Writing formal philosophy

This is a guest contribution by Richard Pettigrew, Professor of Philosophy at the University of Bristol for our series How Philosophers Write:

Stephen Hawking speculated that every formula he included in A Brief History of Time would cut its readership in half. In the end, he included only one, E = mc2, and the book sold half a million copies every year for twenty years.

You often hear a similar complaint from those who work in the more formal parts of philosophy--Bayesian epistemology, philosophical logic, formal semantics, decision theory, voting theory, formal approaches in metaphysics and ethics, for instance. They lament that their work is not so often read, not so often discussed as work on nearby topics that do not use a formal framework.

Now, I don’t know whether the numbers support this gripe, but whether they do or not, there is certainly a lot of work in formal philosophy that is not easily accessible to people who aren’t fully immersed in the formal framework it uses. And while this is no doubt due in part to the difficulty of the mathematical techniques in play, it is also partly the result of the way in which some formal papers are written. And, indeed, mea culpa, I include a number of my own papers in this--I read them back sometimes and hate myself for not doing more to open them up to a wider readership. So I’m writing this not from the smug position of someone who claims to know all the secrets, but rather from the position of someone who’s trying to get better at making their formal work easier to read and digest. I’m not sure whether I’m succeeding, but here are some of the writing techniques I’ve been trying to use. I’d love to hear alternatives or supplements in the comments.

Write like your best teacher We often talk about how our teaching feeds into our research. When we do, we usually mean one of two things: (i) teaching a topic, having to lay it out for your students, makes you understand it better; (ii) having to explain a problem to people who’ve never met it before often makes you formulate a new solution to it. But there is also a third way our teaching can affect our research. It can make us better writers. I think this is true of all philosophical writing, but particularly writing in formal philosophy. For instance, I sometimes write a particularly technical passage in a paper as if I were responding to an email from an engaged and talented student who has heard of the idea but doesn’t quite understand it fully. In my experience, this won’t just engage more philosophers from non-formal parts of the discipline--it will engage formal philosophers better as well. My doctorate was in mathematical logic, and I’ve been working in formal parts of philosophy for over ten years, and I still read papers much faster when they are written more accessibly, even if they’re quite a bit longer than a more succinct, purely technical version might be.

Give informal glosses So how do we write that sort of email to our students? Well, one obvious thing to do is to include informal glosses for the trickier bits of technical material we include. For instance, if I’ve just given the definition of the expected utility of an option in decision theory using the usual mathematical symbols, such as the summation symbol, 𝝨, with its subscripts and superscripts, and so on, I might then give it below in plain English.

Now, it’s something of an art form being able to tell when to do this and when to leave it out. Do it too often and the reader might find it’s taking so long to get through the material that they can’t hold it all in their head throughout; do it too little and those who aren’t accustomed to the symbols will struggle so much they’ll likely give up. Myself, I use a rule of thumb: frontload your glosses. When you are first using a piece of formalism in your paper--a series of definition stated formally, for instance--keep the glosses coming thick and fast. That will help readers understand what the formalism is saying; they’ll get a feel for how it works. Then, later, you can keep the glosses sparser, since readers will be accustomed to what the symbols say.

Of course, remember: you’re not alone. Perhaps you can find a writing buddy from a less formal area of philosophy. You read their drafts and they read yours. Ask them which bits tripped them up; where did they pause and re-read longest; which explanations worked and which didn’t.

I know some philosophers try to provide only glosses, and omit the formal versions of the definitions and theorems. And I sympathise with the spirit. But my own feeling is that we need the precise formal version as well. Perhaps I’m idiosyncratic, but I find I need the gloss to give me the general, often rather fuzzy outline of what’s going on. But I need the precise version to reassure myself that I’ve definitely understood the idea in question with the full rigour that is crucial if I am to evaluate the ideas being presented.

Draw diagrams Sure, it’s a cliché that a picture is worth a thousand words, and it’s often utterly false, but a well chosen diagram can aid understanding enormously. My favourite example of this is the reasonably simple set of diagrams that Lara Buchak uses in her book, Risk and Rationality, to explain how her decision theory works (Chapter 2). Buchak’s risk-weighted expected utility theory is reasonably complicated when it is presented in symbolic form, and certainly not intuitive; and, while the glosses in plain English that Buchak provides definitely help, the thing that turned on the light bulb above my head was this series of diagrams. Like glosses, don’t substitute these in place of precise formal definitions or statements or proofs. Use them as supplements.

Illustrate with examples, analogies, and even metaphors As with diagrams, so with examples, analogies, and even metaphors. No-one thinks that a well described example, a well chosen analogy, or a slightly vague metaphor can take the place of a precise definition or statement or proof. But that’s not what you’re using them for. Rather, you’re using them to aid the reader when they come to consider the precise and general versions. If you have a rather sketchy, hazy, watercolour-wash version of an idea in your head, it’s so much easier to fill it in and sharpen it up and make it a fully detailed line drawing than if all you’d had was a blank canvas.

The appendix is your friend Not all technical material is explanatory. Sometimes, the proof of a central result does not do much to illuminate the result itself; and very often it doesn’t assist with the philosophical work you want that result to do in your paper. If that’s the case, I think it’s probably best to put that proof in an appendix. It avoids interrupting the flow of your paper. Writers are so familiar with the idea they’re presenting, they often forget that their readers aren’t, and they’re having to hold fifteen new ideas in their head throughout the paper. There’s not a great deal we can do about that, except to signpost the dialectical structure of our argument and reiterate its central ideas throughout our writing in order to remind our readers as they journey through our paper. But one thing we can certainly do in formal papers is not to interrupt the flow with a difficult, but unilluminating proof that makes it all the more difficult for the reader to hold all those ideas together.

Motivate the formalism Some philosophers don’t engage so much with formal work because they find the technical parts daunting. But others don’t engage because they’re a bit suspicious of the use of formal models in philosophy. They worry that the formalisms are smuggling in a bunch of substantive assumptions; or they are distorting certain phenomena by making them unrealistically fixed and precise. Given the sort of work I do, I clearly disagree with these philosophers, at least in some cases; but I do understand where they’re coming from. Mathematical models of phenomena of philosophical interest do idealise in various ways, miss out features of the phenomena, misrepresent others, and so on. So the danger is real. I think one way we might assuage these feelings is to talk honestly in our papers about the limitations of the formalism, what assumptions it brings with it, and so on.

Writing these down, I realise I’ve still a long way to go myself to get this right. But since I think that the formal parts of philosophy have so much to offer the discipline, I think it’s important that we try.

Comments

[Brad]

Richard,
Thank you for this very thoughtful post. I work in an area where people often construct model, etc. And not infrequently I am asked to referee papers in formal philosophy. On numerous occasions, I have urged the authors to express their points in nice clear prose. I do so in an effort to help the papers have a greater impact (or at least more readers!). In time, I have seen some of the papers published, and I am dismayed that my advice was not taken. The findings reported in these papers are often very interesting, but few people will attend to them.
You provide some very straightforward ways to address my concern. I think the first point, about writing like you are teaching, may be the most important.

[Shay Allen Logan]

This is great!

Another point worth making: often translating into more intelligible language yields philosophical fruit.

Example: much work in relevance logic (an area I work in) is done in Hilbert systems. These are notoriously impenetrable to non-experts. I'm guilty of using them too, but have recently aimed at instead using the much-more intelligible bunched proof systems that have been developed for these. To my surprise, there's a lot of philosophically interesting things to say about these systems, but since using Hilbert systems makes the formal work easier, there's been much less philosophical work done on the bunched systems. So there's a boatload of philosophically interesting low-lying fruit here, and all it took to find it was looking for a more-intelligible way of presenting things.

[Sophie Horowitz]

Thanks for writing this, Richard! As someone who often tries to read formal philosophy, but writes much more informally, I will admit that I basically only read the English parts of papers unless I am *really* going to work in depth on the topic. So if the English part of paper X is not accessible, I'll probably go read paper Y instead.

I think I disagree with frontloading the formalism. In many cases that is just overwhelming to me. I'd rather have stuff explained in English and then formalism introduced along the way only as it's necessary. (But maybe that's just me, and it's also possible that I'm lazier than the average reader!)

Finally, I'd like to add that whenever there's a ton of symbols in a paper, it's much more reader friendly (to me at least!) to use English words to name things whenever possible. Even using "RAIN" or "TAILS" instead of "P" for the name of a proposition can make it read more smoothly.

[Noah]

Sophie, a small clarification: I think the thing that Richard wants us to frontload is the /glosses/ on the formalisms. That is, we should give English glosses of the formalisms more often in the beginning of the paper.

I don't think Richard wants us to frontload the use of formalisms in general.