The usage of "can not" vs. "cannot" in mathematics

Question

I saw the following passage in Professor West's homepage, and I hadn't noticed this point before. See https://dwest.web.illinois.edu/grammar.html#cannot

"Can not" and "may be". The expression "can not" should not exist in English (I am horrified to hear that it now does appear in some dictionaries). The logical meaning of "can not" is that it is possible for the statement to fail rather than that it must fail, which is the meaning of "cannot". All uses of "can not" must be eliminated, because the reader cannot be sure of what the author intends.

He believes that "can not" can be confusing and goes against the precision of mathematics. So "can not" is bad, and "cannot" is good in mathematical papers.

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.

However, the comments suggest that such statements are not convincing enough, as they are essentially considered to be nearly indistinguishable.

PS: I'd like to extend this question. Besides this phrase, are there any other similar phrases that we should be aware of? That is to say, are there any more examples where there is a significant difference in meaning between "AB" and "A B" where A is a word similar to "can" and B is a word similar to "not"?

Answer 1

Full disclosure, I know Doug West (he’s my Ph.D. advisor’s Ph.D. advisor), so I am probably more inclined to give him credit. But it’s pretty obvious in context that he’s not being curmudgeonly and trying to dictate general idiomatic discourse. He intends his comments to apply only within the context of mathematical discourse, which definitely does call for more precision in expression.

His comments appear in a section he has headed Mathematical Style. Furthermore, in his introduction he explains,

“Mathematical concepts are abstract, without context from everyday experience, so the writing must be more consistent to make the meaning clear. Outside mathematics, imprecise writing can still be understood because the objects and concepts discussed are familiar.”

Answer 2

The good professor is saying that "can not be" is ambiguous.

"X can not be 8" could mean "it is possible for X not to be 8" which then means, by extension, that it is possible for X to be 8 -- at the very least the statement is silent with regard to whether it can be 8.

So, if your goal is to say, unambiguously, that "X is never equal to 8", then you should write "X cannot be 8".

In natural everyday English when speaking about natural things, the context often resolves ambiguity; but in specialized contexts (mathematics, science, law, etc) where there is often nothing to corroborate the meaning, language must be used in a manner that prescribes the meaning. In other words, there are situations where it is necessary to prescribe usage and not rely upon how native speakers speak "in the wild", naturally.

Answer 3

EnglishClub has a balanced overview:

People often ask me whether they should write cannot (1 word) or can not (2 words).

Cannot is a contraction of can not.

In 'British English' cannot is the normal form.

In 'American English' both forms are acceptable but cannot is more common.

In general I would suggest that you use cannot.

However, note that there are times when you really have to use can not. If the word “not” is part of a set phrase, then you have no choice but to use the two-word form can not.

• He can not only play tennis brilliantly, but he can also swim like a fish.

Note, too, that cannot may be contracted to can’t, but in formal written English (such as in an essay or exam) you are not advised to use can’t.

[Josef Essberger; July 2010]

Merriam-Webster concurs:

Both cannot and can not are perfectly fine, but cannot is far more common and is therefore recommended, especially in any kind of formal writing.

As does Thesaurus.com:

The terms cannot and can not are identical in meaning and are typically considered to be alternate forms of one another. However, the form cannot is much more commonly used than can not. Cannot has become the standard form in formal writing (and is typically the form recommended by most grammar resources and style guides).

Answer 4

Can not is ambiguous. Conveniently, it has an alternative spelling cannot which corresponds to the more common of the two possible meanings. It's wrong though to say that this means can not can only have the other meaning ("that it is possible for the statement to fail rather than that it must fail").

The pushback in the comments is not based on denying that can not is potentially ambiguous. It's a response to the exaggerations and overstatements in the quoted passage, such as "I am horrified to hear that it now does appear in some dictionaries". This kind of wording may be effective stylistically as a means of making a memorable point (don't use can not in mathematical writing), but it doesn't accurately reflect what modern dictionary-makers try to do: describe language as it is used, not just how some people think language ought to be. So if you ask a website meant for "linguists, etymologists, and (serious) English language enthusiasts" about these comments, you'll get feedback related to the usage of can not and cannot in the language as a whole (which is described well by Edwin Ashworth's answer). But as PaulTanenbaum says, the passage may need to be read as implicitly restricted to the context of mathematical writing.

Avoiding ambiguity is a useful goal in some contexts, but all natural languages contain some ambiguous constructions. Thus, if you are using a natural language as a medium of communication, it's not practical to try to avoid all ambiguity. (That's why natural languages are not used in contexts where that is necessary, such as computer code.) It is sensible to avoid ambiguity that can feasibly lead to miscommunication, but in many contexts using the spelling "can not" instead of "cannot" will not plausibly cause miscommunication.

"Could not" and "may not" can be ambiguous in the same way as "can not". It would be eccentric to say that means that may not "should not exist in English".

Answer 5

Thanks to Paul Tanenbaum for alerting me to this discussion. In response to these comments, I have modified this item on my grammar page. It now reads:

It appears that some writers of English now use "can not" to mean "cannot." In speech the two cannot be distinguished, so it doesn't matter, but in written mathematics we should avoid ambiguities. The logical meaning of "can not fail" is "may possibly succeed" while "cannot fail" means "must succeed." At the very least, when "can not" is used to mean "cannot" it can be read to have a different meaning, so it is better to use "cannot" to eliminate the ambiguity.
As in many of these items, the abstractness of mathematical statements in contrast to the everyday context of English language begs a higher level of precision and avoidance of ambiguity.

Answer 6

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."

Answer 7

I work more with software specifications than maths, but the problems are similar. The most common problem is with "may not", which is thoroughly ambiguous. Usually the context makes it clear: "The value may not be greater than 1000" means "must not" while "The file may not exist" means "might not". But the ambiguity can always be avoided by writing either "might not" or "must not", so as far as I'm concerned, "may not" is anathema.

Answer 8

This is called an amphibology, an in this case you're describing a first order logic confusion. I'm assuming the math professor has a laundry list of similar terms and phrasing which he finds problematic, because this is very common in the English language.

First order logic is the branch of logic where the basic classical logic statements of is, is not, and, or etc become combined with qualifiers and conditions, such as always, for at least one, possible, some, etc. Classical logic allows us to write precise (yet simple) assertions like "If the ground is not wet, it is not raining"; predicate logic would support quantifiers like "If the ground is not wet, it might not be raining." Predicate logic usually provides a better model for the world as it contains expressiveness around "possibility".

But once you include this concept of "possibility" you potentially hit ambiguity. Mix some "always" and "impossible" and you'll find this is very common. For example:

Every student didn't pass the exam.

Does this mean that No student passed the exam, or Not every student passed the exam? Of course there's no correct answer; you can assume based on context, but it's ultimately an ambiguous statement.

Answer 9

I partially agree with Professor West.

cannot - as in "I cannot do " means I am incapable of doing it but too many people conflate "can" with "may"

"Teacher, can I go to the bathroom?" vs "Teacher, may I go to the bathroom?"

The teacher can give you permission, "may" allow you to, but whether you are capable of doing so, is entirely in your domain.

Responses I often got for asking the "Can I ...?" were "I don't know, can you?"

So with that conflation comes the ambiguity of "can not" which is an incorrect substitute for "may not" which still has two connotations. "may not" means you do not have permission or is another conflation of "might not".

"May not be possible" would be better as "might not be possible" which means that it might be or it might not be possible, it can go either way.

So when keeping these conflations sorted, then in proper English "can not" should be the same as "cannot", but since far too many people don't follow the rules (even in academia) we have introduced the ambiguity.

Thus, without the conflation of "can" with "may", then Professor West's statement is false, as then

The logical meaning of "can not" is that it is possible for the statement to fail

(and that it is possible to not fail) is no longer true.

Answer 10

A similar ambiguous phrase might be...

How many dogs and cats are there?

The sets of 'dogs' and 'cats' do not overlap. Nothing is a 'dog' and a 'cat' at the same time. So if we associate 'and' with the intersection, the set of 'dogs and cats' is empty and the answer is always zero. However, we commonly use 'and' to be the union of the sets rather than the intersection.

In computing, this can lead to mistakes. We might count the set of dogs and cats with...

if (animal.isDog()) ++count;
if (animal.isCat()) ++count;

That would count the animal twice if it was both a dog and a cat. If we wanted the count of dogs and cats, we might write...

if (animal.isDog() && animal.isCat()) ++count;

The logical '&&' operator is called 'and', but this will always return zero. I have fallen down that particular hole because I am thinking what I want to write in words.

if (animal.isDog() || animal.isCat()) ++count;

The logical '||' operator is called 'or'. It also turns the correct value even if a dog-cat existed.

Mathematics has good symbols for the union and intersection of sets. If you try to translate those into common speech, you lose precision.

Answer 11

As others have pointed out, cannot is ambiguous. My Random House Webster's College Dictionary (2001) lists cannot as an alternative form of can not; i.e., they are synonyms. But my American Heritage Dictionary of the English Language, 3rd. Ed. (1992) lists cannot as the negative of can. I find the same difference of opinion in various online dictionaries.

Not only is the meaning ambiguous, but the English language has no final arbiter of what is correct and what is not. The best we have is expert opinion. And the experts themselves disagree. It seems to me that Professor West is trying to impose order on chaos. And he has a good point. But he's fighting an uphill battle.

In cases like this, I tend to follow Postel's Law: Be conservative in what you send, but liberal in what you accept.

The law was first expressed in the context of computer networking. When you send data, try to conform to the specification 100%. But design your receivers to try to gracefully handle common errors or deviations from the specification that have been made for practical purposes.

In that vein, after reading this post, I'll try to choose cannot vs. can not according to Professor West's preferred definitions (because it seems like a helpful distinction). But it would be foolhardy for me to expect everyone else to do the same. When reading the works of others, the onus will be on me to try to resolve the ambiguity based on context.

Answer 12

A dictionary does not define words. A good dictionary should catalogue the current use. A good, big dictionary should include all the different usages, including mathematical or scientific definitions where they exist, and cite examples. A small dictionary such as you might use to check spelling or settle Scrabble squabbles is probably made from a bigger dictionary by omitting these details.

Apologies for the second answer, but it is quite separate from my first one.