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?

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    $\begingroup$ I hope not to see any really good answers to this question, because I'm fairly sure I've written "we have that" lots of times. $\endgroup$ – Andreas Blass Jul 19 '13 at 22:31
  • $\begingroup$ @Andreas: As another long-time user of this phrase, I was just about to comment the same thing :-) $\endgroup$ – Zev Chonoles Jul 19 '13 at 22:32
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    $\begingroup$ @ZevChonoles, listen to your elders, and see my answer. $\endgroup$ – Lubin Jul 19 '13 at 23:09
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    $\begingroup$ I just landed on $60$ million hits by googling ["we have that", mathematical exposition]! It's use is quite widespread. $\endgroup$ – Namaste Jul 19 '13 at 23:57
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    $\begingroup$ @cobaltduck I found this $\endgroup$ – angryavian Jul 24 '13 at 11:22

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.

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    $\begingroup$ This depends on how you read the $=$. «We have that $A$ equals $B$» is perfectly good. On the other hand, if «$A=B$» is really intended to stand for «the proposition "$A$ is equal to $B$"», then «we have $A=B$» is short for «we have the proposition $A=B$», and that is correct too. What some people object to is reading relations as verbs —and that, in my opinion, is just a matter of style. $\endgroup$ – Mariano Suárez-Álvarez Jul 19 '13 at 23:23
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    $\begingroup$ You can’t justify We have $A=B$ in that way unless you’re willing to justify We have that $A=B$ as a short form of We have the assertion that $A=B$, and I rather doubt that you want to do that! (My speech is quite conservative, and I do in fact prefer the that-less version, if not nearly so strongly as you. The fact remains that both are firmly established in mathematical writing, and for a referee to object to we have that is very nearly as inappropriate as for a referee to insist that If x is non-empty really ought to be If x be non-empty.) $\endgroup$ – Brian M. Scott Jul 19 '13 at 23:29
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    $\begingroup$ Well, @BrianM.Scott, we’ll just have to meet behind the bar and duke it out. The mathematical community may be lucky that I’m not asked to referee more often. $\endgroup$ – Lubin Jul 20 '13 at 2:34
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    $\begingroup$ I agree with most of your answer, but the first sentence seems backwards: given that "we have that" occurs only in mathematical usage and not in English usage, and given that as far as the meaning perceived by mathematicians goes there is no difference between "We have P" and "We have that P", I'd say it's precisely a matter of English usage rather than mathematical usage (to which it's irrelevant). But perhaps you used "matter" in some other meaning. $\endgroup$ – ShreevatsaR Jul 20 '13 at 7:48
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    $\begingroup$ You can, with my blessing and officious approval. $\endgroup$ – Lubin Jul 20 '13 at 21:56

"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.

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    $\begingroup$ This is a matter of idiolect. To many people We have that sounds no less natural than the other three. $\endgroup$ – Brian M. Scott Jul 19 '13 at 22:35
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    $\begingroup$ This point seems to be the author's own invention. We have that it need not be followed. $\endgroup$ – daniel Jul 19 '13 at 22:45
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    $\begingroup$ @daniel: It’s not Knuth’s invention, though I don’t think that I’ve seen anyone else make such a point of it. $\endgroup$ – Brian M. Scott Jul 19 '13 at 22:53
  • $\begingroup$ I think we can call this a valid application of argumentum ad auctoritatem. $\endgroup$ – dfeuer Jul 19 '13 at 22:55
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    $\begingroup$ @daniel: Your "We have that it need not be followed" is a good example of a sentence that sounds unnatural in English. Outside of mathematics, "we have that" is rather non-standard. (It is a separate question whether "we have that" is part of the mathematical idiolect and therefore justified, or whether we should try to minimize instances of the mathematical language diverging from the standard one.) $\endgroup$ – ShreevatsaR Jul 20 '13 at 7:42

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!)

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    $\begingroup$ -1 This is merely a declaration of personal preferences. $\endgroup$ – Michael Greinecker Jul 20 '13 at 20:41
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    $\begingroup$ @Michael: No, not merely: It is a claim that there is no reason to think that "we have that" is so bad, and that Knuth went overboard in stating his personal preference as an absolute rule for everybody to follow. So it is an answer to the question. Outside of mathematics, a negative (such as "there is no reason") cannot be proven, so amWhy simply presented evidence: "I find it quite natural to say, read, and hear it." This appears to be evidence significantly beyond personal preference, indicating amWhy's impression of relevant community norms. $\endgroup$ – Matt Aug 23 '14 at 12:16

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.

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    $\begingroup$ Ellipsis of this sort explains a lot about English. Rules are not precisely broken but the temptation to abbreviate is strong. +1. $\endgroup$ – daniel Jul 21 '13 at 0:59

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.

  • $\begingroup$ Hmmm. I've always thought we mathematicians take pride in being the last of a dying breed of people who have a long enough attention span to read and understand a sentence that contains more than three words. $\endgroup$ – user138530 Jul 12 '18 at 1:55
  • $\begingroup$ @ChristianRemling, but/and we also have an inclination to try to reduce the number of words? :) $\endgroup$ – paul garrett Jul 12 '18 at 2:12

"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.

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    $\begingroup$ I personally think of $=$ as standing for some form of the verb to equal: so by itself, "$A=B$" is read as "$A$ equals $B$", while "Let $A=B$" is read as "Let $A$ equal ‌$B$", and "We have $A=B$" is read as "We have $A$ equalling $B$". But I'm sure that's just me. :-) $\endgroup$ – ShreevatsaR Jul 20 '13 at 7:45
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    $\begingroup$ @ShreevatsaR: It's not just you, although I phrase things slightly differently, for example "We have $A=B$" I would read as "We have $A$ equal to $B$". $\endgroup$ – Tara B Jul 20 '13 at 9:41
  • $\begingroup$ @ShreevatsaR: I chose the form "the equality of $A$ and $B$" so that its noun-phrase status was obvious from the outset, but I think that it is grammatically and semantically (if not stylistically) pretty much equivalent to "$A$ equalling $B$". $\endgroup$ – John Bentin Jul 21 '13 at 19:20
  • $\begingroup$ @TaraB: "We have $A$ equal to $B$" is good English, of course, but idiomatic: It is really an abbreviation of "We have $A$ being equal to $B$". This can then be parsed into "We have" and "$A$ being equal to $B$", the latter phrase being equivalent to "$A$ equalling $B$" $\endgroup$ – John Bentin Jul 21 '13 at 19:42

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 $B$."

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    $\begingroup$ Sorry, but this construction is not any kind of relative clause. In any relative clause introduced by “that”, the word “which” (or “who”) may be substituted. $\endgroup$ – Lubin Jul 20 '13 at 2:36
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    $\begingroup$ If I were referring to the sentence "$A=B$" as a syntactic entity (which might actually happen since I'm a logician), then I would certainly omit "that". Most of the time, though, I would be referring not to the sentence itself but to the fact that it expresses, and the use of "that" makes the reference clearer. In most mathematical contexts, especially when the subject is not logic, there is no danger that "We have $A=B$" will be understood as referring to syntax rather than semantics, and then the omission of "that" is a harmless abbreviation. $\endgroup$ – Andreas Blass Jul 20 '13 at 17:15
  • $\begingroup$ I like that we are allowed to add comments like this one. Would you say the previous sentence is "simply not English"? There is an interesting grammatical fact that a sentence can be turned into a noun phrase with "that", as you can see in the appositive in this self-referential sentence. It has nothing to do with what the verb is, although semantically the verb needs to make sense with abstract objects, since the "that <sentence>" construction changes the meaning to be more of a Boolean (a testable statement, a claim). $\endgroup$ – Matt Aug 23 '14 at 11:29

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