Help:Editing/House Style/Linguistic Style

Language
This is an English language website, and so all pages are to be presented in English. Where there is a difference between spellings between US and rest-of-world English, the US version is generally used, with a few exceptions (the spelling of metre is under discussion).

Linguistic Style
During the presentation of a mathematical argument, a formal style is preferred.

For example:


 * Suppose that ...

is preferred to:


 * Let's suppose that ...

and:


 * Hence the result.

is preferred to:


 * ... and we're done.

As an attempt is being made for to appeal to as wide an audience as possible worldwide, using colloquial language (except for example when illustrating logical concepts by means of everyday examples) is discouraged.

"Let" and "Suppose"
It is preferred that "Let" is used to introduce the existence of an entity in an argument, as follows:


 * Let $S$ be a set.


 * Let $x, y \in S: x \ne y$.


 * $\ldots$

However, when introducing an entity whose existence is in question (for example, when constructing a Proof by Contradiction), the word "Suppose" is recommended:


 * Suppose $T \subseteq S$ such that $\card T > \card S$.


 * $\ldots$

"Any"
The word "any" can be ambiguous.

It is recommended that it not be used.

Instead, consider whether "every" or "an arbitrary" can be used instead.

Abbreviations
The difference between "e.g." (exempli gratia - for example) and "i.e." (id est - that is) is sadly falling into obscurity. It is all too common for "i.e." to be used when "for example" is meant, and vice versa.

So as to remove all confusion, such abbreviations are discouraged.

Also, beware the ubiquitous confusion between its and it's. The full version it is should be used instead of it's in any case, so it's should have no reason to appear.


 * Third way.png

Sentence Length
During the course of an argument to present a mathematical proof, follow these rules:


 * Each sentence should be short.


 * Each sentence should convey one step, either:
 * One simple statement, or:
 * One compound statement of the form: $P$, therefore $Q$.


 * Each sentence should be on a separate line.

Compare the presentations:

$(1):$
 * $S$, because of $R$ (we know this from Tom's Theorem), because of $Q$ (from above) which applies when $P$ holds (see Fred's Theorem), but we know $P$ holds because it's what we defined in the first place.

$(2):$
 * Let $P$ hold.
 * From Fred's Theorem, it follows that $Q$.
 * From above, $R$.
 * From Tom's Theorem, $S$.

The following is an example of the style of mathematical exposition which we believe has no place in, and indeed, the entire universe:


 * The ($\implies$) is shown just the same as above, while the other direction easily follows, since $\MM$ satisfying the condition that for every $\LL$-formula $\map \phi {x, \bar v}$ and for every $\bar a$ in $\MM$, if there is an $n$ in $\NN$ such that $\NN \models \map \phi {n, \bar a}$, then there is an $m$ in $\MM$ such that $\NN \models \map \phi {m, \bar a}$, is closed under functions (by directly applying the condition to formulae of the form $\map \phi {x, \bar y} = \paren {x = \map f {\bar y} }$), and hence the universe of a substructure, which reduces it to the statement above.

Parenthetical Notes
It is often tempting to slide some further information or explanation into the middle of a sentence by slipping it into parenthesis.

It is highly recommended that this practice be avoided as far as possible.

Not only does this clash with the house style of short, simple sentences, but also can cause confusion when the parenthesized material contains similar mathematical notation:


 * Let $\map {f_n} x \in \Sigma$ (where $\map {f_n} x$ is a family (indexed by $\map I \alpha$) of functions in $\struct {\Sigma, \mu}$) be a function such that ...

So please don't do this.

A similar stylistic presentation to be avoided is the following


 * The supremum of $f$, $\map \sup f$, is defined as:
 * $\map \sup f = \map \max {\Img f}$

It is clear from the line following that $\map \sup f$ is how the supremum of $f$ is denoted, so there is no need to include it as a parenthetical explanation in the first line.

This is preferred:


 * The supremum of $f$ is defined as:
 * $\map \sup f = \map \max {\Img f}$

If you are uneasy about the ability of the reader to make that connection, feel free to write something like:


 * The supremum of $f$ is defined and denoted as:
 * $\map \sup f = \map \max {\Img f}$

Filler Words
Whether or not filler words are needed (it follows that, we have, hence etc.) is a stylistic decision. Fewer words are preferred, but clarity and completeness override every other consideration.

The general approach is to try to use as terse a form as possible.

Compare:


 * We have that the ordinal subset of an ordinal is an initial segment of it, so it follows that:

with:
 * From Ordinal Subset of Ordinal is Initial Segment:

The latter form is preferred.

Empty Statements and Waffle
It is tempting to fill a page up with statements that do not actually impart any information, but which make the author look and feel good.

Such are to be avoided.

Examples:
 * The first part of the proof is easy.


 * We mention for the interested reader ...


 * This is trivial:

See also the templates handwaving and explain.

Capital Letters begin Sentences
This is raised as a particular point, because it crops up over and over again.

The sentence form in question is:
 * Let (such-and-such) hold, where (so-and-so) means (thus and so).

When (such-and-such) is a statement in mathematical symbols, placed on its own line (as per house style recommendations), the temptation is to present the above sentence as:


 * Let:
 * $\ds S = \sum_{i \mathop \in \N} \frac 1 {2^i}$
 * Where $\ds \sum$ denotes summation.

Just because it starts a new line does not mean that "where" is to be written with a capital W. It is the continuation of the previous sentence, which just happens to have, as part of its main clause, a mathematical expression.

It should be:


 * Let:
 * $\ds S = \sum_{i \mathop \in \N} \frac 1 {2^i}$
 * where $\ds \sum$ denotes summation.

Breaking this linguistic rule can lead to confusion, especially when the "where" clause starts to get complicated:


 * Let:
 * $\ds S = \sum_{j \mathop \in \N} \lim_{x \mathop \to \infty} \cos j x + i \sin j x$


 * Where $\ds \sum$ denotes summation and $\lim$ is the limit as $x$ tends to infinity and:
 * $\cos j x + i \sin j x = e^{ijx}$

In the above, the reader, thinking that "where" starts the next sentence, and therefore a new thought, is left wondering:
 * "Where this applies, and that means that, and this ... then what?"

whereas in fact the only reason for the "where" clause is to amplify the sense of the expression above it.

Similarly:

In the above, the "by definition" phrases in the comment column should not start with a capital letter, as they continue the "sentence" started on the left.

Thus the above structure is better rendered as:

Better still, lose the redundant filler-word "by", and render the entire structure elegantly as:

Now, as there is no filler-word "by", the comment is no longer implicitly part of a sentence, and so the comment is a standalone label which now merits an uppercase presentation.

Here, note that a further evolutionary step has been made: to replace the code  with the template construct   for further streamlining of the source.


 * Beware: This Defof template was designed specifically for the  parameter of the Eqn template.

It is not for using in the body of an exposition, specifically because of the fact that it has been designed to start with a capital letter.