# Some hints for mathematical writing

Two recent blog posts give some useful advice about how to make one's mathematical writing clearer:

Tim Gowers tells us to put examples first. That is, don't just include examples (I hope everyone already knows that much), but include them before the more general case rather than as an afterthought. That way, the readers will be able to have something concrete in mind when they work through the more general concepts you're trying to describe. Illustrated by an example, of course. Updated.

God Plays Dice reminds us that Mathematics is not notation, or maybe better that notation is not language. Even when a mathematical concept can be stated unambiguously and concisely using standard symbols, it's often clearer to spell it out in English words instead.

My own advice, which I'm not following in this post, is to use plenty of illustrations. A figure is not a proof, of course, and often it's not even as detailed as a textual example can be, but I think a well-chosen figure can provide a visual intuition that can go a long way towards making a paper more readable. Additionally, those of us with kids know how much more intimidating it can be to read books without illustrations than those with them, even when the text is the same, and I think the same holds true for professional writing. As a rule of thumb, I like to have similar numbers of figures and pages, both in formal papers and in talk slides.

Now I'm wondering where one might go to find longer and more detailed collections of advice like this on mathematical writing. There's the Wikipedia Manual of Style for Mathematics but it's less focused on general mathematical writing and more on the nuts and bolts of ensuring a consistent house style for Wikipedia. It cites books by Higham and Halmos that look more relevant, though.

I occasionally read math papers, and I'm always struck at how many more examples they have in their papers. In contrast, I've seen math papers introducing a data structure without a single example of how the DS would look like on a given input.

Compared to theoretical CS papers, you mean?

So have I and for the same reasons. Currently there is a trend not to count the bibliography towards the 10 page limit. Perhaps we should add figures and examples to that.

The problem with putting examples first is that you actually have to come up with the examples. Mathematicians, being lazy, don't want to come up with them. The book of Krantz on mathematical writing isn't bad, and its short. You can read it in a couple hours. The notes on mathematical writing by Knuth et al. aren't that good; they seem like they were the notes to a good

Thanks, I'll try to remember to check out the Krantz book the next time I'm ordering stuff at Amazon.

Tim Gowers tells us to put examples first. That is, don't just include examples (I hope everyone already knows that much), but include them before the more general case rather than as an afterthought. That way, the readers will be able to have something concrete in mind when they work through the more general concepts you're trying to describe. Illustrated by an example, of course. Updated.

God Plays Dice reminds us that Mathematics is not notation, or maybe better that notation is not language. Even when a mathematical concept can be stated unambiguously and concisely using standard symbols, it's often clearer to spell it out in English words instead.

My own advice, which I'm not following in this post, is to use plenty of illustrations. A figure is not a proof, of course, and often it's not even as detailed as a textual example can be, but I think a well-chosen figure can provide a visual intuition that can go a long way towards making a paper more readable. Additionally, those of us with kids know how much more intimidating it can be to read books without illustrations than those with them, even when the text is the same, and I think the same holds true for professional writing. As a rule of thumb, I like to have similar numbers of figures and pages, both in formal papers and in talk slides.

Now I'm wondering where one might go to find longer and more detailed collections of advice like this on mathematical writing. There's the Wikipedia Manual of Style for Mathematics but it's less focused on general mathematical writing and more on the nuts and bolts of ensuring a consistent house style for Wikipedia. It cites books by Higham and Halmos that look more relevant, though.

### Comments:

**None**: Examples

**2007-10-23T15:50:16Z**

I occasionally read math papers, and I'm always struck at how many more examples they have in their papers. In contrast, I've seen math papers introducing a data structure without a single example of how the DS would look like on a given input.

**11011110**: Re: Examples

**2007-10-23T17:36:31Z**

Compared to theoretical CS papers, you mean?

*papers introducing a data structure without a single example of how the DS would look*I suppose I'm likely to have perpetrated some of those myself...the ten page limit on typical conference papers (and the tendency for authors not to expand those papers once they're no longer so limited in the journal version) may have something to do with this.

**None**: Re: Examples

**2007-10-24T14:01:36Z**

So have I and for the same reasons. Currently there is a trend not to count the bibliography towards the 10 page limit. Perhaps we should add figures and examples to that.

**madcaptenor**:

**2007-10-23T18:08:14Z**

The problem with putting examples first is that you actually have to come up with the examples. Mathematicians, being lazy, don't want to come up with them. The book of Krantz on mathematical writing isn't bad, and its short. You can read it in a couple hours. The notes on mathematical writing by Knuth et al. aren't that good; they seem like they were the notes to a good

*class*, but they really don't stand alone. I haven't read Halmos or Higham, but I might.

**11011110**:

**2007-10-24T16:14:08Z**

Thanks, I'll try to remember to check out the Krantz book the next time I'm ordering stuff at Amazon.