The *Variations *were not my first attempt to write for wind instruments, but it is my first finished work for winds. (I had previously attempted solo pieces for oboe and bassoon, and “sonata”-like pieces for oboe and piano. I did not think any of these could have been successful, and so I have discarded all such fragments). The piece bears the subtitle *Small Steps and Giant Leaps, *primarily because the foundation of the whole piece is the main theme of John Coltrane’s famous piece *Giant Steps*. The subtitle also plays on Neil Armstrong’s famous words “One small step for man, one giant leap for mankind.” One could interpret this as representative of the fact that the quintet is my first completed work for winds, and hence it represents an important new step in my compositional development. At the risk of disappointing music analysts, the real reason is much more mundane. I think it is simply a rather cool title. If one *has *to interpret it, then I can offer the following suggestion: I tend to write melodic lines with large jumps, while the *Giant Steps* theme heavily features steps of major and minor thirds. It is the combination of these steps and leaps that characterise much of the material in this piece.

If you take a look at my Youtube channel, you will find other attempts at variation form. Perhaps the best one preceding the wind quintet is the *Intermezzo festivo* for string quartet. However, that piece follows the classical variation form more closely than the wind quintet, at least initially. (The piece transitions into a freer form halfway through). In the quintet, the *Giant Steps* theme does not appear until the very end of the work. This is in opposition to the classical form, where one hears the theme at the beginning, and then follow the variations. In this sense, even Arnold Schoenberg’s *Variations for Orchestra* (Op. 31) is classical, although the ways in which he varies the theme are of course more intricate and abstract than what is generally encountered in pre-20th century works. How, then, is my quintet a set of variations? Although the score is divided into sections which may indeed be identified with particular variations, it is not obvious (or at least it is not supposed to be obvious) how they are variations of a theme, which is not explicitly stated until the end anyway. Here, the word ‘variation’ conveys a much more general principle, which I think is similar to the term **developing variation**, often used in connection with Brahms. Another term I like is **thematische Arbeit**, or ‘thematic working’, which is often attributed to Joseph Haydn. I think the second term is more flexible, and hence easier to appropriate into a modern context. The general recipe is to start with a basic idea (the simpler the better), and see how much can be generated from it. Then introduce some embellishments, pertubations, *variations *— this produces a new but related idea. Now consider variations of this second idea, and so on. Of course, this process can happen in a nonlinear way. Moreover, the basic idea need not be a melodic fragment (although this is often a natural choice), but it can be something rather abstract. In my quintet, the basic idea, or **Ursatz** (a gross misuse of a term from Schenkerian analysis), consists of the following pair of elements:

The chord is comprised of the opening bass notes, while the second element is first five notes of the melody in *Giant Steps*. Observe that both elements coincidentally contain five notes, which is perfect for a quintet. An important secondary idea is the following voice-leading pattern, also featuring extensively in the melodic line of *Giant Steps*:

These two ideas comprise the essence of my wind quintet. I noted above that my use of the term Ursatz is a gross misrepresentation. In Schenkerian analysis, the Ursatz is supposed to be the fundamental structure of the entire piece — to put it facetiously, this means that “all of classical music is essentially the chord progression I-V-I.” However, in my current compositional process, the basic idea only needs to affect the ‘surface’ of the piece, and it does not necessarily determine the large-scale structure. (Controlloing large-scale structure remains one of my greatest challenges — you will notice that all the pieces I have written so far are quite short). The fact that the elements presented above do affect the large-scale structure of the wind quintet is the reason for the title *Variations*. These three elements are collectively the ‘theme’ of the composition.

I will offer some comments on the structure of the quintet that may be helpful for both players and listeners. It is easily observed that each variation features one of the instruments of the quintet. The order of appearance is: flute, clarinet, oboe, horn, bassoon. After these five variations, there is a fugal variation, which leads into the coda, where the *Giant Steps* theme is finally present. The way the variations are organised suggests an embedded multi-movement structure. One possible partition of the piece into movements is as follows:

- 1st movement: introduction, flute variation, clarinet variation (“Scherzo”)
- 2nd movement: oboe variation (“Adagio”), horn variation, bassoon variation (“Cadenza”)
- 3rd movement (Finale): fugue,
*Giant Steps*coda

Notice also that the bassoon “cadenza” recalls material from the introduction (namely, the staggered chord-building entries).

]]>Hey, I almost forgot I have a blog! (Since I’m paying for the site hosting, why not make more use of it).

This post will simply be an update, to let the readers of this blog (the number of which is non-zero) know what I am currently doing. It will be a random assortment of thoughts and comments. Right now, I am busy preparing for the semester 1 exams in the Pure Mathematics Honours program at the University of Sydney, but it is nice to take a break from study and write something here. Needless to say, it has been a very challenging semester, but also quite a rewarding one. Mathematics honours students are required to take a total of 6 courses throughout the honours year, as well as prepare a thesis. Many (?) people opt to take 4 of the 6 courses in the first semester, with the intention that more time can be devoted to the preparation of the thesis in second semester. But naturally, this means that one undertakes a lot of coursework in first semester (4 honours level courses at the same time is no joking matter), and as I am prone to procrastination, the time management has been especially challenging. Fortunately, I get along well with my thesis supervisor — who is conveniently also the honours coordinator this year — and he has been understanding and supportive during the periods when I had many assessments to submit and had not worked on the honours project!

**Some thoughts on teaching**

In addition to coursework, I have also been tutoring first-year linear algebra this semester. In all the maths courses at Sydney, students attend (or are supposed to attend) lectures where the essential content is presented, then they attend (or are supposed to attend) tutorials where they work through problems based on the theory. As it turns out, I enjoy teaching both mathematics and music. What both these disciplines have in common is that one can only learn by doing. For a music student, consistent practice is essential, and likewise, anyone who wishes to develop the mathematical skills necessary for a STEM career needs to practise doing mathematics through solving problems. I believe that the key idea is as follows: as a maths tutor, or as a violin teacher, one is faced with the task of *helping the student develop good practise skills*. This is different from delivering a lecture. Taking a tutorial is more personal, and one has to be sensitive to the various difficulties the students face. Although there will be some concepts that are challenging to most or all students, in general the nature of these difficulties will often vary according to the student. Thus it is often necessary to explain the same thing in several different ways, adapting to the responses of each particular student. In this way, one really deepens one’s understanding of the subject.

I think there are two main aspects that must be addressed when one teaches mathematics or music. I describe them roughly as “technical details” and “big picture”, although the union of these two concepts is by no means disjoint (I will explain what this means shortly) . Admittedly this attitude is derived from my music teaching, but I have found it useful for mathematics too. The first is quite obvious: in music, technical details include what is often called “basic technique”, the mechanical aspect of instrumental playing. For any instrument — including the voice! — there is an ample supply of scales, arpeggios, and études (studies) to develop basic technique. For violinists and especially so for pianists, there seems to be an endless supply of such études! In mathematics, the “technical details” are of course the theory and the methods, and the études correspond to the tutorial exercises and assignment problems. In music practice, notice that the mechanical aspects are tied together with the studies and exercises — that is, the studies and exercises are precisely the means by which one develops instrumental technique. In particular, I believe there is nothing in the music lesson that really corresponds to the “lecture”. Of course, the instrumental teacher does impart “theory” to the music student, but this is never done abstractly. It would be an absurd lesson if for example a violin teacher merely describes how to play *legato *and *détaché *while the student dutifully takes notes. Rather, it is all done through the technical studies, and most importantly *with the instrument*. I think there is something valuable for mathematics students and teachers in this perspective: it must be emphasised that *doing* exercises and problems is the primary means of develop mathematical technique. The material delivered in the lectures is of course essential, and without a clear explanation of the basic definitions and theorems, the students will not know how to begin working on the problems. But perhaps we should think of the lectures as a preparation, while the real, *active* learning begins when the student tackles exercises and problems.

The “big picture” aspect is slightly harder to describe, as it is less rigorous and more about developing intuition. In the music lesson, it would be terribly uninspiring if a teacher only focused on the mechanics, without giving the student any guidance on how to develop a sense of what is musically beautiful. After all, the whole purpose of developing good technique is to have the necessary skills to express one’s intentions through sound! (This is what I meant earlier, regarding the intersection of the two main aspects). A great musician not only has a clear idea of what they wish to express through music, but also knows exactly how to achieve it physically (and to do so reliably every time in concert!). In mathematics, I find that one can often get very lost in the details, and it is necessary to step back and simply ask “what is the subject *about*?” This is certainly not a rigorous, mathematically precise question, and hence we do not expect to have a rigorous answer. Similarly, a music student should ask themselves “what is this piece of music *about*?” If the piece of music is a song, a tone poem, or an opera, or anything where the musical process is guided by something extramusical, then you can answer in a definite way what the piece of music is about. But even in abstract instrumental music, this is an important question to ask. Some people will in fact be able to construct some sort of narrative to help their understanding of the abstract work. I happen to have a more abstract imagination, and therefore I prefer to consider the actual musical processes. Some questions I ask include: what are some of the important harmonies? how to the harmonies develop? how do the main motifs develop? what structures arise from these developments? and so on. (Not surprisingly, these are the same questions I consider when I compose music). In any case, the key idea is to break free of the mechanical aspect of instrumental playing for a moment and engage one’s intuition.

The idea is similar in mathematics: the students can churn through problems and memorise definitions and theorems, but that does not constitute an appreciation of mathematics. Quite the opposite is often true: many people develop an aversion or even fear of mathematics for precisely the reason that they experienced mathematics in high school as something tedious and repetitive, and formidably difficult but without reward. To develop a student’s intuition is a real challenge, and I do not claim to have succeeded. However it won’t be for lack of trying. For an example of how it might be done in mathematics, I highly recommend Grant Sanderson’s YouTube channel 3Blue1Brown. You will find that his videos present complex mathematical concepts with visually appealing but carefully constructed animations, and the theory is presented in a way that emphasises intuition over rigorous definition. Grant makes it clear that his channel is not a place to be overwhelmed in technical details, but rather to develop an appreciation for the mathematically beautiful.

**Comments on Joseph Haydn’s chamber music**

I have said to various people on different occasions that as a self-taught composer, I have probably learnt more about composition from Haydn than any other composer. I think about the string quartets very often, in particular the marvellous Op. 76 set (probably the most famous set out of all his quartets). However, may I also suggest the much lesser-known Op. 71 set, and also the unfinished quartet in D minor Op. 103 (I believe it was the last thing he wrote). Another recent source of inspiration is the wonderful set of piano trios which are called the “Bartolozzi” trios (the catalogue numbers are Hob.XV:27, 28 and 29, the keys are C, E and E-flat major respectively). In so much of Haydn’s music, and especially in the works mentioned here, one finds a masterful synthesis of contrasts: simple themes with complex developments; well-behaved, classically elegant phrases set against surprising twists of harmony and uneven phrasese; rigorous, “learned” techniques such as fugue rubbing shoulders with deliberately silly or awkward passages. The last point is important — it is essential to appreciate that Haydn’s music is full of charm, wit, and humour, and not even necessarily a refined sense of humour, but sometimes plainly outrageous. Let me share two of my favourite examples of Haydnesque shenanigans.

- The finale of the string quartet Op. 50 No. 6. This work is nicknamed the “Frog” quartet solely due to this movement. The main theme is a silly “croaking” figure in announced by first violin that is achieved using a technique called
*bariolage.*It is up to you to decide whether this actually sounds like a frog, but my suggestion is that it is even sillier if it does not represent a frog. Here is a lovely recording by the Tokyo String Quartet.

2. The finale of the piano trio Hob.XV:29 in E-flat major (the 3rd in the set of Bartolozzi trios). This movement bears the substitle “In the German Style”, although my feeling is that it is a parody of rustic German folk music, because it is hilariously *way too fast* to be any proper German dance (compare with this Ländler or any of Franz Schubert’s *German Dances *for piano to see what I mean). Suggested image while you are listening: imagine someone recorded a video of a bunch of drunk Bavarians dancing, and now press fast-forward. Here is a delightful recording from the Trio Wanderer.

Of course, Haydn could also produce serious, profound movements. Indeed, in his last (unfinished) quartet Op. 103, the key of D minor seems to refer to a similarly dark and serious quartet, Mozart’s quartet No. 15 in D minor (KV421). I believe he was too frail and sick to complete the quartet. The quartet’s unusually sombre mood perhaps indicates that he felt the end was near. Nevertheless, it is remarkable that Haydn’s characteristic wit still shines through in this work, although more subtly than is expected.

For those who already know a lot about classical music, I encourage you to rediscover Haydn, and for those who are not interested in classical music, then I suggest that Haydn’s chamber music may change your mind.

That’s all for now, and I promise it won’t be another 6 months until the next update! But firstly I must focus on the upcoming exams. Then I intend to continue writing the “Diversions in Mathematics” series, and I also look forward to spending time to compose music and practise violin. Finally, let me express a note of gratitude to all my friends and colleagues in the maths department at Sydney. I feel welcome in this mathematical community, which is not something I imagined when I first made the difficult switch from my music degree. Your support encourages me to maintain focus when things get challenging, and our conversations, whether about maths or otherwise, are invariably delightful.

]]>I would like to discuss the current advertising campaign from the University of Sydney, and in particular the chosen keyword:

**Unlearn. **Why? What is that supposed to mean? My criticism will be concerned mainly with this keyword. Let us first examine the following excerpt:

*“To be brave enough to question the world, challenge the established, demolish social norms and build new ones in their place.”*

I do not object to anything in that sentence, and nothing radical has been expressed either. I think that all universities aspire to be places of innovation and creativity, and this is more or less the usual image a university wants to impress on the general public. However, I do not agree at all with the use of the word “unlearn.” To challenge, question, innovate, re-imagine — none of these necessarily entails a process of “unlearning”. Furthermore, the prefix “un-” is a negation, and thus immediately suggests a negative process — an “undoing” of sorts — while the other buzzwords are more constructive.

For an example definition, I consulted the online Oxford Dictionary:

unlearnv. Discard (something learned, especially a bad habit or false or outdated information) from one’s memory.

To discard something from one’s memory requires an “active forgetting” of the thing to be unlearned. Not only does one make an effort to replace the undesirable habits and thought patterns with a collection of new behaviours, but one also ensures that the undesirable habits do not make a reappearance. In other words, the new behaviours must be *practised* until they override the old ones. If we treat the human brain like a computer (and in many ways, it is just that, albeit an immensely complex one), then we might view unlearning as an analogue to Ctrl-Z, the undo function. Suppose you are working on a PowerPoint presentation, and make some changes to the slides which you find unsatisfactory. It is safe to assume, when you hit undo and rework a new copy, that you don’t want any traces of the bad copy anywhere in your final presentation. The problem is that our brain cannot suddenly “switch on” these changes like a computer. If you have visited this blog before, you will probably have come across my posts on focal dystonia. Drawing from my experience with this condition, which has prevented me from playing the violin for several years already, I will give an example of what unlearning entails, and provide some reasons why I think this does not reflect what the university is trying to promote.

Here is a summary of the key facts: focal dystonia is a task-specific neurological condition which significantly impairs coordination. It is a malfunction of an extensively trained set of movements, such as those required for instrumental playing. A type of dystonia that may be more commonly known (i.e. known outside musical circles) is writer’s cramp. Medical treatments for dystonias are available but results may vary from person to person — unfortunately in my case, medical intervention proved unhelpful in the long term. What is possible, but very time-consuming, is the process of *unlearning *and* relearning*. The plan is to somehow forget the old set of movements which trigger the dystonia, and replace them with new behaviours, which will of course correspond to opening new neural pathways in the brain. This isn’t really magic — after all, this is exactly what the brain does when we learn new things, and although I am no expert in neuroscience, I think I am correct to state that our brains show capacity to learn even in adulthood. Nevertheless, I imagine anyone who has been monolingual for most of their life and then tries to pick up a foreign language in adulthood will probably find it a difficult task at first. Of course, the acquisition of any new skill will require practice anyway, but I like to use the example of learning a foreign language, because in this process one is required in a very obvious way to think* *differently, as the mind grapples with entirely new words and ways of expression. (For example, when learning German, it took some time for me to get used to the fact that verbs in a subordinate clause sit at the very end… and also to navigate the swarm of commas that ensue from using subordinate clauses).

However, a crucial difference with focal dystonia is that by learning a new language, you are not also trying to forget your native tongue. Thus you may now appreciate how truly difficult it is to forget one way of violin playing and replace it by another! What does all this have to do with the university’s Unlearn campaign? The first key point is that unlearning is a drastic option. Afflicted with focal dystonia, I had no other choice but to unlearn, if I wanted to play the violin again. The process of unlearning began by distancing myself from the violin completely. I did not try to play anything at all for many months. Moreover, I enrolled in a Bachelor of Science, which certainly kept me busy and my mind sufficiently occupied by things that were not violin playing. Very gradually, I picked up the violin again, and focused only on the most basic movements, and still without playing anything — it was as if I had to learn the instrument anew. This is where the unlearning and relearning overlap. Going forward in time a couple of years, I am now at a stage where most of the crucial unlearning has been achieved, and I am able to practise scales, etudes, and selections from my favourite repertoire, albeit slowly at first. The focus is now on relearning, to consolidate the new movements so that they eventually replace the old techniques. *This *arduous process is definitely not how we want to learn other things. We do not necessarily want new information to override previously learnt things completely. This leads to another main point, which is a result of my training in the sciences.

Two of the most important subjects in modern science (and arguably in all of human inquiry of the universe) are quantum physics and relativity, both born at the beginning of the 20th century. Both revolutionised the way we see the universe: quantum mechanics gives us the tools to probe the atomic world and ask questions about the fundamental building blocks of the universe; relativity (special and general) takes us to the opposite end of the spectrum, enabling us to explain the dynamics of the universe as a whole, with key concepts such as the expansion of the universe and the curvature of spacetime. However, the discovery of new laws of nature did not render the existing framework of classical physics obsolete. Quite on the contrary, classical physics is still extremely relevant and useful, but quantum physics and relativity extends our understanding of the universe to encompass a much larger class of natural phenomena. Thus, while it is undeniably true that physicists had to “challenge the norm”, introduce new postulates of physics, and “reimagine the world” (to quote from the ad again), I do not think it is true that physicists at the beginning of the 20th century were unlearning in any way. This is part of a more general pattern in science. It is not even desirable to unlearn hypotheses and theories which turn out to be wrong, because very often, to identify the error requires a shift in perspective and exploration of new ideas. To put this more bluntly, problem solving in science can often be about knowing what *not *to do, especially because it is not always clear *what to do* if one is working on a brand new theory! There should be a balance here: on one hand, we should not be overly burdened by older theories and methods, as that may hinder progress, but neither should we obliterate and unlaern everything from the past in order to move forward. As an example of how this may be applied in practice, I attach this extract from an interview with the mathematician Andrew Wiles:

I really think it’s bad to have too good a memory if you want to be a mathematician. You need a slightly bad memory because you need to forget the way you approached [a problem] the previous time because it’s a bit like evolution, DNA. You need to make a little mistake in the way you did it before so that you do something slightly different and then that’s what actually enables you to get round [the problem].

This was taken from this article. The key phrase here is “slightly bad memory”, which, in the spirit of this blogpost, I interpret as saying “unlearn some things, but not all the things!” Furthermore, the comparison with DNA mutations is quite interesting, and reinforces the point that to make progress on a problem, it is not always advisable to throw out a method entirely. Rather it is about making small nudges in different directions until suddenly you stumble on a new path that leads to a solution.

Diversion: I quite like this video, mainly because of the term “chasm of ignorance”.

But perhaps you have objections. Even if unlearning is not the typical pattern of progress in mathematics and the natural sciences, what about in other areas of inquiry such as the social sciences and humanities? Surely, there are harmful ideas and behaviours that should be unlearned and replaced. Wouldn’t it be great if we all unlearned prejudice, bias, and discrimination, for instance? Yes, perhaps, but again, I emphasise a distinction between realising that something is wrong or harmful, and trying to forget it entirely. When the university is advocating to “Unlearn X”, where X stands for various social issues, I do not think that any reasonable person would advocate forgetting the lessons taught by history — or else we are doomed to repeat it, as the popular saying goes. Notice that throughout this blogpost, I have focused all my arguments on the use of the word “unlearn”, and not necessarily on the main ideas of the campaign, which in fact are reasonable goals for a modern university. The real danger with repetition of the word “unlearn” is that it gives the impression that one ought to forget everything one has learned about a certain subject. As I have argued above, at least from a scientific perspective, this is unreasonable, for how can our current state of knowledge be possible without resting on a solid foundation of prior scientific work? To use Isaac Newton’s famous phrase, if we have seen further it is because we have stood on the shoulders of giants. This attitude seems to be in direct contradiction with unlearning. Even in the social sciences and humanities, where advancement of knowledge proceeds in a less rigorous fashion, and it is possible that a highly original thinker or artist produces a work seeminly out of the blue that blatantly challenges the accepted norms of the day, I would not agree that this is necessarily a result of unlearning. We may consider the composer and conductor Pierre Boulez as an example from the realm of modern classical music. In the 1950’s, Boulez became infamous for his avant-garde music and also for some provocative statements regarding the classical tradition, suggesting no less than to “burn down all the opera houses” ! This was part of the “tabula rasa” (Latin for “blank slate”) attitude at the time, when young composers essentially sought to expel all traces of the classical (and largely Germanic dominated) tradition from their work. I wonder if it ever occurred to him that later in his life, he would go on to direct highly acclaimed productions of Wagner operas! Throughout his long career, Boulez was always challenging musical traditions, but his performances of music from the 19th century and early 20th century makes it clear to me that he had not unlearned the heritage of Western classical music. In a strange way, his analytical, modernist approach even lends a hand in rejuvenating the old masterpieces, casting them in a different light.

One final example to reinforce the main arguments:

Out of the all the “unlearn” posters, perhaps the worst is the one above, *Unlearn truth*. As a mathematician, this is complete nonsense. Mathematical truths are simply not unlearned, otherwise they would not be truths in the first place. For example, we may not be doing geometry in the same way as the Ancient Greeks did — so perhaps in a sense, we have unlearned the original methods and terminology — but the theorems still hold. Now thinking more generally, it is still not more reasonable to “unlearn truth”. Reading the description, the intention is to motivate us to think critically, and to prevent ourselves falling into the traps of fake news, propaganda, and conspiracy theories. This is undeniably a positive and worthwile endeavour, but the phrase “unlean truth” seems to be in direct contradiction. Could it not be that we are in fact already in the process of unlearning truth precisely because of fake news and internet bullshittery? If anything, we should be actively relearning truth.

In summary, my main arguments against the use of the word “unlearn” in this advertising campaign stem from my experience with focal dystonia, and the need to unlearn focal dystonia in the hope of performing violin again in the future. I see contradictions between the word “unlearn” and the intended message as explained in the descriptions below the heading, and thus the campaign has potential to be misleading and counter-productive, especially when there are posters around Sydney CBD with just the words **UNLEARN TRUTH **without context. Even if the meaning is clear, then the reader must accept that “unlearn” is somehow synonymous with other education buzzwords like “re-imagine”, “innovate” and “challenge”. Leveraging my own experience with an unlearning process, I hope I have made a case for why “unlearn” should be identified as separate from the other concepts, which I believe are more constructive and productive.

Click here to visit the Unlearn campaign.

]]>At the beginning of the year, I promised that I would try to write more regularly. This has clearly not been achieved! In my defence, studying mathematics full-time requires much dedication, patience, and practice — not unlike learning a musical instrument. But now I have time to write since I have completed my semester 1 exams.

(Main article is below)

A few months ago, I was asked to contribute an article on composition for the October 2017 issue of *Stringendo*, which is the bulletin of the Australian Strings Association. (Incidentally, I have already contributed an article on focal dystonia, which appears in their April 2017 issue). The theme of the upcoming issue is “Add-ons”, which I assume refers to “other” activities in which string players are engaged, besides the usual performing and teaching. Since it is known that I am a self-taught composer, I was asked to write a piece describing my approach to composing for the orchestral stringed instruments. This is far too broad of a topic, so in the end, I decided (with the editor’s approval) to limit the scope of my contribution to writing for the string quartet. Obviously I cannot reproduce the piece on my blog at this time, but I do want to share some thoughts specifically about composing for the violin in this blogpost.

For obvious reasons, writing for violin comes most naturally to me, and as a result, I find it impossible to give impartial advice for composing for the violin. Hence, in this post, I will not attempt to give general “tips and tricks” for writing music for the violin. Instead, let me express what I want to do using this metaphor: imagine “violin music” as a self-contained universe, in which inhabit many violinists and composers. Then I am one of the many tour guides of this world, and what I decide to show you will depend on my (necessarily limited) travel experience in this world, and on what I find most interesting. You only need to ask a different tour guide to see potentially entirely different features and landscapes (or should that be soundscapes?). In this particular tour, I would like to focus on some details of **bowing **and **articulation**.

Anyone who wants to write serious music for orchestral stringed instruments should be knowledgeable about how the bow is used. After all, it is the primary means of sound production, and how well a string player performs is determined by their mastery of the art of bowing. I wonder if non-string players find the proliferation of different terms (mostly in French and Italian) used to describe bowing techniques rather confusing: *détaché, legato, spiccato, sautillé, flautando, portato…* and so on. We will later clarify exactly what these kinds of instruction mean. An essential question to consider is how specific should the composer be with regards to bowing. Often in baroque and classical scores, very little bowing instruction is specified, but that is because there is a well-defined style of playing associated with music of that era, and an experienced string player will intuitively know what to do. For composers nowadays, I suggest giving enough bowing information so that the performer knows, or at least can infer, what kind of sound you wish to achieve. (Of course, if you’re writing something with lots of extended techniques, you’ll need to specify all of these very accurately, but I will be thinking more “classically” in this blogpost). How might one go about this? Instead of just going through the list of bowing techniques, I suggest it is more helpful to think *physically* about what bowing entails, and connect that to the notation. Therefore I must emphasise my opinion that is impossible to talk about bowing without including **articulation**.

Every kid who learns violin starts by playing “separate” notes (*détaché*) and “slurred” notes (*legato*). As a provisional definition, let’s say that *détaché* means something like “each note is played on a separate bowstroke” and *legato* is “several notes are played on the same bowstroke”. So *détaché *is a faster bowstroke (one note per bow), while *legato* is a slower stroke (many notes per bow), right? This is fine in many cases, but the reality is that bowing is a lot more flexible than such a definition would imply. Consider the following simple exercise.

Naïvely, we might identify that the first two bars demonstrate *legato* playing, while the next passage consists of *détaché *playing. This is actually correct, but perhaps not for the reasons suggested by the provisional definition above. Now think about what the bow is actually doing — you should realise that in both passages, the bow does *exactly the same work*! In each passage, it makes a total of two “return trips” from the frog to the tip and back. So this shows that whatever *détaché *and *legato* mean, it cannot just be a bowing pattern, otherwise there would be no difference between the two passages above. A more accurate definition is as follows: *détaché* means that the notes have to be *articulated** separately*, while *legato* refers to *smoothly connecting the notes*. The emphasis has been shifted to how the notes are articulated, and thus the terminology does not necessarily indicate a *specific* bowing pattern. Rather, it describes a manner of “speaking” with the instrument, which is how I think about articulation. Consider the following exercise:

“Now you’re being obnoxious,” I hear you grumble, “what difference does that make? You didn’t even write a slur in the second exercise!” The point is that I expect the violinist to make two different kinds of articulation. In the *détaché* exercise, the notes should be clearly articulated as separate, but in a “neutral” kind of way (i.e. without any kind of accent). I think this would be the default way to play such a passage, if no other context was given, so the designation *détaché *is a bit redundant. But the *legato* marking is definitely not redundant — in this exercise, I expect the violinist to make good effort to disguise the bow change, so that the notes sound smoothly connected, *as if* they could be played all in one bow, or sung in one breath.

This begs the question: what is the point of writing a slurred line then? I find that this quite a fascinating and somewhat difficult point of discussion. The short, unsatisfying answer is that *it depends*. Most of the time, a slur over a group of notes indeed indicates that the group should be played in the same bow stroke, but this is not the only function. There are many examples even in standard repertoire where this clearly cannot be the case:

Given that the tempo for this section is usually around crotchet = 52, have fun trying to fit all those notes in the same bow (especially the cellist)… If you are still wondering, then this more extreme example should clear the doubts:

(These were the two examples that stood out to me. Purely by coincidence, I chose works with consecutive opus numbers!)

Very often in Brahms’ string parts (and also in Richard Strauss and Wagner), slurs indicate **phrasing** as well as *legato*. This is no surprise to a pianist of course, but perhaps it is an overlooked fact when it comes to composing for strings. While we know the phrasing, Brahms does not give explicit instruction on how to bow the passage. In fact, it is not at all obvious in general how to construct a good bowing* — *those pesky sequences of up- and down-bows which have probably sparked many arguments in rehearsals. However, I think both performer and composer can agree that a good bowing is one that finds the best compromise between respecting the phrasing and articulations indicated, and being relatively uncomplicated to execute. In many cases, like the opening passage of Brahms’ First Symphony as shown above, the policy is to *leave the bowing to the performers*. Of course, if you want a very specific effect, then by all means specify a bowing, but in general, I tend to prioritise phrasing, and give the performer some degree of freedom (and hence responsibility!) in selecting a bowing. Let us consider one more example before moving on:

Mendelssohn’s beloved Violin Concerto in E minor needs hardly any introduction, but I draw attention to the opening passage to illustrate further how it is more useful to think about *détaché* and *legato* as describing articulation first (and then the bowing follows). One *could *play the first two detached B’s on separate bows, and likewise the E in the following on a separate bow, but traditionally violinists have performed a more “fluid” bowing, such as the one shown below:

The important thing to note is that the two B’s should still sound detached, even though they are played on the same bowstroke (up-bow). Likewise, if we use the particular bowing above, the down-bow E in the third bar should sound detached from the slurred B to G… although perhaps not too separate. I believe many violinists will not mind including an audible but tastefully executed *portamento *between the G and E… but I will not delve into the details of interpretation for now.

The difficulties discussed above are present due to the overloaded function of the slur — it has evolved to somehow convey *legato*, bowing, and phrasing. The dot above a note is another symbol that is overloaded, especially in violin music, and it can be quite a headache to work out what exactly is meant when there is a combination of slurs and dots! Fortunately the ambiguities are often resolved in context — often by additional instruction from the composer, or appeal to a well-known performance tradition. To discuss these notational difficulties will take us too far from where I intended to go with this blogpost, thus I will now focus the discussion back to bowing technique. As I stated at the beginning, it is instructive to consider the physics (in a qualitative way of course). So far, we have only considered *détaché *and *legato*, which are strokes that are played** on the string** (*alla corda*)**.** I think most kinds of bowing can be characterised by the following fundamental parameters:

- Type of contact between bow and string
- Contact point along the string
- Contact area along the bow
- Bow speed

The first parameter, “type of contact”, actually consists of several “sub-parameters”:

- The amount of friction between the hair and the string. This can vary from hardly any friction (a “floaty”, or airy sound,
*flautando*) to so-called “over-pressure” (excessive friction), which is sometimes found in modern works. The result is a harsh, scratchy, distorted sound (e.g. as shown on the cello in this video). Speaking of modern techniques, you could extend this definition to include*col legno*(playing with the wood of the bow) — that is certainly another type of interaction between the bow and the string. - The length of the contact period, that is, long (e.g.
*legato*) or short (e.g.*spiccato*, a “bouncing” stroke played**off the string**), and everything in between; - The kind of
*attack*on the note. An analogy is often made with consonants in spoken language, so assuming standard English, “k” would be a stronger attack than “b”, which is stronger than “m”, for example.

The second parameter, “contact point along the string”, can vary from being over the fingerboard (*sul tasto*) to being at the bridge (*sul ponticello*). These are special effects, but it turns out that we don’t need any more specifications. The contact point along the string will vary naturally throughout the course of regular playing, in response to different dynamics and expressive markings, so it is unnecessary and probably impractical to prescribe the contact point exactly.

The third parameter refers to where along the bow a particular bowstroke is executed. For slow *legato* playing, for instance, the contact area will often be the whole bow, while for *spiccato*, it is generally a small section of the bow around the middle third to upper-lower-third (i.e. the upper portion of the lower third). I don’t think these designations are exact, i.e. I don’t think anyone has tried to set numerical measurements to define the “middle third”, but string players will know intuitively what is meant. Sorry to everyone else! It is more useful to see it in action:

Finally, variation of bow speed is not usually something composers need to specify, since it should vary naturally, like the contact point along the string. However, I include it here for completeness, as well as the fact that it is a very important aspect of bow technique, and hence useful for composers to be aware of it.

The four parameters above pretty much span the space of all possible bowings (and extended techniques can be incorporated by extending the definitions… this is sounding a bit mathematical now). Certainly we have characterised all the bowing techniques one is likely to encounter in standard violin repertoire, including chamber music and orchestral parts too. We can finally look at the some of the terms given to bowing techniques! What I will show is that any bowing-related term can be easily described in detail using the four fundamental parameters. In this way, I hope that the reader will gain a detailed insight into the how the bowing is executed, as well as the resulting sound.

and*Détaché*we have already discussed at length*legato*— from Italian*Spiccato**spiccare*, “to separate”, which doesn’t give much information. This is in fact a “bouncing” stroke, and generally the contact between bow and string is light. It is played most often in the middle-third to upper-lower-third, as mentioned above. The interaction time between the bow and string is obviously short. The bow starts slightly above the string, then makes contact briefly before being lifted up again. When done well, the notes should sound crisp and clear. A great (and notorious) example is the*Scherzo*from Mendelssohn’s*Midsummer Night’s Dream,*which is traditionally played this way.

— from French*Sautillé**sautiller*, ” to hop”. The key difference here is that the bow is made to jump by its own accord — it may be said that this is a “passive” stroke — and hence this can only be achieved when the speed of the notes is sufficiently fast. See this article for more information. The*spiccato*on the other hand is “active” — each note is articulated separately by the bow arm. The*sautillé*is lighter than the*spiccato*, and the contact area along the bow is even more restricted, since the player has to find the “bounciest” bit of the bow, which is generally around its balance point. Once again, the articulation should be crisp and clear, and due to the speed, this is often a very virtuosic, impressive effect. We have another example from Mendelssohn, this time from the third movement of the Violin Concerto in E minor. As in the preceding example, note that*sautillé*is not an explicit instruction here, but it is traditionally played this way. The speed of the music and also the instruction*pp leggiero*suggest this manner of playing quite naturally.

— self-explanatory (probably?). The player throws the bow onto the string, letting it bounce freely, the result is like a rapid-fire*Ricochet**spiccato*or*sautillé*. However, this stroke is usually played on a few notes at a time, since this action can be controlled precisely. I include a famous excerpt from Rossini’s*William Tell*overture by way of example. Again, there is nothing specifically asking for ricochet in the score, but it is traditionally played that way, and it certainly makes sense due to the speed of the music. Another way this stroke is used commonly is in arpeggiation of chords. With the appropriate impulse from the bowing arm, the bow can bounce across the strings easily. Paganini’s*Caprice No. 1*consists almost entirely of this highly virtuosic bowing.

— Italian for “detached”. This is a very general term indicating that notes are to be played shortened and detached. For stringed instruments, this is done on the string, i.e. the bow is at first stationary on the string, and then released. The initial friction between the bow and string creates the attack, which is like a consonant. With some skill,*Staccato**staccato*can be played pretty much anywhere along the bow.— from French, meaning “hammered” (Italian version is**Martelé**). The physical mechanism is the same as for**martellato***staccato*, but the attack is strong — i.e. start with a significant amount of friction between bow and string. It is often not necessary to write this specifically — for example, if a violinist is instructed to play*staccato*notes at the dynamic*fortissimo*, what will happen is*martelé*. Unlike*staccato,*which has the built-in implication that the notes are short, the*martelé*works very well on long notes too. However (see below), I feel that*martelé*is limited to louder dynamics.— from Italian, meaning “marked”. This is another imprecise term, with the same physical mechanism as*Marcato**staccato*. I think many would agree that it is somewhat stronger than*staccato,*in other words, the attack is sharper, more pronounced, and perhaps the decay of the note quicker. But I don’t think it is as strong as*martelé*. In fact,*staccato*and*marcato*will work in any dynamic marking (e.g. you could write*pp ma marcato*), but to me,*pp martelé*looks very bizarre.

For a modern case-study, many of the above bowing techniques can be found in Stravinsky’s *Dumbarton Oaks* Concerto (see the end of the blogpost).

*Remark*: It is useful to compare the attack of a bowed note to pizzicato. To pluck the string, the finger must “catch” the string firmly, then release to allow the string to vibrate, Similarly, in *staccato* and *martelé*, the bow “catches” the string (with varying amounts of friction) and then releases.

Of course this is not an exhaustive list, but I think I’ve covered the most important ones. I could have just started with this list, but then this entire blogpost would be pointless. My aim has been to equip the reader with a physical understanding of bowing, rather than merely collect a bunch of terms in Italian, French and German. I hope the usefulness of this approach has been made clear. For composers in particular, I suggest that thinking about the four main parameters of bowing identified above enables a closer connection between the composer’s conception and the act of performance. From the performer’s perspective, the worst kind of piece to play is one that is so terribly difficult and unrewarding, because the composer has little understanding of what is *physically involved* in playing the instrument. Where the composer and performer come together is both in real life (yes, composers and performers should actually *talk **to each other*) and, especially if the composer is long dead, in music notation and terminology. If the composer has a good understanding of how to play the instrument, and if the performer has a good understanding of notation and terminology, in spite of the inherent limitations, a meaningful and fruitful collaboration may be established.

There is no better way to learn about bowing than to see and hear it in action. Here are two recordings (I think of the same performance) by the *Passacaglia* (*after Handel*) for violin and viola by Johan Halvorsen, one of them with score, the other with video (so you can see how the bowings are performed). Along with the Stravinsky, these videos contain pretty much all the bowing techniques mentioned in this blogpost — how many can you identify? And can you find examples where a single technique, such as *spiccato*, is played in different ways?

**Bonus example!**

Some of the most effortless ricochet I’ve ever seen — and the left-hand pizzicato is bordering on magic.

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

In the previous part, we looked at two of the most fundamental objects in all of mathematics: numbers and sets. We ended with the inductive definition of the natural numbers, which consisted of a pair of statements:

- 0 is a natural number
- If
*n*is a natural number, then*n*+ 1 is also a natural number

We concluded that infinity is a consequence of this definition, and proved it via a simple contradiction argument: suppose that there is a “largest natural number” N, then by the inductive definition, N + 1 must be a natural number, but this is larger than N. So there is no such “largest natural number”; in other words, the set of natural numbers is infinite.

Before we begin our visit to Hilbert’s Hotel, an important definition to know is the following:

Definition 2.1The

cardinalityof a setXis the number of elements inX. It is denoted |X|.

Informally speaking, cardinality is simply the size of the set. For example, if *X *= {1, 3, 6, 10}, then |*X*| = 4. If *X *is the set of all letters of the English alphabet, then |*X*| = 26.

How do we know this? This seems like a trivial question, but only because we take something very important for granted. Indeed, we were able to compute the cardinalities in the examples above quickly because we can **count** the number of elements, and counting is inextricably linked to the natural numbers*. *The importance of this is reflected in the very language that mathmaticians use — this will become clear later! I mentioned in the previous Diversion that the indigenous Australian language Warlpiri can only make a distinction between “one” and “two”, and all quantities greater than two can only be described as “few” or “many”. Nevertheless I imagine that a Walpiri person could still tell which of the following sets contains more elements:

They could simply take one element of A and match it with some element of B. At the end of this procedure, they will notice that B has “a few” elements with no partner from A, and hence conclude that B contains more elements. This is a seemingly innocuous exercise, but it introduces the concept of a **mapping** — a relation between sets.** **In fact, I think I should now introduce the following enormously important definition:

Definition 2.2aA

functionormappingis a relation from one set A to another set B, such that every element of A is uniquely assigned to an element of B. We write:

f: A → BThe set A is then called the

domain,and B is thecodomain.If the element

a∈ A is mapped to the elementb∈ B, we denote this:

f(a) =b

Informally, a function is a rule for assigning elements of A uniquely to elements of B. (In high school, we almost exclusively study functions which have a nice, simple formula, but this is not necessary). The “uniquely” bit is very important. Here is an alternative definition (taken, with a slight adjustment, from Michael Spivak’s textbook *Calculus, 3rd ed.*) which emphasises this point:

Definition 2.2bA

functionis a collection of pairs of objects with the following propery: if (a,b) and (a,c) are both in the collection, thenb=c.

Using the notation in first definition, this means that if *f*(*a*) = *b*, and *f*(*a*) = *c*, then necessarily *b* = *c*. In the diagrams below, the left hand side is a representation of a function, while the right hand side is not, since one of the red dots is mapped to two different elements of B. Notice that not everything in B needs to be paired with something in A.

We also allow elements of B to correspond to multiple elements in A. This actually happens quite a lot: for example, take the function defined by squaring a number, . For simplicity, we take both the domain and codomain to be the set of integers, . Then* *, and also , but of course 3 is not equal to -3, despite both being mapped to 9. There is also the trivial example of a **constant function**, for example: *f(x) *= 5. For any input value *x, *the function outputs the number 5 — a perfectly legitimate but otherwise boring function.

**Thinking time:**The**image**of a function*f*: A → B is the set im(A) = {*f*(*a*) | all*a*in A}, i.e. the set that you get from applying*f*to everything in A. (This is also called the**range**of*f*)**.**In the example above, we have . What is the image of*f*? Is it the same as the codomain?

OK, with this important knowledge in mind, let’s visit Hilbert’s Hotel!

*As you approach the Grand Hilbert Hotel, you are astounded by the magnificence of the building, and cannot wait to check-in to your room and relax. Its location is ideal for your vacation — nestled in a lush Bavarian forest, yet only a pleasant half-hour walk to the banks of the Obersee. Moreover, it was advertised as being infinitely spacious. You had initially thought this to be a strategic exaggeration for marketing purposes, but walking up to the front entrance now, you think perhaps you were too quick to judge.*

*Just as you are about to step into the revolving doors, you notice a large sign displaying: *NO VACANCY (Please enquire within). *Unbelievable! Your room had been confirmed some days ago, surely there could not have been a double booking? “So much for ‘infinitely many rooms’,” you grumble as you ring the bell for reception. Within seconds, an elegantly dressed, bearded man with spectacles emerges from the office behind the reception desk. You notice his nametag: *Bernhard Riemann, concierge. *He speaks quietly, yet somehow with great authority: “Guten morgen, how can I help?” You indicate to the “No Vacancy” sign outside, and ask about your room.*

*“Ach ja, we are operating at full capacity. But that is no problem! We can certainly accomodate you today, but please wait a little, as I will have to organise for some guests to move around…” Slightly impatient, you interject: “Wait, sir, just how many rooms do* *you have?”*

*“Why, we have countably infinitely many! Just as it says on our brochures.”*

*Countably infinite? How can something be infinite *and *countable? But you suspend your disbelief for now, as Riemann is rummaging through a pile of papers. He produces a form for you to sign.*

*“Here you go, I will put you into room 1. It is not the room you booked, but you know how it is, many of the rooms are the same up to isomorphism. And room 1 does have a nice view of the Obersee, as you requested.” *

*While you did not understand exactly what was meant by “isomorphism”, nevertheless something has aroused your curiousity. “Herr Riemann, let me get this right,” you enquire, “the hotel is full, but you can accommodate me… just by moving people around?”*

*“Exactly right. I guess that means Mr. Ramanujan will finally be able to move into room 1729, he should be happy about that!”*

*Your expression must have betrayed your confusion, as Riemann continues:*

*“It is a simple affair, my colleague Georg Cantor has devised an excellent procedure to solve such problems. Our hotel contains countably infinite rooms, that is, the number of rooms is the same as the cardinality of the natural numbers. Today we are completely full, so every natural number has been assigned to a guest. But if a new guest arrives, we can simply ask the guest in room *x *to move to room *x + 1. *So, the guest formerly in room 1 will move to room 2, and the guest originally in room 2 will move to room 3, and so on ad infinitum, giving you a vacant room 1. Is this not ingenious?” he remarks proudly.*

We have just heard from Mr. Riemann the concept of a set being *countable. *Of course, you know already that any finite* *collection of things can be counted, but you see now that an *infinite* set can also be countable!

** Definition 2.3**

A set *X *is **countable** if either:

*X*is a finite set, or- |
*X*|

In the latter case, *X *is said to be **countably infinite.**

This is why I wrote earlier that counting is inextricably linked to the natural numbers. Actually, this definition is not quite complete (which is why I haven’t put it in a box). To complete it, let’s analyse how you were able to be accommodated at the Grand Hilbert Hotel, despite it being full. We will put into practice our knowledge of functions! (Note: there is some discrepancy amongst mathematicians whether or not zero belongs to the natural numbers. The following discussion should convince you that this discrepancy does not matter).

Consider the natural numbers = {1, 2, 3, …}. This set is also the set of all room numbers at the Grand Hilbert. Initially, before you arrived at the hotel, all* *the rooms were occupied, so that tells us there were guests. Now add yourself to the set — for reasons which will be clear very soon, let’s assign you the number zero. So now we need to **map** the set of guests X = {0, 1, 2, 3, …} to the set of room numbers = {1, 2, 3, …}. Can you find a simple rule to do this?

Indeed, the rule that Riemann described is the simplest solution. Let be a function, and define for every . Then you can see that *f*(1) = 1 + 1 = 2, *f*(2) = 2 + 1 = 3, and so on, exactly as Riemann stated. In particular, you have been assigned to room 1, or in our function language, *f*(0) = 1. This function has mapped the set *X* to the natural numbers, and there is no element left out in either set. Every element in *X* has a unique partner in , *and* vice versa — that is, every element of the codomain corresponds to a unique element in the domain. This is a special kind of function, called a **bijective function, **or **bijection** (or the delightfully clumsy “one-to-one and onto function”).** **In particular, it tells us that the cardinalities of the two sets are the same! This was probably obvious to you for finite sets — you know when two sets contain exactly 10 elements, for instance — but now we have defined the notion of “same size” for two infinite sets. So let us now complete the definition above:

Definition 2.3A set

Siscountableif either:

Sis a finite set, or- |
S|is the cardinality of the natural numbers, i.e. there exists abijectionfrom the setSto the natural numbers.In the latter case,

Sis said to becountably infinite.

Here’s where the weirdness of infinity becomes apparent. Intuitively you might think that the set *X *is “larger” than . Indeed, *X* actually contains all of , since *X* has the element 0 in addition to {1, 2, 3, …}. We can write . But since we have found a bijective function , this shows that the two sets have the *same* cardinality — the same size! We can in fact do better. Suppose you bring along *n *– 1 other friends on holiday, so you have *n *people checking in. Then Herr Riemann can simply ask the guest in room *x *to move to room *x *+ *n* (for all *x* = 1, 2, 3, …), leaving the rooms* *1 to *n* vacant. This is another bijective function from the set of guests to the set of rooms, and once again, everyone is happily accommodated. Loosely speaking, as long as *n *is a finite number, you can “add” *n *elements to the natural numbers, and its cardinality remains the same! This is the reason why infinity cannot be a number. If you treat it like a number — let’s use its proper symbol, ∞ —* *it makes no sense to write something like “∞ + 127 = ∞*“*. There is nothing wrong with infinity or 127 for that matter, but rather, the operation “+” becomes meaningless. However, in the language of sets and functions, there is no problem. Take the set , and another set X containing 127 elements, and then you can certainly construct a bijection , and thus, . You can now also appreciate why mathematicians insist on precise, rigorous definitions of the objects we study.

**Thinking time:**Can you see why*subtracting*a finite number of elements from also does not change its cardinality?

*“Ah yes, I see how it works!” you remark. “As long as a finite number of new guests arrive, you can always shift along the existing guests by the appropriate number?”*

*“Well observed, that is correct,” Riemann says with a subtle smile. “I am glad that it is clear to you, many other guests become terribly confused whenever they ask me about such a thing.”*

*You are distracted by a commotion from outside, which quickly grows louder, until it begins to sound like a busy marketplace. You see an enormous congregration of guests piled up at the front door. Herr Riemann had also noticed, and exclaimed all of a sudden, “Ach Gott, I do apologise! Perhaps you will not object to staying in room 2 instead? I assure you, it is isomorphic to room 1, and…” he trailed off as he went back into his office, but then emerged quickly again with the keycard room 2. “As you have noticed, we have a countably-infinite busload of new arrivals.”*

*You have only just understood how the Grand Hilbert can accommodate finitely many new arrivals, but how could it possibly fit *infinitely *many additional guests?*

*“This is most extraordinary, but surely, you cannot simply use the shifting method again?”*

*“Again, well observed. Thankfully, Herr Cantor has also solved this problem, and the solution is not any more difficult. We can simply ask the guest in room *x *to move to room *2x*. You were previously assigned room 1, but now you will move to room 2, the person in room 2 will go to 4, and so on.”*

*You pause for a little while, and suddenly realise the implication. “Aha, so now, all the odd numbered rooms are free, and since there are infinitely many odd numbers, everyone will have a room!”*

*“That is indeed correct. You are quite astute, I must say. Are you perchance a student of mathematics?” Riemann seems genuinely impressed.*

Well, look at you, smarty-pants. Let’s see why this solution is able to accommodate countably infinite new guests into the hotel. Once again, our target set (codomain) is , which is also the set of room numbers. This time, our domain contains what is essentially two* *copies of : we have the existing guests at the Grand Hilbert, plus the newly arrived countably-infinite busload. Let’s write this as:

- Set of existing guests =
- Set of new guests =

and so, we are looking for a bijection . This time, let’s try out the Spivak defintion of a function (Defintion 2.2b), and write down the ordered pairs.

Firstly, we move the existing guests according to the rule “guest in room *x *goes to room 2*x*“. Thus we obtain this set of ordered pairs {(1, 2), (2, 4), (3, 6), (4, 8), … (*x*, 2*x*), …}. Then we accommodate the new guests into the now-vacant odd numbered rooms: {(1′, 1), (2′, 3), (3′, 5), … }. The existing guests are assigned to only even numbered rooms {2, 4, 6, …} and the new guests occupy odd numbered rooms {1, 3, 5, …}. Indeed, the union of these two sets gives us all the natural numbers, and everyone is happy!

**Thinking time:**what is the rule/formula that assigns the new guests to their rooms? You can drop the ‘ from the numerical labels, I used them initially to distinguish the two copies of .

How can we check this is a bijection? It is clear that each guest has been assigned to a separate room, but let’s check the “other direction”: given any room number, can you work out which guest has been assigned to it? For example, room 8 is now occupied by guest 4, and *only *by guest 4, and (verify this!) room 17 is now occupied by guest 9′ from the new arrivals. You should convince yourself that given any room number, it is possible to “reverse” the procedure and work out which guest has been assigned that room.

I mentioned earlier how arithmetic operations are meaningless when dealing with infinities. Naively, what we have just done is essentially “∞ + ∞ = ∞”, which looks ridiculous. In the more precise language of sets and functions, we have created a bijection from a set containing two copies of the natural numbers to the natural numbers themselves. There is no contradiction at all — this reinforces the point that our everyday intuition of concepts such as “size” do not work when dealing with infinity.

*“Well, truthfully sir, I was not so keen on mathematics before, but I must admit I am developing an interest,” you say, as Riemann hands you the keys to your room.*

*“I am glad to hear that! I presume that you will enjoy many of the mathematical treasures this hotel has to offer. May I recommend the Museum of Differential Geometry on the 4th floor, west wing. I curated it myself, you see,” Riemann smiles awkwardly. You get the impression that he really wants to promote his own work, but at the same time feels uncomfortable doing so. “I am always unsure about my own work, but a certain Herr Einstein seems to be *very *interested in my theories of curved spaces… but I am rambling now. I must get back to work, lots of new guests to attend to!”*

*“Evidently! I won’t keep you any longer. It seems you are quite an integral part of this business.” You collect your key and papers, and head towards the elevators.*

*“Danke, most kind of you! Oh, and dial * *for room service.”*

Updated version: 2 September 2017

**Thinking time:**Consider the set of integers , or in other words, all the natural numbers with their negatives and zero. Show that this set is also countably infinite, i.e. has the same cardinality as the natural numbers.**Further reading/Googling:**Remarkably, the set of all rational numbers (loosely speaking, anything that can be expressed as a fraction*p/q*, where*p*and*q*are integers) is also countably infinite, but the argument is not so simple. Cantor did pioneering work on this problem of countable and uncountable sets.- What I have introduced here is merely the beginning, there are many extensions to this story of the Grand Hilbert Hotel. I am more concerned with introducing a general audience to the language of mathematics. But check out this solution for accommodating
*countably infinite many busloads each with countably infinite guests*!! plus.maths.org/content/hilberts-hotel - I wrote above that statements like “∞ + 127” are meaningless — this is only in the context of the basic arithmetic operations, the main point being that ∞ should not be treated naively as a number. In more advanced mathematics, we use what is called the extended real line, where it is perfectly fine to write ∞ + 127 = ∞, but it is still important that infinity or minus-infinity is not treated as a real number. The Wikipedia article has been linked for those who are interested, but I avoided getting into these details for the general reader.

I am very pleased to announce that my *Three Concert Pieces *for piano will receive its première performance in the Utzon Room at the Sydney Opera House on **9 February at 7 pm**. The pianist is a good friend and colleague, Nicholas Young, with whom I collaborated frequently during our time at the Sydney Conservatorium. It is a pleasure to work with him again, but this time as a composer!

For more information about the concert and ticketing, click here to be directed to the corresponding page on the Sydney Opera House website.

For some insight into Nicholas’ project, and the rationale behind such a concert, you may be interested in this interview for CutCommon magazine.

The program notes for the concert are available online now.

The *Three Concert Pieces *represent my first attempt at a ‘serious’ work (for lack of a better term) for solo piano. However, I had been writing for piano for quite a long time. Being a self-taught composer, I developed my craft primarily by imitation of composers I admired. This involves writing short pieces or variations of a melody based on the the various styles of the composers I studied. Although I do not play the piano myself, the instrument is nevertheless ideal for experimentation, for reasons I have outlined in a previous blogpost. A composer can develop their understanding of such fundamental tools as melody, harmony, counterpoint, texture, and tone colour all with the one instrument. Moreover, the ubiquity of the piano not only in classical music, but also in jazz and popular music, allows a tremendous freedom of expression. No matter what style of music you want to write, the piano has something important to say — it is perhaps the most versatile instrument of all.

In the classical tradition, pianists pride themselves on their ability to make the piano “play anything” — from the most intimate melodies to thundering orchestral textures or complex fugues. In my piano pieces, I have endeavoured to capture the richness and diversity of musical expression the piano has to offer.

The first piece, **Allegretto**, is very short, lasting just over 2 minutes. I was inspired very much by the piano works of Schoenberg and the piano sonata (Op. 1) of Alban Berg, and I believe this is evident in the dense contrapuntal textures, economical motivic working, and concision in structure.

However, there is an element that looks ahead to the following two pieces, and is a result of my study of Elliott Carter’s music. The opening presents two* *contrasting characters: a *dolce, legato* motif, and a witty *staccato* motif. With this setup, I can move away from the classical melody-plus-harmony texture, and instead create two distinct layers, which may or may not move in the same way. Thus it creates the opportunity for some interesting counterpoint in melody, as well as in character!

The second piece is an expansive **Adagio, **and is very remote from the obsessive motivic working and condensed structure of the first piece. Indeed, it contains virtually no melody in a classical sense. The entire piece is built by overlapping layers of pulsating chords, which are free to resonate and interact in a way that naturally produces rich and complex harmonies. The layers also partition the range of the piano, and moreover, the period of pulsation in each layer is different. The result is complex, as exemplified in this excerpt near the end of the piece:

I like to imagine large pendulums (*) with different oscillation periods, all swinging slowly and independently. However, I do not assign a certain speed to each layer, and then proceed mechanically to write out the resulting pattern. Particularly from the second half of the piece onwards, I take more liberties with rhythmic invention for the sake of interest, while keeping the overall effect of being ‘suspended’ in time. By this I mean that, with the exception of the climax, you will never hear a clear downbeat anywhere! This is the most overtly Carteresque out of the three pieces, in my view.

The third and final piece is the **Allegro diabolico (alla toccata), **and is essentially a virtuosic *moto perpetuo*: a continuous stream of fast notes from beginning to end. The first half of the piece is playful and witty, making some use of abrupt contrasts in dynamics and harmony. As the material develops and increases in complexity, the ‘diabolical’ character becomes more apparent. However, just before the music approaches a violent character, there is a surprising intrusion of calmness:

The tranquility is short-lived, and the music grows in intensity once again, this time without interruption all the way to a ferocious coda. Towards the end, I insert a highly dissonant chord which breaks up the stream of semiquavers at irregular intervals. The violence of this chord suggests a great force trying to oppose the momentum of the semiquaver notes — as if trying to stop an imminent disaster. The final ‘push’ is marked with the almost ridiculous (**) instruction *sfffff* — that’s five *f*‘s —* *and the work concludes with a musical explosion and disintegration!

While the pieces were not initially conceived as being part of a whole, it makes sense to put them together as a suite of Concert Pieces. They share the same harmonic language, one that I have gradually developed over the years, and collectively represent a style which I can consider my own. They are very different in character and form, but all three share the same attitude of discovery and exploration, of constant development, and a preoccupation with finding the right balance between contrast and continuity.

**Footnotes**

(*) As anyone familiar with pendulum physics will know, the mass of the pendulum bob does not affect the oscillation period. But for the sake of musical imagery, let’s imagine that the pendulums are also very large and heavy, as well as being attached to a long rod, or being in an environment with small gravitational acceleration…

(**) I say that this is *almost *ridiculous, because I know for a fact that more extreme dynamics are found in the works of György Ligeti.

**A note about Berlin**

I spent a very enjoyable five weeks in December 2016 to January 2017 visiting Oslo, Munich and Berlin, meeting up with friends, walking around town, enjoyable good food, seeing some concerts, and just in general appreciating being in those cities.

Fortunately I was not in Berlin when the terrorist attack at Breitscheidplatz happened on the evening of 19 December. But being in Germany nonetheless, there was still the feeling of a numbed shock when something hits “too close to home”. This was exactly like the hostage crisis at the Lindt café in Sydney back in December 2014 (in fact, I remember that day I had wanted to go to the library at the Conservatorium of Music, which would have brought me dangerously close to the situation). Returning to Germany: on the evening of 19 December I was enjoying a concert at the Gasteig in Munich, where the violinist Renaud Capuçon performed wonderfully the Mendelssohn Concerto with the Academy of St-Martin-in-the-Fields. After the concert, I turned on my phone to browse the news as I travelled on the U-Bahn, and then immediately learnt of the situation in Berlin. It is difficult to find the right words to say, but I have found something which I believe is appropriate. If you walk around the Mall of Berlin on Leipziger Platz — which is near Potsdamer Platz, where there are remnants of the Berlin wall — you will find gold plaques on the floors with quotes from famous people. One such quote is the following from a speech given by Barack Obama in 2008:

(Image from German Wikipedia). The original English is: “People of the world — look at Berlin, where a wall came down, a continent came together, and history proved that there is no challenge too great for a world that stands as one.” This quote captures quite well one of the many reasons I admire the city of Berlin. A note about the use of language here: I think it’s rather neat that *to stand as one* is encapsulated in the single verb *zusammenstehen*!

**Some notes about composition**

Admittedly I have been lazy in regards to writing music. However, I have recently produced a Christmas-themed composition: the *Intermezzo festivo*, which was performed in a concert at the Vigeland Museum, Oslo, on 8 January this year. You can enjoy a “studio” recording of the piece here (more information in the video description):

After the Rhapsody No. 2 for solo violin, I’ve actually felt a bit stuck when it comes to writing new music. The main problem is that a Rhapsody is a freely-structured piece, which is fine for a solo work. I am sure I could write in a similar style for other instruments, but I am unwilling to proceed this way, as I have ambitions to write works on a larger scale, and also I am in principle against doing the same thing over again just because it is convenient and easy. In particular, I remain firmly focused on my goal to produce a piano trio someday, and although I have made concrete plans for the work and even started sketching some things out, I have yet to find the ‘right’ connections between the ideas floating around in my mind to build the work. In short, the problem is one about structure, form, and how the music progresses.

I am proud of my *Intermezzo, *as it represents something new in my approach, despite being written in a mostly classical harmonic language. I think it is a step in the right direction to solving the problem I have outlined above. I have actively tried to emulate the variation techniques found in the chamber music of Beethoven and Brahms for a while now, but with this new piece I feel that I have begun to grasp the essence of it, rather than merely imitating my favourite pieces. Listen to this excerpt (12:56 to 14:50 in the attached Youtube link) from the second movement of Beethoven’s Op. 127 quartet, for example, to see where I got inspiration for the *poco scherzando* section in my piece (and when you’re done, you might as well listen to the whole of Op. 127, because it’s such a masterpiece). The key idea is that the variations do not adhere to the structure of the theme as the piece progresses. The theme is seen as a flexible entity, perhaps like a piece of soft clay which I am free to mold as I see fit.

As early as the first variation, I start to deviate from the harmonies of the theme, and towards the end, the harmony has a life of its own. What prevents the variations from falling apart completely is the motivic integrity. It may seem trivial, but I am particularly pleased with bars 92-94, where I subtly bring back material from the very opening in order to close* *the greatly expanded variation beginning at bar 73. It is this kind of ‘long-range’ thinking that I need, if I am to write more ambitious works. This is also an example of re-contextualisation and re-harmonisation — the same melodic material is seen in a different light, or can be made to play another role (such as using the opening motif as a closing motif), by using different underlying harmonies. In contrast, my Christmas variations for piano from 2014 sound academic and student-like compared to the *Intermezzo*. While there are some nice figurations in that piece, overall there is nothing interesting structurally, and each variation follows the theme almost exactly in harmony.

**A mathematical remark**

I am quite certain that my studies in mathematics have influenced my approach to writing music. I do not like the idea of using rigid mathematical structures or algorithms to create the piece (call me conservative, but I think music should be developed primarily according to musical processes!), but there are nevertheless interesting and useful analogies. A mathematical concept that has occupied my thoughts frequently during the previous semester is the **group homomorphism** (studied in any introductory algebra course). It is not the place to go into details, but here is a rough idea. A **group** is a set of mathematical objects (numbers, functions, symmetries of a geometrical figure, etc.) equipped with an operation, and some simple rules that describe how to combine elements from the set using that operation. An easy example is the set of integers with the operation of addition. A **homomorphism **between two groups, say G and H, is a **map** (a mathematical rule) that connects the groups while *preserving their structure*. In layman’s terms, this means that performing the group operation of G is “analogous” to performing the group operation of H, and a homomorphism is any rule which can facilitate this “analogy”. Here is an example to aid the intuition. You will probably recall from high school the following rule involving logarithms: log (xy) = log x + log y, where x and y are any positive real numbers. Once again, I won’t go into details, but this can be understood in terms of a homomorphism (in this case, the logarithm function). The important thing to notice is that there is addition on one side, and multiplication on the other. In more complex examples, it could well be the case that a certain operation in a group G is tricky to compute, but a homomorphism between G and another group H could be constructed so that the tricky operation in G becomes “analogous” to an easier operation in H. We can do a lot more than merely computation though! Homomorphisms give us deeper insight into the mathematical objects being studied, and crucially provide a way of studying how they relate to one another. (Note to self: here’s a topic for a future Diversion in Mathematics blogpost…)

Why should this be an attractive idea to a composer? My answer is the term *structure preserving, *which has a natural interpretation in music. The composer can apply elaborate transformations to a particular motif or theme but still retain the same fundamental (musical) structure of that motif. In music as well as in mathematics, there is an essential desire to seek meaningful relationships between apparently different phenomena. This can be achieved via homomorphisms in abstract algebra, and via the principles of variation and motivic development in music composition. I like being able to connect together ideas which may be completely different on the surface, but nonetheless share a common deeper structure. Such transformations may be applied to a single musical phrase (motivic development) or indeed entire sections of a piece (which is what generally happens in variation forms).

**An update regarding focal dystonia**

I purposefully did not take my violin on my recent travels in Norway and Germany. It was the first time I had been overseas without my violin! I must say that it was a welcome change, as I was able to relax and simply have a good time. There is also a psychological factor: every previous trip to Europe I had done was due to performances or professional development courses. As I would inevitably visit concert halls, see concerts, and meet up with musician friends and colleagues, I did not want to be reminded of something I used to do, but could not do now.

Back home, I have begun to practise again, and the improvements are noticeable. Having had time to de-stress and look at the problem with a refreshed mind, the left hand is behaving better and better, and I can actually feel the coordination slowly being restored. Most reassuringly, the *sound *I am able to make has progressed from, well, utterly crap to decent. The left hand is now stable enough so that I can practise vibrato exercises effectively. Overall the results are promising, and I look forward to building upon these developments in my practice.

For string players and teachers living in Australia, I would like to announce that I have contributed a short article for Stringendo magazine, which will appear in the April issue. If you have already read my posts on focal dystonia on this blog, then you will know most of what I have to say, but the article is much more concise, and I do not ramble about my personal life as much, so it is more useful to students and teachers (which is the whole point).

Finally, here is an inspirational article which was sent to me from my violin teacher Ole Bøhn, about the oboist Alex Klein, who suffered severely from focal dystonia for many years. He has made a stellar comeback, playing once again in the Chicago Symphony Orchestra!

http://www.chicagomag.com/Chicago-Magazine/February-2017/Oboe-Alex-Klein/

As always, thank you for reading. I appreciate all the messages of encouragement I receive via Facebook, and also via the occasional email. I am determined to write more regularly here from now on, as it seems to hold interest for many people, and it will also help to focus and elaborate my thoughts about practising violin, composing music, or doing maths!

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When this particular student came to me for the first lesson, I pointed at one of the pieces he was studying for the exam: the beloved *Etude No. 2 *by* *Rodolphe Kreutzer (depending on your attitude, “beloved” can be interpreted as genuine or sarcastic…). His mother had contacted me saying that her son had no prior training in music theory. To test exactly what she meant, I asked the student “What key is this piece in?”

The student wasn’t able to give a reply! He could not identify C major, and so I had to start at the beginning. Over the next 4 lessons, I introduced the concept of major and minor scales, the rudiments of the Western system of tonality (key signatures and the circle of fifths), and also the names of intervals. He had particular trouble grasping the fact that the terms major and minor can describe a scale, a key, and also an interval (which might be a comment on the language we use to describe music… or my teaching skills, or both).

Playing scales is a mandatory part of all AMEB exams, so there was no doubt that he could play a C major scale, but I realised that he somehow lacked the concept of C major as a sort of “separate entity” — that is, divorced from the specific action of playing the scale on the violin, in the manner prescribed by the AMEB Technical Work book. To give a concrete example of this, consider the following snippets of music:

An experienced musician will know at a glance that all 6 excerpts are just different manifestations of the basic C major chord. The notes C – E – G form the C major chord regardless of the order of the notes, the register, and the rhythms used. Numbers 1 to 5 are just examples I’ve cooked up on the spot to illustrate some commonly used figurations, but number 6 is actually the opening of a piece of chamber music by Mozart (+10 cool points if you know which one!). One does not need to have practiced those *particular* figurations in order to execute them, it is about recognition of a broader pattern, and being able to adapt to variations on the basic pattern. This brings us to a brief discussion of **sight-reading, **another component of the AMEB exam.

One of the most remarkable features of Western classical music is the complex system of notation that has been developed over many centuries. Improvisation *used* to be an integral part of Western classical music, but generally it has lost its prestige in our era. Although there have been many successful efforts (e.g. historically informed performance, modern compositions which include improvised sections, and cross-overs with jazz and folk traditions) to reintroduce it as part of the classical musician’s skill set, I think it is safe to say that most classically-trained students are not taught improvisation. There is hence a strong emphasis on being able to read and interpret notated scores. Sight-reading is the practice of performing a score which has not been prepared beforehand. Since playing a musical instrument is so demanding, sight-reading is not a trivial exercise, and techniques must be developed. The minimum requirement is simply to be able to reproduce the notes and rhythms faithfully, within some reasonable margin of error. At higher levels of examination, students are expected to also pay attention to different articulations, dynamics, and expressive markings on the score. I would argue that the first step is the most difficult.

Translating a single note that is written on paper to a sound on the instrument (this includes the human voice too!) requires the musician to identify what pitch is represented by the notation, then to engage whatever physical actions are necessary to produce that pitch on the instrument. However, in order to play a passage of music, one must also be able to take into account the rhythm (roughly speaking, the relative durations of notes), as well as the succession of pitches, and translate all of that into a fluent process on the instrument. There is no hope of attaining the fluency required by considering single notes at a time. Experienced musicians will be able to internalise larger chunks of music, say an entire bar or several bars at a time, and also be able to read ahead, so that while they are playing a certain passage, they are mentally prepared for what comes next. Thus, to successfully perform even a simple piece of music at sight, the reproduction of the notated pitches and rhythms should be second-nature, as effortlessly as a literate person can read and recite written text (which is the motivation behind my choice of words “translate” and “fluent”). Attention to articulations, dynamics, and expressive markings can be trained later and often comes naturally with experience, but it is the fundamental, near-instantaneous connection between notation, sound, and physical action that is difficult to master, and requires diligent practice and time commitment.

I find that it helps tremendously if the student already has some knowledge of the Western tonal system — the rudiments of scales and keys — and a decent sense of rhythm. In this case, the bare essentials of a piece of music can be quickly internalised, and sets a rough framework or guideline during the sight-reading. As the student further develops their sight-reading skills, they will be able to transform the “passive” knowledge — e.g. *recognising *that a piece is in a given key — quickly into “active” knowledge, that is, knowing how to *realise* the notation as sound on the instrument. When I see a notated pitch, I can instantaneously hear the said pitch (unless it’s in some strange transposition!), and if it is violin music I am reading, I also immediately ‘feel’ the correct position of the fingers even without the instrument on hand. This is the fundamental connection I described above, and I’m sure all highly-trained musicians can experience it.

Unfortunately for my student, before our lessons, he lacked the knowledge even to recognise basic features like tonalities and intervals, and hence, as his mother had described to me, was practically unable to do sight-reading. After the limited number of sessions we had before his exam, I feel confident that he can now recognise key signatures and tonalities appropriate for the grade 4 level, but unfortunately we did not have enough time to make significant progress in putting this knowledge into action and developing his “inner ear” (referring to the connection between the notation and sound, not the anatomical inner ear). Nevertheless, I hope he has grasped the basics quickly enough to allow him to score some points in the sight-reading component of the exam. It is at this point that I bring in the relationship with studying maths.

There is nothing inherently wrong with the AMEB exam format. After all, music competitions and professional auditions all require the candidate to prepare selections from a set list of music. However, there is the unfortunate tendency to view the grade progression as the definitive* *way to study music, as if all it takes to become a good musician is to “level up” your music skills (like in the Sims computer games). This is hazardous, as I saw in the case of my student. It was clear that he knew his chosen examination pieces well and could perform them competently, but he was unable to adapt and extend his existing knowledge to sight-read a piece he had never seen before. There is a similar situation in high school maths. Let’s look at the example of solving quadratic equations. A typical “drilling” exercise might be as follows:

**Solve the following equations for x using factorisation:**

(The last one isn’t as obnoxious as it looks. What are the divisors of 55? and 21?)

While solving quadratics is a very important skill, this is not a particularly inspiring exercise. Now consider this GCSE exam question, which went viral for apparently being “unfair” and too challenging:

There are *n* sweets in a bag. Six of the sweets are orange. The rest of the sweets are yellow. Hannah takes a random sweet from the bag. She eats the sweet. Hannah then takes at random another sweet from the bag. She eats the sweet. The probability that Hannah eats two orange sweets is 1/3. Show that

This is actually more of a probability question, but I think part of the reason it was considered difficult is that the resulting quadratic equation appears (at first glance) to have no connection with the other information. In fact, the question could have been *more* challenging if the students were simply asked to “find the value of *n*“, it was quite kind of the examiners to provide the correct quadratic equation! The point is that the question combines two topics — basic probability and quadratic equations — that when considered individually should have caused no problems for a student who has adequately prepared for the exam. But of course, interesting mathematical problems are interesting precisely because the techniques needed are not handed to you on a silver platter, and the road to the solution is not paved nicely and marked with flashing signposts. There is no huge conceptual leap from typical textbook exercises to Hannah’s sweets, but students who are too accustomed to the textbook fail to adapt the basic techniques to tackle more interesting problems that require a more involved process. (Previously, I have discussed briefly the role of creativity in mathematics).

As I continue my mathematical studies at the University of Sydney, I come to realise the utmost importance of complementing the theory explained in lectures with a rich variety of problems to tackle, so that I begin to appreciate the myriad of ways the theory is used in practice. Interesting and challenging problems will often require creative manipulations, finding connections between different concepts, expressing the same quantity in different ways, combining the results of various theorems in a clever way, and so on. The temptation is to consider yourself a master after getting all the textbook exercises correct. Sure, you understand the basic theory, but this is only the beginning! It is likely that most of the students who complained about the GCSE question were perfectly capable of solving quadratic equations, but floundered when the technique was disguised in a more creative way. The converse seems to be true in the case of my AMEB student. He had diligently prepared four *specific* pieces, but lacked the techniques required to appreciate and process music more generally, as demonstrated by his initial inability to sight-read music. In mathematics as in music, it is most satisfying when you begin to appreciate the interaction between theory and practice.

As is common in many problems, it helps to express the same quantity in two different ways. We want to compute the probability of getting two orange sweets in a row. At the first draw, there are *n* sweets, 6 of which are orange, so the probability of getting an orange sweet is simply 6/*n*. At the second draw, there are now *n *– 1 sweets in total, 5 of which are orange (remember Hannah ate the first one!), so the probability of getting orange on the second draw is 5/(*n *– 1). Now, by the multiplication principle, it follows that the probability of getting two orange sweets in a row is:

where the right hand side of the equation is the probability of getting two orange sweets as provided in the question. This simplifies to the desired quadratic equation (I’ll leave the details to you, dear reader ). To take the problem one step further, we can factorise and solve for *n:*

There are two solutions (as expected from a quadratic), *n* = -9 and *n* = 10, but clearly you can’t have minus 9 sweets (unless all the sweets in Hannah’s bag are stolen and she owes someone 9 sweets!), so the only valid answer is *n *= 10.

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.

Let us start at the very beginning: 1, 2, 3, 4, … This is presumably how you learnt to count when you were still a baby. Furthermore, as you may remember from Sesame Street (or similar shows for kids), counting and the learning of numbers always corresponds to pairing the number word, say **eight**, with a visual aid. To give an explicit example, we know that the following picture contains **eight penguins:**

Notice that I write ‘number word’, because it is only later in life that we learn to treat numbers abstractly. By this, I refer to the separation of the number word from the visual aid, so that ‘eight’** **need not be understood only in the context of ‘eight penguins’, but stands by itself as an abstract entity. We learn to do this when we are introduced to arithmetic in primary school. We all know that it is possible and meaningful to say “two plus seven is nine” without referring to any physical objects, and so over the course of millenia, humans have devised symbols to express numbers. For example, **seven** corresponds to the symbol **7**, if we use the Hindu-Arabic system, or **VII **in the Roman system, or **七** if you use Chinese characters, and so on. In fact, the labels actually don’t matter, the important thing is to have a system for **counting**. Now we know intuitively what the word ‘counting’ means, but how do we really define counting? No, there’s no need to reach for the dictionary, we are doing maths here. Let me suggest a definition of sorts. Grab any object, let’s say a plush-toy penguin. We can all agree, provided we use the English language and the Hindu-Arabic numerals, that a single object will always correspond to the number called **one** with the associated symbol **1**.** **As it turns out, we only need one other condition: to agree that getting another** **plush-toy penguin (or whatever your object of choice) will correspond to an action we call **adding one (+1) . **Now I claim this is all you need, mathematically speaking!

Suppose you meet a representative from an intelligent alien civilisation, and clearly neither one of you speaks the language of the other. Nevertheless, you are eager to start collaborating on mathematics (maybe they have proved the Riemann hypothesis?), and the natural place to start is… well, with the **natural numbers**, which is what mathematicians call the counting numbers. [An important side note: it was not obvious to our ancestors that **zero **is a number, and in fact, the discovery or invention of zero is one of the most important results in the history of mathematics. Many mathematicians include zero in the natural numbers, but for now, we’ll stick to counting starting at ‘one’]. As long as you manage to convey to the alien visitor the meaning of **one **and the concept of **adding one**, the rest is simple. Convey the meaning of **two **by **adding one **to **one**, and likewise, show that **three **is the result of **adding one **to **two**, which you have already defined. To see this in action, watch this inexplicably hilarious Khan Academy video:

Now it is your job to convince your intergalactic colleague to repeat the same process. Suppose you learn that **one **corresponds to *un*. Then by adding *un* each time, you can learn the labels for each successive natural number in the foreign language. Maybe it goes a bit like this:

- 1:
*un* - 2 = 1 + 1:
*deux* - 3 = 2 + 1:
*trois* *4 =*3 + 1*:**quatre*

and so on. (At this point, I hope any French readers out there have a decent sense of humour). Of course, this gets tedious after a while, and both of you will inevitably need to learn each other’s **number system **to do anything useful, but in principle, you can learn the label for any natural number this way. Furthermore, you have exploited an important feature of the natural numbers: **order**. As you have probably observed, any natural number not equal to one has a **unique** **successor **and **predecessor **(remember, we are not considering zero for now). Why is the uniqueness important? In this case, it ensures that our operation of adding one** **is **well-defined**. When you add one to nine, for example, you can only get ten. Likewise, there is a unique number that comes before twenty, namely nineteen, and so on. This makes it possible to say, for example, that five is less than seven, which is written 5 < 7. Equivalently, we can say that seven is greater than five, and write 7 > 5. It is this ordering of the natural numbers that allows us to count.

You might be thinking, ‘what a waste of time, I certainly knew that!’ But think about how important it is in our daily lives to be able to discern not only what a specific quantity is, but also how different quantities relate to each other. Take a look at the following figure. Which set, A or B, contains more black dots?

It is immediately obvious, isn’t it? You can count how many dots are in each of the sets, and hence determine which is larger as a result of the ordering property of the natural numbers. But this is because we also possess the number system to describe exactly that set A contains 4 dots, and set B contains 7. Now suppose you were a child of the Warlpiri indigenous Australians, whose language only makes distinctions between what is essentially “one” and “two”, and all other quantities are described as “few” or “many”. You won’t be able to describe in words that “set A has four dots”, but you can certainly work out which set has more. Simply take one dot from A and pair it with a dot from B, and you will end up with some quantity of dots in B without a matching partner from A. Then you must conclude that B has more dots than A, although you won’t be able to express exactly how much more. But you certainly can tell when B has *exactly one more* than A, and once again we return to the fundamental notions of “one”, “adding one”, and the ordering of natural numbers. Keep this example in mind, as later on (in a future post), we will discuss more formally the concept of taking a thing from one set and pairing it with an element from another set.

It seems I’ve just taken you back to kindy class (that’s kindergarten, for the readers who may not be aware of the Australian custom of abbreviating everything), but in fact I have disguised some very important mathematical notions in this section. Experienced mathematicians will also notice that I have been rather naughty, since I have introduced various concepts with no rigorous definitions, instead appealing to the general reader’s intuition. But I hope you will understand that this blog isn’t the place to give a full treatment of Peano axioms and other such formalisms. Nevertheless, we have discussed enough for the general reader to proceed to the next section. So far, I have stressed that the number labels (1, 2, 3, 4, …) are not actually important, all you really need are notions of “one” and “adding one”. This suggests that we can *construct the natural numbers* from even more fundamental concepts than numbers…

We will now introduce what is perhaps one of the most fundamental of mathematical concepts.

Definition 1.1

- A
setis a collection of objects.- If some object
xbelongs to a set A, we sayand writexis an element of A,x∈ A.- The set with no elements is also a set, and is called the
empty set, denoted by ∅.

That was easy, wasn’t it? In fact, I have already sneaked the term into the previous section. You will already have intuitive notions of sets. For example, consider the set A = ‘set of all birds’, then obviously owl ∈ A, penguin ∈ A, but llama ∉ A (notice the crossed-out symbol denoting ‘not an element of’). Sets can also contain other sets, and this is in fact a very fundamental concept in mathematics. For example, let B = ‘set of all flightless birds’. Then B is a **subset **of A, and in mathematical notation, we write B ⊂ A. Because mathematicians love symmetry, you can also write this as A ⊃ B. (Notice how similar this is to the notation 1 < 2 and 2 > 1, I am anticipating the discussion ahead…). Another way of saying this is that every element of B is also an element of A, and in our example, this says nothing more than ‘all flightless birds are birds’. One more fact before continuing: when writing down elements of sets, duplications are not counted, so the set {1, 2, 3} is the same as {1,1, 2, 2, 3, 3, 3}.

Sets are quite naturally visualised in Venn diagrams. It is a bit of a mystery to me why some basic set theory isn’t taught in high school. Given the proliferation of Venn diagram memes on the internet, I am sure it can be approaced in a fun way:

In the first example… I have absolutely no idea how that relates to maths, I just really like that meme. But in the second example, we could define A = set of all symbols of chemical elements, and B = set of all abbreviations of US states. The elements that are in the overlapping region are elements of both A and B, so is there a way to denote them? And can we also denote the set consisting of everything in A and B? Indeed we can, and the definitions are quite natural:

Definition 1.2

- Let A and B be sets. Then
xis in the intersection of A and Bifx∈ A andx∈ B. We denote thisx∈ A ∩ B. It should be clear that A ∩ B is a subset of A and also of B. Hence the set A ∩ B is the set of all elements that are in both A and B, and in mathematical notation:A ∩ B = {

x|x∈ A,x∈ B }[The curly brackets collect elements of a set. The stuff in between can be read as follows: “(all)

xsuch thatxis an element of A and also of B.” The comma (,) often represents “and” in the language of maths. The bar | represents “such that”, and equivalently may be written with the colon : instead.]

- The
unionof A and B is denoted A ∪ B, and is defined as the set consisting of all elements in A and all elements in B:A ∪ B = {

x|x∈ A orx∈ B }

Earlier I hinted that we could construct the natural numbers, and indeed, one of the ways to do it is using sets. The following approach is paraphrased from Paul Halmos’ text *Naive Set Theory*. By the way, he takes 42 pages to get to this point, so you can thank me for saving you the trouble. (But in all seriousness, given how rigorously Halmos treats the subject, it’s actually surprisingly concise. Halmos’ approach is based on something called Zermelo-Fraenkel set theory).

Take the empty set ∅, and then consider the **set containing the empty set **{∅}. There is a world of difference between the two. The empty set clearly has no elements, while the other contains one element: the empty set! (Recall that sets can contain other sets). Now here is the exciting bit. For any set *X, *we define its **successor **to be the set obtained by joining *X *with the **set containing X,** namely

- ∅+ =
- + =
- + =
- + =

and by now, I think it’s quite clear what this looks like! Moreover, we have the following relationship between the sets:

which should instantly remind you of 0 < 1 < 2 < 3 < 4 < 5 < …, the ordering of natural numbers.

Notice that this construction uses nothing but the basic definitions of sets. It does not even require any notion of arithmetic, since we have effectively defined “adding one” by adjoining sets, which is why I liked Halmos’ suggestive notation *X*+. It also quite naturally includes zero (hooray!), using its set-theoretical counterpart, the empty set. For a rigorous treatment, there is still a bit of housekeeping to do before we can claim to have defined the natural numbers properly, and so I attach an extract from Halmos’ book if you wish to see how to complete the argument (I do not find it to be an easy conclusion). At this point, perhaps you’ve had enough of counting like a pure mathematician, and I can’t blame you for that. However, there is still one more thing to do. Using more familiar notation, we can rewrite the construction above using numbers (now that we’ve “defined” them):

Definition 1.3: Inductive construction of the natural numbersThe set of natural numbers

Nis a set with the property that:

- 0 ∈
N- if
n∈N,thenn+ 1 ∈N

[Note: the *N* should really be a fancy , but I can’t get LaTex working within those quote boxes].

Hopefully this definition is quite clear by now, as it is completely analogous to the construction with sets. But is it clear that this process of “adding one” can be continued indefinitely? In fact, it can be logically concluded from the above constructions that the set of natural numbers must be infinite. We can show this as follows. Suppose there is a “last natural number”, we call it *Z.* Then , but by the definition above, we *must* also have . By the ordering of natural numbers, Z + 1 is certainly larger than Z, which is a contradiction. To explore infinity further, we must pay a visit to Hilbert’s Hotel — our topic for next time!

Butterworth, B., Reeve, R., Reynolds, F., & Lloyd, D. (2008). Numerical thought with and without words: Evidence from indigenous Australian children. *Proceedings of the National Academy of Sciences of the United States of America*, *105*(35), 13179–13184. http://doi.org/10.1073/pnas.0806045105

Halmos, P. (1974). *Naive Set Theory*. New York, Springer-Verlag.

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 ).* *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…

So what about popular *mathematics*? Undoubtedly there are many who enjoy recreational mathematics, but my feeling is that the average person will not suddenly decide, “gee, I really feel like reading something about abstract algebra today!” In fact, I wonder if there are any books on abstract algebra intended for the general public! Sciences such as astronomy and astrophysics, biology and psychology tend to capture the public imagination better. For lack of a better word, they are simply more relatable (and not in the internet meme “omg so relatable” sense). Maths demands more abstract thinking, and understanding of mathematical concepts is inseparable from understanding mathematical notation. We write equations not because we want to be deliberately obscure — quite the opposite is true! Equations are extremely precise, and represent relationships between mathematical objects that would be tedious, cumbersome, perhaps even impossible to describe in words. By way of illustration, take a look at this beautiful equation:

In fact, this presentation is already quite verbose, in some texts you’ll see it written simply as:

If you’re curious, this is *Laplace’s equation*, and some mathematicians and physicists have devoted signficant parts of their career studying the properties and applications of this equation. Clearly, there’s a **lot **more than meets the eye (for example, what do those triangles mean?). It is often the case in mathematics that simple things turn out to have profound consequences.

Hopefully, *Diversions* will become a recurring feature of my blog. It is my intention to write about mathematics in an informal way and with the general public in mind, but I will not attempt to hide the ‘real’ mathematics. I will present equations, guide the reader through calculations, and discuss complex concepts. But if I succeed, hopefully I will not intimidate the reader either, and ideally there should be an element of fun. I use the word ‘diversion’ in this sense too, similar to the French musical term *divertissement* (or *divertimento *in Italian). At the time of writing, I am studying an introductory abstract algebra course at the University of Sydney. It is a challenging course, the material is (not surprisingly) very abstract, there is a lot of new (mathematical) language to learn. However, our lecturer Dr. Stephan Tillmann has emphasised the idea of having fun in maths. Frequently, we are asked to “play around” with a new concept, his assignment questions are geared towards exploration, and we are encouraged to ponder “what happens if…?” It is in this spirit that I write these blogposts. If you are interested, then watch this space!