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I received a comment from a colleague that starting a sentence with "let" is not good practice. I know that starting a sentence with "let" is grammatically correct, so that is not the question. My question is if starting a sentence with "let" is good practice and acceptable in technical writing.

Here are a couple examples:

  • Let scalars be denoted by lower case letters.
  • Let x be a real number greater than zero.
  • Let the set of real numbers be denoted by R.
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These are all acceptable. My personal preference would be to reserve "let" for statements that are (essentially) assignments, like the second one in the question. The other two establish notation conventions. I might say something like

We use lower case letters for scalar variables.

or

Lower case letters represent scalar variables.

and

We write R for the set of real numbers.

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    Me too. The first and the third sound distinctly non-English. – TonyK Dec 31 '20 at 22:36
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    @TonyK both are fairly standard in scientific papers – Nihar Karve Jan 1 at 6:08
  • When you get right down to it, those are all notational conventions, and all are also assignments. I don't see any reason other than personal preference to treat them differently. – jpaugh Jan 1 at 9:23
  • I could be splitting hairs, but OP's second example strikes me as an assignment (as it presumably introduces a previously unused quantity in a proof) rather than a notational convention. Ethan makes a good point; starting with "let" is something of a technical term within proofs so outside uses might detract from ease of reading. – johncip Jan 3 at 4:11
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    Agree. In mathematics, sentences like "Let x be a real number greater than zero" or shorter "Let $x>0$" are very common. For notation, like the first and third sentences, there might be better ways that using "let", e.g. "Scalar variables will be denoted with lower case letter: $a,b,c,x,y,z$." – md2perpe Jan 3 at 12:01
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These sentences will appear normal to anyone educated in mathematics in the English language.

They could, perhaps, be made more concise if written without "Let" but I would not think too hard about it.

For such trivial issues, it is often wise to give your colleague what they want.

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    How could it be more concise without the word "let"? What's the more concise phrasing? I would've thought the "let" phrasing is the most concise way to phrase this – vijrox Dec 31 '20 at 16:08
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    @vrijox Active verbs: "Lower case letters denote scalars. R denotes the set of real numbers." – eric_kernfeld Dec 31 '20 at 17:53
  • @eric_kernfeld Putting the first in active voice changes the meaning. But you can avoid "let" simply by getting rid of the "let": "Scalars are denoted by lower case letters." – Especially Lime Dec 31 '20 at 18:40
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    Yes, the meanings are different. "Lower case letters denote scalars" implies that all lower case letters denote scalars, not other mathematical objects, and it allows for some scalars to be written in other ways. Though this is not exactly the same meaning, both statements will likely hold in this use case (mathematicians declaring conventions to the reader). And it's more useful to the reader to have a map from what they see (a lower case letter) to what it means (a scalar) than the reverse. This is a great example of how improving concision also improves clarity. – eric_kernfeld Jan 1 at 16:55
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Yes, this is common in math writing. It is hard to imagine a book or long paper that doesn't use the convention to some extent.

But, I would only add that too much repetition of a phrase become tedious and even annoying to your readers. So while

Let x denote a real number.

is perfectly fine,

Assume x is a real number.

is an alternative.

Make the writing lively if possible. But clarity in math is the more important thing.

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    To me, "Let x be a real number" and "Assume x is a real number" have distinct logical meanings. The first is a type declaration, the second is part of a case distinction. In the statement of a theorem it wouldn't be wrong to use "assume" instead of "let", but "let" seems cleaner (at least if you dislike untyped variables). – Arno Dec 31 '20 at 15:16
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    A better synonym for "let" would be "suppose", or a construction like "If x is a real number, then (something)". IMO "assume" should be reserved for making an assumption about the properties of some object which will later be proved either true or false. – alephzero Dec 31 '20 at 16:10
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    @alephzero, there are a lot of possibilities, but I doubt that any working mathematician would be misled by any of them. The message here is "lively writing" not the specifics. And if you want to use "Let" and "Assume" and "Suppose" etc. in some narrow technical sense, you should explain your usage. – Buffy Dec 31 '20 at 16:13
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    I would interpret "assume x is a real number" as meaning "We already know that x is a complex number, now for the duration of this [sentence/paragraph/section], assume it's also a real number." – Kevin Jan 1 at 16:03
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In your first example I would actually agree that the "let" construction is not such great style. I wasn't sure why at first. But as Kimball pointed out in his comment, "let" is for definitions. You aren't really saying "let a, ..., z be 26 scalars", since you'll use those letters to stand for different scalars in different places. Besides you could just say "lower case letters denote scalars" or something like that.

The other two examples are certainly typical mathematical English.

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    The first example is not of the form "Let X be [definition]" because you are not defining scalars but just your notation for them. A cleaner way to say this with let, though certainly more old-fashioned would be "Let it be known that scalars are denoted by lower case letters." – Kimball Dec 31 '20 at 17:55
  • @Kimball, +1 and question updated to agree with you! – Simon L Rydin Myerson Jan 1 at 12:02
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The immediate, facile answer is that "yes, it's fine, since so many people have done it already"... and, also, since your audience will certainly understand the content.

A secondary point is that quite often a more effective-and-efficient sentence structure is available, as exampled in other answers and comments. In some ways, these secondary things don't truly matter, but they can distract the reader, as well as allowing a critical reader to wonder about what you think is happening... which is probably not good.

An exaggerated case is "Let the real numbers be complete." What? Well, yes, they are, but what is the point of saying this? Or "Assume $x$ is a real number." Wait, what? What if it's not? Does one mean "here let $x$ be a real number..."? Probably. And, yes, experienced readers can decrypt the weird syntax, but there's no point in making them do so.

Alternatives to "Let $x$ be a real number..." are, for example, "For real numbers $x$, ..." and such.

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    "An exaggerated case is "Let the real numbers be complete." What? Well, yes, they are, but what is the point of saying this?" Well, presumably you're making a mathematical argument that's based on that premise - possibly involving producing a result that shows that the real numbers aren't complete, thereby producing a proof by contradiction of whatever the main thrust of your proof is. – nick012000 Jan 1 at 1:01
  • @nick012000, for the example you mention, I'd be inclinced to say "assume" or "suppose" [thing to be shown false], rather than "let". But, once again, perhaps language drifts even over a few decades... – paul garrett Jan 1 at 18:55
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One of the most lauded writers in mathematics in Jean-Pierre Serre. The first sentence of his book A Course in Arithmetic is "Let K be a field". The first sentence of André Weil's book Basic Number Theory begins "Let F be a finite field..." The first sentence of Section 1 of Chapter 1 of Wiles' proof of Fermat's Last Theorem is "Let p be an odd prime."

Conclusion: it is perfectly fine to start a sentence with "Let" when you are talking about fields or odd prime numbers. Maybe you can generalize this observation.

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I see no problem with saying, for example, "Let x be a real number", but I often say "Choose a real number x", as well as "Take" or "Consider".

I don't really have a reason for saying it one way versus another way, but I like to mimic the language that my math professors use when going over proofs during class. They know better than me.

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Obviously a matter of opinion. LGTM.

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    Expand a bit, please? – jakebeal Dec 31 '20 at 14:54
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    @jakebeal Google says that's "looks good to me". Who knew. – Ethan Bolker Dec 31 '20 at 20:04
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    I really don't think this should be an answer (no offence intended, of course). – YiFan Jan 1 at 7:20
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    @YiFan Several of the longer (and upvoted) answers really don't say more than this. – jpaugh Jan 1 at 9:29
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    @jpaugh I disagree with that interpretation of those answers. – YiFan Jan 1 at 10:31

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