There’s a good book for beginning computer programmers I recently came across. All young kids wanting to write code professionally should check out Robert Martin’s book, “Clean Code: A Handbook of Agile Software Craftsmanship” (Ideally get your kids to read this before the more advanced “Design Patterns” books.)
But is there such a guide for writing clean mathematics?
I could ask around on Mathforums or Quora, but instead here I will suggest some of my own tips for such a guide volume. What gave me this spark to write a wee blog about this was a couple of awesome “finds”. The first was Professor Tadashi Tokieda’s Numberphile clips and his AIMS Lectures on Topology and Geometry (all available on YouTube). Tokieda plugs a couple of “good reads”, and this was the second treasure: V.I. Arnold’s lectures on Abel’s Theorem, which were typed up by his student V.B. Alekseev, “Abel’s Theorem in Problems and Solutions”, which is available in abridged format (minus solutions) in a translation by Julian Gilbey here: “Abels’ Theorem Through Problems“.
Tokieda’s lectures and Arnold’s exposition style are perfect examples of “clean mathematics”. What do I mean by this?
Firstly, what I absolutely do not mean is Bourbaki style rigour and logical precision. That’s not clean mathematics. Because the more precision and rigour you demand the more dense and less comprehensible it all becomes to the point where it becomes unreadable and hence useless.
I mean mathematics that is challenging for the mind (so interesting) and yet clear and understandable and visualizable. That last aspect is crucial. If I cannot visualise an abstract idea then it has not been explained well and I have not understood it deeply. We can only easily visualize 2D examples or 3D if we struggle. So how are higher dimensional ideas visualised? Tokieda shows there is no need. You can use the algebra perfectly well for higher dimensional examples, but always give the idea in 2D or 3D.
It’s amazing that 3D seems sufficient for most expositions. With a low dimension example most of the essence of the general N dimensional cases can be explained in pictures. Perhaps this is due to 3D being the most awkward dimension? It’s just a pity we do not have native 4D vision centres in our brain (we actually do, it’s called memory, but it sadly does not lead to full 4D optical feature recognition).
Dr Tokieda tells you how good pictures can be good proofs. The mass of more confusing algebra a good picture can replace is startling (if you are used to heavy symbolic algebra). I would also add that Sir Roger Penrose and John Baez are to experts who make a lot of use of pictorial algebra, and that sort of stuff is every bit as rigorous as symbolic algebra, and I would argue even more-so. How’s that? The pictorial algebra is less prone to mistake and misinterpretation, precisely because our brains are wired to receive information visually without the language symbol filters. Thus whenever you choose instead to write proofs using formal symbolics you are reducing your writing down to less rigour, because it is easier to make mistakes and have your proof misread.
So now, in homage to Robert Martin’s programming style guide, here are some analogous sample chapter or section headings for a hypothetical book on writing clean mathematics.
Keep formal (numbered) definitions to a minimum
Whenever you need a formal definition you have failed the simplicity test. A definition means you have not found a natural way to express or name a concept. That’s really all definitions are, they set up names for concepts.
Occasionally advanced mathematics requires defining non-intuitive concepts, and these will require a formal approach, precisely because they are non-intuitive. But otherwise, name objects and relations clearly and put the keywords in old, and then you can avoid cluttering up chapters with formal boring looking definition breaks. The definitions should, if at all possible, flow naturally and be embedded in natural language paragraphs.
Do not write symbolic algebra when a picture will suffice
Most mathematicians have major hang-ups about providing misleading visual illustrations. So my advice is do not make them misleading! But you should use picture proofs anyway, whenever possible, just make sure they capture the essence and are generalisable to higher dimensions. It is amazing how often this is possible. If you doubt me, then just watch Tadashi Tokieda’s lectures linked to above.
Pro mathematicians often will think pictures are weak. But the reality is the opposite. Pictures are powerful. Pictures should not sacrifice rigour. It is the strong mathematician who can make their ideas so clear and pristine that a minimalistic picture will suffice to explain an idea of great abstract generality. Mathematicians need to follow the physicists credo of using inference, one specific well-chosen example can suffice as an exemplar case covering infinitely many general cases. The hard thing is choosing a good example. It is an art. A lot of mathematician writers seem to fail at this art, or not even try.
You do not have to use picture in your research if you do not get much from them, but in your expositions, in your writing for the public, failing to use pictures is a disservice to your readers.
The problem with popular mathematics books is not the density of equations, it is the lack of pictures. If for every equation you have a couple of nice illustrative pictures, then there would be no such thing as “too many equations” even for a lay readership. The same rule should apply to academic mathematics writing, with perhaps an reasonable allowance for a slightly higher symbol to picture ratio, because academically you might need to fill in a few gaps for rigour.
Rigour does not imply completeness
Mathematics should be rigorous, but not tediously so. When gaps do not reduce clarity then you can avoid excessive equations. Just write what the reader needs, do not fill in every gap for them. And whenever a gap can be filled with a picture, use the picture rather than more lines of symbolic algebra. So you do not need ruthless completeness. Just provide enough for rigour to be inferred.
Novel writers know this. If they set out to describe scenes completely they would ever get past chapter one. Probably not even past paragraph one. And giving the reader too much information destroys the operation of their inner imagination and leads to the reader disconnecting from the story.
For every theorem provide many examples
The Definition to Theorem ratio should be low, for every couple of definitions there should be a bundle of nice theorems, otherwise the information content of your definitions has been poor. More definitions than theorems means you’ve spent more of your words naming stuff not using stuff. Likewise the Theorem to Example ratio should be lo. More theorems than examples means you’ve cheated the student by showing them lot of abstract ideas with no practical use. So show them plenty of practical uses so they do not feel cheated.
Write lucidly and for entertainment
This is related to the next heading which is to write with a story narrative. On a finer level, every sentence should be clear, use plain language, and minimum jargon. Mathematics text should be every bit as descriptive and captivating as a great novel. If you fail in writing like a good journalist or novelist then you have failed to write clean mathematics. Good mathematics should entertain the aficionado. It does not have to be set like a literal murder mystery with so many pop culture references and allusions that you lose all the technical content. But for a mathematically literate reader you should be giving them some sense of build-up in tension and then resolution. Dangle some food in front of them and lead them to water. People who pick up a mathematics book are not looking for sex, crime and drama, nor even for comedy, but you should give them elements of such things inside the mathematics. Teasers like why we are doing this, what will it be used for, how it relates to physics or other sciences, these are your sex and crime and drama. And for humour you can use mathematical characters, stories of real mathematicians. It might not be funny, but there is always a way to amuse an interested reader, so find those ways.
Write with a Vision
I think a lot of mathematical texts are dry ad suffer because they present “too close to research”. What a good mathematical writer should aim for is the essence of any kind of writing, which is to narrate a story. Psychology tells us this is how average human beings best receive and remember information. So in mathematics you need a grand vision of where you are going. If instead you just want to write about your research, then do the rest of us a favour and keep it off the bookshelves!
If you want to tell a story about your research then tell the full story, some history, some drama in how you stumbled, but then found a way through the forest of abstractions, and how you triumphed in the end.
The problem with a lot of mathematics monographs is that they aim for comprehensive coverage of a topic. But that’s a bad style guide. Instead they should aim to provide tools to solve a class of problems. And the narrative is how to get from scratch up to the tools needed to solve the basic problem and then a little more. With lots of dangling temptations along the way. The motivation then is the main problem to be solved, which is talked about up front, as a carrot, not left as an obscure mystery one must read the entire book through to find. Murder mysteries start with the murder first, not last.
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That’s enough for now. I should add to this list of guides later. I should follow my own advice too.
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