It's funny, but I happened to run across Professor West's little web page myself just a few weeks ago, soon after I got a copy of his textbook Introduction to Graph Theory. Right now there are just four books on my desk: West's book, a couple discrete mathematics texts, and a Merriam Webster dictionary published about 30 years ago.
Yes, I looked up cannot in that dictionary. It defines the word one way only: as can not. So the idea that the two terms are synonymous is not some novel horror recently visited upon the English language.
Language can be sloppy. Context is often as critical to correct understanding as syntax, grammar, or one's choice of words. Taking care in constructing phrases and sentences is always important. Mathematicians in particular will use symbols when absolute clarity and precision are needed.
Professor West's opinions on this matter, unfortunately, are being addressed to "non-native speakers" of English, but as far as I'm concerned my 30-year-old dictionary settles the matter; that is, cannot and can not are largely equivalent. When Professor West says the logical meaning of "can not fail" is "may possibly succeed," he's ignoring the fact that human languages are not always logical. If they were, symbolic logic might never have been invented.
I want to address a thing or two in this passage:
Here are some examples illustrating my understanding of West's words.
When we want to say x ≠ 8, if we say "x can not be 8," it can be
misinterpreted to mean that x can be 8 and not be 8. However, West
suggests that when we say "x cannot be 8," it's clearer and less prone
to misunderstanding because it implies that x must not be 8.
From my own personal experiences as a student and later a professor of mathematics, it feels in no way natural to interpret "x can not be 8" as meaning "x may not be 8" or "x might not be 8" or "x has the ability to assume a value other than 8." It's terribly clunky sounding. Something Lieutenant Commander Data might say in his early days. Consider this: "x can't be 8." Certainly can't is just a contraction of can not, yet there ain't no way to bleed a "may not" or "might not" out of a "can't"! This third contender, can't, joins the party late, but it further complicates any attempts to put even this tiny backwater of the English language into perfect one-to-one correspondence with the formal constructs of symbolic logic.
In mathematics I believe it is better to phrase things in ways that conform more closely to the literal translations of the relevant mathematical symbols. So, translate x ≠ 8 into English as "x is not equal to 8" or "x does not equal 8." Always be extra careful when using modal verbs, because in everyday language they get tossed around all too carelessly. Example: asking "Can I go to the bathroom?" when really "May I go to the bathroom?" is more appropriate.
Here, let me crack open Professor West's graph theory textbook (2nd edition) and pull a couple sentences out of exercise 1.3.53:
Each game of bridge involves two teams of two partners each. Consider
a club where four players cannot play a game if any two of them have
previously been partners that night.
So what if "cannot" were replaced with "can not"? Would this truly cause confusion? I think not, for the simple reason that if by "can not play a game" one meant "have the ability or option to not play a game," it would be immediately recognized as a bizarre and clumsy way of phrasing things. There are far, far better ways to convey the idea that the bridge players have the choice to play a game or not. And of course if this exercise were being read to a blind person by the KNFB Reader app, "cannot" and "can not" would sound the same, and therefore (according to West's own rules) we have a potential semantic disconnect between the written and spoken words. I'd say "can't" would be the word of choice if we were really worried about ambiguity, since it has the advantage of sounding distinct from "cannot" and "can not."