Can not vs cannot

I have just discovered that for many years I have been using “can not” where “cannot” would be clearer. For example, concerning Riemann sums, I wrote

The Riemann lower sum for f corresponding to a partition P of \null [a,b] can not be greater than the Riemann upper sum for f corresponding to a partition Q of \null [a,b], even if P and Q are different.

Concerning uniform convergence of sequences of real valued functions defined on a domain D, I wrote

Let D be a non-empty subset of \mathbb{R}^d and suppose that f is a bounded function from D to \mathbb{R}, i.e., f(D) is a bounded subset of \mathbb{R}. Let (f_n) be a sequence of functions from D to \mathbb{R}. Suppose that all of the functions f_n are unbounded on D. Then (f_n) can not converge uniformly on D to f.

It looks like I haven’t used “can not” all that often in my second-year analysis notes. But it certainly never worried me when I did use it.

I had been ignoring the clue given me by Microsoft’s automatic spelling and grammar correction, which kept telling me to use “cannot” instead of “can not”. But recently this emerged as a difference of opinion with a colleague, and I am now convinced that “cannot” really is safer.

The difference is that “cannot” only means “is not able to”, while “can not” can have that meaning, but can also mean “is able not to”. Although I was aware of this second meaning, I was convinced that “can not” was common usage, and that it would be sufficiently rare to think of the other usage that I should not worry. However, various Google searches have convinced me that, at least in maths, I am in a relatively small minority with my use of “can not”. So I am going to change to “cannot” from now on.

But did McEnroe say “You can not be serious”, or did he say “You cannot be serious”?


5 responses to “Can not vs cannot

  1. According to my walking, talking grammar oracle I am informed that many people are/were taught at school that “can not” is just not permitted (except obviously in phrases such as “He can not take the medicine, if he chooses” and the like where the “not” part is part of some other phrase and so can’t be adopted by the “can”).

    So perhaps the answer is that “can not” is indeed not even permitted in the senses that you’re talking about at all!


    • Well, I’m not completely convinced about that yet.
      See for some (also not fully convincing) discussion.
      For example, it says there

      The Washington State University language site says:

      These two spellings [cannot/can not] are largely interchangeable, but by far the most common is “cannot” and you should probably use it except when you want to be emphatic: “No, you can not wash the dog in the Maytag.”


    • On the other hand, I often do use it when I am trying to be emphatic! But you can’t (hah!) tell this in the written version.


      • Looks to me like all these references are consistent with the following situation:
        (i) historically cannot didn’t exist
        (ii) once cannot was invented it was used exclusively in the natural sense of forbidding something. i.e. the technical “can not … ” construction to permit cases where something may not occur was entirely left out of cannot’s meaning
        (iv) nowadays if you want to say that something doesn’t need to happen then you have to use “can not” and it’s logical and sensible (to everyone except pedants like us) to use cannot in all other cases (i.e. forbidding cases).

        Those references look like they say “can not” is permitted but it’s only really been maintained because of the need to be able to say things like:
        (i) “Paul can not only sing well, he also paints brilliantly.” where the “not” forms part of some other phrase and isn’t really the normal word “not”, OR
        (ii) Cases where logicians want to make clear that something isn’t forced to happen given some earlier conditions.


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