Mathematical writing: why should we not use the phrase "we have that"?

6,651

Solution 1

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.

Solution 2

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!)

Solution 3

"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.

Solution 4

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.

Solution 5

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.

Share:
6,651

Related videos on Youtube

angryavian
Author by

angryavian

Updated on July 21, 2020

Comments

  • angryavian
    angryavian over 3 years

    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?

    • Andreas Blass
      Andreas Blass over 10 years
      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.
    • Zev Chonoles
      Zev Chonoles over 10 years
      @Andreas: As another long-time user of this phrase, I was just about to comment the same thing :-)
    • Cameron Williams
      Cameron Williams over 10 years
      @AndreasBlass I just wrote a paper and used the phrase "we have that" a lot more than I care to admit (though to be fair, it's a lengthy paper). I agree with your sentiment. Haha.
    • Lubin
      Lubin over 10 years
      @ZevChonoles, listen to your elders, and see my answer.
    • amWhy
      amWhy over 10 years
      I just landed on $60$ million hits by googling ["we have that", mathematical exposition]! It's use is quite widespread.
    • Julien
      Julien over 10 years
      FWIW, "on a que $x=y$" sounds like very awkward French and (hopefully) most people would say "on a $x=y$". So I would have never thought of saying "we have that $x=y$". Fun question. What about Spanish, @Peter Tamaroff?
    • Pedro
      Pedro over 10 years
      @julien What are you referring to?
    • Julien
      Julien over 10 years
      @PeterTamaroff If you literally translate "we have that $x=y$" and "we have $x=y$" in Spanish, do both sound ok?
    • Pedro
      Pedro over 10 years
      @julien Well, one is "tenemos que $x=y$" while the other is "tenemos $x=y$". They sound OK, yes.
    • angryavian
      angryavian over 10 years
      @cobaltduck I found this
    • Theodore Norvell
      Theodore Norvell almost 10 years
      Knuth is guilty of changing two variables at once, the verb and the object. To me "We have A is a group" sounds as wrong as "Assume A is a group". On the other hand, "Assume A=B" doesn't sound all that bad to me. Conclusion: it's more the nature of the object than the verb that determines whether "that" is helpful in parsing the sentence.
    • Steve Jessop
      Steve Jessop almost 10 years
      I think "we have that" is in effect a contraction of "we have the fact that". The former may be grammatically right or wrong. The latter is grammatically correct but ugly.
  • Brian M. Scott
    Brian M. Scott over 10 years
    This is a matter of idiolect. To many people We have that sounds no less natural than the other three.
  • daniel
    daniel over 10 years
    This point seems to be the author's own invention. We have that it need not be followed.
  • Brian M. Scott
    Brian M. Scott over 10 years
    @daniel: It’s not Knuth’s invention, though I don’t think that I’ve seen anyone else make such a point of it.
  • dfeuer
    dfeuer over 10 years
    I think we can call this a valid application of argumentum ad auctoritatem.
  • Mariano Suárez-Álvarez
    Mariano Suárez-Álvarez over 10 years
    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.
  • Brian M. Scott
    Brian M. Scott over 10 years
    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.)
  • Lubin
    Lubin over 10 years
    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.
  • Lubin
    Lubin over 10 years
    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.
  • ShreevatsaR
    ShreevatsaR over 10 years
    @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.)
  • ShreevatsaR
    ShreevatsaR over 10 years
    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. :-)
  • ShreevatsaR
    ShreevatsaR over 10 years
    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.
  • Tara B
    Tara B over 10 years
    @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 $B$".
  • Andreas Blass
    Andreas Blass over 10 years
    I agree with your second paragraph up to and including "Maybe a few other kinds of verbs." I believe the permitted verbs include "know" (as in one of the examples you mentioned) and "infer". The mathematical meaning of "we have" is ordinarily "we know" or "we infer" (or perhaps "we assume"). So I'll probably continue writing "we have that" without fear that it's wrong, but probably with a new awareness that I usually have the option of being more specific (and avoiding conflict with you and Knuth) by using other verbs like "know" and "infer" instead of "have.
  • Andreas Blass
    Andreas Blass over 10 years
    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.
  • Michael Greinecker
    Michael Greinecker over 10 years
    -1 This is merely a declaration of personal preferences.
  • Michael Greinecker
    Michael Greinecker over 10 years
    How about "It follows that..."? "Follows" is as unrelated to sensing, thinking, or saying as "have".
  • Lubin
    Lubin over 10 years
    Ah, @MichaelGreinecker, but what is the subject of the verb “follows”? In the formal grammar that I learned, the it is an “expletive” (look it up) and stands for the true subject of the sentence, which is the noun clause beginning with the word “that”.
  • Michael Greinecker
    Michael Greinecker over 10 years
    So can I write "It follows that A=B."? I'm not a native speaker.
  • Lubin
    Lubin over 10 years
    You can, with my blessing and officious approval.
  • daniel
    daniel over 10 years
    Ellipsis of this sort explains a lot about English. Rules are not precisely broken but the temptation to abbreviate is strong. +1.
  • John Bentin
    John Bentin over 10 years
    @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 $B$".
  • John Bentin
    John Bentin over 10 years
    @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 $B$"
  • Matt
    Matt about 9 years
    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).
  • Matt
    Matt about 9 years
    It is not wrong, wrong, wrong. Your argument (par 2) is that there is a semantic problem; a verb of sensing, thinking, or saying is required. But my previous sentence shows that you need to add verbs of being, and this sentence shows you need to add verbs of demonstrating -- where will it end? What's wrong with verbs of having? You claim "sensitive native speakers" just don't say it. But they do, if the situation is right. Imagine a movie where cops investigating a crime are discussing it: "What do we have?" "We have that he wanted her inheritance, and that she trusted him." Not at all wrong.
  • Matt
    Matt about 9 years
    @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.
  • Lubin
    Lubin about 9 years
    We’ll just have to disagree on this.
  • Admin
    Admin over 5 years
    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.
  • paul garrett
    paul garrett over 5 years
    @ChristianRemling, but/and we also have an inclination to try to reduce the number of words? :)
  • amWhy
    amWhy almost 2 years
    -1 @MichaelGreinecker your comments are mere opinion, across this post.
  • Michael Greinecker
    Michael Greinecker almost 2 years
    @amWhy I'm afraid you cannot vote on comments, even if they are 9 years old.
  • amWhy
    amWhy almost 2 years
    Dear @Michael, I only returned to it after an update today. Of course I can't downvote a comment, but I did so figuratively to reflect that you voiced an opinion only, while a moderator, and downvoted, that it was your opinion to accuse me of posting an opinion. In any case, I hope you are enjoying the non-moderating life! But I do thank you for your service, and, I wish you the best, and I sincerely mean that!! :-)
  • Michael Greinecker
    Michael Greinecker almost 2 years
    @amWhy I do enjoy the post-modding life. I was just a bit flabbergasted by the timing. I remember you as already outspoken back then.
  • amWhy
    amWhy almost 2 years
    Hah! Yes, indeed, @Michael! Glad you're still still enjoying these parts! :-)