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I commonly want to use the phrase "we have" when writing mathematics, to mean something like "most readers will know this thing and I am about to use it". My primary question is whether this is too colloquial. My secondary question is what the alternatives are if it is too colloquial.

For example, right now I have a sentence "Given a point $P\in X$ we have the residue map ${\text {res}}_P \colon \Omega_{K(X)} \rightarrow k$, as defined in ...". I don't feel saying "there exists the" is quite right. Even if it is grammatically correct, I don't think this conveys the implication that it will be almost certainly familiar to the expected audience.

I have seen this question but I feel this is slightly different. If not then my apologies.

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"We have" is fine. "One has" is also fine, if you are at least 50 years old. –  TonyK Feb 8 at 23:00
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@MarcvanLeeuwen "We have 'one has'" –  Oliver Feb 9 at 1:45
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I think 'one has' would be better, 'we' does imply that most readers will know what you are talking about and thus it has the potential to make readers who don't know about it feel alienated or stupid. Also it might possibly sound patronising, like a teacher to a 5-year old: "Now Timothy, we don't eat the crayons now, do we?". –  Pharap Feb 9 at 5:19
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"Now Timothy, one doesn't eat the crayons now, does one?" –  Joel Reyes Noche Feb 9 at 15:20
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"We have" is also perfectly acceptable when the author of the paper is a king or queen. –  Arkamis Feb 9 at 19:07
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11 Answers

up vote 82 down vote accepted

In my opinion it is even good style. You are involving the reader somehow to the discussion, if you write phrases like "we have", "we consider", "we may assume", "one can see that" etc.

By the way, it is common in other languages as well and you can read similar phrases in papers of famous mathematicans all over the world.

Some examples:

Noi sappiamo (...) $\tag{Fubini, 1903}$ (we know that, ... Ref, p.6 )

or

(..) nehmen wir an, dass (..) $\tag{Minkowski, 1900}$ (We assume that ... Ref)

or

However in the reducible case (..) we have to consider (..) $\tag{Wiles, 1999}$ (Ref, p.4)


Furthermore I found a guide by MIT in which is said

Be forthright: write in an unhesitating, straightforward, and friendly style, ridding your language of needless and bewildering formality. Be wary of awkward and inefficient passive constructions. Often the passive voice is used simply to avo id the first person. However, the pronoun “we” is now generally considered acceptable in contexts where it means the author and reader together, or less often, the author with the reader looking on.

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OK, how some examples that aren't more than a hundred years old? ;-) –  vonbrand Feb 8 at 22:06
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@vonbrand you could take almost every actual paper and find this kind of phrases. With my examples I was about to point out that this is even well-established all around the world and for decades now –  user127.0.0.1 Feb 8 at 22:12
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@vonbrand however, I added a newer reference by A.Wiles, who proved i.a. Fermat Last Theorem –  user127.0.0.1 Feb 8 at 22:36
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+1 for other languages. –  qwr Feb 8 at 22:37
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I have a huge objection with using 'we have'. The reason is something you seem to take for granted. To me it is not clear if 'we' means all of the authors and the reader or just the authors. Clearly you assume it is the first case, but I see no reason to suppose it is so and due to this ambiguity I avoid using it. –  Git Gud Feb 9 at 21:25
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It is not too colloquial -- it is used all over the place in papers. In your example, "we have" is preferable (IMO) to "there exists the," which is verbose and a bit ugly.

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Okay, thank you. I think I am at least a bit paranoid due to general bad writing and a pedantic supervisor (as all are). –  Joe Tait Feb 8 at 17:48
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But it does not mean "Most readers know this and I am about to use it" (as suggested in the question). –  Michael Hardy Feb 9 at 4:39
    
@MichaelHardy I meant that it implies this (perhaps quite subtly), not that it says precisely this. –  Joe Tait Feb 9 at 11:07
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(EDIT:) The basic difference is that 'we' suggests two minds, specifically a teacher/student paradigm.

This is perfect for any teaching context.

But academic peers exchanging information will go to absurd lengths to avoid insulting their colleagues by insinuating such a relationship.

The object of maths is understanding.

The object of mathematical writing is communicating that understanding with the utmost clarity.

And at the heart of communication is dialogue. It is one person explaining to another person.

Look at the Greek dialogues (e.g. http://www.sacred-texts.com/cla/plato/laches.htm), and observe how easy it is to understand the material when both parties are voiced, even if the student's part is little more than a device.

This duality is critical; question and answer compliment one another; the question creates a space into which the answer may manifest, the answer provokes further questions, and so on. This creates an interconnected web of understanding.

And if the 'we' is lost, then the dialogue is also lost.

Look at Einstein's famous papers (e.g. http://milesmathis.com/five.html) and notice how you feel he is looking over your shoulder and explaining something to you.

It is unfortunate that academic papers feel the need to be absolutely impersonal, because it makes them dry. I have to read through a lot of them, and the conventional style is a hindrance; you should break every rule in the pursuit of clarity.

Interestingly, as YouTube videos are gaining popularity over written text as teaching resources, we are seeing a resurgence of the Socratic Dialogue paradigm! As a video is typically made in real-time, the presenter doesn't have the luxury of formalising the presentation. This is a good thing!

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I am more and more aware that the point is to communicate, and with this I am more and more aware of the fact that I often do not do this well. I guess that if you stare at anything long enough it doesn't look right... –  Joe Tait Feb 9 at 16:09
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"We have" is just an expression that is meant to involve the reader. It's a little less dry than "there exists"; it's a bit of a colloquialism, and a little friendlier than the passive voice or third person. I think it's a good stylistic choice.

Here's an example to illustrate the point, from a different field entirely:

"Looking at the top-right corner of Van Gogh's Starry Night, there is a bright-yellow moon, a great contrast that draws one's attention."

vs.

"Looking at the top-right corner of Van Gogh's Starry Night, we can see a bright-yellow moon, a great contrast that draws our attention."

I think the latter sounds more pleasant than the former, because it is less stilted and less distant. The case for wanting to use "we can see that" or "we have that", etc. in mathematical writing is similar.

For reasons similar to the above, I sometimes even use expressions like "then our sequence converges" instead of "then the sequence converges", which is perhaps a little more controversial.

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A nice example, thank you :) –  Joe Tait Feb 8 at 18:01
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First person plural does seem a little bit chummy; for the first, how about "...a bright yellow moon draws viewers' attention with its contrast"? –  supercat Feb 8 at 20:54
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Passive voice is generally considered a no-no in writing. –  vonbrand Feb 8 at 22:07
    
@vonbrand: Where is this "passive voice" that you speak of? The only passive voice I see, in this answer and all of its comments, is "is meant". –  TonyK Feb 8 at 22:50
    
@vonbrand: Apart, of course, from your "is generally considered". If that was deliberate on your part, then you are too subtle for the internet :-) –  TonyK Feb 8 at 22:54
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Aside from its effectiveness as a rhetorical tool, as discussed in all the other answers, and argued in the Knuth paper in the question you cite (point 6), there is a much deeper — and to me more compelling — reason for using "we".

Mathematics, like all the sciences (and arguably all human endeavors, when done right), is a collective enterprise: we advance and appreciate and critique its contents together, over centuries. No individual owns it, and no authority controls it.

Given "$P\in X$" we — truly, all of us together — have "the residue map...". No one is excluded from this statement; each of us can confirm it, and has the potential to refute it; many, stretching back to ancient times, have contributed to establishing it; and all of us together will contribute to the elaboration of its implications, for as far as math is true and "we" have descendants.

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That is a much better way of phrasing what I was trying to say, thank you –  Joe Tait Feb 9 at 16:10
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"We have" not only sounds reader friendly and nice, it also seems much more appropriate to me than "there exists", because "existence" is a complicated thing when dealing with theoretical entities. We sure have numbers. But do numbers "exist"?

But I must confess, I'm a philosopher, no mathematician.

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We don't have numbers. Numbers have us. 8^) –  Zane Feb 11 at 11:17
    
I can't help but feel the most common understanding of 'existence of mathematical entities', at least for the mathematicians for whom the paper is written, will be in the sense that axioms have implied the existence of the object, as opposed to whatever wishy washy definition that philosophers choose to attribute to the term. –  Daniel Rust Feb 11 at 19:06
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I would replace " we have" by ", then" or just ", "

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+1. Why say "given $P \in X$, we have the residue map, as defined in ..." when you could just say "given $P\in X$, the residue map is defined as..."? –  sjy Feb 10 at 5:38
    
Well, if I think that anyone reading the paper should already know something then I am not going to "is defined as...". Obviously it would be nice to explain everything, but this is unrealistic if you want to have a paper that is of a readable length. –  Joe Tait Feb 10 at 8:58
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In all disciplines, there are rules. I think there's an unwritten rule about precision:
An impersonal and precise way of writing is important if we want to seem rigorous.

Unless you are communicating in a casual context, divulging informally, I would always use an impersonal speech and writing style. But that's only my opinion.

"I have" or "we have" should be replaced by "there is" or "there exists" in a formal mathematical context.

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"Impersonal and precise" is the realm of automated proofs (yes, there are journals publishing such; some 10 times longer than human-oriented proofs and totally unreadable for non-initiated). When you write mathematics, you want the reader to understand, and that (if you want it or not) has a large non-rigurous, psychological component. We like to consider ourselves "rational animals," when the thruth is that we aren't. –  vonbrand Feb 9 at 14:48
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Most style guides recommend the active voice over the passive voice, and most readers prefer it. But in some academic fields, especially the sciences, authors use a stilted and awkward style that replaces clear concise sentences like, "We performed the experiment," with circumlocutions like "The experiment was performed." Asked why they write like that, many scientists admit that they don't like it, but they are under the impression that journals require it. They are wrong. Of the journals that have style guides, the vast majority explictly ask authors to write in the ACTIVE VOICE. –  Sjoerd C. de Vries Feb 9 at 16:02
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Maybe I should have a look at new academic style guides. Is the Chicago manual of style the one you most recommend? What about european publications? –  Stella Feb 9 at 16:14
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This is irony, right? You are using the construction yourself "if we want to seem rigorous" –  Zane Feb 11 at 11:19
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interesting meta at "An impersonal and precise way of writing is important if we want to seem rigorous." [italics mine] Either the author is unaware or fiendishly clever!

I think "we have" and so on is already an established, rigorous math norm. Also, check out the Russian "imejem" which is the "have" of "we have", without "we". What is the Greek word?

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It's a matter of standard practices. Euclid didn't say "we have" in making a conclusion, but he did say "I say that" to declare a claim just before proving it.

Personally, I'd like to see technical papers written in a less formal style. We dropped the subjunctive and future tense some time ago. We no longer write "If $x$ be a real number, then $x^2$ will be nonnegative." Language marches on, and it makes sense to keep with the times.

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Agree with all others about the value of friendliness in math education when it doesn't reduce precision. But in this particular case you could write a little more directly:

"Given a point P∈X we can define a residue map resP:ΩK(X)→k as..."

Still has the "we" concept, just ditch the verb "to have" and go straight to "define".

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I was assuming the definition, and just referencing where someone can look it up in case. However, there certainly are ways that I could restructure the whole thing to varying degrees. Indeed, I was once told that if you have to change a sentence, you should start the entire paragraph again, and that would certainly lead to different phrasing –  Joe Tait Feb 9 at 17:46
    
Ah, I didn't quite grasp the larger context. To be honest, I grew up hearing teachers use the "we have" preface to a lot of facts and coming at it from the 'less is more' ethic is a bit of a rebellion on my part –  Dee Dub Feb 9 at 21:23
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