Oops. I had intended to alternate serious (you may substitute long-winded or boring) posts with lighter (read mindless, insubsantial) ones. Yet somehow I seem to have managed to follow up a long post on an esoteric piece of mathematical machinery with an equally long one on art history and ... yawn ... something or other. Hmm.
Anyway, here's a corrective.
I'm going to share a secret with any of my readers who happen to be men. Actually, I'm sharing it with all of you indiscriminately, but if you're not a man, it shan't be of much use to you.
In two words: Shaving soap. (In three words, 'use shaving soap'; but I can do it almost as well in two.) Try shaving with proper, old fashioned shaving soap.
I think that peoplle gave up using shaving soap because it takes slightly longer than modern gels and mousses. However, if you are at all used to shaving soap it may take less than one extra minute. And boy, is it ever worth it. You get a far closer shave. Far, far, far closer. There's just no comparison. Have you ever shaved against the grain just to get a closer shave? Well, if you shave with shaving soap you can get that finish by just going with the grain. And if you go against the grain too; well, wow!
You'll need a brush to whip up a nice rich lather on the surface of your soap. The purists recommend one with badger-hair bristles, but these are astonishingly expensive. (A couple of years ago a decent-sized badger hair brush was £40 in England.) Anyway, there's nothing like trying it out for yourself, so off you go. Run along and have shave.
But not at the same time - weren't you ever taught not to run with blades?
Friday 13 July 2007
Civilisation
Welcome back.
I'm about halfway through watching an outstanding BBC documentary series called Civilisation. It was written and narrated by Kenneth Clark, who, I recenetly found out, was the father of Alan Clark.
Clark is magnificent in the series. The cinematography itself would make this series worthwhile, but his narration is beyond praise. He is absolutely clear, always intelligent, and almost never patronising.
The series is also impressive for the coherence of the thoughts and arguments expressed in it. Some of his slightly more xenophobic comments may be offensive to those who are easily offended; but they should also serve to remind one that the ideas and the narrative are the result of a single man's thoughts. That alone is worth applauding in an era of collectively produced, almost corporate entertainment.
It is also astonishing that there are only two signs of age shown by this work, apart from the obvious signs of the times in background, such as cars. The first is the occasional comments, mentioned above, which do not conform to the bland opionlessness of modern art documentaries. Secondly, the complexity of the thoughts expressed is significantly greater than anything else I have ever seen on television. It is a reminder that television can be educational and intellectually stimulating without failing to be entertaining.
However, the greatest quality of this documentary is its intellectual honesty. The presenter motivates his series with questions, and proceeds to attempt to answer them. But he does so in a scupulously open and honest way. There is none of the disingenuity which makes so many television programs look patronising, or even like propoganda. Where the evidence supporting his point is ambiguous he lets us know, and then explains his opinions, rather than prsenting an unbalanced narrative.
So many articles are written, or documentaries made, by people whose minds are already made up. Their contributions to the public discourse are an attempt to sell their position, and they do so without any regard for this sort of intellectual honesty.
This quality has all but dissappeared today; from political discourse where all issues are reframed by each side in ridiculous black and white, downwards (or upwards, according to one's tastes) through so much of society. It reflects a mind-set which is truly rare even in academic circles. It shows a man who is prepared to make his arguments seem weaker, in order to make them more honest.
Of course, Kenneth Clark was an emminent and distinguished scholar at the time he made the series; and this quality may have been extremely rare at the time. So this quality may never have been widepread in mankind. Nonetheless, I was glad to have caught this glimpse of it.
I'm about halfway through watching an outstanding BBC documentary series called Civilisation. It was written and narrated by Kenneth Clark, who, I recenetly found out, was the father of Alan Clark.
Clark is magnificent in the series. The cinematography itself would make this series worthwhile, but his narration is beyond praise. He is absolutely clear, always intelligent, and almost never patronising.
The series is also impressive for the coherence of the thoughts and arguments expressed in it. Some of his slightly more xenophobic comments may be offensive to those who are easily offended; but they should also serve to remind one that the ideas and the narrative are the result of a single man's thoughts. That alone is worth applauding in an era of collectively produced, almost corporate entertainment.
It is also astonishing that there are only two signs of age shown by this work, apart from the obvious signs of the times in background, such as cars. The first is the occasional comments, mentioned above, which do not conform to the bland opionlessness of modern art documentaries. Secondly, the complexity of the thoughts expressed is significantly greater than anything else I have ever seen on television. It is a reminder that television can be educational and intellectually stimulating without failing to be entertaining.
However, the greatest quality of this documentary is its intellectual honesty. The presenter motivates his series with questions, and proceeds to attempt to answer them. But he does so in a scupulously open and honest way. There is none of the disingenuity which makes so many television programs look patronising, or even like propoganda. Where the evidence supporting his point is ambiguous he lets us know, and then explains his opinions, rather than prsenting an unbalanced narrative.
So many articles are written, or documentaries made, by people whose minds are already made up. Their contributions to the public discourse are an attempt to sell their position, and they do so without any regard for this sort of intellectual honesty.
This quality has all but dissappeared today; from political discourse where all issues are reframed by each side in ridiculous black and white, downwards (or upwards, according to one's tastes) through so much of society. It reflects a mind-set which is truly rare even in academic circles. It shows a man who is prepared to make his arguments seem weaker, in order to make them more honest.
Of course, Kenneth Clark was an emminent and distinguished scholar at the time he made the series; and this quality may have been extremely rare at the time. So this quality may never have been widepread in mankind. Nonetheless, I was glad to have caught this glimpse of it.
Wednesday 11 July 2007
Everything you've ever wanted to know about Etale Cohomlogy but were afraid to ask
This is going to be a short post to attempt to explain the 'point' of etale cohomology and the etale fundamental group. The latter is often called the algerbaic fundamental group.
In case you don't know what any of these things are, they are important in the study of arithmetic algebraic geometry. It is an area of mathematics somewhere between algebraic geometry and number theory. Lang called it Diophantine geometry. If you don't know what those words mean, what follows will not make sense to you.
To begin with, the most readable introduction to this area across which I have come is in Milne's online notes. They are incomplete and lack many proofs, but they give one the general idea. I don't aim here to teach anyone about etale cohomolgy or the fundamental group, but if you are learning about them and you happen to stumble across this site (a concurrence which is mind-bogglingly unlikely), then this may help you to see the big picture.
First of all, start by understanding the fundamental group. Consider two cases. If our base space is a field, the fundamental group is its absolute Galois group. If the base is a complex variety, the etale fundamental group is closely related to its regular fundamental group. For any other scheme, it is different, but you should understand that it generalises these very different notions, one purely geometric and the other purely algebraic.
Its importance in the study of etale cohomolgy lies in the following statement: Connected etale covers of a scheme correspond to transitive sets acted on by the etale fundamental group. The analogy with the geometric case is perfect, because connected covering spaces of a nice (if these comments are making sense to you so far, you'll know what nice means) space correspond to transitive sets acted on by the regular fundamental group.
Then the basic construction of the machinery of etale cohomology is very similar to that of regular sheaf cohomology, but extra algebraic input is often need for standard proofs.
However, the cohomology groups produced by etale cohomology are very different from those produced from Serre-style quasi coherent chomolgy theory. The real value of etale cohomology lies in these differences. They are sufficiently many and large to require little comment.
It is often said that etale cohomology (especially in characteristic zero) has two main ingredients: Galois cohomology and clasical topology. This is made most concrete by comparing the cohomlogy of a k-scheme to that of its base change to the algebraic closure of k. The relationship is broken down by Grothendieck's spectral sequence theorem into precisely the algebraic and geometric data for which one would hope.
If you make it this far, please leave a comment. I'd love to discuss this with someone.
In case you don't know what any of these things are, they are important in the study of arithmetic algebraic geometry. It is an area of mathematics somewhere between algebraic geometry and number theory. Lang called it Diophantine geometry. If you don't know what those words mean, what follows will not make sense to you.
To begin with, the most readable introduction to this area across which I have come is in Milne's online notes. They are incomplete and lack many proofs, but they give one the general idea. I don't aim here to teach anyone about etale cohomolgy or the fundamental group, but if you are learning about them and you happen to stumble across this site (a concurrence which is mind-bogglingly unlikely), then this may help you to see the big picture.
First of all, start by understanding the fundamental group. Consider two cases. If our base space is a field, the fundamental group is its absolute Galois group. If the base is a complex variety, the etale fundamental group is closely related to its regular fundamental group. For any other scheme, it is different, but you should understand that it generalises these very different notions, one purely geometric and the other purely algebraic.
Its importance in the study of etale cohomolgy lies in the following statement: Connected etale covers of a scheme correspond to transitive sets acted on by the etale fundamental group. The analogy with the geometric case is perfect, because connected covering spaces of a nice (if these comments are making sense to you so far, you'll know what nice means) space correspond to transitive sets acted on by the regular fundamental group.
Then the basic construction of the machinery of etale cohomology is very similar to that of regular sheaf cohomology, but extra algebraic input is often need for standard proofs.
However, the cohomology groups produced by etale cohomology are very different from those produced from Serre-style quasi coherent chomolgy theory. The real value of etale cohomology lies in these differences. They are sufficiently many and large to require little comment.
It is often said that etale cohomology (especially in characteristic zero) has two main ingredients: Galois cohomology and clasical topology. This is made most concrete by comparing the cohomlogy of a k-scheme to that of its base change to the algebraic closure of k. The relationship is broken down by Grothendieck's spectral sequence theorem into precisely the algebraic and geometric data for which one would hope.
If you make it this far, please leave a comment. I'd love to discuss this with someone.
Explanation
Some people share their lives through their blogs. I wouldn't presume that anyone would rather read about my life than get on with their own. So instead I intend mainly to share my thoughts. Hopefully someone or other will find them entertaining, thought-provoking, or otherwise worth reading.
Tuesday 10 July 2007
Indian (English) and American (English)
Welcome back. I would like to talk to you this evening about two languages that I know and ... well, perhaps love is a little too strong, so let us say, in which I take more than a passing interest.
They are American English and Indian English. I make some claim to impartiality in this discussion because I was raised in England. I can make some claim to knowledge too, because I have frequently visited India, and currently live in the USA. Objectivity and knowledge are essential when analysing a topic. Alas, they are also often illusory.
Let's start at the beginning, which is a very good place to start. We consider the development of these two languages. The vast majority of the (modern-era) settlers of this country were native English speakers. They arrived, however, before the modern fad for widespread literacy had truly taken off. They therefore arrived knowing how to speak, but not how to spell, the English language. This had the following two consequences.
Firstly, the standard American accent is, objectively, somewhat similar to the standard English accent. (These are both, of course, entirely fictitious beasts; but this is not a sholarly article, only a pretentious blog aping the mien of one. We proceed, then, free from the possibility of serious scholarly scrutiny.) One could argue that the American accents one hears today are descended directly from accents once heard in England, as people learnt English from their parents and grew up in a society where English was spoken.
Yet American English exhibits drastically different spelling conventions from English. Furthermore, where these differences appear, the American spelling is almost invariablysimpler or more natural. Standard examples include flavour becoming flavor; skilful becoming skillful; analogue becoming analog; and (my perennial favourite) plough being transmogrified into plow.
Indian English is somewhat widespread in India today. Yet even a generation ago it was far less common, something of an achievement. Those who spoke it several generations ago would mostly have been well educated and extremely well read.
However, the number of native speakers of English English in India was never high enough to teach all, or even most, of those learning English in India. Thus most Indians learning English have, for generations, been very carefully how to read and write, but not at all how to speak, English. Again, this may have had two consequences.
Indinan speakers of English would have grown up in a society where other languages were much more common, and this would undoubtedly have affected the development of the Indian English accent.
Hence an objective judge (probably another fiction, but ably substitued for by the author) would probably judge the Indian English accent to be further from English than is American English. And yet variations in spelling are remarkably few.
Aside from the radically different pronunciation, an interesting difference between Indian English and English is created by a certain ruthless Indian logic. The English appear willing, and even happy, to accept irregularities in their language, but the Indians do not. For example, the English word 'postpone' dates back to at least 1496 (according to the OED). It lived a happy life for many centuries. When it moved to India, however, Indians seem to have become unhappy that such a word has no opposite, and very soon created 'prepone'. It was an entirely logical, and thouroughly welcome addition, but very un-English.
A final, comical, example is provided by the word 'pyjamas'. In English the word is spelt with a 'y', but pronounced as if it were an 'a'.
In American English, it is thus pronounced 'pajamas', and, this being so, it is only reasonable for it to be also spelt with an 'a'.
The word probably reached India in written form, so it is still spelt with a 'y', but it is now pronounced 'pie-jamas'. I find that rather funny.
They are American English and Indian English. I make some claim to impartiality in this discussion because I was raised in England. I can make some claim to knowledge too, because I have frequently visited India, and currently live in the USA. Objectivity and knowledge are essential when analysing a topic. Alas, they are also often illusory.
Let's start at the beginning, which is a very good place to start. We consider the development of these two languages. The vast majority of the (modern-era) settlers of this country were native English speakers. They arrived, however, before the modern fad for widespread literacy had truly taken off. They therefore arrived knowing how to speak, but not how to spell, the English language. This had the following two consequences.
Firstly, the standard American accent is, objectively, somewhat similar to the standard English accent. (These are both, of course, entirely fictitious beasts; but this is not a sholarly article, only a pretentious blog aping the mien of one. We proceed, then, free from the possibility of serious scholarly scrutiny.) One could argue that the American accents one hears today are descended directly from accents once heard in England, as people learnt English from their parents and grew up in a society where English was spoken.
Yet American English exhibits drastically different spelling conventions from English. Furthermore, where these differences appear, the American spelling is almost invariablysimpler or more natural. Standard examples include flavour becoming flavor; skilful becoming skillful; analogue becoming analog; and (my perennial favourite) plough being transmogrified into plow.
Indian English is somewhat widespread in India today. Yet even a generation ago it was far less common, something of an achievement. Those who spoke it several generations ago would mostly have been well educated and extremely well read.
However, the number of native speakers of English English in India was never high enough to teach all, or even most, of those learning English in India. Thus most Indians learning English have, for generations, been very carefully how to read and write, but not at all how to speak, English. Again, this may have had two consequences.
Indinan speakers of English would have grown up in a society where other languages were much more common, and this would undoubtedly have affected the development of the Indian English accent.
Hence an objective judge (probably another fiction, but ably substitued for by the author) would probably judge the Indian English accent to be further from English than is American English. And yet variations in spelling are remarkably few.
Aside from the radically different pronunciation, an interesting difference between Indian English and English is created by a certain ruthless Indian logic. The English appear willing, and even happy, to accept irregularities in their language, but the Indians do not. For example, the English word 'postpone' dates back to at least 1496 (according to the OED). It lived a happy life for many centuries. When it moved to India, however, Indians seem to have become unhappy that such a word has no opposite, and very soon created 'prepone'. It was an entirely logical, and thouroughly welcome addition, but very un-English.
A final, comical, example is provided by the word 'pyjamas'. In English the word is spelt with a 'y', but pronounced as if it were an 'a'.
In American English, it is thus pronounced 'pajamas', and, this being so, it is only reasonable for it to be also spelt with an 'a'.
The word probably reached India in written form, so it is still spelt with a 'y', but it is now pronounced 'pie-jamas'. I find that rather funny.
Sunday 8 July 2007
Aims
I'm not sure what made me write that. It looks fairly pretentious, and entirely pointless, now that I have finshed it.
Most of my posts will not be that long. If I have a thought that I think is interesting, and do not change my mind in the time it takes me to come to my computer and post it here, I may post it. Otherwise, this may be a sort of diary.
Most of my posts will not be that long. If I have a thought that I think is interesting, and do not change my mind in the time it takes me to come to my computer and post it here, I may post it. Otherwise, this may be a sort of diary.
Materialism
I think that it is clear to most of us that there are, at least, two forms of materialism. Let's be simplistic for a moment and consider only one extreme, to show that there are many forms. And rather than calling the sort that I believe that I display, Good, and the other, Bad; I shall, in a radical departutre from my usual form, strive for an approximation to objectivity.
In my bathroom sits a spongebag. It is a truly beautiful spongebag. Made of a rich, deeply coloured, lustrous brown leather, it holds my attention every time I see it. (Lest you think too poorly of me, I may add that it is somewhat new.) It is a distinct pleasure to use it now; and even with the passage of time, I don't doubt that it will be more pleasurable to use it than it was to use my old, rather ordinary one.
Now, to dwell upon the purely physical qualities of such an object may be called materialistic. However, there is now venality, no greed in my thoughts. I am happy to have it, no doubt, but I would have absolutely no use for another.
This is not the end of the moral aspect. A digression, into waters of which I know little. Hopefully the ship of my allegory shall not run aground on the rocks of my ignorance.
At the beginning of the Renaissance, Humanist architects and artists produced work on a human scale. One identifies this period with small, but beautifully proportioned, rooms in small, but beautifully proportioned, houses.
Later, towards the Baroque, this sense of proportion was lost or discarded. Palaces became bigger (or rather, bigger palaces were made), works of art became more ornate, involved, and extravagant. Did this immediately lead to a decline in quality? No, but we can crudely characterise this progression as one from the building of wonderful houses to the building of, admittedly more, wonderful palaces; from achitecture designed for the enjoyment of individual human beings to architecture designed to overawe and to overwhelm.
So how on Earth does this relate to my spongebag? The point is that I don't intend to show it to anyone. I don't intend to compare it to other people's spongebags, and so the pleasure that it gives me is, in the sense above, humanist and even humane.
It is also reassuringly physical. Its seductive smell, and it smooth, sensuous finish are, of course, entirely irrelevant to its function. But they are to be enjoyed by only me, and so have a different impact on my thoughts than would the beauty of an object designed for ostentatious display, or indeed any sort of display.
The final measures, I feel, are given by how a material object may affect my actions, and my happiness.
The fact is that I didn't covet this spongebag before I bought it. I didn't dream of, one day, buying such a bag. Thus I was not at all unhappy without it, yet I was made marginally more happy by its acquisition. And doubtless, when it is gone, I shall not shed tears for its passing, nor attempt to replace it.
It entered my life unasked, its tenure in my life is marked by a slight increase in my happiness, and at its passing I shall be unmoved. If only I could feel this way about all of my possessions.
In my bathroom sits a spongebag. It is a truly beautiful spongebag. Made of a rich, deeply coloured, lustrous brown leather, it holds my attention every time I see it. (Lest you think too poorly of me, I may add that it is somewhat new.) It is a distinct pleasure to use it now; and even with the passage of time, I don't doubt that it will be more pleasurable to use it than it was to use my old, rather ordinary one.
Now, to dwell upon the purely physical qualities of such an object may be called materialistic. However, there is now venality, no greed in my thoughts. I am happy to have it, no doubt, but I would have absolutely no use for another.
This is not the end of the moral aspect. A digression, into waters of which I know little. Hopefully the ship of my allegory shall not run aground on the rocks of my ignorance.
At the beginning of the Renaissance, Humanist architects and artists produced work on a human scale. One identifies this period with small, but beautifully proportioned, rooms in small, but beautifully proportioned, houses.
Later, towards the Baroque, this sense of proportion was lost or discarded. Palaces became bigger (or rather, bigger palaces were made), works of art became more ornate, involved, and extravagant. Did this immediately lead to a decline in quality? No, but we can crudely characterise this progression as one from the building of wonderful houses to the building of, admittedly more, wonderful palaces; from achitecture designed for the enjoyment of individual human beings to architecture designed to overawe and to overwhelm.
So how on Earth does this relate to my spongebag? The point is that I don't intend to show it to anyone. I don't intend to compare it to other people's spongebags, and so the pleasure that it gives me is, in the sense above, humanist and even humane.
It is also reassuringly physical. Its seductive smell, and it smooth, sensuous finish are, of course, entirely irrelevant to its function. But they are to be enjoyed by only me, and so have a different impact on my thoughts than would the beauty of an object designed for ostentatious display, or indeed any sort of display.
The final measures, I feel, are given by how a material object may affect my actions, and my happiness.
The fact is that I didn't covet this spongebag before I bought it. I didn't dream of, one day, buying such a bag. Thus I was not at all unhappy without it, yet I was made marginally more happy by its acquisition. And doubtless, when it is gone, I shall not shed tears for its passing, nor attempt to replace it.
It entered my life unasked, its tenure in my life is marked by a slight increase in my happiness, and at its passing I shall be unmoved. If only I could feel this way about all of my possessions.
Subscribe to:
Posts (Atom)