The phrase "can not" may mean "cannot" or "can't," but it also may mean "able not to," which is much different than "not able to" as it conveys the ability to not do something rather than the inability to do it.
Why is “cannot” spelled as one word whereas other similar constructions such as “do not,” “will not,” “shall not,” “may not” and “must not” are spelled as two words (unless they are contracted as “...
So, "cannot" means something (denoted S) is impossible, while "can not" means the inverse of that something (denoted !S) is possible. These two statements do not imply each other. Of course, the second statement could be rendered much more clearly as "The variable can be not initialized," by moving the "not" next to the word being negated.
So here it’s about minimizing ambiguity: can not permits two interpretations, while cannot permits only one. Such care is critical in the language of mathematics. As another example, West would tell you that in mathematical discourse, both “x is a minimum” and “x is a minimal” are valid utterances, but they mean different things.
8 Both are acceptable, but cannot is now more common. OED has this much to say about cannot: (ˈkænət) the ordinary modern way of writing can not: see CAN v. Notwithstanding, in some situations ambiguity may arise if you write can not, and the difference might not be a minor one. Compare: I cannot make love to you.
9 "cannot not say" would only rarely be used in English, and only in very specific circumstances. In particular, this is not a simple double negative. "cannot not" does not mean the same as "can", it means "must". It also carries a connotation that not saying is the expected or default action.
The irony of the idiom is that one would expect to be able to eat the cake that he or she owned. In that case, possession of the cake would logically come before usage of the cake: You cannot have your cake and eat it, too. Some other interesting observations: Paul Brians, Professor of English at Washington State University, points out that perhaps a more logical or easier to understand ...
Cannot is the only negative form that contains not rather than -n't. Theoretically, since it is a single word, you can say why cannot you... without a problem. My theory is that modern English speakers don't want to put cannot before the subject because it contains not and sounds like can not.
1 There is no correct way to use "cannot help but." There are two traditional and synonymous idioms, "cannot but [do something]" and "cannot help [doing something]." Those who say "cannot help but" are conflating the two (in much the same some people conflate "irrespective" and "regardless" to create the nonsensical "irregardless").
