# Starting sentences with mathematical symbols.

I apologise if this is a duplicate in any way or is too opinion-based.

To what extent is it best not to start a sentence with a mathematical symbol?

I find that when trying to solve a problem or prove something it’s an unnecessary distraction to care too much about forming proper sentences and so forth, but when writing things up, I just can’t bring myself to start a sentence with a symbol. It doesn’t look right.

I remember in my first year as an undergraduate I was told by a PhD student that it’s “bad form” to do so and I somewhat agree, but I’ve seen it frequently in lectures, seminars, and even in papers.

Consider the following example I cooked up:

Definition: Let $L$ be a Lie algebra. $\color{red}L$ is solvable if there exists an $n\in\mathbb{N}$ such that $L^{(n)}=\{0\}$ in the derived series of $L$.

This, to me, is a word away from how it “should” be written; just stick “Then” at the start of the second sentence.

Now, I am aware that mathematical concepts are difficult enough to write about without worrying over such things. (I agree with Stephen Fry when it comes to language.) But what’s the convention? Does it matter?

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You might be interested in the essay Halmos, How to write mathematics and the lecture Serre, How to write mathematics badly. Both suggest to avoid starting a sentence with a symbol. In my personal experience from reading mathematical texts and having my own texts corrected by professors, they all agree with this convention. It’s just not pleasing to read a sentence that starts off with a symbol.

Update: I removed the noise from the recording of Serre’s lecture and changed the link to the improved version.

My two-pennyworth …

Starting sentences with lower case letters from the Roman alphabet “looks wrong” in general. We don’t expect sentences to start that way, and this habit of the well-trained eye (if we can call it that!) naturally carries over to some extent — doesn’t it? — to the case where the lower case letters are italic letters used as mathematical symbols in a sentence of mathematical English. So I’m conscious (though it isn’t a strict rule) that I’ve tried to avoid starting such sentences with lower case letters when writing books or papers. But I wouldn’t make a fetish of this. It’s just a default practice which can occasionally be overridden if it isn’t going to distract the reader.

And I’ve not worried at all about occasionally starting sentences with upper case italic letters (so your displayed usage is just fine by my lights). And I see — getting one of my books off the shelf! — that I’ve not worried too much about starting sentences with lower case Greek letters either.

So, in sum, because we expect sentences not to start with lower case letters from the usual alphabet (even if in italics), it is probably a good thing to avoid doing that in maths too, other things being equal. Otherwise it is surely liberty hall, as long as we remained guided by the general principle of not unnecessarily distracting the reader by perverse symbolic choices.

I believe there is a sort of convention going around that no sentence should start with a mathematical symbol, even in a case like the example you provide. But as far as I know it’s a typographical convention and not a semantic or grammatical one: if you start a sentence like

$G$ is a simple group…

then the typesetting can lead to some confusion depending on if there are other mathematical symbols floating around. For example if the previous sentence ended like

…so $O(H)=n^2+k/2$. $G$ is a simple group…

then upon a quick glance the $G$ might appear to be a part of the equation. So this is why some authors prefer not to start sentences with mathematical characters at all.

My opinion: starting a sentence with mathematical characters (even lowercase Roman!) doesn’t bother me, but you might want to watch out for typesetting issues like the one I described. In that sort of situation I like to put in a regular English word or two to clearly separate the two instances of mathematical expressions. But almost everything else is fair game, I think. (Personal aesthetics should govern things like whether “$r$ is idempotent” is ok at the beginning of a sentence.)