Diversions in Mathematics #2: Hilbert’s Hotel

In this instalment, I introduce the concept of infinity in a simple and (hopefully) entertaining way, which puts into practice the counting concepts introduced in the previous Diversion. In fact, Hilbert‘s infinite hotel was one of the ‘stories’ that got me seriously interested in mathematics in the first place, and so it is a pleasure to share it here. This is a very well-known piece of story-driven mathematics. I hope that experienced mathematicians who happen to come across this blog do not tire of hearing (reading) it again, and that they see the value in telling the story to the general public.

Just before we start: I assume knowledge of the definitions and notations introduced in the previous instalment, namely, the very basics of set theory.

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Sight-reading, AMEB exams, and studying maths

This would seem like an awkward mix of topics, but hopefully I can convince you of the similarities. The connection occurred to me recently, as I have been tutoring a student for theory and sight-reading in preparation for a grade 4 AMEB violin exam. (The AMEB is the Australian Music Examinations Board, a bit like the local Aussie version of the ABRSM, which is a world-wide music education organisation). In addition to preparing a set of pieces for performance, the student sitting the exam must also answer questions relating to music theory and history. I have only ever taken one AMEB exam in my life, so I don’t know exactly what kinds of questions are asked during a typical exam. Based on this student’s learning materials, I can deduce that, at this early stage in the progression of grades, they are likely to be questions regarding the fundamentals of music history and analysis, such as: “What is a concerto?”; “What is the form of this movement?” (binary, ternary, ritornello, and “through-composed” are the expected possible answers at this grade — sonata form comes later!); “The music of Mozart is representative of which period of music?”; “You just performed a piece by Handel, can you name some other pieces by Handel?”; “What does allegro moderato mean?; and so on. This is not particularly challenging. A student who is curious and motivated will probably know the answers already, via searching on Wikipedia and other sources on the internet. These basic concepts of Western classical music may also be covered in high school music classes, if the school is fortunate enough to provide them. For the average student, these facts can be imparted easily by the teacher during a lesson, with the additional advantage that explicit examples from the music being practised may be used. With a little more effort, the fundamental concepts of musical analysis and theory can be similarly acquired, or else taught in the lesson too. Remember, at this early stage (grade 5 or below), the student only needs to recall the basic facts.

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Diversions in Mathematics #1: How to Count like a Pure Mathematician

If you want to know what this is all about, read my introduction to the series.

If you have already read the introduction, then welcome to the first Diversion in Mathematics! I emphasise, as I did in the introduction, that I will write with the general public in mind, so don’t be worried if you don’t consider yourself a “fan” of mathematics, or if you’ve totally suppressed all memories of maths classes from high school. And if you are a keen mathematician, whether recreationally or studying seriously at college/university, hopefully you will also find these blogs to be of some interest.

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Diversions in Mathematics #0

Introduction to the series

I have always been interested in maths, and not only in the subject itself but also the ways in which maths is explained and taught. In general, a crucial part of studying and researching is to be able to communicate one’s findings to other people, who may or may not be knowledgeable in your field. For this reason, I’m all for popular science books and magazines, which (provided that it is done well) serve to explain scientific research in an accessible way, and to promote scientific awareness and appreciation amongst the general public. However, in my opinion, popular science books too often simplify, and even completely skip the mathematics behind the science. There is certainly a cultural aversion to mathematics — at least from my perspective as an Australian, and from my awareness of similar attitudes in the US and UK — which may be part of the reason for the lack of ‘real’ mathematics in popular science writing. Here is an anecdote: apparently Stephen Hawking’s publisher advised the great scientist that every equation he included in his A Brief History of Time would result in reduced sales. (There is one equation though: Einstein’s E = mc^2). Of course, this is one of the bestselling science books ever, and sits atop many a coffeetable, but I wonder how many people have seriously read it…

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Composer’s Notebook #3

I haven’t posted written anything here for a long time, since I’m busy with Uni study, but here’s a quick piece!

This is a fascinating video of the 2nd movement of Webern’s Variations. I highly recommend listening to the whole piece — after all, even with all the three movements combined, it’s only 5 minutes long! I want to point out a few important features here.

First of all, when I first came across this piece some years ago, I thought it was very beautiful. In case you’re interested, the recording was by Mitsuko Uchida (the album also features the Schoenberg Piano Concerto). Played by a computer here, it sounds rather ugly to me, which dispels the misconception that somehow serial music is completely emotionless.

Secondly, the binary form of this movement is perfectly visible in this animation, you can even tell where the repeats are! The pitch symmetry is also very clear from the graphics, and we appreciate the almost mathematical precision of the structure.

Thirdly, those of you more familiar with 12-tone music might hear that the harmony is very saturated. I haven’t studied this piece in detail at all, but I think one can discern by ear the inherent symmetries in Webern’s tone row manipulations. Indeed if you look at the score, you will see many palindromic phrases. In order to construct such highly symmetrical music it is necessary to restrict the set of “allowable” chords — this is what I mean by “saturation.” (There are probably more precise ways of expressing this idea, but I don’t wish to get too technical, and in any case I’m no expert in 12-tone theory). This practice is predominant in much of Webern’s mature style, but I’m thinking especially of the Concerto Op. 24, whose tone row has so many symmetries, it is the musical equivalent of a magic square. Click this link to the Wikipedia article if you want a brief overview of the Concerto. Here I cannot help but use mathematical terminology: because the harmonic ‘domain’ is so restricted, eventually we begin to feel that the harmony is static, since we have practically exhausted all the permutations of the particular elements in the set. Note that I’m not trying to be obscure — ‘chord’ in 20th century music theory can be virtually any group of notes, and in more abstract contexts is usually called a pitch-class set. This tendency towards saturation might be one of the reasons why Webern’s compositions are so short, yet each is exquisitely crafted according to a rigorous logic.

Of course, the most important dimension is missing in this discussion: the human performer. While it is fascinating to discuss Webern’s music in abstract and theoretical terms, it is altogether a different experience to listen to it, or indeed to perform it. But that’s probably a discussion for another time.

A Brief Ramble on Mathematics and Creativity

I recall that I was interested in mathematics from a relatively early age, probably from primary school. I was never a particularly talented mathematician — that is, I was certainly not a Wunderkind who won Olympiads, nor did I accelerate through school — but I always did well in maths, and most importantly, I have always had great curiosity and interest in the subject. More than likely my good marks were the result of my curiosity and passion, not the other way round. Oh alright, stereotypical Asian parenting also plays a role, I’ll happily admit. You may have read news articles or psychological studies about the phenomenon of math anxiety. I was rather skeptical about this when I first heard of it, but it seems to be a real psychological condition! You can read this rather dramatic article about this so-called phobia of numbers. Needless to say, this was not a problem for me. I enjoy maths, and I generally do well in maths, which only serves to increase my interest and confidence. When I don’t do well, there is still the enjoyment of a good challenge, of having grappled with a difficult problem or concept, even as I fall miserably short of the desired solution. Incidentally, if you ask a musician why they devote so much of their daily routine to practice and rehearsal, usually the response is along these lines: I enjoy it; music inspires me and I hope to inspire others; it is rewarding and fulfilling; this is my passion, and so on. In both cases, music or mathematics holds intrinsic interest for certain individuals, and they are motivated to master the skills or understand the concepts in their area of interest, and perhaps go as far as to extend current knowledge by exploring and developing new techniques and ideas. In short, I would like to propose that creativity is essential in mathematics. Obviously this is not the same kind of creativity necessary for artistic development, but nevertheless some form of it is needed. Now this is not a theorem which can be proved rigorously, but I hope you will allow me to conjecture a little!

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