To boldly split...

Cadet. "'SCUSE ME, SIR—ARE YOU A DOCTOR? THERE'S A BOY FAINTED."
Doctor. "AH—FATIGUE, I SUPPOSE?"
Cadet. "No, SIR. THE SERGEANT SPLIT AN INFINITIVE."
Punch, Vol. 156, April 30, 1919

Johnson bemoans the rule in the Economist style guide, which reads:

Happy the man who has never been told that it is wrong to split an infinitive: the ban is pointless. Unfortunately, to see it broken is so annoying to so many people that you should observe it.

The ban seems to have appeared during the 19th century, after at least six centuries of happy infinitive splitting. Since it made its way into school grammar texts, it has been memorised as a fixed rule, even though it makes no sense and isn't part of formal English grammar.

Theodore Bernstein said hopefully in Miss Thistlebottom’s Hobgoblins (1971) that “within the near future the split-infinitive bugaboo will be finally laid to rest.” Merriam-Webster’s Dictionary of English Usage describes the resistance to the construction as having “established itself in that subculture existing in the popular press and in folk belief.” Garner’s Modern American Usage includes the split infinitive in the entry Superstitions. John Bremner wrote in 1980, “This so-called rule has no foundation in grammar, logic, rhetoric or common sense.”(John E. McIntyre: You don't say]

So, for more than 100 years, style books written for professional writers give similar advice to the Economist: there is nothing wrong with splitting an infinitive, but don't do it because some people will think you've made a mistake. Johnson refers to an article on Language Log, 'Crazies Win'. Arnold Zwicky comprehensively demolishes both the split infinitive 'rule' itself, and the style guide ban. He says:

Whether teachers proscribe split infinitives because they believe in [No Split Infinitives] NSI or because they think that's the easy route around [Split Infinitives as Last Resort] SILR, their students grow up thinking that split infinitives are a Bad Thing. If, later in life, they come across SILR in the advice literature, what are they to make of it? What was once banned is now permissible, but in what circumstances? How to know when the alternatives are worse?

The advice givers have tacitly acquired their own tastes in the matter, but they can't assume that their readers (already) share these tastes. In fact, by formulating principles like NSI and SILR, they have failed to teach their readers anything useful about how they might judge different versions of infinitival verbal phrases.

In other words, Zwicky believes that the ban perpetuates the myth that there is something wrong with split infinitives. He finishes:

The objective fact is that split infinitives are standard English. So my advice is: split an infinitive if it suits you (or don't, if that suits you). Good writers do it. And you don't even have to have a defense for it; do it because it sounds right for you. Don't let the crazies win.