How to use "given" in expository writing.

Question

In "How to write mathematics", Halmos says the following.

Two digressions about “given”. (1) Do not use it when it means nothing. Example: “For any given $p$ there is a $q$.” (2) Remember that it comes from an active verb and resist the temptation to leave it dangling. Example: Not “Given $p$, there is a $q$”, but “Given $p$, find $q$”.)

I understand his first point, but the second one confuses me. As far as I understand, his examples are ways of conveying that

$$(\forall p)(\exists q)\, ... \,,$$

but what is exactly the problem with "Given $p$, there is a $q$"? Furthermore, how is the meaning of “Given $p$, find $q$” equivalent?

Answer 1

I think Halmos is being a little too pedantic in his second point. It would be more technically correct to say, "If you are given $p$, there is a $q$ . . . ." But I'm of the belief that if your writing is easy to read and to understand, that's more important than grammatical niceties, provided that the grammar doesn't inject unintended ambiguity to your writing.

Answer 2

Halmos dislikes:

• For any given 𝑝 there is a 𝑞.
• Given 𝑝, there is a 𝑞.

Halmos likes:

• Given 𝑝, find 𝑞.

Why? As others have noted the "given" in the first sentence is superfluous, and Halmos hates wasted words. Makes sense. Why does he dislike the second sentence? Because the reader is superfluous. It is a subtle point, and he obscures it by complaining about "dangling" rather than what he really dislikes: whereas the reader is commanded to do something in the third sentence (good), the reader does not exist in the second sentence (bad). The active, imperative usage he prefers has ample historical precedent, illustrated by some books on my shelf:

• Heath's Elements (1908): "Given two numbers not prime to one another, to find their greatest common measure."

• Euler's Algebra (1840): "Given 24x=13y+16, to find x and y in whole numbers."

• Niven's Theory of Numbers (1960): "Given any constant c, prove that there exists..."

These statements instruct the reader to accomplish something - the reader exists. I just skimmed the books and I only find given in these imperative forms. If the authors need to assert the existence of something, they seem to do so declaratively or using if/then statements, only. Casey's Elements (1885) even states "A Problem is a proposition in which something is proposed to be done, such as a line to be drawn, or a figure to be constructed, under some given conditions", emphasizing the active context for given.

So I think the core of Halmos's objection in his second point is that given is being used in a way that doesn't suit the word well.

N.B. Halmos had a passion for teaching and writing and communicating. He wrote almost conversationally yet had zero filler so his books are short and seem like they should be easy to read. But they are not easy reads, or, at least, the apparent simplicity is deceptive. Much is demanded of the reader because of his terseness - he explains a concept or position with minimal verbiage and that is your one chance to find out what he meant. For he will not elaborate, that would lengthen the exposition. I think this is a deliberate strategy on his part to make the reader chew on every sentence. This post is (to me) a classic example of the effort needed to parse his tersely stated position.

Answer 3

Two digressions about “given”. (1) Do not use it when it means nothing. Example: “For any given $p$ there is a $q$.” (2) Remember that it comes from an active verb and resist the temptation to leave it dangling. Example: Not “Given $p$, there is a $q$”, but “Given $p$, find $q$”.)

I understand his first point, his examples are ways of conveying that $$(\forall p)(\exists q)\, ... \,,$$

You misunderstand: Halmos's first point is just that "given" is a redundant filler in “For any given $p$ there is a $q$ such that $P(p,q)$”, because after all there is some q such that P(p,q) holds for any $p$—regardless of whether $p$ has specifically been given to the reader.

but what is exactly the problem with "Given $p$, there is a $q$"? Furthermore, how is the meaning of “Given $p$, find $q$” equivalent?

Here, Halmos is not saying that they're equivalent, just that the former example, “Given $p$, there is a $q$” is, as explained, inferior to its alternative phrasings, and also feels like an incomplete instruction; then he gives the latter example to illustrate one way to appropriately use the word "given".

Answer 4

It's a question of the evolution of the mathematical 'language' from Euler in Math-Latin to todays Menglish, where for an article one copies the wording from a given TeX source in one's environment and fills it with some new symbols and relations.

Halmos - a philosopher with mathematics as secondary field - and the other Martians were at pivotal moment, when Mathematics, written and thought in Latin switched to national languages.

The group of Hungarians were educated in the kuk Gymasium in Budapest in Latin and Greek and probably a very poor mathematics course with fixed formulae in Latin bought from the corpus of the laws.

At that time, after Riemann, Cantor, etc, the traditional course at universities was useless wrt to the new axiomatic approach.

In 1930 nothing could be assumed to be 'given'.

The latin term does not mean a gift or an assumption set at the higher spheres; 'datum' is more like the start value of a process, an idea bought from process law.

After the Nazis expelled the most important German-speaking mathematicians from Germany in 1933, they made the conversion of the outdated mathematics courses at American universities their passion project.