Well done, youtube-algorithm…

In the past few weeks I have encountered a youtube channel that gives several nice ideas about art and about some artistic ideas work. Apparently, youtube (which is to mean: Google) knows enough about my musical preferences to present a video called “Lord of the Rings: How music elevates story”, which de-constructs the musical leitmotifs in the score of Peter Jackson’s Lord of the Rings trilogy:

In particular, I had recognized the different themes in the score by myself, but the way they were intertwined had stayed hidden from my conscious thinking. A very nice clip that most certainly made my day – and in particular, I found out that the nerdwriter-channel had plenty of other very insightful ideas to discover: understanding Picasso or Edvard Munch (he didn’t just paint The Scream, you know, even though it is his very most famous piece, and rightly so), looking deeper into Bob Dylan’s lyrics, analyzing the speeches and tweets of President Donald Trump, something on how great directors work, as in Sherlock, or Saving Private Ryan, even about the Beatles’ cover of Sgt. Pepper’s Lonely Hearts Club Band (which would tell me little new, but I wouldn’t expect that in a non-specialist video). A very nice channel, challenging me to think deeply about many different topics, both old and new to me.

A slight introspective.

There are basic differences between the categories in which this blog is organized. On the one hand, there is maths, often rather technical and detailed, highlighting some certain aspect of a topic that has caught my interest. Usually, those texts are written immediately after I have spent enough time with this particular topic – as I often switch the topics of interest and as I tend to forget the technical details pretty fast. As a matter of fact, this is exactly why there are these mathematical texts: I can get amazed by remembering the details when I re-read the blog, and especially when I have understood fundamental things about something, I wish to keep some record and some hint of how things work.

On the other hand, there are the deeper texts on literature, like book reviews or texts on topics that I have spent a lot of time with (usually a lot more than with the particular math topics: those switch faster). In some respects, I find those texts not only deep and long, but also somewhat more mature. I have taken a lot of time to develop an understanding of these things, and even if someone was to disagree with me, I would feel ready for discussion, even years later. This is why my stack of topics to be covered changes rather slowly. I have a text on Sebastian Haffner in the back of my mind, which still needs to be written, and which can’t be written in just half an hour of leisure time; same goes for a text on the TV-series Sherlock, which itself may be still work in progress, anyway. I was tempted to start a text on ancient Greece and democracy (today compared to the past), but I didn’t feel to have any definite view yet – this topic needs some more maturity. Time and again, texts like the one on The Beatles just drop out of my head and appear here, which then strikes me as a nice event, as I have managed to put down my present view of some aspect that I have spent very much time with.

And then, there are the posts like this one. Somewhat short, lacking the really deep insights, but in some way serving as a blog in the original sense of the word: as my “web-log”. They deal with topics that sparked my immediate interest, they sometimes deal with my lack of putting my insights in writing, they sometimes even focus on my personal situation of sorts. I am quite aware of the problem that this post doesn’t bring any benefit to humanity, and that this kind of self-centration is one of the tombstones of modern media. But, if my deeper insights still have to mature, then so be it. Let us stick to Gauss: Pauca sed matura. And, as the phases come and go, my lack of spare time has returned and makes it harder to actually let my insights mature as much as I want them to. But, on the up-side, new music by Judith Holofernes is about to show up, I encountered the most amazing unplugged music by the wonderfully talented Taylor Swift, I’ve had a look at the 4th season of Sherlock, and I retrieved a great book with short-stories by Andreas Eschbach. All this made my day, several times over.

Maybe, in some way, having thought about the structure of what can be found here, has been quite an insight for myself. Considering where this post started, that’s not nothing.

Loreena McKennitt’s music and lyrics

In the winter, when the dark evenings are long and I enjoy a cup of tea while looking out of my window to the stars, it is the time for me to listen to Loreena McKennitt’s music. This is by no means “easy listening” but it relaxes me a lot. It is music that I can sink into perfectly, especially when it’s dark outside and I can dream away.

Loreena McKennitt is a Canadian soprano singer and songwriter. Her music is inspired by mystic melodies, by celtic folk songs and middle eastern influences. She accordingly uses many unusual instruments: besides her singing, she performs many different instruments herself, such as a harp, an accordion and the keyboard. Many of her songs have been written by herself, but she also re-arranges classic folk music and even Elizabethan (resp. Shakespearian) songs.

My enjoyment of this music stems from her clear soprano voice, which has a marvelous, dreamy and still variable sound to it. Here is a very special singer who can transport her love for music to her listeners. I once had the distinct pleasure to listen to her in a live performance in an open-air concert – an evening that most certainly made my day.

I find it tricky to point out a few special songs here, as I can’t find any failures in her nine studio albums at all. They may all be recommended, and each has a certain feel to it, and I can’t put my finger on it – but different moods inspire me to listen to different albums. In particular, I refrain myself from posting links to any videos of her recordings – I wish to avoid cherry picking here.

My first encounters with her music came from the album “The Visit”, which contains the most beautiful rendition of the poem “The Lady of Shalott” – a poem known to every fan of Miss Marple’s detective stories, of course (“The mirror crack’d from side to side”) – which made me spend some time with English poetry for the first time. But then there is also the joyously sounding “All souls night”, the sad “Bonnie Portmore” and the ancient “Greensleeves” which apparently was only recorded in a short break in one take and still sounds wonderful on the album.

After I had been hooked, the album “Elemental” caught my attention with the harmonic duet “Carrighfergus”, a classic Irish song which is performed by a male singer and Loreena McKennitt rather contents herself with dreamy backing vocals. “Blacksmith” and “Stolen Child” are similar but the lyrics are performed by herself marvelously. Very notable is “She moved through the fair” for being entirely a cappella – there is not a hint of accompanying instrumentation to her crystal-clear voice. I am hard-pressed to think of any musician who would show this much courage on a recording (especially since this recording is about 30 years old now and couldn’t have been digitally enhanced).

“Parallel Dreams” was an album that I bought on a vacation trip many years ago, when there was too few information to know which albums were available – when I saw this during my trip to France, I bought it on the spot, the price certainly higher than what I’d have paid at home, but I wouldn’t regret that. With “Annachie Gordon” she performs a traditional Scottish song telling a sad love-story. And “Dicken’s Dublin” is a most beautiful and yet heart-breaking Christmas Song, speaking of the hope of a homeless child to find a home, and being voiced-over by a child narrating the story of the birth of Jesus. What beauty lies in this composition.

For “The Mask and the Mirror”, she wrote about her travels in the booklet, and how she was inspired for her various songs on this album. Here appear references to oriental and middle eastern music for the first time. “The dark Night of the Soul” has been her encore in her live performance that I attended, and the evening only got complete with this song. Never before have I witnessed a crowd that was so focused and so mesmerized by a single person performing on the stage.

“The Book of Secrets” appears to be one of her most famous albums, and rightfully so. With “Skellig”, “Dante’s Prayer” and “The Highwayman”, she again takes classical pieces uses this influence and produces wonderful art. I do not wish to make this whole post tedious by enumerating all the marvelous songs that can be found on “An Ancient Muse” and on “The Wind that shakes the Barley”, but let it be said that those most recent albums are by no means any worse than their predecessors. Wonderful music all over the place.

The sad thing is, that since 2010 there has not been any new music from her. But as she still travels the world and performs her music on stage, I have not given up hope that we’ll have new music eventually. And wouldn’t that be beautiful?

Other blogs have nice entries, too.

This time, I have to recommend another blog that brings up a wide variety of topics and has made me think thoroughly about many of them ever since I got aware of it. About half a year ago, a friend of mine brought my attention to Wait But Why, saying something to the effect of “that’s gonna be your kind of thing” – and he was quite right…

Tim Urban is the main writer of the texts on WBW, and he brings several properties with him that make his texts click with me:

  • he’s a fine writer (in the sense of producing easily readable texts; certainly much more easily readable than mine)
  • he’s curious on various topics
  • he can be mesmerized by ideas and concepts to the point that he can transfer his own fascination to his readers
  • he can be hilarious, both in his texts and his self-drawn figures

On top of that, he is a horrible procrastinator; a property that he has been writing and talking about quite often. He seems to have accepted that trait of his, and he acknowledges it openly.

There are, of course, some posts that caught me more than others did. For instance, the series on Elon Musk (“the world’s raddest man”) was really interesting, as I knew almost nothing about this man and his projects – in particular, I had no idea that both SpaceX and Tesla were companies of his. On WBW these projects are considered to be the hottest spots in town, and they are described in much detail. Expect to bring some time when you wish to read those. But don’t expect a fully neutral description – Tim Urban says himself that he is some sort of fan of Musk in general and Tesla in particular. The texts are biased towards these projects and to why they are deeply necessary. Personally, I tend to agree, especially since it is important to have someone rich and interested in the world such as Musk who brings attention to certain problems and drive into their solution. In the post, the reader can learn not only about what Tesla and SpaceX do, but why it is actually important: there’s a lot on the finiteness of fossil energy, where energy comes from and where it goes, and so forth. Highly interesting, even if you might start at a slightly higher level of knowledge at the beginning – the post will catch you.

Other posts may shift your perception of the world, or of history.

For example, the post on artificial intelligence is both exciting and scary about the prospects of humanity; same holds for the post on the Fermi Paradox (“Where is everybody?”) – it made me think really deep about where humanity stands and where it goes.

The post on sound gave me thrills, when I read that “our brains send information to other brains through complex patterns of air pressure waves. Have you ever stopped and thought about how incredible that is?” – Well, no, I haven’t. It is incredible indeed.

Similarly thrilling is the post on numbers – up to Graham’s Number (one of the largest numbers used in a serious mathematical argument). In some way, numbers don’t scare me, being a mathematician: I am well aware of the concept that there are infinitely many numbers after any given number. But the twist was: considering how unfathomable numbers can get – I don’t want to live forever, I do want to die at some point; even though death comes way too early (thinking of the “tail end“-post), living forever would be really scary.

The posts on the American presidents, however, is not scary at all – but it is highly funny and insightful. Quite the same holds for the post on “horizontal history“, giving a wholly new perspective of what happened somewhere in the world at the same time (which is not totally new to me – I own a “Kulturfahrplan” which is a bit more thorough than WBW, but lacks some small highlights :-)).

I couldn’t even start to agree on the post on cryonics – it doesn’t catch me content-wise (which holds, of course, for several other posts, for that matter). But I learned many things nontheless.

I have to mention the stick figures briefly – they are put into the posts and certain places to underline the text and to produce some comic relief. So simple and so effective… I, personally, know my limits and therefore omit any drawings that might or might not happen while I think about new posts…

And what totally makes my day, that’s how WBW can create memes and “mind-shortcuts”, not just the stick figures but also the metaphores like the “instant gratification monkey” in the procrastination post, for instance.

For all those reasons: check out WBW 🙂

Sinhtgunt und Aelfgyva

Kürzlich habe ich mich durch den Wikipedia-Artikel über den Teppich von Bayeux faszinieren lassen. Der Teppich stellt die Eroberung Englands durch die Normannen im Jahr 1066 dar, und erzählt fast comic-artig von der Begegnung der späteren Könige Harald und Wilhelm, vom Tod Edward des Bekenners, von der Invasion der Normannen in England und der Schlacht bei Hastings. In Auftrag gegeben wurde der Teppich von den normannischen Herrschern nach ihrem Sieg, sicher auch um ihre Herrschaft zu legitimieren (etwa sichtbar daran, dass der Vasalleneid Haralds für Wilhelm aufgeführt wird, sodass nur Wilhelm der Eroberer Herrscher über England sein kann und nicht sein Vasall Harald). In der Schlachtdarstellung bricht der Teppich ab, er war ursprünglich noch etwas länger und endete vermutlich mit der Krönung Wilhelms zum König von England.

Die Texte sind in lateinischer Sprache geschrieben und scheinen von einem mittelenglischen Muttersprachler zu stammen, wie sich aus Namens- und Städtebezeichnungen ablesen lässt, ebenso wie aus der Benamung der Normannen als „Franzosen“ – sicher keine Selbstbezeichnung der Eroberer, auch wenn sie französisch sprachen. Die Angelsachsen und insbesondere Harald werden in recht positivem Licht dargestellt, etwa hilft er Ertrinkenden aus dem Meer, er erscheint nicht machtgierig als er zögert nach der angebotenen Krone zu greifen, und in der Schlacht stirbt er einen als ehrenvoll erachteten Tod (allerdings gibt es auch die Lesart, dass er wie ein Verurteilter Eidbrecher starb, nämlich durch einen Lanzenstoß durch das Auge). Man erfährt aus dem Teppich einige historische Details in Bezug auf Kleidung, Rüstung, Barttracht oder Schiffbau, es hat sich sogar rekonstruieren lassen, dass der Halleysche Komet erwähnt wird (unter dem lapidaren Text “isti mirant stella”). Der Teppich ist außergewöhnlich interessant für die Geschichtswissenschaft und die Kunstgeschichte. Und auch vom ästhetischen Standpunkt ist es ein sehr ansprechendes Werk, in das eine Menge Arbeit geflossen ist.

Ein interessantes Detail ist die Anekdote von Aelfgyva und dem Kleriker. Es ist unklar, was die entsprechende Szene darstellt – in der lateinischen Beschreibung, die mit eingestickt ist, fehlt das Verb: „Ubi unus clericus et Aelfgyva“. So ist es unklar, ob der Kleriker Aelfgyva schlägt oder ihr über die Wange streichelt. Beides ist aus dem Bild heraus denkbar. Offenkundig ist die Szene in irgendeiner Weise anzüglich, da auf der Bordüre eine nackte männliche Figur sitzt, die den Kleriker zu imitieren scheint. Es ist unbekannt, wer Aelfgyva ist, aber offenkundig war die dürre Andeutung auf dem Teppich für die Zeitgenossen mehr als ausreichend, um Bescheid zu wissen. Insbesondere war die Geschichte wohl nicht belanglos – und doch wissen wir heute nichts weiteres darüber.

Eine kurze Suche bei google hat gezeigt, dass sich viele intelligente Leute bereits damit befasst haben, aus der Szene und dem dürftigen Kontext möglichst viel Information herauszulesen (nur zwei Beispiele). Das ist außergewöhnlich interessant, aber es wird immer ein Stochern im Trüben bleiben, da niemand mehr die wahre Geschichte erzählen kann.

Um thematisch sehr weit zu springen – ein ähnliches Stochern im Trüben findet bei der Interpretation der Merseburger Zaubersprüche statt. Es handelt sich dabei um einen der ältesten Texte in deutscher Sprache (“von höchster sprachlicher und mythologischer Altertümlichkeit”), geschrieben um das Jahr 900. Mangels vergleichbarer Texte ist nicht einmal eindeutig festzustellen, in welchem Dialekt die Verse aufgeschrieben wurden. Die Handschrift erscheint auf den Buchdeckeln eines Kodex und ist daher möglicherweise nur eine Fingerübung für den Schreiber in seiner Muttersprache (im Gegensatz zum sonst geschriebenen Latein), eines Textes, den er entweder auswendig kannte oder von einer verlorenen Vorlage abschrieb. Es handelt sich um einen Heilungszauber und um einen Zauber, der Fesseln lösen soll, jeweils eingeleitet durch eine kurze mythische Rahmenhandlung.

In dieser Rahmenhandlung werden heidnische Götter genannt, darunter Wotan und Freia, aber auch eine Sinhtgunt. Diese ist sonst nirgends belegt und unbekannt, sie taucht nur hier auf. Und schon beim Namen stochert man im Trüben, da die Buchstabenfolge -nht- nirgends sonst belegbar ist und nicht zu den Lautgesetzen der althochdeutschen Sprache passen will. Daher wird zumeist stillschweigend dieser Name verbessert zu Sinthgunt, allenfalls mit markierter Stellung der veränderten Buchstaben.

Auch hier gibt es die wildesten Theorien, die praktisch alle ohne Verankerung an Fakten auskommen müssen, oder die ihre Verankerung erst in sehr großer Entfernung haben. Und auch ansonsten sind viele der Beteiligten unbestimmt: die Handschrift könnte anstelle von „Friia“ auch „Frua“ bedeuten und damit nicht die Göttin Freia, sondern einfach eine Frau; „Phol“ könnte einen Gott dieses Namens bezeichnen oder eher ein Fohlen, mit dem Wotan geritten wäre (im Text: „Phôl ende Wuodan fuorun zi holza“ – auch hier wird stillschweigend das h in „Phol“ eingefügt, das in der Handschrift oberhalb der Textzeile steht, möglicherweise schon in der Handschrift eine Korrektur).

Zwei immens interessante Details aus dem Mittelalter, die sich nie mehr klären lassen werden und die jeweils nur eine einzige Quelle haben, über die wir sie überhaupt kennen. Ein Indiz, wie vielschichtig die Geschichtsschreibung ist und wie vielschichtig das Wissen der Menschheit ist, das wir verloren haben. Ich war sehr an die „Haarteppichknüpfer“ von Andreas Eschbach erinnert und muss dennoch ergänzen: beide genannten Details sind für das zentrale Verständnis des Mittelalters im wesentlichen belanglos. Spannend sind sie trotzdem.

Math is important

There have been many instances when people told me about how much they disliked math at school – especially when I speak highly of my studies and that I do abstract math frequently even today. On top of that, I have talked to many students (either B.Sc. students or teachers-to-be) who speak similarly about math at university-level. And finally, I know several math teachers who chime in to that tune as well. Among the most usual comments that I get are:

  • math at school was so boring and dry, it doesn’t have anything to do with what happens in real life
  • it would be sufficient if I had just learned the basic computations (addition, subtraction, multiplication and division) and to deal with percentages – that’s all I’ll ever need
  • math at school was so hard, I never understood why I should have come up with these particular tricks
  • the exercises at the university were so hard, I never managed any one of them on my own
  • it’s no use to learn so many hard topics in math that I will never use in my later work (and, oh boy, I had to go to so freaking weird lectures)

Here’s my random rant about why none of these comments is valid – or why any of these comments would apply just the same to any other high school subject. I just had to collect those arguments once in a single place, to get them off my heart. And for further reference. None of these arguments is originally mine, which might just show how deep this anti-math feeling is rooted.

Before I start, let me make something clear: I do not intend to say that the whole universe of mathematics is interesting in and of itself and that everybody should make themselves acquainted with that just for the sake of it. There is a lot of mathematics in which I have lost my interest as time went by (or in which I never had any interest in the first place). On top of that, most of what I have to say applies accordingly to other sciences or topics: I myself would never have become much of a biologist or theatre critic. In what follows, I will rather talk about things that should be standard knowledge to anybody who consider themselves an educated person: things that are part of what mankind has achieved, be it the Mona Lisa, be it Romeo and Juliet or be it Newton’s Law of Gravitation. In some way, many people tend to say that the first two are “must-knows” but the last one is not. You can live your life perfectly at ease if you’re unaware of Newton’s Law of Gravitation, but if you don’t recognize the Mona Lisa or have never seen it, some would think you’ve been living with eyes closed – how’s that? A painting may be easier to approach than a theorem, but both require some sort of education to properly grasp them and both will be (aesthetically) pleasing to the educated beholder. And there also is really deep beauty in maths, there is elegance, there are unexpected connections. But it is hard work, no doubt about that.

On top of that, there are many people who work in some shade of a scientific profession. That not only applies to scientists but also to engineers, programmers, actuaries, and – yes – especially (math) teachers. All these people must possess a certain way of thinking to do their job properly: an analytic, logical train of thought which allows them to cut through a problem and to make their reasoning clear to other people (it hardly matters whether you have to explain your work to your boss or to a bunch of children). Besides, these people have learnt to actually tackle a problem and not run away if they don’t understand it at first sight. In practical professional uses these trains of thought will be what people have learnt – this is what employers look for in scientists: they have learnt to think and they can’t be scared away by some hard and unsolved problem. This way of thinking is being taught by considering mathematical problems, but the actual skill is generally not mathematical. If the way of logical thinking could as well be taught by applying for a truck driver’s licence, then employers would also look for truck drivers, why not.

So, this is one of my first points that I’d like to make: in learning or studying math, you’re supposed to learn to think. The fact that you learn math is just incidental, if you will. You’re supposed to be suspicious about what people just tell you without proof – you’re supposed to think for yourself and to convince you of the truth for yourself. In social studies, in theology, in psychology, the teacher might shoot down any of your questions with a phrase like “I’m right because that’s the general opinion and you’re not yet aware of it”, but in mathematics and in the sciences any student is able to convince the teacher of mistakes that he’s made. I am aware that I’m over-simplifying here, since there is no such thing as an absolute truth in psychology and social sciences (theology is somewhat tricky in this respect) and there is a living culture of discussions – those professions will teach you a different, albeit similarly important way of thinking. Discussions are equally important in math, but in the end there is only one answer (apart from very special cases that usually aren’t taught) and the students are supposed to learn arguments towards that answer. In philosophy, no-one needs to be convinced by arguments since no-one is wrong; maybe different or on shaky ground, but not wrong. My point is not to diminish the non-mathematical sciences, my exact point is that you can learn argumentation and discussion on topics that actually have universal truth hidden inside them. This is how you can uncover false arguments, how to fix arguments gone wrong, how to think twice at crucial points – and how to wonder about unconvincing statements other people may make (and how to question these statements).

Another issue made about math is that it is considered hard. Well, that’s right. Math is hard and exercises are hard. But if your mind is to be trained to a purely logical way of thinking, then how else would you do the training? If you want to run a marathon, you also need to do your training (and yes, of course, running a marathon is hard as well). On top of that: problems in work are harder than the math exercises. Applications of mathematics are even harder than abstract math, since the abstraction is gone. There are no proper circles in the real world with a circumference of 2\pi r. In applications you always have to account for measurement errors, lack of data quality – and you have to stop dealing with those inaccuracies at some point because you couldn’t possibly get all of them right (but at least you should give an estimate then about how big your error is). Your math exercises are supposed to give indications about what can and what cannot be done, and how you can deal with that. About using math successfully, to discuss and to convince others by your logic. And without training, you can’t do anything properly. Learning is when you do things that you can’t do yet.

Now, some people will say they don’t have to deal with circles or anything similar in their everyday life, so this example doesn’t apply. Maybe so. But then they have to deal with something else and time and again something changes. People are not robots who receive their programming once and then do the same thing day after day – and every time something changes, you will have to adapt and to think about that. It might be useful to be able to think about that properly: what am I supposed to do now, how do I do that, how can I optimize that (i.e. have less trouble or effort with it). This doesn’t need to be mathematical thinking – it’s just that math and exercises may have taught you how to deal efficiently with “new things that you don’t yet understand”.

The other point of view to this is: math is universal because it’s abstract. Some things and techniques have been invented for a special purpose, but when you take the special situation away from it, the technique can be used more generally. I won’t start the long line of examples here, it would probably just distract from my point. Abstraction is hard, but it unveils the underlying principles and once you’ve understood those, many things will just fall into your hands like that. Or, to phrase it differently: you should make things as simple as possible – but not any simpler. Applications are not simple because they may be concrete – in this respect, math is simple because it is not concrete. And to give a quote from one of my high school math teachers: “The task of mathematics is to make life as simple as possible.”

About math exercises, some complain that, sometimes, tricks are required and you can’t come up with the trick in the first place. That’s right. But no-one expects you to invent a mesmerizing trick on your own. You’re supposed to look at the exercise and think about it, try to solve it. If you have spent some time with the exercise, sometimes you will come up with a solution (and sometimes a neater solution uses a dirty trick that you didn’t see). But there will always be exercises that you can’t solve. That’s not a problem. When you are given the solution, you are supposed to think about that, too. You wouldn’t understand it properly, if you hadn’t thought about the question in the first place – and you will understand the trick, so it will be part of your own quiver of tricks from now on. If there is a similar exercise later, you can use the trick since you’ve been taught it now. And, yes, there are many tricks. It takes experience and effort to know which can be used in a given situation and which can’t. This, in turn, enables you to deal with more complex problems that you couldn’t tackle without that quiver of tricks. That would bring us back to the training-argument from above: you can’t run a marathon, if you just walk around your house once a week. If you want to become better in something, this will take effort and time.

Much of what I said about some profession or science as a given example can be replaced by something else and still keep the same meaning. I actually intend to say, that a broad education is essential for anything an educated person does – you need to be really good in your special subject, but you also need to try and see the whole picture (or understand why you can’t see it). It’s rather my own point of view that made this posting start about mathematics. It actually applies to most other educated professions I can think of, and many people shun away from these “alien” educated professions when a discussion crosses those. I myself am often confronted with the mathematical angst of many people, which made me write this text.

Seeing the bigger picture (whatever your personal profession and specialty is), will also help you in another aspect. You always need to know a little more than you actually have to use. Thus, you will know the boundaries of what you’re doing (and the shortcuts that you may have taken). It will be immensely helpful to have an understanding of why the world works as it does, what the principles are – that doesn’t mean you have to properly understand each and every bit of it, your time is finite of course, but the basics should be standard knowledge. That makes it possible to appreciate the really hard stuff (e.g. it’s not an easy thing to encrypt your online banking access data; it’s not easy to write a song that will rank top in the charts; it’s not easy to understand why everyone actually puts any trust in paper-money instead of gold, bread or shells).

Especially teachers must know more than their students, and they must know more than what they actually teach their students. On top of that, teachers need to know how to deal with children and how to keep them awake and attentive in their classes. That’s why this job is so hard, after all.

Similar reasoning applies to languages. I can hardly overstate the need to speak foreign languages. You will understand the culture of a foreign country much better (if only at all) if you can make yourself familiar with its language. And that also applies to dead languages such as Latin or Ancient Greek. German educator Wilhelm von Humboldt once said “Having learnt Greek can be no less useful to a carpenter, than making tables is useful to a professor”. You learn to think analytically by learning the grammar and you learn about another culture. How can that be entirely useless, even though no-one will have a conversation with you in Latin? Of course, your time is finite and you’d rather deal with something else – but apart from very legitimate indulgent things like watching some football game or spending time with your friends: is a soap-opera really the better way to kill your time as opposed to learning a new language, playing chess, doing a painting or thinking about a science problem? What do you want to do with your time?

Finally, one paragraph about the boundaries of math. Mathematics is not a description of the world. Mathematics is a set of assumptions (called axioms) which are taken at their face value because no-one doubts them and you can’t prove anything without having a starting point. From those axioms, things are proven by logical conclusions. The construction is powerful enough that people have drawn useful and sensible conclusions from it to describe the world. But mathematics is not the world. It is a model (albeit quite a good one) and there are boundaries. Dealing with mathematics allows you to find out where those boundaries are and where your model assumptions break down (do they break down in your application?). You mustn’t use math without a sufficiently deep knowledge, and this is why economists, psychologists and many others have to deal with statistics, even though their computers will tell them about the significance of their test statistics. Still, you have to know the limits and deal with them.

That’s why people are supposed to deal with math.

One Moment of Clarity

On Star Trek, Jadzia Dax said she experienced two moments of clarity in seven lifetimes. On Pulp Fiction, Jules experiences a moment of clarity in a near-death situation. I just had one this night when I lay awake in my bed. This was the final piece to a puzzle over which I have been brooding for more than four years now. Wonderful. For the first time ever, I feel I have understood everything about this one. It’s haunted me and even though I managed to connect the biggest parts two years ago, there has always been this one aspect that escaped me.

Now, I can understand. How marvellous.

Actually, the next posting is already in the pipeline, but it cannot hope to compare with this thought that came to my mind tonight. Made my day.

Turn off your Mind, relax and float downstream. Surrender to the Voice.

Recently, John Lennon would have celebrated his 75th birthday. This is reason enough to have a look at his musical oeuvre and that of the group that made him famous: The Beatles.

In the 1960s, John Lennon, Paul McCartney, George Harrison and Ringo Starr experienced their way to fame, and they grew from good-looking fine musicians to the best that modern music had to offer. Their early songs are by-the-book love songs, well executed and still fine today – but not memorable. In the late 1960s, they explored new ways to go and produced revolutionary music. Never again has music been so well done: new sounds, experimental designs, all kinds of styles, fine lyrics – and still: awesome to hear today. The Beatles have stood the test of time. By 1970, they had evolved so far as to seek new ways for themselves as well, and the group split up. None of the four ever came back to the top level at which they had been together, even though they tried it independently.

I myself have had a phase of discovering the Beatles several years ago. Before that, I was vaguely aware of them and of course I knew those songs that everyone knows: Yesterday, She Loves You, Yellow Submarine, Let it Be. All well and good, but I wasn’t caught. Then, out of the blue, I tried to find out about the fuss about the White Album: many people talk about it, it’s quoted in several movies – so, what’s the deal?

The White Album (which is actually called “The Beatles”) was a shock of many sorts. It is long: it has 30 tracks, and it can’t be put in a category. There are all kinds of styles, from gems (Revolution 1, Julia), fine rock (Helter Skelter), down to the weird (Glass Onion, Bungalow Bill, Piggies) and the abyss (Revolution 9). It was the very wrong album to go to in the first place – without background knowledge, I couldn’t make heads or tails of it. Later, when I returned to it, I knew how it came to be: that would shift my appreciation of the whole album a lot. I’ll return to that later.

The famous samplers which are appropriately called “the Red Album” and “the Blue Album” proved to be better as a starting point. From them, I got a survey of what the Beatles have achieved in musical terms. The Red Album, covering the first half of their career as a band, starts off with the rather uninteresting love songs (even though those are fine songs, don’t get me wrong). The real deal comes halfway through: songs like Norwegian Wood, Nowhere Man, Girl and We Can Work it Out nearly flashed me. Awesome stuff. Ticket To Ride is one of my very favorite songs ever and I shall wish that In My Life be played at my funeral. All these songs really touched me, both the lyrics and the music. For the most part they are still rather conventional, but you can already see the band grasping new ideas and trying things out (I’ll just name the Indian sitar that George plays instead of a guitar).

Those songs pushed me to the original albums on which they were released in the first place: with Rubber Soul and Revolver the Beatles really have their kick-off. Both albums have been highly influenced by drug abuse, but the music that came from there is amazing. Here, There and Everywhere has been named a favorite by both John and Paul in retrospect; Tomorrow Never Knows is an experience in weird music effects that obviously is a milestone in what can be done in a studio – I wouldn’t name it as a favorite of mine, but I can hear the genius there (and the line from the title of this posting is taken from Tomorrow Never Knows). Of course, it is songs like these that made The Beatles stop having concert tours, since the music was no longer reproducible on a live stage.

After that, they published the first so-called concept album Sgt. Pepper’s Lonely Hearts Club Band. Actually, there is very little holding the album together apart from the intro and the outro (which was defining this concept anyway for later bands). One can already distinguish the very different roads that the group was taking, but they managed to assemble a great set of songs here (many of which then found their way to the Blue Album as they could be considered as highlights). This is my personally favorite album, and not just because of the cover. Here originated the gems like Lucy in the Sky With Diamonds, She’s Leaving Home, A Day in the Life and For the Benefit of Mr. Kite. Especially A Day in the Life is one of the finest examples of how John and Paul worked together to produce astonishing music: this is actually a sampler of several snippets of musical and lyrical ideas of both, and none of those snippets really kicked in to make them a stand-alone song. But John and Paul connected those bits to a masterpiece. As a side-note: as long as The Beatles were a working group, John’s and Paul’s songs were always labelled as “Lennon/McCartney”, no matter who really wrote them.

Two Songs that have only been released as singles are about as old as Sgt. Pepper, again, two favorites of mine: Strawberry Fields Forever and Penny Lane. Remarkably, they have never been listed No.1 in the single charts (thus breaking a long row of Beatles’ singles that came out top in the lists) – but it was a close call. Since they could not agree which of the songs should have been the A side, it was labelled a double A side and the sales numbers were split half for both songs, which made it still to rank 2 and 3 of the charts. Those songs are highly introspective of John and Paul with rich, sometimes nearly unintelligible lyrics – and yet they can always cheer me up when I listen to them. I have listened to both these songs countless times and they are fresh every single time. When I discovered the Beatles, I learned a lot about what technical details make Strawberry Fields work and yet there is more to it. The magic still works today. In his last interview, John said he would re-do many of his songs if given the chance, reflecting all of his experiences – “especially Strawberry Fields”. It must have been a very special song for him, and it is most special for me.

Now, we return to the White Album. Having learnt about the whereabouts of the Beatles until 1968, I could understand a lot more of this, and appreciate it, too. The reason for its huge size is that the group had developed enough conflict with one another that no-one agreed on dropping one of his own songs – so instead of having one album with purely amazing music, the ended up with a double feature of all kinds of stuff. In particular, had I purchased this album in 1968, I would have either thought my record-player was broken or that Lennon/McCartney had gone insane. The second-to-last song is Revolution 9, a highly experimental, avant-garde piece: no lyrics, just tapes being wound forward and backward, time and again being tuned up in volume to yield a mixture of indescribable effects – very strange. But, Lord help me, I spent some time thinking about this, and this is not chaos. John has thought hard and deep about this, there is a lot of structure there (for instance, the “Number 9” text appears strictly at the start of each full minute) and it tells a story (just using a very unusual way), apparently one of war and peace and without telling it straight from start to finish but jumping through time. Even though I can hear my way through this by now, it is somewhat unsettling to hear it and to listen to John and George reading what seem to be random lines from newspapers in an apparently random fashion. Not to mention Yoko Ono talking about being naked (at least she has no singing verse as in Bungalow Bill).

As I mention Yoko Ono: the different development of the group members is today often connected to her name. She is supposed to be the women who brought the Beatles down. Well, she wasn’t. But she wasn’t totally innocent either. In the end, she was the perfect wife for John, she saw in him the artist and not the star, being an artist herself. When you look at pictures of the couple, they really look happy. So: no bad feelings about Yoko.

The White Album has had a huge influence on popular culture and beyond. Sadly, Charles Manson had put the thought in his mind to seek inspiration in the lyrics of the White Album (in particular from Revolution, Piggies and Helter Skelter), from which the Beatles obviously distanced themselves. If the Beatles ever had a common message, it was a peaceful one. Especially John and George tried to use their influence to bring more peace and less suffering to the world.

Their final musical project as a group was Abbey Road, and you can tell it has been crafted to say good-bye. Many consider the medley on Abbey Road a masterpiece (I, personally, don’t – it’s a sample of tiny ideas that didn’t grow to be complete songs), and you will find some fine music in it, such as Octupus’ Garden or Here Comes the Sun. Legend has it that the album should have been called Everest and the cover would have shown the group on top of the Mount Everest – but Ringo Starr objected to it because he didn’t want to travel around the world for this one picture. Instead he suggested “We might as well call it Abbey Road or whatever this street is called”. Hence originated one of the most famous album covers in the history of music – spontaneously within ten minutes and without a larger concept. Later, this cover really sparked the ideas of “Paul is dead”, but I won’t even address this here.

The final album Let It Be has been made after the fact, if you will: the group had already split and their producer used the existing unpublished tapes to make an album of it. Many things have been contrary to the wishes of the Beatles themselves (such as the choir in The Long and Winding Road), and Let It Be feels already strange when I listen to it. But it does contain footage from the legendary last rooftop concert that had to be stopped by the police, since traffic broke down around the place. So, it has some historical value, if nothing else.

After their break-up, each Beatle tried to be an independent musician. Only John succeeded and brought the unforgettable song Imagine to the world (next to some more avant-garde together with Yoko). In 1980, he was shot, aged 40. I will not mention the name of his murderer, since this person shot John to make his own name eternally remembered – this is the best reason not to memorize this name.

Paul had been the best songwriter of the group, at least as far as the mainstream songs go. Each of the songs I mentioned at the begining (those that I knew even before I got acquainted with the Beatles) have been written by Paul. Today, it’s different: when I think of my favorite Beatles’ songs, I think of Strawberry Fields, In My Life, Nowhere Man and others – all of which are John’s songs. Paul never got back to his top performances as a solo artist. Possibly, he lacked his counterpart John, they pushed each other to new highs as long as they worked on common projects, and they tried to switch each other’s styles (for instance in Helter Skelter, Paul tried to make a rock song similar to John’s Revolution).

George seems to be the Beatle that I have least to say about (especially considering that I only scratched the surface of what I have in mind about John and Paul). This is sad, in a way, since he really pushed the boundaries of what was done in western music these days. His ideas of Indian instruments and many great compositions make the later Beatles songs really wonderful. But he could never step out of the shadow of his band-mates.

Ringo was the last member to join the band, and in the end, he was “only” the drummer. Apart from Ringo and Phil Collins, no drummer ever got famous (most bands even switch their drummer from time to time – I know of hardly any case in which they switch their lead singers). Musically, he was quite limited, but the band always managed to write one song for each album in which Ringo would be the lead singer. This has two effects: Ringo actually grew better and the songs are rather easy-going, so you can sing along really nicely: Yellow Submarine, With a Little Help From my Friends and Octopus’ Garden are plainly wonderful songs. Besides, Ringo was a catalyst for many ideas of the other three: the songs Tomorrow Never Knows and A Hard Day’s Night were inspired by him. Phil Collins once considered Ringo to be a vastly underrated musician and one of the best drummers ever (and he must know it), and the other Beatles knew what they had with him – even though John once said sarcastically that Ringo might not be the best drummer in the world, he might not even be the best drummer in the band (since Paul had played the drums on some tracks in Ringo’s absence).

Some people have been called the 5th Beatle, having been part of the band earlier or having had some significant influence on them. In Hamburg, the place where their career started, there is even a statue for the Beatles including Stuart Sutcliffe who had been with them these days (and who was the first to wear the remarkable Beatles haircut of their early days).

There is so much more that you could say about the Beatles, about their music, where they came from and where they went – but I can’t hope for completeness and so I shall stop here. All I can do is to strongly encourage everyone to listen to their music. They are in there, and quality has no age.

A brief recapitulation

I have been idle for a little while now, which has to do with a severe loss of my spare time lately and some stuff I have to deal with in the little spare time that I get. But, fortunately, the stressful times will end some day, even though I can’t tell when I will actually be able to return to my usual “academic” topics.

One thing that has happened is that I abandoned my quest for the sorting algorithms in Knuth’s book. At least, for now. I have found that many things aren’t definite in this field – possibly because it’s rather hard stuff and not because no-one would care. But the heuristics behind the standard algorithms such as heapsort are 1) well-known and 2) not as deeply interesting as the thorough analysis. In my delving into Quicksort, for instance, I have learned a lot about the algorithm itself and I have learned the recursion-solving technique that is applied there. Knuth does show some sophisticated mathematics for the other sorting algorithms as well – but he is unable to achieve a definite goal with this. That disappointed me a little. Of course, this is no disgrace to his books, but I have found other topics to focus on in the meantime (and this doesn’t exclude me from returning to the Sort some other time – when I’m more in the mood for “academic finger exercises”). I am still at the elementary stage, but similarly to what I experienced when I returned to Galois theory, I have come to a deeper understanding of the subject. Even though I knew and understood the relevant stuff from start. And for the time after the elementary things, I already found some non-trivial subjects to turn my attention to.

Besides, Confused Matthew has published a video on Interstellar which is certainly worth watching. He does not go into details as not to spoil anyone – so he is rather cheerful in his review. He does not address the ending (which I found rather meh), but briefly says that he liked it – a point where I can’t actually follow. Begging for a longer review will hardly be of use there, sadly. By the way, he also addresses Inception, which he didn’t like after all. Well, to each his opinion.

This leads to a preview of my present historical interest: I have started re-reading Christopher Clark’s book on Prussia, which I have already read several years ago. At first, I wanted this to be a warm-up for the Sleepwalkers by the same author, but I got stuck… and I still am digging through Prussia’s history. My whole text on this will come here in several months, but I really adore Clark’s writing. As a parallel, I am using two other of my books on the subject (or, more precisely, some selected chapters from these books on the corresponding subjects from Clark’s book), about one of which I will definitely have to say more. For now, let it be said that Sebastian Haffner‘s historical essays should become a standard for lessons on German history – not because each and every sentence in there was the pure truth (it isn’t), but because I have never read any text on a historical subject that is as lucid and as clear and as knowledgeable than this. Haffner has found the ideal of “narrating” history (for my lack of a better English word – I am aware of the meta-joke here). More on this some other time.

Tiny Pines

Recently, I did one of the three things everyone is supposed to do in his life: I have planted a tree. Well, it’s not a tree yet, it’s several small plants that will grow to be pines some day. I am not yet sure what I’m going to do about them, maybe I’ll keep some in their pot and some will go outside when they’ve grown enough. But I am amazed indeed what those tiny pines look like. You can even see the needles already.

Let’s see how long it’s going to take me until they stop growing… 🙂

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