This looks pretty good mostly, but I'd emphasize more that the most important thing is to be understood. When it comes to a difficult mathematical concept, this is actually really really hard. If you can nail that, then worry about stylistic stuff, but you probably didn't nail it.
With that in mind, this advice is really problematic:
> Try to avoid jargon, buzzwords or overly technical language. And don’t use the same word repeatedly — it’s boring.
Boring is better than incorrect! I love when authors clearly define a term and use it over and over, because I know exactly what they mean every time. Jargon may be necessary to communicate precisely (but define your jargon, avoid assuming it's familiar). E.g. mathematically, there is a difference between a ball and a sphere. And if your L2 ball becomes an L2 balloon in the middle of a proof, readers will be very confused.
Also, this is just dead wrong:
> Avoid placing equations in the middle of sentences. Mathematics is not the same as English, and we shouldn’t pretend it is.
I can see this being useful advice when a proof is full of dense lines of inequalities, but the reasoning is still incorrect. Mathematical notation is nothing more nor less than shorthand for English (or any other language). The following is fine for example: If f(x) >= 1, then either x = 0 or x \in [1,2]. So is this: Noting that x < 5, Pr[f(y) + x >= 3] <= 0.1. Perfectly grammatical.
For mathematical writing, the first thing I was taught was don't begin a sentence with a formula. That rule generally applies that to other parts of sentence structure. I'd rewrite your second example to something like, "Noting that x < 5, we have Pr[f(y) + x >= 3] <= 0.1."
I drew a stronger conclusion from this point on page 5.
13. Many readers will skim over formulas on their first reading of
your exposition. Therefore, your sentences should flow smoothly when
all but the simplest formulas are replaced by “blah” or some other
grunting noise.
My writing has benefited greatly by treating all mathematical
expressions within sentences as nouns regardless of their relational
operators. My version of the sentence would be "Noting that x < 5
holds, we infer a probability of no more than 0.1 for f(y) + x taking
values of 3 or more.".
These two stuck out. Also, the advice about commas (for pauses) is not entirely wrong but not quite right either, and the advice about dashes (for emphasis) is simply wrong. If you've read Strunk and White, or most any style guide, nothing in this article will strike you as both new and correct.
The advice I would give most science writers today is:
1. Have something to say.
2. Say it.
Many of the problems with overuse of jargon come from skipping directly to step 2.
Mathematical notation is not a shorthand for human language.
It is a rigorous grammar with properties that make it possible to express very unambiguous qualities of a system.
Mathematical notation can be spoken out loud. Hence, you can include it in a sentence without breaking the flow of that sentence.
For the more complicated equations, it becomes harder to unambiguously pronounce them, and the ideas are more separate anyway. Hence we need to break them out.
Moreover, I disagree that mathematical notation is a rigorous grammar. Notation is used to be consise, to represent ideas so we can work on them.
Precision is only added to resolve ambiguities that are hard to resolve based on the surrounding text.
One can make notation fully unambiguous, but that often comes at the cost of consiseness, and thus clarity.
It actually isn’t a rigorous grammar. Something like Coq or Agda or Mathematica is, but traditional maths notation is full of ambiguities and unclearness that we’re just used to. There’s lots of discussion of this around, including previous HN threads. But if you want a really good example, check out Sussman et al. discussing the Euler Lagrange equations in Structure and Interpretation of Classical Mechanics.
Actually here you go, they repeat the same discussion in the preface of this free book:
I think you're on to an important point, but with some confusion. Mathematical notation is absolutely a shorthand for human language. But the human language it is a shorthand for does have very specific grammar rules governing it for the sentences to be mathematically correct.
Agreed. For what it's worth I think Atiyah's advice on writing mathematical papers, written for the Princeton Companion to Mathematics, is well worth a look.
With that in mind, this advice is really problematic:
> Try to avoid jargon, buzzwords or overly technical language. And don’t use the same word repeatedly — it’s boring.
Boring is better than incorrect! I love when authors clearly define a term and use it over and over, because I know exactly what they mean every time. Jargon may be necessary to communicate precisely (but define your jargon, avoid assuming it's familiar). E.g. mathematically, there is a difference between a ball and a sphere. And if your L2 ball becomes an L2 balloon in the middle of a proof, readers will be very confused.
Also, this is just dead wrong:
> Avoid placing equations in the middle of sentences. Mathematics is not the same as English, and we shouldn’t pretend it is.
I can see this being useful advice when a proof is full of dense lines of inequalities, but the reasoning is still incorrect. Mathematical notation is nothing more nor less than shorthand for English (or any other language). The following is fine for example: If f(x) >= 1, then either x = 0 or x \in [1,2]. So is this: Noting that x < 5, Pr[f(y) + x >= 3] <= 0.1. Perfectly grammatical.