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.
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.
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.
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.
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.
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.
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.
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.
Obviously a matter of opinion. LGTM.
