Use soft words and hard arguments. (Proverbial)

Mathematical notation is a wonderfully useful tool, and it can be exciting to learn for the first time the meaning of mysterious and arcane symbols such as $\forall$, $\exists$, $\emptyset$, $\implies$, etc. However, just because you can write statements in purely mathematical notation doesn’t mean that you necessarily should. In many cases, it is in fact far more informative and readable to use liberal amounts of plain English; if used correctly and thoughtfully, the English language can communicate to the reader on many more levels than a mathematical expression, without sacrificing any precision or rigour. In particular, by subtly modulating the emphasis of one’s text, one can convey valuable contextual cues as to how a statement interacts with the rest of one’s argument.

An example should serve to illustrate this point. Suppose for instance that P and Q are properties that can apply to mathematical objects x and y. The mathematical statement

$P(x) \wedge Q(y)$,

which asserts that x satisfies P and y satisfies Q, is a well-formed and precise mathematical statement. But there are many possible ways one could express that mathematical statement in English, for instance:

1. P(x) and Q(y) are both true.
2. P(x) is true. Also, Q(y) is true.
3. P(x) is true. Furthermore, Q(y) is true.
4. P(x) is true. Therefore, Q(y) is true.
5. P(x) is true. However, Q(y) is true.
6. P(x) is true. In particular, Q(y) is true.
7. P(x) is true. More interestingly perhaps, Q(y) is also true.
8. Since P(x) is true, Q(y) is true.
9. P(x) is true (which implies for instance that Q(y) is true).
10. P(x) is true. Unfortunately, Q(y) is also true.
11. P(x) is true. Equivalently, Q(y) is true.
12. x satisfies P, but y satisfies Q.
13. x satisfies P, and thus y satisfies Q.
14. x satisfies P. Meanwhile, y satisfies Q.
15. x satisfies P; y, in contrast, satisfies Q.
16. x satisfies P. More generally, y satisfies Q.
17. x satisfies P. In other words, y satisfies Q.
18. x clearly satisfies P. A little more thought also reveals that y satisfies Q.
19. x satisfies P (because y satisfies Q).
20. x satisfies P. For future reference, we also observe that y satisfies Q.
21. x satisfies P. Fortunately for us, y satisfies Q.
22. P is satisfied by x. Similarly, Q is satisfied by y.
23. P is satisfied by x. On the other hand, Q is satisfied by y.
24. x (resp. y) satisfies P (resp. Q).
25. P and Q are satisfied (by x and y respectively).
26. x and y satisfy P and Q respectively.
27. etc., etc.

From the viewpoint of formal mathematical logic, each of these English statements is logically equivalent to the mathematical sentence $P(x) \wedge Q(y)$. However, each of the above English statements also provides additional useful and informative cues for the reader regarding the relative importance, non-triviality, and causal relationship of the component statements P(x) and Q(y), or of the component symbols P, x, Q, and y. For instance, in some of these sentences P(x) and Q(y) are given equal importance (being complementary or somehow in opposition to each other), whereas in others P(x) is only an auxiliary statement whose only purpose is to derive Q(y) (or vice versa), and in yet others, P(x) and Q(y) are deemed to be analogous, even if one is not formally deducible from the other. In some sentences, it is the objects x and y which are indicated to be the primary actors; in other sentences, it is the properties P and Q; and in yet other sentences, it is the combined statements P(x) and Q(y) which are the most central.

Thus we see that English sentences can be considerably more expressive than their formal mathematical counterparts, while still retaining the precision and rigour that mathematical exposition demands. By using such humble English words as “also”, “but”, “since”, etc., a sentence conveys not only its semantic content, but also how it is going to fit in with the rest of one’s argument (or in the wider theory of the subject), giving the reader more insight as to the overall structure of that argument. The paper may become slightly longer because of this, but this is a small price to pay for readability (which is not the same as brevity!).

On the other hand, one should not try to excessively “improve” the paper by using overly fancy or obscure words (from English or any other language), especially since such words can be mistaken for some sort of technical mathematical terminology. In many cases, one can replace complicated words by plainer equivalents, thus increasing the readability of one’s text without compromising the message. The primary purpose of mathematical writing is to communicate and inform, not to impress.

Finally, there is one situation in which it does make sense to use the terse language of mathematical notation rather than a more leisurely English equivalent, and that is when you are performing a tedious and standard formal computation. In those cases, the reader should already know in general terms what is going to happen (especially if you flag the computation as being standard beforehand), and will only be distracted by superfluous explanation or digression. (See also “give appropriate amounts of detail“.)

Naturellement, la discussion ci-dessus s’applique également à d’autres langues, telles que la langue française.