After a long time, I have decided to resurrect the mathematics part of this blog! I started *Diversions in Mathematics *as a way for me to try to explain mathematics to the general public. This continues to be the main goal of this series of blogposts — for a more detailed introduction, please read the introductory remarks. I wrote two posts in this series, but then abruptly stopped. Part of the reason was that my studies got in the way, but I was also unsure exactly what material to present, and how to present it. I wrote down some of my thoughts on this matter in a previous update post.

One of the main issues as a writer is to consider the readers’ background in mathematics. For posts targeting the general public (like the previous two Diversions), I have tried to assume as little as possible while maintaining the discussion at an intelligent level, i.e. without “dumbing down” anything) However, this already assumes familiarity with many mathematical concepts taught in high school, or at least, some level of maturity in regard to abstract reasoning. Consequently I have decided to relaunch Diversions in Mathematics with high school mathematics as a foundation.

In this blogpost, I will introduce the *Cauchy-Schwarz inequality*, one of the most fundamental results in mathematical analysis, with the aim of connecting various topics that are typically studied in the Year 12 HSC maths curriculum in NSW. Click below to read the main article.

Much of the high school maths curriculum, for better or worse, is devoted to methods of solving equations. Do the following exercises ring a bell?

**Homework Exercise** (due on Monday): *For each of the following equations, find all solutions* in the real numbers.

I hope this doesn’t induce traumatic memories. As an unfortunate consequence, I suspect that many people have the impression that maths is just about solving equations. This is far from the truth! I am rather fond of this cartoon from Ben Orlin:

The cartoon comes from this article, which I recommend to read.

Of course, learning to solve equations is extremely important and useful. However, the key point I want to raise is that tackling *inequalities* often requires a set of tools that is quite different from those used to solve equations. A solution to an algebraic equation, as in my hypothetical homework exercise above, is simply a number. This is a *quantitative *statement: we are affirming that when the variable is equal to a certain number, it satisfies the desired formula. On the other hand, the study of inequalities gives the opportunity to consider *qualitative* statements. For example, if is a given function, then the statement

for all

says that the absolute value of the output is bounded above by 1, for real number input . An equivalent statement is given by

for all

It is conceivable that we don’t know the value of for some input, or even for many inputs . For example, could be the solution to a differential equation that is extremely difficult or impossible to solve by hand, and hence we can only estimate the values. Depending on the situation, we may not even care to know the precise values of the function. We could fill many blogposts with discussions on the utility of qualitative information in many aspects of science and humanities. Instead, I want to focus on just *one* famous inequality in this blogpost. Before we see its proper form, let’s discuss **positivity**.

Given any expression involving an inequality sign , we can easily transform it into a statement about positivity: the fundamental principle is that if and are quantities such that , then this is equivalent to the statement that , i.e. is *positive*. [Before we continue, a quick note on terminology. Our convention differs slightly from other texts, where positive means . However, I prefer to avoid double negatives such as “non-negative”. Hence, if , we will say that “ is **strictly positive**” to emphasise the difference from ].

The basis of many inequalities is the following fact:

(for all real numbers )

Since this is true for *all* real numbers, we can substitute any other expression for a real number in place of to obtain many other inequalities. This is a common algebraic trick that ought to be familiar to many students.

As a simple but important example, consider substituting in place of . Then we can expand out the brackets to obtain , which implies the following classic inequality:

If are positive numbers, we can replace by their square roots in the above inequality, rearrange a little, and obtain

This is a special case of another famous result, namely the arithmetic-geometric mean inequality (often abbreviated to AGM inequality). Now we are ready to introduce the Cauchy-Schwarz inequality in its simplest form.

Let be an integer. Given any two sets of real numbers and , the following inequality holds:

This is definitely an unwieldy statement — we will see a dramatic improvement towards the end of this blogpost! We will prove this result by induction (another important HSC topic). However, it is worth examining some simple cases first.

In this case where , there is almost nothing to prove. The summations reduce to single terms: we have on the left hand side and on the right. However, recall that if is any real number, then

So the right hand side of the Cauchy-Schwarz inequality is equal to , and we can clearly see that this is larger than or equal to .

For , the Cauchy-Schwarz inequality reads

Suddenly, things are not so obvious! How could we demonstrate the truth of this inequality? The presence of squares and the way that the terms are paired up on the left hand side suggests that we use our starred inequality from earlier. Then we obtain

Unfortunately, this is not quite right: the right hand side is a bit too large. Indeed, upon a moment’s reflection, perhaps you spot the connection to the AGM inequality . By setting and , we deduce

Clearly we need to look elsewhere… but we need not go too far. Recall that the simple AGM inequality above was a consequence of followed by various clever substitutions. Could we not deduce the case of the Cauchy-Schwarz inequality by considering the positivity of a certain quantity?

As an initial investigation, let us square both sides of the Cauchy-Schwarz inequality and expand out the brackets:

At first glance, this appears to have made things worse, but fortunately, many terms cancel out. We find that the above inequality is equivalent to

At this point, it is hopefully clear that lurking in the background is the positive quantity

We can now prove the Cauchy-Schwarz inequality in the case where . Starting from inequality (2b), this immediately implies (2a). Adding to both sides of (2a) yields (1b), which can be factorised to produce (1a). Finally, since both sides of (1a) are positive, we may take the square root of both sides, and using the fact that any number is always smaller than its absolute value, we conclude

which is the desired result. Observe that we have discovered the steps in the backwards order! This can happen when proving inequalities — very often, a lot of investigation “around” the problem is required before the solution strategy becomes clear.

It is about time we tackle the general case. We will proceed by mathematical induction, which is a staple of the HSC examinations (and does not involve electromagnetism). As it turns out, the discussion above has given us all the tools needed to complete the proof!

**Base case**:

In this case, the Cauchy-Schwarz inequality reduces to a statement about real numbers and their absolute values:

and we have verified the truth of this statement in the previous section.

**Induction step**

Now assume that the Cauchy-Schwarz inequality holds for some integer . We will show that it holds for ; that is, we will prove

Since we have assumed the result to be true for terms, it follows that

At this point, you should be wondering, how can we get the to appear inside the square roots? This seems like a very messy operation. However, upon further reflection, you would probably stumble across the idea of using the case of the Cauchy-Schwarz inequality, which we already proved in the previous section! Notice that the right hand side of the result directly above is of the form , where and are the somewhat unwieldy square roots of sums. We know already that

and consequently

This completes the inductive step, and hence proves the Cauchy-Schwarz inequality.

The proof of the Cauchy-Schwarz inequality we have just developed is purely algebraic: as a setup, we used nothing more than a few elementary inequalities, and the conclusion was achieved using the power of inductive reasoning. The x’s and y’s appearing in the inequality can be taken as arbitrary real numbers. However, as I remarked above already, the square root of the sum of squares that appears on the right hand side of the Cauchy-Schwarz inequality is rather unwieldy algebraic expression. Surely, you suspect, there must be something deeper. That expression must have a neater interpretation?

If you suspected this, then you would be absolutely right: the Cauchy-Schwarz inequality is most useful in the context of **vectors**.

**Remark 3.2. **The study of vectors has been introduced to the Extension 2 mathematics syllabus in NSW only this year (2020), although it has been in the mathematics syllabus in the Victoria and South Australia examinations for some time already.

Vectors are the foundational tool for mathematics in higher dimensions, and vector spaces are the natural setting for many problems throughout mathematics and hence in all areas of science and engineering. Although the concept of a vector space in general is quite abstract, there is a particular example that is familiar to everyone: the 3-dimensional space we live in. Every point in 3-dimensional space can be described by a set of 3 real numbers — its *coordinates* — denoted by , where is an arbitrarily chosen reference point called the *origin*. In a similar way, mathematicians have no difficulty thinking about points in -dimensional space, which are of course described by a set of coordinates, .

In this setting, a *vector* can be represented as an arrow connecting two points in the space. A vector that is drawn from the origin is called a *position vector*, since it specifies uniquely the position of some point, say . Then we can label the vector with the same coordinates as we used for the point . In the picture below, we have a position vector in 2D space (the plane) with coordinates .

One of the fundamental properties of a vector is its *length*, or *magnitude *or *norm* (the last term is used in more abstract situations). We denote the length of a vector using the symbol . For position vectors in the plane, it is easy to see that the length can be calculated using Pythagoras’ theorem. If , then

The generalisation to an arbitrary number of dimensions is quite natural:

Thus, given two vectors and , the right hand side of the Cauchy-Schwarz inequality is simply , the product of the lengths of and .

Naturally, you may wonder if we can interpret the left hand side in terms of vectors. Indeed, it does appear that there is *some *kind of “product” operation going on: notice that the left hand side is obtained by taking the product of each pair of coordinates , and then summing up everything. As we will see in the next blogpost, such an operation is called a **dot product**, which is a special case of the very general concept of an **inner product** on a vector space. The Cauchy-Schwarz inequality demonstrates its full power when generalised to a statement about inner products. In this regard, the inequality has a nice geometric interpretation, and moreover can be derived as a consequence of an optimisation problem. We will discuss these things in the next blogpost!

For the curious reader, I have written a set of notes on the essential vector concepts taught in the HSC syllabus.

]]>There are many more conversations with other musicians who are dealing with various forms of focal dystonia. I highly recommend checking out the Youtube playlist: https://www.youtube.com/user/HeldenCors/playlists

Katie’s website: https://www.focalembouchuredystonia.com/

]]>To clarify what I mean by “elementary”: no, it wasn’t a silly error in the sense of a missing minus sign or multiplying two numbers incorrectly. Mathematicians say that something in their field of study is “elementary” if it is a basic fact in that particular field. However, to paraphrase my supervisor, “basic does not mean easy”, so if I were to describe the mistake to a non-mathematical audience, it would still be extremely difficult. Another interesting point is the following. When you do research in pure mathematics, quite often you deal with abstractions upon abstractions, and it is easy to forget the humble origins of these abstract concepts. For example, the concept of a linear operator is fundamental to many areas of maths, and as cool as it is to be working with linear operators on infinite dimensional Banach spaces or whatever, it is also worth checking a statement about abstract linear operators on, let’s say, a 2-by-2 matrix as encountered in 1st year university maths (or even high school, depending on where you were educated). This comic *sums up* the sentiment quite well (excuse the pun).

Now this brings us to the main point of this blogpost. The wrong calculation I described above is *not* the mistake I want to write about; rather, I want to focus on what happened next. I assume that most people do not enjoy looking at their failures. I am not sure what I did with those few pages of calculations: either I threw them out, or I shelved them away somewhere I had forgotten. In any case, recently I wanted to return to the same problem with a different perspective, but now I have lost my point of reference. Because I had misplaced or thrown out the incorrect calculations, I cannot really remember what I had got wrong the first time! To be fair to myself, I have a vague idea of the *kind *of error it was, and it is very unlikely I will make exactly the same error again, but nevertheless it would have been more efficient to keep the original calculations, wrong as they are, so that I can remember *exactly* what the lapse of judgment was. As I have learned by now, mathematical work is not only about putting one true statement after another. It is equally important and even illuminating to know what kind of result *cannot* be true. In hindsight, throwing away those calculations was a bigger mistake than the calculations themselves.

There are certain similarities with the performing arts, although I have to admit that the comparison gets fuzzy on closer inspection. For a start, mistakes in mathematics are objective errors: you are objectively wrong if you claim that 1 + 1 = 3, for instance. On the other hand, speaking from a background in Western classical music, even though there *are* certain objective errors (e.g. playing a wrong note), usually we are unhappy with a performance because we feel that we could have done certain things *better*. One could say this is a “subjective” error. Despite this fundamental difference though, both mathematicians and performing artists undeniably improve by learning from their mistakes, objective or subjective. Indeed, I would say that learning from mistakes is an essential part of development in the two disciplines.

As a violin student, I used to dislike making recordings — what a way to highlight one’s faults and weaknesses! However, in hindsight I should have taken more opportunities to record my playing, and to embrace whatever mistakes (objective or subjective) occurred during a performance. One should not take this viewpoint too pessimistically. The idea is not to focus only on failures; rather it is about using those failures strategically to improve one’s understanding or problem solving technique in the case of mathematics, or one’s expression capabilities in the case of the performing arts.

I end this post with a nice little comic, which comes with some wise words from the great mathematician Paul Halmos:

]]>- I am now in a masters program in the School of Mathematics at the University of Sydney! My honours year went quite well overall, and my supervisor was happy to take me as a postgraduate research student.
- Since March, I have also been working casually for Matrix Education, a private tutoring company that is well-known in particular for its HSC preparation courses.

From the two reasons listed above, you can appreciate that most of my time is spent doing mathematics in some way — reading articles and learning new theory for my research project, working on exercises from textbooks, or preparing tutorials and classes. The third reason is really a blend of two:

- My own laziness coupled with a lack of direction about this blog.

This update post will explain this last point further.

For a while, I have been contemplating what kind of content to feature on this platform. The blog started essentially as a way of coping with my focal dystonia, as I was making a transition from studies in music to studies in mathematics. You will find that the earlier blogposts are mainly about my musical activities. However, at some point I decided to make the presence of mathematics more prominent, and ambitiously began a series titled “Diversions in Mathematics”. Currently, there are only two proper articles in that series (not counting the introductory Chapter 0), and it appears all but abandoned. This is only partially true: although nothing more for the series has yet been produced, I haven’t forgotten about it either. In fact, I am thinking all the time about writing mathematics! The problem is that I became indecisive about *how *to present the material — there was certainly no shortage of topics that I wanted to present. If you take a look at the current entries in the Diversions in Mathematics series, both contain formal definitions of mathematical concepts, and both attempt essentially to “teach” the reader some proper mathematics: the first piece deals with basic set theory, the second deals with countable infinities. While I am happy with the two existing pieces, I realised that the blogpost is simply not the right medium to “teach” mathematics, for the following simple reaon: I think that casually reading a blogpost isn’t a good way to *learn *mathematics in the first place! Hence I knew I had to change the direction of the mathematical part of my blog.

I am still committed to the goals set forth in Chapter 0: I intend to present mathematics in a way that combines the formal and the informal, and seek a balance between rigour and fun. As my current viewpoint stands, to present what is basically lecture material (as in the case of the extant Diversions) is not the way to achieve this. I have been planning the next installment of the series, and I believe it will be a blogpost of a very different kind than has so far been featured — please stay tuned if you are interested! Moreover, the mathematics and music have been rather separate entities on my blog, with a few exceptions (like the blogpost about mathematics and music exams). In the future, I will be more confident in letting the two worlds mix.

Unfortunately, I have devoted very little time to composing and arranging this year. Of course, I am always thinking about new music, and I play with little fragments of music in my mind constantly. But in order to write something coherent, even if it is only a short piece of 5 minutes or so, I find it necessary to make a consistent effort. However, doing so tends to disrupt my mathematical activity too much, which is not desirable at this time. Perhaps I will eventually develop a way to juggle these two demanding mental activities, but for now, it is very much the case that the activities of writing music and doing mathematics are mutually exclusive. I find that my listening habits have also changed. Although I have never lost my passion for music, these days I don’t listen to a lot of music, but rather choose very specific pieces when I feel like listening to something. It is possible that this is an effect of studying and teaching mathematics, which requires a focused, organised mind. When I was studying music, it was not uncommon for me to listen to a lot of very different music throughout the course of the day. Nowadays I often just have a single piece of chamber music by Haydn, Mozart or Beethoven, or a violin piece by Bach, which will persist in my mind for a few days, before getting replaced by another specific piece. But recently, I have also renewed my appreciation for Stravinsky and Ligeti — it is important to be connected to “newer” music too.

Perhaps I’m going about this the wrong way. If you look at my Variations for wind quintet (which is probably the piece I am most proud of, to date) you will see that I enjoy a certain amount of complexity in my music. However, I have also thought about writing some simple pieces — even tonal pieces! — for the sake of simply “keeping things going”. I know all too well that it is a mistake to wait for inspiration to strike. So it is likely that you will be hearing simpler compositions from me in the near future, for example, compositions that are suitable for younger students to play. I am certainly eager to try it out!

If you have reached this point, I want to thank you for taking the time to read my blog. I find that I express myself better in writing anyway, so you can imagine that the things I write on my blog are the things I would like to share with you in person, if only I had the conversational skill and lacked the social awkwardness to do so. It means a lot to me that a non-zero amount of people read the words I write!

]]>First of all, I remark that it is a happy coincidence that the summer school was held at Barker College, where I was a former student. It was at Barker that I first experienced playing chamber music — specifically, the string quartet. As it turned out, I played string quartets throughout my entire education at Barker College, and it was one of the most important factors in my development as a musician — not only as a performer, but also as a composer! The main point that I will emphasise is this: playing chamber music makes great demands on the musicians involved, in technical aspects (i.e. the physical part of instrumental playing) as well as conceptual (i.e. intellectual understanding of the music). These demands are quite different from those encountered in solo playing, and furthermore, I will try to argue that chamber music should play a significant role in a student’s musical development.

A piece of (well-written) chamber music will make good use of the possible interactions between different members of the ensemble. This kind of composition is clearly different — let’s say for concreteness — from a piece written for a solo violin, and thus the manners of playing implied by a violin part in a work of chamber music will be different from the solo violin composition. I like to use the analogy of a play, or a similar dramatic work like a film script, in which there are typically several main **roles***. *A solo violin piece is more like a dramatic reading of a poem, for instance, performed by a single actor. On the other hand, reading a single part of a string quartet is like reading the script of a play but seeing only the lines spoken by a single role. I think it is obvious how absurd it would be if an actor playing, let’s say, Macbeth only studied Macbeth’s lines without reading the whole of Shakespeare’s play. Regardless of the actor’s talent, it would make little sense to deliver a Macbeth soliloquy without knowledge of the context in which that portion of the play sits. It is similarly absurd to practise a chamber music part (and orchestral parts too!) without consulting the full score, which corresponds to the entire playscript in my analogy. Of course, my analogy is highly flawed, when you consider that in chamber or orchestral music, the various parts are mostly playing simultaneously. Nevertheless, in both music and drama (and indeed in any artform), as soon as we begin to contemplate the relation between the part and the whole, we have already started an investigation of **form***, *i.e. how the work is organised and built. Very often, one of the first problems when studying a new piece of chamber music is simply to understand how your part fits into the whole. Of course, when one is studying a violin concerto, it is still very important to have a good knowledge of the accompanying orchestral parts, but in this case it is clear that the violin occupies the leading role, whereas in chamber music, the parts can interact in a multitude of ways, and it can be a challenge (especially in modern works) to determine which part plays the leading role in a particular section of music. In general, one of the most essential and often one of the most challenging tasks in the study of chamber music is to find the right balance between the parts throughout the course of the work. Let us now explore some of these issues in more detail.

As I already mentioned above, one of the essential aspects of chamber music is the understanding of how each part fits into the whole. For this reason, playing chamber music is an excellent way to develop basic skills of musical analysis and comprehension. For young students who are new to chamber music, it can be quite a challenge. They would be accustomed to working on solo pieces, or pieces with piano accompaniment in which they play the main role. I am not demanding that every music student should have a firm grasp of the complexities of sonata form from a young age. However, I think even beginner students should be trained to be able to describe basic features of a piece of music. These include:

**Keys and modulations**: Teachers should always ask “what key is this piece in?”, or “what key is this section in?” and the students should know! Moreover students should be able to identify modulations, i.e. key changes. This is the first step in developing an understanding of harmony. It is also a good way to motivate the practice of scales and arpeggios.**Repetition**: e.g. noticing when a theme or musical motif is repeated; noticing when a particular theme returns after a contrasting episode; identify**sequences**(when a motif is repeated several times in succession, but either ascending or descending in pitch).**Contrast**: in classical music, a single theme is often stated in many different ways throughout the piece. Students should be able to identify different keys, different dynamics and articulation, different instrumentation and textures, and so on.

In all of the chamber music sessions I took, a common question I asked my students was: “who has the melody?” or “who has the main role in this section?” It is a very simple question, but the students are often unsure how to answer. Thus each student must place their own part in the context of the group, and when that happens, it is gratifying to the students and the teacher to hear the piece “taking shape”.

An essential exercise in chamber music (for players of all skill levels) is to rehearse sections of the piece while leaving out one or more members of the ensemble. For example, in a string quartet, there may be a passage where the two violins are playing in thirds or sixths. Then this exercise is a good way for the violinists to polish their intonation and consolidate their sound to be more harmonious; or if there is some tricky counterpoint between the viola and cello, then these members should rehearse together to improve (for instance) clarity and rhythmic precision. Meanwhile, the non-playing members should be actively listening, giving constructive criticism to their colleagues, as well as following their respective parts. Taking things apart is not only a great way to solve problems, but through this process everyone also gains a better understanding of how the piece is constructed, and hence a better understanding of their role in the piece.

An obvious technical challenge arises when the writing is overtly virtuosic. There are many such passages in the first violin parts of classical string quartets (Haydn, Mozart, Beethoven), and certainly in modern works there are examples of virtuosity in all parts. However, this is not what I have in mind when I refer to “chamber music technique”. Most of the time, the manner of playing required for chamber music is entirely different from that which is needed to perform a solo concerto. Where this is most apparent is when a part does *not* carry the main role — second violin and viola parts come to mind!

It is true that in classical chamber music for strings, the first violin has the leading role for most of the piece. Pieces such as Mozart’s *Eine kleine Nachtmusik* are instantly recognisable, and whether you like it or not, such pieces have become the “face” of classical music to the general public. Perhaps this is one reason for the misconception that only the first violin has anything important to play, and the rest of the “band” are merely the equivalent of “backing vocals”. Worse still, there is the impression that it takes less skill to play accompaniment roles in classical music — I’m sure collaborative pianists will have a lot to say here! The fact is that this couldn’t be further from the truth. In the lessons I gave at the Zhang Violin School, I looked for opportunities to show the importance of a good accompanying or supporting role in a string ensemble, and some of the techniques involved. I believe the main qualities at play are: *precision*, *dexterity*, and *sensitivity* or *subtlety*. Let me illustrate these qualities with Beethoven’s Quartet in C minor Op. 18 No. 4. We will look at some excerpts from the first movement.

The reference recording here is from the Dover Quartet — not only is this a great performance, but the sound engineering is superb, allowing us to appreciate all four voices in the quartet. At the very opening, the cello plays a pedal point with repeated quavers on a single note, C. The writing couldn’t be simpler, yet the performance should be anything but simple-minded. Indeed, the excitement of this opening is entirely dependent on the momentum generated by these repeated quaver notes. Furthermore, running quaver accompaniments are found throughout the entire movement but in highly contrasting situations. Compare the opening with these two other moments:

- 0:59 (second main theme of the movement, announced by the second violin with playful accompaniment from the viola and counterpoint in the first violin)
- 2:39 (a passionate exchange between cello and first violin, with suitably energetic quavers from the viola and counterpoint in the second violin)

Precision of articulation and rhythm is essential. Moreover, it requires some finesse in bow technique to be able to adapt the staccato stroke to the many different characters and contrasts in dynamics and tone colour throughout the piece.

Dexterity is usually associated with fast, agile playing — passages with fast bowstrokes, or rapid left hand movements, or a combination of both. However, at around 4:01 in the recording (this is the recapitulation section), a different kind of dexterity can be observed in the second violin part. This is one of my favourite places in this quartet! The second violin has to alternate between accompaniment and melody — one moment, it is playing a syncopated rhythm accompanying the first violin, only to jump out with a little countermelody in the very next bar. This requires clear differentiation in the tone colour, and the changes happen more or less instantaneously. Notice that all four parts are playing continuously, so the countermelody in the second violin almost sounds like an additional fifth voice, increasing the complexity of the overall texture. This kind of seamless juggling between roles is difficult for inexperienced players, but is nonetheless a vital part of chamber music.

Finally, I make a comment about the qualities of *sensitivity *and *subtlety. *Here I do not mean to encourage a discreet, timid kind of playing in chamber music. On the contrary, if the music calls for it, one must be prepared to offer the full range of dramatic expression — in this sense, Beethoven’s string quartets must surely be considered among the finest “dramas” ever created. In chamber music as well as in solo playing, one should seize the opportunity to develop the widest possible dynamic range, from the barely audible *ppp* to the most ferocious *fff*. Sensitivity and subtlety are good qualities to have in solo playing as well — the more shades of tone colour, the better, so why make a song and dance about it now? I simply think that these qualities are especially important in chamber music. Furthermore — and here is the key difference — in chamber music, it is not only about the individual tone colour, but it is also about understanding how one’s sound fits in with the group. Despite the saying “too many cooks spoil the broth”, in some ways, chamber music is like several people cooking the one dish simultaneously. Everyone contributes their part of the recipe, and hence there is a responsibility to understand what kind of “flavours” each part brings to the overall dish. I don’t want to extend this analogy too much, so let’s get back to music. As well as achieving all the contrasts required within their own part, the individual player must also understand how their tone colour fits into the overall sound of the ensemble. It is possible for the players in an ensemble to execute their individual parts excellently, and the result could still be unconvincing if each player is not also aware of the sound of *the others*. This is the *extra* bit of sensitivity and subtlety that I think distinguishes chamber music playing from solo playing. This extra bit also allows room for spontaneity in performance: in a great ensemble, if someone decides to something a little different, the others can adjust accordingly to support this creative decision.

By now, I have mentioned some rather advanced technical concepts that would not usually be mentioned in a workshop attended by students who are chamber music beginners. Nevertheless, we can encourage our students along the road to technical mastery and show them the first steps. This brings us to the next section.

As musicians, we develop our technique so that we have more refined tools to express ourselves musically. However, our ears must be sensitive to sound, just as a master painter is sensitive to colour, so that we know what to listen for in the rehearsal room. Hence all this talk of technique is useless without good aural skills, because if we do not have some idea of what good technique is supposed to sound like, our efforts are likely to be in vain. Of course, there are many ways to develop aural skills. However, in my experience I find that playing chamber music is some of the best aural training a musician could ask for. Playing chamber music places many demands on the ear. One obvious challenge is intonation — indeed, for those who do not play a fixed pitch instrument, refining intonation is one of the most basic tasks, yet it is a lifelong challenge! In chamber music, intonation is particularly difficult. The players must be able to execute their own part in tune, of course, but this is not sufficient, because the entire ensemble must also sound in tune! This requires a keen awareness of harmony and voice leading — not necessarily from a theoretical/analytical point of view (although this is helpful) but at least with regards to aural perception. Consistently good ensemble intonation is the result of much effort and experience — some would even say painful effort!

When one works with students who are inexperienced in chamber music, I would say that it is almost never a good idea to be very pedantic about intonation. Of course, if something is horribly out of tune, then one could suggest to the students that they should refine their intonation individually; it is probably not the most helpful thing and certainly not the most enlightening thing to spend an hour tuning a single bar of music, especially in the context of a short course like the Zhang Violin School. However, there is something the students can work on immediately, even in the span of an hour’s workshop: simply *playing together*. It sounds like a no-brainer, but it is more challenging than it seems. Very often, chamber music beginners are quite content to continue playing even when everything has descended into chaos. This is not meant to be a judgemental observation — it is understandable because they have yet to develop the skill of listening to other parts while playing at the same time (which is not a trivial task). One possible reason that a runthrough of a piece has fallen apart is that someone has made a mistake, but instead of continuing in tempo, they instinctively start again from the beginning of the measure where the error occurred. This is quite a natural thing to do when one practises alone, so again, this is a merely a product of the lack of experience in ensemble playing.

One way to remedy this situation, and to encourage good listening habits, is once again to focus on fewer parts at once. For example, there are often passages where two parts are playing the same melody an octave apart. The two players involved could then rehearse this melody together to consolidate intonation and achieve unity in rhythm, while the others listen. The advantage of playing in octaves is that it is quite obvious when it is out of tune! Another common situation is when a subgroup of instruments play the same rhythm — an accompanying figure, for example. If the problem is lack of rhythmic organisation, then isolating such a passage is a good way to develop the required rhythmic discipline. However, perhaps the problem is one of intonation, and the rhythmic aspect is fine. Then another viable exercise is to practise only the harmony in the passage. If, for example, the passage consists of repeated quavers, then it is a good exercise to ignore the repeated notes, and instead play sustained notes in order to hear the underlying harmony better. In both cases, the strategy is to group together parts by similar features. In this way, the ear is trained to understand this grouping as one object, e.g. two parts playing the same melody an octave apart can really be perceived as one part. From the performers’ perspective, this is a great simplification, and it also creates a kind of “landmark” in the piece. For example, the cellist might make a mental note: “ok, when I get to this section, I’m together with the viola,” or perhaps something like “when the first violin has this melody, I’m accompanying with the same rhythm as the second violin.”

The two examples I have given above are common features of classical chamber music works, but undoubtedly, exercises more specialised to the piece under consideration will be needed. The teacher should then use their knowledge and experience to decide what would make an effective exercise, taking into the account the ability of the students as well. In any case, however, the key idea is to use chamber music training as an opportunity to demonstrate good listening strategies, which is directly linked to effective practice techniques. Even in very simple pieces, there is a lot of worthwhile listening to do, and at the very least, the students will become more sensitive to sound quality.

In the sections above, I have outlined how chamber music training refines instrumental technique, improves aural skills, increases understanding of basic music theory and analysis — in short, how chamber music develops all the skills that are essential to good musical development. However, I have left the most obvious feature of chamber music to last: above all, chamber music is about *playing together*. When humans do anything in a group, there is usually (necessarily?) a consideration of leadership and teamwork, and music-making is no exception. A natural question arises: in a chamber music ensemble, who is the leader, if there is one at all? In my opinion, this is a bit of a trick question, as the answer is highly dependent on context. In the rehearsal room, there may very well be one person who tends to take charge. In string quartets, this role is traditionally assumed by the first violin, who is traditionally the leading role after all. But of course, in a healthy working relationship, the other members of the quartet should not hesitate to make their concerns heard as well. However, from a purely musical point of view, this question of leadership is slightly absurd. Clearly, the person who plays the leading role in a particular section of the music should be the leader of that section! If the viola, let’s say, has the main melody (yes, this does actually happen!), it is probably a good idea to let the violist lead. Therefore, during the performance of a piece of chamber music, leadership is always changing in accordance to what is required by the work itself. All of this is just a roundabout way of saying that everyone has a chance at exercising leadership when playing chamber music. It can be a good way to instil some confidence in the more timid students. On the other hand, for students who have strong personalities and are used to playing solo, chamber music is a good way for them to develop a more refined technique, as they must reconcile their playing with the ensemble.

Despite the length of this blogpost, I have merely scratched the surface of the subject. In no way do I claim to be an expert chamber music coach — these are merely some key observations and insights I have gathered in my limited experience as a chamber musician. However, I hope that my enthusiasm for chamber music comes across, and that I have made some coherent arguments for the importance of chamber music in general musical development. To conclude, let me make one final remark. Let’s consider music more generally, away from the realm of so-called “Western art music”: pop, rock, metal, jazz, Indian classical music, African drumming, Mongolian throat singing, Irish folk music, etc. It seems that such a large portion of the totality of music comprises of music-making in small ensembles. Chamber music is just one of thousands of ways humans have developed to make music together, and perhaps it is this universality which is the most wonderful thing of all.

]]>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. (Controlling 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 unlearn 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 2.1**

The **cardinality **of a set *X *is the number of elements in *X*. 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 2.2a**

A **function **or **mapping **is 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 → B

The set A is then called the **domain, **and B is the **codomain.**

If the element *a *∈ A is mapped to the element *b *∈ 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 2.2b**

A **function **is a collection of pairs of objects with the following propery: if (*a*, *b*) and (*a*, *c*) are both in the collection, then *b* = *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 2.3**

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

*S*is a finite set, or- |
*S*|is the cardinality of the natural numbers, i.e. there exists a**bijection**from the set*S*to the natural numbers.

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