Does 'hence' (thus, therefore) need a comma after it?

Question

In mathematical papers, when "hence" is placed at the beginning of a sentence, I have observed that some authors use a comma while others do not. The same issue can arise with "therefore" or "thus" as well.

For example:

• Hence, by Lemma 5.2, $G$ has at most $3c+ 3n− 8$ triangles, where $c$ is the number of crossings of $ϕ$. (Counting cliques in 1-planar graphs)

• Assume first that $t$ has at least two incident uncrossed edges in $\Gamma^{\prime}$. It obtains 12 charges from these two edges, and at least $3\left(\operatorname{deg}_G(t)-2\right)$ further charges from the remaining edges that it had in $G$. Hence $c(t) \geq 12+3\left(\operatorname{deg}_G(t)-2\right)=3 \operatorname{deg}_G(t)+6$. (Matchings in 1‐planar graphs with large minimum degree)

• By $\left[2\right.$, Theorem 2.4.1] $\phi^{\prime}$ and $\phi$ are of the same order and type. Therefore we can find $C>0$, depending on $\rho$ and $\alpha$, so that $\left|\phi^{\prime}(z)\right| \leq C e^{\alpha|z|^\rho}$ for all $z$ in $\mathbb{C}$. (Superposition operators between weighted spaces of analytic functions

• Therefore, $G_1$ cannot contain more than six crossings.(1-Planar Lexicographic Products of Graphs)

• Thus, the natural questions here are: For which entire functions $\phi$ do we have $S_\phi\left(\mathcal{H}_1\right) \subset \mathcal{H}_2$ ? When is $S_\phi$ bounded?

• Thus we can suppose that $f_n$ converges uniformly on compact subsets of } $\mathbb{D}$ to a holomorphic function (Bounded superposition operators between weighted Banach spaces of analytic functions)

Can both of these approaches be used? Is it just a matter of style, or is adding a comma more accurate? I'm a bit confused because my text editor keeps suggesting commas after these words.

I also noticed that Professor West mentioned this issue on his homepage (see Item 14), but I'm not sure if I correctly understood his point that a comma should be added after "therefore" but not after "hence" and "thus."

[14] Words of conclusion: "Hence", "Thus", "Therefore" A long proof does not fit in a single sentence; hence often one needs a word to start a sentence that states a conclusion. Among the choices are "Therefore", "Hence", and "Thus". Purists (and copy editors) desire a comma after every such introductory word or phrase (as they do after "Finally", "On the other hand", "In 1965", etc.). This can make language overly formal.

Among these choices, I treat "Therefore" as the most formal, introducing a major conclusion and hence taking a comma. Because "Hence" and "Thus" are single syllables, I use them without commas to indicate the flow of argument without making the writing choppy. This choice modifies strict English punctuation in the service of mathematical understanding. It is not incorrect to put commas after all these introductory words, but it enhances mathematical communication to omit the commas after short words introducing short conclusions that are just a step along the way. Copy editors put in the commas, and I insist that they be removed again.

Are there specific guidelines in mathematical papers regarding this matter?

Answer 1

In your first bullet, the two commas function like parentheses, and are necessary.

In the remaining bullets, your hence/thus/therefore all begin their sentences, so I typically prefer a comma after each one (though not always: I usually omit commas in the sentence “...and therefore...”), but this is a matter of taste (I agree with Sarvaes that this can make a text sound overly formal).

Ultimately it also depends on the desired effect of the sentences and the overall piece of writing, that is, it is also a matter of style/wit.