin" appeared on the new HumanistLife website on my 70th birthday, 8th February, but has not attracted any comments. Perhaps this means that it is perfect as it is and doesn't need any further comments? Probably not! I'm glad to see that more articles are appearing with a greater frequency now. There are strong disagreements between Humanists on a number of issues, for instance the assisted dying question, and whether the burka should be banned. These have attracted the most comments.

Thursday, 11 February 2010

The Religion of Infinity

I happened to notice that the Horizon programme on BBC2 TV this evening was about "Infinity and Beyond". Hoping to learn of some new research I tuned in but was sorely disappointed. The programme was aimed at about the intellectual level of a five-year-old. The commentary was given by an Aleister Crowley lookalike who was filmed in murky black and white endlessly walking up stairs and reappearing again, Escher-like, on the bottom landing, and making pompous and portentous-sounding statements and poses. Half-way through he even renamed Georg Cantor "Gregor".

All the usual elementary illustrations of infinity were included, such as Hilbert's Hotel, Cantor's diagonal argument and monkeys typing Shakespeare, followed by speculation about whether the universe might be infinite. There was one chap who didn't believe in infinity, but all he could say was that there was a largest number, but no-one knows what it is, and it is followed by zero.

My argument for finitism runs as follows: It is true that we can generate symbols for numbers in a systematic manner using the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and the positional convention, but this does not mean that the set of all such symbols 'exists' already until we actually construct it. Nor does the mere construction of a symbol, such as n+1 for a number imply that the number 'exists' in this sense.

The mathematical term 'finite' applies to sets of things and the numbers of things in those sets: a set is said to be finite if it has the sensible property that it cannot be placed in one-to-one correspondence with a part of itself; a number is finite if it describes the size of a finite set. The Finitist maintains that all sets, and therefore all numbers, are in fact finite.

In order to introduce infinity into mathematics it is necessary to postulate that it exists, or to assume some other axiom that implies this, for instance Peano's axiom that every number has an immediate successor. Further the properties of infinities depend on the axioms that are chosen. For example Paul Cohen proved that, under the usual axioms for arithmetic, it is impossible to say whether there is an infinity between that of the integers and the real numbers.

On the other hand the properties of finite sets and numbers are a matter of physical fact, at least within the 'realisable' realm, where they can be applied to material objects. Statements about 'all' numbers, such as Goldbach's conjecture, may not be realisable.

What do we mean by saying that something 'really exists'? The simplest definition is that something exists if it is material, that is if it has measurable mass. On this basis it might be argued that 'ideas' like numbers do not exist since they are immaterial. But are they? Ideas exist in the minds of people, and presumably therefore they exist materially in the form of electrical or chemical energy in the brains of those who think about them. By Einstein's equation, E = mc², anything that has energy has corresponding mass. So if mathematician's brains really contained the infinite set of all whole numbers they would have infinite mass and implode into a black hole!

By a similar argument, the universe is finite in mass, since if it were infinite there would be infinite gravitational force at every point in the universe (a version of Olbers' paradox).

Even if we discount the argument by weight, so long as we accept that ideas exist in the form of electrical or chemical configurations in the brains of thinkers, there can still only be a finite number of ideas in existence, certainly of human ideas, held by human beings, because there is only a very finite number of human beings extant, and their brains contain only finite numbers of neurons.

EDIT: In contrast to the puerile "Horizon", Melvyn Bragg's "In Our Time" on Radio 4 this morning was an adult-level programme about "unintended consequences" in mathematics, on how ideas developed purely out of mathematical interest later prove to have practical consequences: such as prime number theory in cryptography, complex numbers in alternating electrics, and non-euclidean geometry in relativity. Why does TV have to dumb-down, while Radio does ideas so well?

Friday, 5 February 2010

More Chess

More chess this Friday evening, a six-player all-play-all with twenty minutes on the clock. I was given a low grading of 80 and had 13 minutes to 7, or 15 minutes to 5, depending on opponents' gradings. This time I won two, though my opponents in those games were either distracted by the time handicap, or thought I needed a win. Both ended in a straightforward queen checkmate. I did deliberately try to play in a more attacking style compared with last time.

I've been feeling rather tired in the afternoons lately, and unable to rouse myself to get much done. Perhaps I need to keep more regular hours, or perhaps I will come out of hibernation when the weather warms up a bit.