Using "we have" in maths papers 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.
 A: 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.

A: "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.
A: I would replace " we have" by ", then" or just ", " 
A: Another characterization of the use of "we" can be found in the introduction to  William Bloch's The Unimaginable Mathematics of Borges' Library of Babel:

This should not be construed as a “royal we.” It has been a construct of the community of mathematicians for centuries and it traditionally signifies two ideas: that “we” are all in consultation with each other through space and time, making use of each other’s insights and ideas to advance the ongoing human project of mathematics, and that “we”—the author and reader—are together following the sequences of logical ideas that lead to inexorable, and sometimes poetic, conclusions.

A: 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? 
A: 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.
A: 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.
A: (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 complement 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!
A: 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.
A: "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. 
A: 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.
A: 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".
