Mathematical writing: When to write definitions in text and when in a definition environment

Question

In mathematical writings (in journals), I noticed that sometimes definitions are listed in a paragraph form, while sometimes they are listed in a definition environment " \begin{definition}

\end{definition}" form.

For good mathematical writing, are there any guidelines on when one should put definitions in paragraph form, and when one should put them in an environment? Purely using paragraph form will lead to very wordy long paragraphs, while putting all in environment may lead to a "disjointed" look, where it is one definition after another.

For theorems, lemmas and propositions, it is clear that the environment format will be better, but for definitions I am not so sure.

Thanks for any help.

Answer 1

I would say it depends on the purpose of the definitions.

If it's to introduce something new then I prefer the definition environment (and ideally a non-trivial example of what's being defined), while if it's to refresh the readers's memory on things that are being assumed I'm happier with a (well-written) paragraph. The more definitions there are (which is more the case with a refresher section) the quicker the definition environment format turns into a shopping list.

By way of example (I guess I have to stick to my own guidelines...): a journal paper on metric measure spaces might want to list the definitions of doubling measure, Radon measure, length space, geodesic space, and rectifiability with a reference to where more detail can be found and these can be neatly summarised in a paragraph or two. However, if the same paper goes on to define a new measure for these spaces with specific properties that it then studies I would want that definition to have the definition environment.