Well these days (as opposed to a hundred (good to one sig fig) year period of conservatism around which the language is fluid as hell) that’s maybe not a useful question. We do what we please with English and the language is sort of famous for surviving it. For a long time, however, and currently amongst the sort of pedant that has a strong opinion about Oxford commas, the split infinitive was Not Allowed.
But English is really good for splitting infinitives.
The infinitive form of a verb is its naked form, unconjugated. So in English the infinitive “to go” is conjugated as “she goes, we go, they go, you go”. That infinitive is apparently never allowed to have a word inserted between “to” and “go”. It’s to be treated as though it’s un-fucking-divisible. A single word with a space inside it that apparently acts like a letter.
This is, I think, mostly an effort at linguistic political correctness to avoid drawing attention to the fact that many (maybe most) lesser languages do not have this feature. Their infinitives (aller, for example, en Français) are really one word. Which means they do not have the tonal equivalent of “to boldly go” which delivers a mood distinct (to my ear anyway) from “boldly to go” or, worse, “to go boldly”. It’s perhaps the proscription itself that lends this tone (which totally undermines my argument by making the proscription necessary in order to have the feature) by undermining the formality of the “correct” structures. Kirk in the Star Trek opener is established by his linguistic choice as an everyman who doesn’t give a rat’s ass about ancient style guides nor, by extension, Robert’s Rules of Order. We know in our viscera before we even see him that he’s a hero we get to aspire to be. He shirked his way through college and the academy (which later we find out is true). He must have.
And some infinitive busting structures don’t even have correct variants. Consider “I’m going to fucking shoot you in the face.” It’s distinct from “I’m going to shoot you in the fucking face” in that the rude word modifies face instead of shoot. And obvious you can’t say “I’m going fucking to shoot you in the face.” Then you just get laughed at. You’ve descended below the low bar of lovable rogue to incomprehensible villain. “I’m fucking going to shoot you in the face” is weird acceptable, modifies the wrong word, and seems like a grammatically worse choice than splitting the infinitive even though it’s fine. It’s more of a hipster bandit move; an attempt to get you to argue with their usage so they can produce evidence it’s correct. Before shooting you in the face.
So let me suggest that we need not be polite to our compatriot languages who are stuck with indivisible verbs. Our verbs are naturally divisible and this division begs for modifiers. Every space is a possibility for a slightly different tone. It does not invite confusion but rather establishes the writer’s intent clearly and efficiently. The space in the middle of our infinitives is a tool to be wielded however we like to use tools.
Of course, once we get to this point we have to wonder what the “to” is for anyway. What does “to go” mean, decomposed? What work does the “to” do? In the phrase “I’m going to go” it seems to have more to do with “going” than “go” to my ear. That is, as the sentence proceeds, “I’m going to…” is still sensible — I’m certainly going somewhere and to is a somewhere word. I’m going to the store. I’m going to outer space. I’m going to sleep. The “to” is independent — it doesn’t need a verb at all to be useful.
So rather than knuckle under to linguistic equivalentists who would hobble English in order to put it on equal footing withe French or, heaven forbid, Latin, let’s instead celebrate the feature of the English infinitive. Split it at will. It’s already split.