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In Knuth's Mathematical Writing, he writes on page 2, at number 8:
Don't omit "that" when it helps the reader to parse the sentence.
Bad: Assume $A$ is a group.
Good: Assume that $A$ is a group.
The words “assume” and “suppose” should usually be followed by “that” unless another “that” appears nearby. But never say "We have that $x=y$," say "We have $x=y$."
The last bit seems to contradict the previous sentences; is it just an exception? What makes "we have that" so bad?
This is absolutely not a matter of English usage, but a matter of mathematical usage. I once said to a colleague of mine in the Anthropology Department that mathematicians often say, and write, “We have that $A=B$”, and his mouth dropped wide open. Now I have to admit that the Great and Blameless Serge Lang often wrote the offending words, but after all he was not a native speaker of English, and I think it’s wrong, wrong, wrong. Here’s my analysis of what’s going on:
The conjunction that is used to introduce a dependent noun clause of indirect statement: “I know that my Redeemer liveth”, “I’ve heard that you were sick.” My firm conviction is that such a clause can be introduced by, can be the object of, only verbs of sensing, thinking, or saying. Maybe a few other kinds of verbs. But it can’t be the object of the verb “have”. Syntactically it’s fine, “have” takes a noun as object, the clause of indirect statement is a noun clause, it all fits. It’s just that sensitive native speakers of English don’t complete that verb with that kind of clause.
And I agree with Knuth, that it’s perfectly all right to say “We have $A=B$.” Here the equation as written stands as a thing-in-itself, maybe you can explain the construction as a short form of “We have the relation $A=B$.” So the object of “have” is either the whole equation “$A=B$” or the noun “relation”, with the equation standing in apposition to the noun.
One more remark, before I get off my high horse: When refereeing a paper, I always say that I strongly urge the author to change the offending construction, but I do not insist. The usage is too well established to try singlehandedly to change things. But it’s still wrong, wrong, wrong.
Here endeth the sermon.
"We have that" is just poor use of language. "We know that" or "We see that" or "We have assumed that" all sound much more natural.
Edit: I won't claim never to have used the construction, but I don't like reading it and I'd take pains to avoid it in a formal setting.
I regard the form “We have that X” as an elliptical form of “Thus, we have established that X...” or “Thus, we have proved X”. In other words, it emphasizes that it has been shown that X holds. “We have X” may say the same thing, but does not spell out if X has been proven or merely supposed.
Because the form sounds and appears slightly clumsy, Knuth is within his rights to inveigh against it. On the other hand, the problem does not seem so serious that people should lose sleep over it. It is a style issue moreso than a grammar issue.
Personally, I'm all for parsing sentences, and agree with the first remark made by Knuth. But I disagree with Knuth regarding his caution/exception. I think there are times when using the phrase "we have that" can be used to help parse sentences and enhance the clarity of an argument or proof.
So it seems to me that Knuth is voicing his preference regarding the phrase "we have that", and that is all well and good. But I think he went a little "overboard" in warning readers to never say it. There is nothing very wrong with saying "we have that $x = y$" or "we have that $x \gt y$". I find it quite natural to say, read, and hear it. I would prefer that Knuth save his vehement objection to the phrase to address more mathematically significant issues.
(Admittedly, I am writing this, in part, in defense of my rather widespread use of the phrase!)
I'd tend to second @Lubin's remarks, but/and also that "We have x=y". is shorter than "We have that x=y", whatever grammatical opinions one has. Shorter is better, all other things the same.
As other answers and comments have remarked, there are many entrenched habits in English-language mathematics that are not functional, whatever their provenance or justification. Verbosity doesn't help anything. Non-functional phrases are not purely innocent, since they consume a bit of time to scan, parse, and dismiss. A needless or functionless "that" is a small thing, but is a relative of much worse noisy, pointless phrases: "in a certain sense" (well, duh), for example. Or the propensity of mathematicians not to think a sentence through well-enough to be able to organize it without self-interruptions by comma-separated prepositional phrases. Not even as good as Faulkner. Better to aim at Hemingway's style?
I'd claim that we mathematicians have become too willing to claim that we have special dispensations about use of English or other natural languages, leading to corruptions that are at best pointless and at worst counter-productive.
"We have $A$ equals $B$" is wrong. (The sentence has two competing verbs: "have" and "equals".)
"We have the equality of $A$ and $B$" is fine.
The word "that" is needed in the first sentence to convert "$A$ equals $B$" into a noun clause, while its intrusion into the second sentence would be quite wrong. Thus "we have that $A=B$" is right or wrong depending on how you read "$A=B$". It seems that Knuth adopts a prescriptive position over English but a liberal one over mathematics, since he surely interprets "$A=B$" as "$A$ equals $B$" in other contexts.
The concern is probably that omitting the complementizer that (giving rise to a "reduced relative clause") will create some ambiguity, or at least a seeming ambiguity which is resolved toward the end of the sentence.
According to the reduced relative clause Wikipedia page:
"Reduced relative clauses often give rise to ambiguity or garden path effects, and have been a common topic of psycholinguistic study, especially in the field of sentence processing."
It's not easy to think of an example relevant to mathematics, though.
"We have that S" is simply not English. The verb "have" does not take a relative clause as an object. Such usage may be a word for word transliteration from another language.
"We have $A = B$" is not a case of "$A = B$" corresponding to the relative clause "$A$ equals $B$". Here, possibly, the $A = B$ serves as quoted sentence, which behaves as a noun, as in "we have (before us the equality) '$A = B$'".
This is very similar to, for instance, an utterance like "Albert Einstein was famous for $E = mc^2$" (that is, "Albert Einstein was famous for 'E equals em cee squared'") where the formula is a quoted item that serves in a noun-like way. Another example: "$\frac{GMm}{r^2}$ comes from Newton." Here, the formula doesn't even have a verb in it that would make it eligible to be a clause.
Another reading of "we have $A = B$" might be "we have $A$, (which is) equal to $B$."