“Let for each 𝑗” vs. “For each 𝑗 let”

Question

I wrote

Let for each 𝑗 < 𝑛 a permutation ℎ_𝑗 : 𝐿 ↪ 𝐿 be given.

A proofreader (whom I can no longer ask) changed it to

For each 𝑗 < 𝑛, let a permutation ℎ_𝑗 : 𝐿 ↪ 𝐿 be given.

This made me stumble. I think that we introduce all the symbols ℎ_𝑗 once rather than separately for every 𝑗. Also, if you were to go for pure non-mathematical grammar, the prepositional phrase “For each 𝑗 < 𝑛” is probably short enough such that it might not require a comma after it. In mathematical writing, you may agree or disagree on these two matters.

1. Why the change? What could be better™ in the changed formulation compared to the original one?

2. What is worse™ in the changed formulation compared to the original one?

Answer 1

Mathematician here.

Your proofreader is right.

The revised version is how this is normally written. I would understand your original text, but it would make me stumble. You should avoid wording that distracts your mathematician readers.

You could avoid the passive construction and replace some symbols by words by writing

For each j < n let h_j be a permutation of L ...

Answer 2

Your proofreader is correct. Another alternative I might accept would be:

Let a permutation ℎ_𝑗 : 𝐿 ↪ 𝐿 be given for each 𝑗 < 𝑛.

I’m less likely to Write a sentence with mathematical notation this way, because in formal notation, I’m used to specifying the ∀ and ∃ quantifiers from outermost to innermost. Your readers are probably used to seeing the “let,” “for all” and “there exists” clauses in the order you used, too.

But “Let for each” is not standard English grammar. It’s just one of those arbitrary rules.

Here is a document with many good examples of how to write quantifiers in your math papers, in English.

Answer 3

You ask what is wrong with it. Nothing is "wrong", but the proofreader's version is definitely much better.

If you break up both phrases into their clauses, you get this:

Let
for each 𝑗 < 𝑛
a permutation ℎ_𝑗 : 𝐿 ↪ 𝐿 be given.

For each 𝑗 < 𝑛
let a permutation ℎ_𝑗 : 𝐿 ↪ 𝐿 be given.`

The basic "natural phrase" that most easily guides the user's focus would be "Let a permutation be given". We insert two further clauses, one specifies the permutation itself, which fits naturally into it ("Let a permutation ℎ_𝑗 : 𝐿 ↪ 𝐿 be given"). The other specifies the scope of the definition, so to speak: "For each 𝑗 < 𝑛".

By inserting this as you did, the user has to break off the "flow" of the clause, then resume it. As modified, they don't have to do this.

The effect is a bit like this:

For every Stack Exchange question, let there be one great answer!

(Easy to read and flows well)

Let, for every Stack Exchange question, there be one great answer!

(Not so easy or flowing)

Answer 4

The "default" order is "Let a permutation ℎ_𝑗 : 𝐿 ↪ 𝐿 be given for each 𝑗 < 𝑛." We have a verb ("let") and an object (the rest of the sentence). The object is itself a phrase that contains an infinitive ("be") that is modified by an prepositional phrase ("for each 𝑗 < 𝑛"). When a prepositional phrase precedes the verb that it modifies, it should be followed by a comma. That is why there's a comma in your proofreader's proposed correction; it's not about length, it's about a modifier preceding the thing it modifies. In this case, there's also the bonus that the comma separates j < n from the rest of the sentence, making it more clear that "< n" is modifying "j".

The prepositional phrase "for each" is modifying the verb phrase "be given", not "let", so the default order is for it to be placed next the the former, not the latter.

I think that we introduce all the symbols ℎ_𝑗 once rather than separately for every 𝑗.

If you want to conceptualize it as a single act of giving, you would have to phrase it as a single object, such as "Let a function ℎ: j → Sym(L) be given".