Mathematics - Writing for Computer Science - page 127

Metric, measure . “Metric” is sometimes used informally to mean measure ,but
both have specific meanings in mathematics. In particular, when used in a formal
context a metric is expected to satisfy conditions such as the triangle inequality.
While “measure” also has a formal meaning, it is usually the less confusing of the
two words, as it also has an appropriate informal usage. In mathematical contexts,
use “measure” unless metric is intended.
Theorems
When you submit a paper containing a proof of a theorem, you should be satisfied
that the proof is correct. However, the details of the proof may not be important to the
reader and can often be omitted. Steps in the logic of a proof should be simple enough
that the gaps can be completed by a reader mechanically, without too much invention.
Acommonmistake is to unnecessarily includemechanical algebraic transformations;
you need to work through these to check the proof, but the reader is unlikely to find
them valuable.
Theorems, definitions, lemmas, and propositions should be numbered, even if
there are only two or three of each in the paper, and you could consider numbering
key examples. Not only does numbering allow reference within the paper, but it
simplifies discussion of the paper later on. It is much easier for a correspondent to
refer to “Definition 4.2” than “the definition towards the bottom of page 6”. Many
readers skim papers to find theorems (or other results such as illustrations or tables).
For this reason, and because they may be quoted verbatim in other papers, statements
of theorems should be as complete as possible.
In some theoretical contexts, authors choose to end a paper with a proof of some
theorem or lemma. This style of writing can be unsatisfying for the reader. All papers
can sensibly have an introduction and a conclusion, and it is worth reminding the
reader of the main lessons of the paper in its final paragraphs.
Some presentation problems are not easily resolved. For a theoremwith a complex
proof, if the lemmas are proved early they appear irrelevant, and if they are proved
late the main proof is harder to understand. One approach is to state the main theorem
first, then state and prove the lemmas before giving the main proof, but in other cases
all that can be done is to take extra care in the motivation and make liberal use
of examples. Explain the structure of long proofs before getting to the detail, and
explain how each part of the proof relates to the structure.
When stating your proof in a paper—that is, making it comprehensible to a
reader—remember that you are presenting a reasoned argument. Use any available
means to convey your argument with the greatest possible clarity; a diagram, for
example, is perfectly acceptable. The end of each proof, example, or definition can
be marked with a symbol such as a box. Alternatively, proofs and so on can be
indented to set them apart from the running text.