Why do single-author math papers use "we" instead of "I"? Even when math papers are authored by one person, you will see stuff like "We now prove so and so" or "In the following section, we show that...". 
How come papers with one author use "we" instead of "I"? 
 A: Zev's answer is excellent, so let me just summarize it: In mathematical papers "we" is used instead of "I" by convention. There may have been a reason for this convention in the past, but by now it's just a convention, and nothing beyond that.
Some of the other answers offer some explanations to this convention, but these are "folk-explanations"; the real explanation is that it's a convention. Given the convention, using "we" or "I" can have certain connotations, but these arise out of usage rather than the other way around.
A different (and interesting) question is when and why did the convention arise, a question to which unfortunately I do not know the answer.

Let me add another example: the use of gendered personal pronouns. In some mathematical prose there are actors participating in a game, and these are usually anthropomorphized. A few decades ago, other than in isolated examples (such as "battle of the sexes" in game theory), these people would invariably be males. Nowadays, it is becoming common to use persons of both genders (the prototypical example being Alice and Bob), and even uses the female gender as default. 
At this point this is a conscious decision on the part of authors of mathematical papers, influenced by feminism. But in the future these rules could become standard practice, and at that point they would be a mere convention. People might come up with folk explanations which may be on-mark (we use feminine pronouns to represent all the populations) or slightly off-mark (Alice is named after Alice in Wonderland, and a generic person is considered Alice since this is the commonly used name starting with A).
The folk explanations for using "we" might similarly be true or false from a historical perspective, but they are mostly based on popular perception, though there is always the possibility of a line of mathematicians which have transmitted the reasoning of the earlier debate on "I" vs. "we" to present-day practitioners.
A: Here is Steven Krantz's answer in A Primer of Mathematical Writing:



A: Sometimes they include you in their "journey" through the mathematics and they use "we". I guess others, including me, just use "we" out of habit.
A: I used to write we instead of I to match with the standard tone in any academic document, but I now I do it for a different reason.
Everything I accomplish in my field is based on the work of others.  In some sense, I would certainly have trouble doing calculus without Isaac Newton, but more to the point I wouldn't be able to test my hypotheses without the contributions from the active, living members of my research community.  Even when I physically do something myself on paper or in the lab, it is always in the context of the people I work with, researchers I succeeded, or contacts from related fields that I asked advice of.
