1. Audience and Ideal Reader
  2. Pitching Your Work
  3. Addressing Your Ideal Reader
  4. Be Kind to Your Audience
  5. PDF Version

“I'd do anything for you.”
“Would you please please please please please please please stop talking?”

—Ernest Hemingway, Hills Like White Elephants.

Even technical writers write to be read and not so someone can tell them to shut up.

You do not want to bore your “Ideal Reader”.

In this lecture we discuss who your “Ideal Reader” is and how to pitch your work so that they best appreciate it. More specifically, we discuss how to determine your Ideal Reader depending on the type of text your are writing, and how to fine-tune your work accordingly. We then turn to the writer's presence—which should be felt, but not overwhelmingly so—and how the writer is meant to address themselves in a text. Finally, we give some self-editing tips that should improve the readability of your text.

Audience and Ideal Reader

Who are they?

Mathematical truths exist without us, but if we already choose to study them and convey them to other humans, we ought to conform to the basic tenets of clear, precise, and directed communication.

That is, we have to always consider the existence of a reader—a person who will be spending their precious time trying to understand our words and we have to tailor our work to suite them.

You’re not a journalist or fiction writer, sure, but you should be writing for the same reasons:

  • to be read,
  • to be understood,
  • and—let's face it—to be appreciated.

Appreciation comes in many forms: as a grade, as an acceptance letter, as a coveted research position or industry job, or (cough) simply as the pleasure of having your ideas contribute to humanity's knowledge of the sciences.

So how to gain that appreciation?

A useful analogy

The following is a useful analogy to have in mind. Imagine you are waiting for the train and in front of you...

  • A person stands on a platform saying the same thing to every passerby. The passersby understand nothing, care not, and walk on.

This is the mathematician who writes for nobody in particular, and therefore nobody in particular reads their work.

  1. Their work is wasted

Now imagine you are waiting for the train and in front of you...

  • A person stands on a platform talking to themself. Most passersby walk on, but even if someone stops to listen, they understand nothing because the speech is only intelligible to the speaker.

This is the mathematician who writes for themself and therefore nobody else can understand their work.

  1. Their work is wasted

Finally, imagine you are waiting for the train and in front of you...

  • A person has invited all their mates from work, and is standing on a platform, beginning to address them:  “Listen friends, I’ll tell you something that'll tickle your fancy.”

This is the mathematician who writes for a particular set of people, aiming to have all of them pay attention.

  1. This is the mathematician you want to be, directing your work at a concrete audience.

Pitching Your Work

Directing your work at a concrete audience is called pitching to that audience.

An appropriate pitch excludes what the audience already knows, and presents the rest of the material in a relevant way.

You can’t pitch to a motley bunch, so your audience needs to be homogeneous.

But once your audience is homogenous you can pick one member, any member, call them the Ideal Reader and pitch your work to them.

Then there are two things you need to determine:

  • What is the background of your Ideal Reader?
  • What is their mathematical maturity?
Extreme representatives of Ideal Readers
  • A fifty-year-old professor of geometry probably has “infinite” mathematical maturity, but no background in a specialised field of statistics.

This means you need to give specialised definitions but do not need to explain any mental “tricks” or standard mathematical patterns of thought.

  • If the same professor happens to be the founder of your specialised field, then they have both infinite mathematical maturity and infinite background.

This means you need to explain only the very latest, cutting-edge, original research that you have been doing.

  • A whizz-kid of sixteen might have specialised knowledge of algorithms, but very little mathematical maturity.

This means that you need to focus on explaining the “big picture”, and on mathematical patterns of thought that are gained only through experience.

  • First year undergraduates usually have neither the background nor the maturity in any field.

This means you need to start from scratch: specialised definitions and patterns of thought.

As you are starting to realise, the Ideal Reader resides in your head (as do all ideal things in maths).

Let us look at some examples of audience and pitch in practice.

Theses and semester projects

With some exceptions, all of these will only be read by an adviser who is already familiar with the material.

Therefore, your adviser is not your Ideal Reader.

Your Ideal Reader is someone of similar mathematical background to yourself as you were before you started the project or thesis.

Think of a colleague and think of how you would quickly and clearly explain your work to them.

The role of the adviser as the actual reader of your work is to decide whether you have done a reasonably good job at addressing your Ideal Reader. You are graded not for the maths you learned, but for the maths you were able to pitch correctly and explain coherently.

For example, this means that if you wish to discuss a certain class of manifolds in a third-year semester project, you need to define a manifold, but you can assume the Ideal Reader is familiar with sets, topological spaces, calculus and so on.

In contrast, in a PhD thesis on the same subject you would assume the Ideal Reader is familiar with classical manifold theory and you would spend a paragraph defining this special class and stating a few of the standard results regarding this class.

  1. Your Ideal Reader is your classmate.

Starting at the level of a Master's thesis, and certainly that of a PhD, your work may be read by a wider audience and not just your adviser. This changes little in your approach: you are still writing for a colleague who is at the same level as you were prior to commencing the work.

As you mature mathematically and your work becomes more specialised, your Ideal Reader will mature with you and also specialise up to a point.

You will also start taking other considerations into account when imagining your Ideal Reader.

If your preprint is primarily in number theory but you think it might have some applications in statistics, you might pitch your writing so it includes sufficient number-theoretical background that statisticians can read and understand the preprint too.  

Your goal is to increase your core audience while staying relevant.

Remember the standing on a platform scenario from earlier: you do not want to be speaking to the whole world, nor speaking only to yourself. This is a classical maximisation problem.

  1. Your Ideal Reader is a colleague in your field.
Application essays

Applications for further studies may require a motivational essay or personal statement. We will discuss these statements in detail next lecture, but here are some pointers.

These essays will be read by members of the application committee, who are professional scientists, though perhaps not in your precise field or interest area. Before writing the essay, your task is to determine, as closely as possible, who will be reading it.

In most of Europe, you generally apply to a particular professor with which you wish to do your PhD. Anything you write should be directed the person you are applying to.

Your Ideal Reader is your desired future adviser who will be impressed by a specific, advanced theorem in her field that you name-dropped in the essay.

In America or the United Kingdom, you generally apply to a university department and so the motivational essay should be kept general.

Your Ideal Reader is a some mathematically minded scientist who will not be impressed by any specific name-dropping.

  1. Your Ideal Reader is the person who will be reading the essay.

Talks are not read by an audience, but delivered to it. That said, they have to be prepared, usually written out by hand or digitally, and they most definitely have to be pitched at the correct level.

Suppose you do not pitch it correctly.

Imagine entering a room of ten-year-olds and starting to tell them about Banach manifolds. You will get puzzled looks, questions, and eventually laughter. Any prepared slides or materials will prove useless. You will have to invent, on the spot, a new talk and it will not be fun or productive.

So when preparing a talk you should figure out who your Ideal Listener will be. It is a graduate seminar? Is it a group seminar? A department seminar? A job interview?

  1. Your Ideal Listener is a typical attendee of the seminar series you are invited to speak at.

Addressing Your Ideal Reader

Set the tone

In a formal piece of writing at university and above you do not address the reader directly, though your aim is to establish a professional tone.

This means you want to be courteous, but not machine-like rigid, and you want to be friendly, but not gushing with camaraderie.

Avoid the following extreme cases:

The machine:

  1. Let $f$ be a smooth function. Let $X_f$ be the set of stationary points of $f$. Let $g$ be a smooth function. Let $X_g$ be the set of stationary points of $g$. Let $h$ be a smooth function.

The enthusiastic best friend:

  1. Let’s think about these smooth functions $f$ and $g$ and their stationary points. You must remember how cool a concept this was back in high-school, when we would draw a graph and look for its “special points”. But now we have better tools to study graphs. Hey we do not even have to draw a graph! We can just differentiate. It’s lit.

Establishing a professional tone also means avoiding statements that are loaded with more subtle emotion or attitude.

We have already mentioned some of these examples in passing, but here’s a fun list.


  1. The remainder of this proof is obvious if you just think about it.


  1. We believe the reader is smart enough to complete the proof.


  1. My proof is as sleek as Riemann's.


  1. After much lucubration I have alighted on this most elegant of proofs.


  1. Where every other version of this theorem requires an additional assumption we require none, leading to a shorter, slicker proof that can be explained to anyone with a basic grasp of graph theory.


  1. All readers in possession of a working brain should by now have worked out the end of the proof.

And so on and so on. The sins of a tone-deaf writer can take many forms.

Aim for an amicable, but strictly professional tone.
Where tone meets pitch

All the examples above have at their core an emotion, attitude, or opinion that does nothing to further the reader's understanding of the content.

Conversely, if such content does further the reader's understanding it should be included. For example, you might say:

  1. It is surprising that in four dimensions there exist infinitely many exotic Euclidean spaces.

Here the inclusion of the word surprising draws their attention to a mathematically salient point.

Telling the Ideal Reader what to do

Now let us zoom in on the actual mode of address.

How do you tell the reader to pay attention?

We discussed last time how maths often uses the imperative mood of a verb:

  1. Let $k$ tend to infinity.
  2. Set $n := 2$.
  3. Suppose that the conjecture is true.

The imperative mood may seem like it is commanding the reader to do these actions, when of course it is not; writing in this way is merely a convention.

Secondly, depending on the style of a text, you may have come across a liberal application of the personal pronoun you. This pronoun is a natural extension of the imperative mood:

  1. If you let $k$ tend to infinity.
  2. When you set $n := 2$.
  3. Therefore you could suppose that the conjecture is true.

In modern texts you is understood to be a warmer, more humane version of the pronoun one (e.g. If one lets $k$ tend to infinity).

The Writer’s Persona

Striking the correct tone when addressing the Ideal Reader involves choosing the correct presence for yourself within the text.

Recall from the previous lecture that you have four options when writing maths sentences:

  • The active voice, with yourself as the subject.
  • The active voice, with a maths term as the subject.
  • The passive voice, with yourself as the object.
  • The passive voice, with a maths term as the object.

But how do you refer to yourself in the first option?

Here is your choice:

  1. I prove the theorem in the Appendix.
  2. We prove the theorem in the Appendix.
  3. The authors prove the theorem in the Appendix.
  4. One proves the theorem in the Appendix.

We discuss the pros and cons of each of these options separately.

Option 1: Using I

Using the first person singular may seem the most natural if you are the single author of a text.

However, in mathematics using I is distracting: it draws the readers attention away from the content.

It is best to reserve using I for a motivational essay where you discuss your personal history or for a dedication at the beginning of a formal work.

Option 2: Using We

Using the first person plural may seem an odd choice for a single author of a text, but regardless of the number of authors, it has become the norm.

Traditionally we is taken to mean writer and reader and is therefore a less off-putting position to be in as a struggling reader (the burden of the struggle is shared, so to speak).

However, nowadays this is simply the option everyone expects, and so it is least obtrusive in a text.

Option 3: Using The author

Referring to yourself in third person is useful in situations where you must differentiate yourself from the reader and would be much tempted to use I; this is usually in situations where you must humble yourself and admit your ignorance in some respect.

For example, in a conclusion section, you might say:

  1. At this time we do not know how to prove the Conjecture, but we are hoping to tackle it using a modification of the methods developed in Section 4.

The reader will not misunderstand what is being said, but it would not be wrong to emphasise the researcher's role at this stage and say:

  1. At this time the author does not know how to prove the Conjecture, but is hoping to tackle it using a modification of the methods developed in Section 4.
Option 4: One

Sentences using the indefinite pronoun one are impersonal and formal.

They feel halfway between the active voice (which they utilise) and the passive voice.

Note this is different than the previous use of one as a stand-in for you (now you are referring to yourself, not the reader!)

In everyday English—unless you are careful—the use of one can sound a bit pretentious:

  1. One might develop this argument further, but one could not be bothered.

Or it can sound quaint.

  1. One is careful when developing such an argument, because so many other papers have failed at the task.

However, occasionally one can come in handy in a formal setting where you might tempted to address the Ideal Reader using the less formal you but it is not appropriate to do so.

  1. One can think of the 3-dimensional sphere $S$ as union of two solid tori $T_1$ and $T_2$.

In summary, go for Option 2 most of the time:

When writing formal, technical mathematics refer to yourself as we.

Be Kind to Your Audience

Do everything you can to make your content stand out, rather than the words transmitting it.

Here are a couple of tips (more are in the accompanying lecture notes).

Order words for flow

Consider the following sentence on the topic of symplectic diffeomorphisms[1]:

  1. Since symplectic linear transformations have determinant 1, we can conclude using several-variable calculus that a symplectic map is always locally volume preserving and locally invertible; roughly speaking, this means that the map $\phi \colon A \to \phi(A)$ is invertible whenever $A$ is a sufficiently small subset of $U$, and $\phi(A)$ has the same volume as $A$.

Now isolate the second part of the sentence, and try shuffling around the phrase roughly speaking. Note the changes in sentence flow.

  1. This means, roughly speaking, that the map $\phi \colon A \to \phi(A)$ is invertible whenever $A$ is a sufficiently small subset of $U$, and $\phi(A)$ has the same volume as $A$.

The pattern at the beginning of the sentence is: two words, phrase, three words, formula. This is choppy writing.

  1. This means that the map $\phi \colon A \to \phi(A)$ is invertible, roughly speaking, whenever $A$ is a sufficiently small subset of $U$, and $\phi(A)$ has the same volume as $A$

The phrase severs the sentence in two. Also choppy.

  1. This means that the map $\phi \colon A \to \phi(A)$ is invertible whenever $A$ is a sufficiently small subset of $U$, roughly speaking, and $\phi(A)$ has the same volume as $A$.

This is nonsense. The phrase interrupts two conditions.

  1. This means that the map $\phi \colon A \to \phi(A)$ is invertible whenever $A$ is a sufficiently small subset of $U$ and $\phi(A)$ has the same volume as $A$, roughly speaking.

The phrase appears to be an afterthought. The reader is warned too late that the preceding statement is approximate.

Order words for emphasis

Whenever you can, you should reorder your sentence so that the words of greater importance come first (which is the location of greatest importance in a sentence).

Consider how the emphasis changes with the word order:

  1. An important example of a knot is the trefoil. (The focus is on a special kind of knot.)
  2. The trefoil is an important example of a knot. (The focus is on the trefoil.)

As the complexity of a sentence grows, more variations emerge. What you decide to emphasise is up to you but this decision will drive the reader's attention on a small scale (which will be felt cumulatively).

  1. The trefoil is the only knot with crossing number 3. (The focus is on the trefoil.)
  2. Only the trefoil knot has crossing number 3. (The focus is on the uniqueness of the trefoil.)
  3. The only knot with crossing number 3 is the trefoil (The focus is on knots with crossing number 3.)

In summary:

  1. This sentence is taken from the Princeton Companion to Mathematics, and was written by Gabriel P. Paternain. ↩︎

PDF Version

Here is a PDF version of these notes.

Comments and questions?