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 , , , , 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

,

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:

- P(x) and Q(y) are both true.
- P(x) is true. Also, Q(y) is true.
- P(x) is true. Furthermore, Q(y) is true.
- P(x) is true. Therefore, Q(y) is true.
- P(x) is true. However, Q(y) is true.
- P(x) is true. In particular, Q(y) is true.
- P(x) is true. More interestingly perhaps, Q(y) is also true.
- Since P(x) is true, Q(y) is true.
- P(x) is true (which implies for instance that Q(y) is true).
- P(x) is true. Unfortunately, Q(y) is also true.
- P(x) is true. Equivalently, Q(y) is true.
- x satisfies P, but y satisfies Q.
- x satisfies P, and thus y satisfies Q.
- x satisfies P. Meanwhile, y satisfies Q.
- x satisfies P; y, in contrast, satisfies Q.
- x satisfies P. More generally, y satisfies Q.
- x satisfies P. In other words, y satisfies Q.
- x clearly satisfies P. A little more thought also reveals that y satisfies Q.
- x satisfies P (because y satisfies Q).
- x satisfies P. For future reference, we also observe that y satisfies Q.
- x satisfies P. Fortunately for us, y satisfies Q.
- P is satisfied by x. Similarly, Q is satisfied by y.
- P is satisfied by x. On the other hand, Q is satisfied by y.
- x (resp. y) satisfies P (resp. Q).
- P and Q are satisfied (by x and y respectively).
- x and y satisfy P and Q respectively.
- etc., etc.

From the viewpoint of formal mathematical logic, each of these English statements is logically equivalent to the mathematical sentence . 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.

## 9 comments

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16 July, 2007 at 3:20 pm

Some new pages « What’s new[...] I also added a new page to my advice page on writing and submitting papers, on “taking advantage of the English language“. [...]

27 August, 2007 at 1:13 pm

Printer-friendly CSS, and nonfirstorderisability « What’s new[...] or by using the tremendously convenient O() and o() notation of Landau. One then takes for granted that one can eventually unwind all these phrasings to get back to a sentence in formal, first-order logic. As far as analysis is concerned, this is a fairly safe assumption, since one usually deals with objects in very concrete sets such as the real numbers, and one can easily model all of these dependencies using functions from concrete sets to other concrete sets if necessary. (Also, the hierarchy of magnitudes in analysis does often tend to be rather linearly ordered.) But some subtleties may appear when one deals with large categories, such as the category of sets, groups, or vector spaces (though in most applications, one can cap the cardinality of these objects and then one can represent these categories up to equivalence by an actual set). It may be that a more diagrammatic language (perhaps analogous to the commutative diagrams in category theory, or one based on trees or partially ordered sets rather than linearly ordered ones) may be a closer fit to expressing the way one actually thinks about how variables interact with each other. (Second-order logic is, of course, an obvious candidate for such a language, but it may be overqualified for the task. And, in practice, there’s nothing wrong with just using plain old mathematical English.) [...]

24 April, 2008 at 4:10 am

上同调论讲义（林金坤）Chapter 1 « Liuxiaochuan’s Weblog[...] 林教授的讲义仅仅80页，每次两课时仅仅讲三五页而已。这样的书是很难读的，充满了严格的数学符号，没有文字上的解释。原本我是不大喜欢这样的书，但是我最近又体会到了这种书的优点，那就是可以比较快速的进入这门学科，虽然要下些功夫。但是为了今后更进一步的学习这门学科做些准备，我想不出更好的方法了。 [...]

6 September, 2008 at 5:11 pm

Anonymous> From the viewpoint of formal mathematical logic, each of these English statements is logically equivalent to the mathematical sentence P(x) ^ Q(y).

Going to have to disagree here. 4, 8, 9, 13 read like material implication to me.

I’d avoid 11, 24, 25, 26 since they’re confusingly worded.

I’d avoid 18 like the plague. Nothing angers me more than a paper saying “clearly”. I interpret that as either the author saying “if you can’t immediately see why, you’re a moron” or “please don’t ask me why, I’m going to use the word ‘clearly’ to motivate you not to because you’ll feel like a moron if you do ask”.

8 September, 2008 at 9:43 am

Terence TaoDear anonymous:

Statements 4, 8, etc. affirm the antecedent P(x) in addition to the material implication . Of course, by modus ponens, is logically equivalent to .

I definitely agree with you, though, that even if the logical content of these various sentences are the same, their semantic connotations to the reader can be totally different, and one should take some care in selecting which of these formulations to use in any given context; this is actually one of the main points of my article above.

(Incidentally, while it is true that the

~~adjective~~adverb “clearly” can be abused if applied to a statement which is not, in fact, clear, I find such tags to be actually rather informative when used properly, for instance to signify that P(x) is indeed a much simpler statement to prove than Q(y), and in particular does not require any high-powered machinery that the author has already introduced in the text.)14 December, 2008 at 2:26 am

IndianDear Prof Tao,

“Clearly” is an adverb, and not an “adjective”—- as stated by you in the above reply to someone’s query; I referred to an advanced dictionary that was present on the web, and it says “clearly” is used as an adverb, and not as an adjective.

Do correct me if I inadvertently provided some wrong bit of information. Thanks.

18 July, 2009 at 9:38 pm

fsActually, it’s used here as a disjunct. Lexically, the word “clearly” is considered an adverb.

22 November, 2009 at 8:24 am

Advice on writing paper « Computer Vision[...] first time authors especially, it is important to try to write professionally. One should take advantage of the English language, and not just rely purely on mathematical [...]

3 January, 2013 at 12:34 am

thoughtsVery nicely done.