Archive for the ‘literature’ Category

Rereading ‘To the Lighthouse’

August 23, 2015

There are some statements I started to believe at a certain distant time in my life and which I have continued to accept without further examination ever since. One of these is ‘the English-language author who I admire most is Virginia Woolf’. Another is obtained by replacing ‘English-language author’ by ‘author in any language’ and ‘Virginia Woolf’ by ‘Marcel Proust’. At one point in her diary Virginia Woolf writes that she has just finished reading the latest volume of ‘A la Recherche du Temps Perdu’ which had recently been published. Then she writes (I am quoting from memory here) that she despairs of ever being able to write as well as Proust. Perhaps she was being too modest at that point. Until very recently it was a long time since I had read anything by Woolf. I was now stimulated to do so again by the fact that Eva and I were planning a trip to southern England, including a visit to St. Ives. For me that town is closely associated with Woolf and it is because of the connection to her that I was motivated to visit St. Ives when I spent some time in Cornwall several years ago. (Here I rapidly pass over the fact, without further comment, that the author with the widest popular success whose books have an association with St. Ives is Rosamunde Pilcher.) The other aspect of my first trip to Cornwall which is most distinct in my memory is missing the last bus in Land’s End and having to walk all the way back to Penzance where I was staying. We visited Land’s End again this time but since I did not want miss the bus again I did not have time to visit the ‘Shaun the Sheep Experience’ which is running there at the moment. As a consolation, during a later visit to Shaun’s birthplace, Bristol, I saw parts of the artistic event ‘Shaun in the City’ and had my photograph taken with some of the sculptures of Shaun.

When I go on a holiday trip somewhere I often like to take a book with me which has some special connection to the place I am going. Often I have little time to actually read the book during the holiday but that does not matter. For Cornwall and, in particular, St. Ives the natural choice was ‘To the Lighthouse’. That novel is set in the Isle of Skye but it is well known that the real-life setting which inspired it (and the lighthouse of the title) was in St. Ives. This lighthouse, Godrevy Lighthouse, cost a little over seven thousand pounds to build, being finished in 1859. In 1892, on one of two visits there, the ten year old Virginia signed the visitors book. The book was sold for over ten thousand pounds in 2011. So in a sense the little girl’s signature ended up being worth more money than the lighthouse she was visiting. Of course, due to inflation, this is not a fair comparison. Looking on my bookshelves at home I was surprised to find that I do not own a copy of ‘To the Lighthouse’. On those shelves I find ‘The Voyage Out’, ‘Jacob’s Room’, ‘Moments of Being’ and ‘Between the Acts’ but neither ‘To the Lighthouse’ nor ‘The Waves’. Perhaps I never owned them and only borrowed them from libraries. I have a fairly clear memory of having borrowed ‘To the Lighthouse’ from the Kirkwall public library. I do not remember why I did so. Perhaps it was just that at that time I was omnivorously consuming almost everything I found in the literature section in that library. Or perhaps it had to do with the fact that lighthouses always had a special attraction for me. An alternative explanation for the fact I do not own the book myself could be that I parted with it when I left behind the majority of the books I owned when I moved from Aberdeen to Munich after finishing my PhD. This was due the practical constraint that I only took as many belongings with me as I could carry: two large suitcases and one large rucksack. I crossed the English Channel on a ferry and I remember how hard it was to carry that luggage up the gangway due to the fact that the tide was high.

I find reading ‘To the Lighthouse’ now a very positive experience. Just a few paragraphs put me in a frame of mind I like. I have the feeling that I am a very different person than what I was the first time I read it but after more than thirty years that is hardly surprising. I also feel that I am reading it in a different way from what I did then. I find it difficult to give an objective account of what it is that I like about the book. Perhaps it is the voice of the author. I feel that if I could have had the chance to talk to her I would certainly have enjoyed it even if she was perhaps not always the easiest of people to deal with. Curiously I have the impression that although I would have found it extremely interesting to meet Proust I am not sure I would have found it pleasant. So why do I think that I may be appreciating aspects of the book now which I did not last time? A concrete example is the passage where Mrs Ramsay is thinking about two things at the same time, the story she is reading to her son and the couple who are late coming home. The possibility of this is explained wonderfully by comparing it to ‘the bass … which now and then ran up unexpectedly into the melody’. I feel, although of course I cannot prove it, that I would not have paid much attention to that passage during my first reading. The differences may also be connected to the fact that I am now married. Often when I am reading a book it is as if my wife was reading it with me, over my shoulder, and this causes me to pay more attention to things which would interest her. A contrasting example is the story about Hume getting stuck in a bog. I am sure I paid attention to that during my first reading and it now conjured up a picture of how I was then, perhaps eighteen years old and still keen on philosophy. After a little thought following the encounter with the story it occurred to me that I knew more of the story about Hume, that he was allegedly forced to say that he believed in God in order to persuade an old woman to pull him out. This extended version is also something I knew in that phase of my life, perhaps through my membership in the Aberdeen University philosophy society. On the other hand this story does come up (at least) two more times in the book and it is a little different from what I remember. What the woman forced him to do was to say the Lord’s Prayer.

I came back from England yesterday and although I did not have much time for reading the book while there I am on page 236 due to the head start I had by reading it before I went on the trip. The day we went to St. Ives started out rainy but the weather cleared up during the morning so that about one o’ clock I was able to see Godrevy lighthouse and look at it through through my binoculars. They also allowed me to enjoy good views of passing gannets and kittiwakes but I think I would have been disappointed if I had made that trip without seeing the lighthouse.

Bootstrap arguments

November 22, 2009

A bootstrap argument is an analogue of mathematical induction where the natural numbers are replaced by the non-negative real numbers. This type of argument is a powerful tool for proving long-time existence theorems for evolution equations. For instance, it plays a central role in the proof of the stability of Minkowski space by Christodoulou and Klainerman and the theorem on formation of trapped surfaces by Christodoulou discussed in previous posts. The name comes from a story where someone pulls himself up by his bootstraps, leather attachments to the back of certain boots. This story is often linked to the name of Baron Münchhausen. In another variant he pulls himself out of a bog by his pigtail. This was a person who really lived and was known for telling tall tales. In later years people wrote various books about him and incorporated many other tall tales from various sources. The word ‘booting’ applied to computers is derived from ‘bootstrapping’ in the sense of this story. There are also bootstrap methods used in statistics. They involve analysing new samples drawn from a fixed sample. In some sense this means obtaining more knowledge about a system without any further input of information. It is this aspect of ‘apparently getting something for nothing’ which is typical of the bootstrap. In French the procedure in statistics has been referred to as ‘méthode Cyrano’. Unfortunately is seems that in PDE theory the French have just adopted the English term. I say ‘unfortunately’ because of a fondness for Cyrano de Bergerac. As in the case of Münchhausen there was a real person of this name, this time a writer. However the name is much better known as that of a fictional character, the hero of a play by Edmond Rostand. The non-fictional Cyrano wrote among other things about a trip to the moon. There is also a Münchhausen story where he uses a kind of inverse bootstrap (could there be a PDE analogue here?) to return from the moon. He constructs a rope which he attaches to one of the horns of the moon but it is much too short to reach down to the ground. He climbs to the bottom of the rope, reaches up and cuts off and detaches the part ‘which he does not need any more’ and ties it onto the bottom. He then repeats this process. Returning to Cyrano, he describes seven methods for getting to the moon of which the sixth is the one relevant to the bootstrap. He stands on an iron plate and throws a magnet into the air. The iron plate is attracted by the magnet and starts to rise. Then he rapidly catches the magnet and throws it into the air again. I should point out that Cyrano does not believe in the nonsensical stories he is telling – his aim is a practical one, holding the attention of the Duc de Guiche so as to delay him for a very specific reason.

Now I return to the topic of bootstrap arguments for evolution equations. I have given a discussion of the nature of these arguments in Section 10.3 of my book. Another description can be found in section 1.3 of Terry Tao’s book ‘Nonlinear dispersive equations: local and global analysis‘. A related and more familiar concept is that of the method of continuity. Consider a statement P(t) depending on a parameter t belong to the interval [0,\infty). Let S be the subset consisting of those t for which the statement P is true on the interval [0,t). If it can be shown that S is non-empty, open and closed then it can be concluded that the statement holds for all t, by the connectedness of the interval. The special feature of a bootstrap argument is the way in which openness is obtained. Suppose that, starting from P(t), we can prove a string of implications which ends again with P(t). This is nothing other than a circular argument and proves nothing. Suppose, however, that in addition this can be improved so that the statement at the end of the string is slightly stronger than that at the beginning. This improvement is something to work with and is a typical way of proving the openness needed to apply the continuity argument. It is more convenient here to work with the open interval (0,\infty) since we want to look at properties of solutions of an evolution equation defined on the interval (0,t). Let P(t) be the statement that a certain inequality (1) holds on the interval (0,t) and suppose that P(t) implies the statment Q(t) that a stronger inequality (2) holds on the same interval. Things are usually set up so that Q(t) implies by continuity that (2) holds at t and that the the property of being ‘stronger’ then shows that P(t') holds for t' slightly greater than t. This shows the openness property. I think the best way to really understand what a bootstrap argument means is to write out a known example explicitly or, even better, to invent a new one to solve a problem which interests you. The key thing is to find the right choice of P and Q. What I have described here is only the simplest variant. In the work of Christodoulou mentioned above he uses a continuity argument on two-dimensional sets.


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