Now that, it seems, Spring is at last here, I've started waking up earlier, or at least not going back to sleep. So there is something to the idea that one tends to go into hibernation in the winter. I was up at six this morning and went for a swim at the baths between seven and eight. Whether I will be able to keep this up regularly remains to be seen.
One of the sounds that now greets me on waking is that of the blackbird, who seems to start singing before the noisy gulls get going. I've always liked the song of the blackbirds. They seem to be speaking to me personally, they often sound as if they are saying "What'ya doin' Georgie". Not that I like any one else being that familiar. My father was also named George, so I got the diminutive version.
It looks as though the British Chess Variants Society will close down this year, since John Beasley is retiring and no replacement has come forward to act as secretary and editor. Also Peter Fayers will not be able to carry on as treasurer and publishing manager. I will probably try to keep the magazine Variant Chess going in some form on the web, but not produce a printed version.
I've been looking into the costs of registering suitable internet addresses, to reorganise my web content, including the magazine. Because my "ntlworld" site was closed I've had to cram all my stuff onto the "mayhematics" site, which was not my original plan. There are also moves afoot to form some sort of International Variant Chess Society. This would be a welcome development, but needs a new generation of internet-savvy enthusdiasts to develop it.
Wednesday, 17 March 2010
Saturday, 13 March 2010
Chess and Mathematics
Last Monday I took part in a chess match, playing on behalf of the Hastings club against a team of four from Kent. I was on the third board. The time allowance was quite generous, which suits my slow play, and I managed a draw. Most of the time I was a knight down, but with the advantage of a passed pawn, so it was a matter of trying to get the pawn promoted. There were a lot of interesting tactical situations that arose the game. The opposing team won overall by 2.5 to 1.5.
On Wednesday I received a letter from Professor Donald E. Knuth of Stanford University. We have previously corresponded on knight's tours, but I hadn't had a letter from him for several years. He is putting together a book of his Selected Papers on Fun and Games which will include several chapters on tours, among much else. I had to look up what "potrzebie" was all about. It seems it's a Polish word adopted by MAD Magazine as a running joke back in the 1960s.
The topic Prof Knuth was asking about concerned the results obtained by Robin H. Merson on non-intersecting knight's paths. As a result I have now placed PDF versions of Robin Merson's two main letters to me, dealing with open and closed paths, on the knight's tours page of my mayhematics website. They haven't scanned very clearly; for instance the background graph lines have not come out, but that's the best I can do at present.
Prof Knuth also likes to collect the middle names of everyone whose work he cites, but I was unable to locate what Robin Merson's "H" stood for. He worked for the Royal Aircraft Establishment at Farnborough on the use of satellites for mapping the Earth, among other activities.
On Wednesday I received a letter from Professor Donald E. Knuth of Stanford University. We have previously corresponded on knight's tours, but I hadn't had a letter from him for several years. He is putting together a book of his Selected Papers on Fun and Games which will include several chapters on tours, among much else. I had to look up what "potrzebie" was all about. It seems it's a Polish word adopted by MAD Magazine as a running joke back in the 1960s.
The topic Prof Knuth was asking about concerned the results obtained by Robin H. Merson on non-intersecting knight's paths. As a result I have now placed PDF versions of Robin Merson's two main letters to me, dealing with open and closed paths, on the knight's tours page of my mayhematics website. They haven't scanned very clearly; for instance the background graph lines have not come out, but that's the best I can do at present.
Prof Knuth also likes to collect the middle names of everyone whose work he cites, but I was unable to locate what Robin Merson's "H" stood for. He worked for the Royal Aircraft Establishment at Farnborough on the use of satellites for mapping the Earth, among other activities.
Friday, 5 March 2010
Puzzle Addiction
Every week, though not every day, I buy a few newspapers, mainly for the puzzles rather than the news. On Friday for instance the Guardian has a sudoku and kakuro that are usually a bit harder than during the week. The Guardian on Saturday always has a good prize crossword, often by Araucaria, but I don't always buy it because it has far too many sections, on subjects such as sport, travel, finance, fashion and so on, that I'm not really interested in.
I also tend to get the Times once or twice a week, mainly for the crossword and the killer sudoku. However this week the Times has started to include a four-page puzzle section every day! This will mean that I have to avoid buying the Times in future, because once I start on the puzzles I have the compulsion to solve them all, and waste most of the day when I should be doing something productive. Why they have changed to this new scheme I don't know, I thought their previous policy was just the right balance.
I also tend to get the Times once or twice a week, mainly for the crossword and the killer sudoku. However this week the Times has started to include a four-page puzzle section every day! This will mean that I have to avoid buying the Times in future, because once I start on the puzzles I have the compulsion to solve them all, and waste most of the day when I should be doing something productive. Why they have changed to this new scheme I don't know, I thought their previous policy was just the right balance.
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.
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.
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?
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.
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.
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