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In this essay, we discuss on the technical side:

They say that success is all in the mind. So we’ll also discuss psychosocial aspects of writing a paper, including:

We conclude with some general advice:

The first duty of the introduction is not to excite, fascinate, inspire, and engage the reader—​that is its second duty.
The first duty of the introduction is to give the reader an idea to latch on to.

The reader will approach your paper asking three simple questions, which you must answer as well as you can:

I had this in mind when I wrote the introduction to the paper to the right. Check out the four paragraphs highlighted (click to open in a new window).

If your introduction enables your reader to understand and remember pithy and compelling answers to these questions, then your introduction has done its job—​and you’re half-way to getting published.

Let me say this again. To illustrate my point let me give you an example of two ways of describing a window:

I see (and write) too many introductions like sample 1 above. It’s not because authors are stupid: if you spent three years inventing something, you may lose perspective on your work.

Note the last bit of sample 2 above. This addresses the "how does it make the world better?" part of Test 1 above. Ideally, you need your introduction to make it clear, in a non-threatening way, how your paper advances the state of the art. Yes, this can be hard. That’s why talking with people is so very important. You learn how to …​ well …​ communicate.

Here’s one of my poorer introductions (since the paper is co-authored, let me repeat that I take responsibility for it). See how I failed to express what the paper is about, what makes it special, and so on and so forth.

It got published, but I just got lucky; I didn’t make it easy for the referees.

The good example was published in 2011, the lemon was published in 2005. I’ve got better! I wouldn’t have written the text to the right, today, because I’d have applied the test above and seen that it fails.

If you can’t get the introduction right, then that’s the text trying to tell me something. You just need to be sensitive and understand what it’s trying to say. Here are some possibilities I consider:

Here’s a little trick: make your text so it’s really nice. Leave it for a few days. Now read it through hurridly. Go on …​

Now …​ you see that paragraph that you skimmed over? The one where you read the first sentence and then jumped to the end or to the start of the next paragraph? That’s a weak paragraph and something’s wrong. The sentences are flabby and/or the ideas aren’t clear.

Work on it.

Now that you’ve aced the first few paragraphs, I suggest that you tell the reader about the specific structure of the paper to come.

Now is no time for technical details; just give a framework to help the reader ease into the paper. To the right is part of a page 2. Have a look.

It’s just like the introduction, but perhaps with specific technical reference to what has preceded or what will follow.

The reader has landed in this part of the paper and has probably not read the paper linearly. He’s been attracted to this part of the text because the section heading created a visible break in the flow of text, and perhaps because the title of the section interested him.

Just help the reader to get oriented and find his way around. Mention what the main definitions and theorems of this section are. At the moment I often write "The main definition is Definition X, the main results are Proposition Y and Theorem Z, which depend on technical Lemma Q".

I’ve slipped in a couple of recent examples, just to illustrate what I mean.

I think layout is important.

Here’s an example. The material is the same, but the layout has evolved (and improved).

Aside from the obvious, like choice of colour and improvements in notation, note that in the second paper above I don’t line up the arrows in Figure 2 with the arrows in the examples. They don’t belong together. The examples, however, do belong together, and this is expressed by the alignment. The reader’s eye understands this and is drawn to the three examples as a unit, in a way that does not happen so much in the first paper.

It’s not much. It just helps that little extra bit. Lets look at another example:

Amazing, isn’t it?

The papers mentioned so far are all conference papers. For journal papers the pressure to grab the reader’s attention is less intense.

I enjoy the challenge of writing for conferences, but I do prefer journals; there’s more space for manoeuvre and you can communicate with referees via the editor. The only disadvantage is that sometimes, a paper vanishes for 12 months and then comes back summarily rejected.

(That happened to me once: I submitted a survey paper and the journal didn’t publish surveys, but it took the editor a year to tell me. I had enough sense to stay polite, which is good because it turns out that that editor had been bereaved and really wasn’t thinking straight. But the journal should have structures in place to handle that. We’ve got real problems in academia with the way we’ve set things up—​some kind of Amazon or eBay like points system for journals, might not be a bad idea.)

In Two-level nominal sets I open with a pithy paragraph (see the highlighted text to the right), but there then follow several pages of discussion—​you couldn’t do that in a conference paper.

But look at page 5. I boil the whole discussion down to a single line, in a quoted box so the reader absolutely can’t miss it. Above that line, I pick holes in previous work (including my own) while giving the reader specific and mathematically precise metrics of this paper’s success. Below that line, I sum up exactly what the paper does and the two ideas underlying it.

Though I do say so myself, it’s a beautiful page. Layout, ideas, and text all come together very nicely.

Notice the footnote. I’m breaking one of my own rules: this is a direct critisism of previous work. The only mitigating factor is that I was one of the authors. I felt so strongly about this point that I had to raise it, even though I knew I was running a risk of rejection if the paper hit the wrong referee when he was feeling touchy.

You might also like to take a look at page 6. Because the introduction is so long, I condensed it down for the reader to an executive summary in Subsection 1.4.

I personally don’t have a standard format for my conclusions. I do whatever suits the material and the space available.

Conclusions are retrospective. You’re reminding the reader what you’ve done. So:

My current practice is to write the introduction in the present or future tense, and the conclusions in the past or present tense.

I start with a brief summary of the paper, and then segue into a discussion of significant challenges and difficulties, and/or related work. I conclude with future work. And, I put in precise references so the reader can check up on me.

(Talking about having "shown it sound and complete" without references is OK because there was a section called "soundness" and another called "completeness". Still, precise references might have been better.)

If there are any likely misunderstandings on technical matters that would have confused the reader in the introduction, now is a good time to head them off (also do this at the point of definition, of course). Often, you only find out about these likely misunderstandings after the paper is refereed. This might cost you a resubmission or two. That’s another reason to prefer journals: you can get into a back-and-forth discussion with the referee which converges quicker to a fixedpoint than a sequence of conference rejections (best, of course, to just get it right first time; but one can only do ones best).

The example to the right is a case in point. Summary, a few pointers into the text, a few nice intuitions, and heading off a potential misapprehension (and yes, that came from a previous version of the paper).

It is often suggested that you should write the introduction, abstract, and title last. This is because you should write the introduction, title, and abstract last. By all means write them first too, perhaps to get the ideas straight in your head—​just do so in anticipation of tearing it up repeatedly as the material evolves. The titles of my papers may undergo revision up until the last minute.

What happens with the abstract?

Sometimes, you might want your abstract to be short and scary. An example is below left. I sent this in to a conference and I wanted to make sure that it would go to referees who were comfortable with the material in the paper. That material was quite technical, so I needed to make sure it would be picked by a referee who enjoyed a mathematical challenge. Being a conference, space was also at a premium.

Sometimes, you might want your abstract to be long and friendly. The longest abstract I’ve ever written is below right. It was for a survey journal paper; it had already been accepted so I didn’t have to worry about referees. I had plenty of space. The editor rightly asked me to make the abstract more descriptive so I thought: "what the hell, I’ll write an essay". So I did, and it worked fine.

So: the abstract is like a flower. Design it to attract the right pollinators.

Titles should be descriptive. Doh!

At the moment, my titles are long (about two lines). I tend to split them up with a colon. Why? Because in this day and age of Google, papers are not taken off the shelves of libraries; they are indexed by computers. Lists of papers are not compiled and perused; they are auto-generated and skimmed.

The questions I ask myself are:

So the third title above would presumably show up in searches for "nominal meta-variables", or "nominal terms", or "nominal sets", or "sets meta-variables"--to all of which this paper might be relevant. You get the idea.

I try to put as much information in the title as I possibly can. That’s just my current policy based on observing my own search behaviour at the moment. It might change.

After all, academic search tools are improving all the time, albeit from a low baseline. It’s going to get better, right?

Look at Proposition 4.11 to the right, and Remark 4.12 immediately following it. I spend two lines on the result, and then eight lines explaining to the reader why it is not trivial.

Why?

There’s a very similar result in the literature (it’s cited in the extract). I know that the expert reader will skim over 4.11 thinking "Oh yes, that’s just like Corollary 5.2 of [15]".

It isn’t; the proof is very different and it’s actually a little bit surprising that it works at all. But if I don’t say that, the reader won’t figure it out for himself, will he?

In my early days, I imagined the reader poring over every sentence of my paper, as I had. I’d have left Proposition 4.11 unadorned, to leave the reader the pleasure of figuring out for himself the sublime difference from Corollary 5.2. That flattering illusion didn’t last long: all my papers just got rejected.

This is very easy to get wrong. You spend half a year with your nose jammed up against a theorem, and you lose track of …​

If there’s something in your paper that looks simple, but isn’t—​then say so. Explain it in the introduction or conclusions and also discuss it at the appropriate point in the paper.

If you don’t, the reader won’t know it’s there.

At some point, you’re going to have to mention previous work. Maybe you wrote that previous work, maybe somebody else did. Here’s what not to do:

There’s lots wrong with this.

Let’s try this again:

There. Now you’re happy, the reader’s happy, and Professors X and Y …​ well, they may or may not be happy, but at least it’s all about the maths and you’ve given the reader honest evidence for your position.

Look at it from the point of view of Professors X and Y (who, perhaps, had good reason for setting things up as they did and see the subsequent developments very differently). God willing you’ll be in their position one day, if you’re not already. It’s no fun seeing some smug little bastard who doesn’t understand what he’s talking about trashing your work in public. This is not just for the humiliation of it, but also for the crushing sense of despair that for all that hard work—​people still don’t understand what you wrote. So be nice and try to talk to everybody concerned.

But, I digress.

A few rules of thumb:

So you’ll get referees saying "this is clearly wrong, because Y has properties that X does not" (but you never intended to model all of Y, only the bits you’re interested in). You’ll get referees saying "X just isn’t equal to Y", or "How dare this stupid author claim that we don’t need Y?".
So I tend to conclude by saying something like "I’ve just built a model of Y. There are bits missing, and I’m not claiming that X=Y. Nevertheless, by studying X we can better understand Y, because of Theorem Q".

So if possible I conclude by adding "…​ and I don’t claim that X necessarily replaces Y. X has the following advantages …​ but Y is easier to read / shorter / more flexible / less flexible (delete as appropriate)".

I don’t mean to be cynical here. We can’t see the future, so try to play to all sides of the audience wherever mathematically possible.

You never know; your next paper might contradict this one. Then you’ll look pretty silly if you said (as I see people writing, without substantiation) "This author is absolutely definitely sure that what he is doing is absolutely right". Explain your reasons and allow the reader to draw his own conclusions.

I covered this one above. If you don’t explain where the hard bits are then the reader will assume that there aren’t any. It’s not a matter of bragging; you just need to tell the reader the plot. So I end up writing something like "It is not evident that Widgets should equal Wombats, and Definitions X and Y require some careful mathematical design", or "Most of the cleverness is in getting Definitions X and Y right".

I have found that my papers are generally better for being written with coauthors. Even if the coauthor doesn’t do much, having somebody to talk to makes a huge difference.

My advice is: find good coauthors and treasure them, and if you can’t, then find bad ones and be grateful for what you’ve got! Don’t worry if you end up doing all the work: what matters is producing the best document you can. Two people are better than one, at this game.

If a coauthor is slow, as is often the case, then chase him up politely. I call this a "standard pester".

One week later:

Dear Professor X,
This is another standard pester, just to make sure I stay near the top of your stack. You were going to spend five minutes looking at two weeks of intense effort on my part that I sent you last week. Do you have any idea when you’ll be able to do this?

Next month:

Rules of thumb:

If a couple of e-mails don’t do the trick, pick up the phone.

Sometimes, you just have to deal with difficult characters. I had one coauthor who I knew from experience would procrastinate. I set a deadline and when he missed it I removed his name from the paper and told him I’d done it. Tee hee, that made him sit up. He needed publications so that really got his attention.

I had one coauthor who disappeared for a year. No reply to e-mail. No answer to phone or messages. I couldn’t finish the paper without him, so I was just stuck. I bumped into him in a conference and asked "Look, I’m really sorry for whatever I did to offend you, but what did I do or say to drive you away?". He replied "Oh no, you didn’t offend me. I just got busy." I finished the paper with him, got it published …​ and (quite rationally) chose to work with coauthors other than him thenceforth. But we did finish the paper, and it was better for his participation!

I don’t mind the nonsense above. But I’ll tell you what does annoy me. A coauthor who vanishes, doesn’t respond to e-mail, lets me painstakingly polish a paper to publishable standard all on my own, and then suddenly reappears perhaps a week before the submission deadline and starts tearing into the paper.

I hate that. What a lack of consideration! Polishing a text is hard work: forcing your coauthor to do it twice (once on his own, once with you) is either self-centred, or a power-trip at my expense. Yet even so, when this happened to me the paper ended up better for the second pair of eyes on it. I considered carefully whether to work with the person concerned again (to their subsequent detriment, I like to think), but after we had finished the paper.

I’ve twice got involved with coauthors that were just a waste of time. One was an idiot, the other couldn’t be arsed. If this happens to you then I suggest that you leave their name on that paper (if it ever gets published). If you don’t then you’ll spend the rest of your life looking over your shoulder, terrified for the comeback: "Did this paper get rejected because X was on the programme committee and he hates me?". Grin, be polite, chalk it up to experience, perhaps hold back a bit on the results, and later on write another and a better paper without them.

If you are reading this guide for help with your own work, then probably that would be you.

Whoever is the junior partner in a collaboration will typically do most of the legwork (writing out / checking details of theorems, trying out design options, etc).

It’s not a hierarchical thing: it’s a matter of who is

Junior people typically satisfy a, b, and c, whereas senior people typically don’t. Junior people also typically need to work ten times as hard to get the same results (if they’re lucky) because they lack overview, technique, and experience, so a senior partner can pay for himself many times by providing ideas, guidance, advice, and knowledge of how to present the material to a broader audience.

Everybody brings expertise to the partnership. Everybody is equal. But, if you’re junior and you really want that paper, then be prepared to sweat for it. Conversely if you are junior and your senior co-author is putting in most of the hours, then ask yourself what is in it for him and …​ will he work with you next time? Just a thought.

Productivity grows with practice. Personally, I relish the opportunity to do most of the work. It gives me a deeper understanding of the underlying maths, which I can then bring to the next paper. So if your coauthor leaves you in the lurch then remember …​ there are no problems, only opportunities!

My advice: alphabetic order. No egos = no complications.

You think that your contribution was most valuable? What if your coauthor disagrees (bear in mind that he might not say this to you, and you might suffer for it only later)? Do you argue over this? No. Keep it simple. If it wasn’t worth changing your name to "Aarnon, Aliyah", then that’s your fault and it’s not worth losing a collaboration over it now.

Conversely, if your coauthor is determined to have a particular order, let him have it. Avoid direct conflict. What matters most is who develops the material later; the issue here is not position but velocity. If such a disagreement does arise, then your coauthor is insecure about his contribution—​and that suggests you may overtake him later. Anyway if you’re arguing about priority then there are probably deeper problems with your relationship. Weigh up whether you want or need to write another paper with him. Give him what he wants today, and get out.

We project ourselves onto other people; I tend to assume that you’re just like me — and you tend to assume that I am just like you. It’s only human.

When people forget this they can get very annoyed. "Why don’t they understand? The only possibilities are idiocy or deliberate malice!"

Wars have been fought, and many books written. Let me try to discuss briefly how this phenomenon affects academic communication.

Your reader is not equal to you.

As an author of an academic paper you are somebody who loves the material, or at least has laboured intensely to create it. You will be tempted to tell your reader how hard it was to do this; all the false leads; also all the limitations. You will tend to assume that:

At least, you will tend to assume this if you are just like me.

But no. Even for close colleagues, this is usually not the case. What was obvious for you may be a revalation to the reader if you are lucky, or perverse if you are not. Significant distinctions to you may (rightly or wrongly) seem minor details to the reader—​conversely, the reader may be fascinated by what seems a minor curiosity to you. As for awareness of research context, your other papers, and the underlying motivations—only if you tell him, and if your reader has any prior awareness of these things it may be very different from yours.

Most likely, your reader just wants to get the idea: "what is in this maths for me?". Or, your reader will be after a specific piece of information: "does he encounter problem X which I’m familiar with, and if so how does he solve it?"; "what’s his syntax?"; "does he use Widgets or Wombats?"; "can he model Y, Z, and U?". Once these questions are answered, if you’re lucky, some very few readers may go beyond that.

Your reader may be a future version of you, by the way. It’s amazing how quickly one can forget the minutiae of the material, so by writing a good, clean document you are also helping your future self.

Let me give you a real-life example. A PhD student asked to give a 30 minute lunchtime seminar spent most of it describing the technical limitations of his work. At the 20 minute mark I interrupted him: "But the overview of your work is this, correct?" (I had extracted it from the 20 minutes of technical data; it took me about 30 seconds to sum up). "Yes", he replied, "that’s right". "That’s an incredible achievement, you know", I said encouragingly. He was stunned and silent.

My point is not about PhD students giving bad talks. My point is that this PhD student had spent 20 minutes talking to himself. He was a modest, sweet man, very involved in his work, aware of his own shortcomings and deeply concerned about getting stuff just right—so he projected this onto his audience and spoke accordingly. I intruded with a different point of view, which he had not considered, and he was surprised.

Your readers are a strange, amazing, alien species. Get out there and get to know them.

One paragraph = one idea. Make sure that idea is clear from the first couple of sentences of the paragraph. Do not introduce the central idea of a paragraph sandwiched in the middle of the text.

It’s very simple. For every paragraph in your paper ask yourself: what is the point of this paragraph and is that point clear? If not, then why is that text sitting there?

If you find yourself writing a really long paragraph that does not naturally split in two, then odds are you’re really writing an itemised list.

I could go online and find you a dozen great examples of how not to do it, but I promised to slag off only my own work so I’ll try to rewrite the text above as I wouldn’t normally do it. This goes against every fibre of my being __…​

Each sentence in the excerpt above is reasonable, but they are assembled into a bad paragraph. I take three sentences to get to the point, and than ramble on. You can easily lose your reader with this, before you even get to the hard maths.

If your text is lively, direct, straightforward, engages the reader and does not waste their time, then your text is more likely to be favourably refereed and once published is far more likely to be read and cited.

It’s not rocket science: the clearer your text, the clearer your point will be and the more clear it will be to your reader that they have achieved clarity in understanding your work. And yes, I did use the root "clear" four times in that last sentence; that was deliberate.

There is plenty of just plain bad writing out there, but it can also be quite subtle. Poorly-edited text does not necessarily look dreadful. Consider:

The cumulative effect of awkward phrasing, unnecessary circumlocutions, and waffle, can sap a reader’s energy and enthusiasm. Don’t do it.

As a writer, I may imagine myself trying to impress a girl (or lady) at a party, or speaking with a new colleague in the common room. Do I sound smart, or do I sound like I’m trying to sound smart? Am I making sense, or am I nervous and droning on a bit? Are my reader’s eyes glazing over? If I get it wrong, my reader might give up and find another source to read (and cite).

Note that technical text is not necessarily like normal prose. You’re not writing a novel and you don’t get points for smart language. It’s much more like journalism than fiction.

I note that in many continental languages, it is considered good style to write complex sentences (lots of "the former" and "the latter", and paragraph-long sentences). I suspect this is descended from Latin. I tell foreigners this is not good English and am universally ignored. I try to be understanding: their whole culture is at odds with me.

Examples of fine English style are Mark Twain (I haven’t read his fiction but I adore his essays) or that great English essayist George Orwell. For a logician, try reading anything by Paul Halmos; his clarity is amazing.

A few random things that dropped out of my fortune cookies:

This is about publishing clear text that people
can read,
will want to read, and
will be better for having read.

Your papers won’t go stale, they won’t corrode, and thanks to the internet they’ll stay online until the end of civilisation—​and, hopefully, until long after you and I are gone.

Get it right, and through your papers you might have the opportunity and privilege of being a colleague, a guide, and perhaps even a friend and inspiration to people you’ve never met.

That’s worth a bit of work.