The golden standard of any piece of scientific research is, in the current value system, complete objectivity. In hard sciences, it goes all the way to the supression of the researcher in the finished article.
Coming across the first person, in a maths article, feels weird. Mathematicians will usually prefer, when needed, to use the passive form ("A is defined as ..."), or when there is no choice, use a general plural ("we"). In languages where it is permitted, indefinite pronouns (like the French "on") will usually be chosen.
The use of the first person is in direct opposition to the sacrosanct objectivity. What the reader wants to find out about is the object of the scientist's research, and he probably feels more comfortable when that object is already detached from any enunciative situation.
However, this approach has many inconvenients - on the research validity side, and on the reader's side. It also brings in a heavy dose of hypocrisy. This is why I am putting a case for first-person science forward.

The advantage of first-person science to the researcher
When producing a piece of academic work, the researcher usually tries to distance himself from his research. Some will refuse to say "I" and will talk of themselves in the third person, in order to put that distance between themselves and their object of research.
This behaviour is based on a confusion between subjectivity and bias, or partiality. I believe it is important, for the research's validity's sake, to avoid bias and prejudice as much as possible. It is always possible to "see" trends where they're not, just because the original research hypothesis was going in that direction - and this should indeed be suppressed.
However, suppressing the researcher himself, goes in the same direction as negating his role and influence in the research (the direction the research is taking, but also his direct influence on the topic). By admitting that he has a role in the research, the researcher can take his influence into account. He becomes, in a way, more accountable - which greatly improves his work's external validity.
In which ways does a researcher influence his work? Well, the obvious issue is that of the questions asked. Research will be made in a certain direction - depending on the researcher's preconceptions. It may seem to be more important in social science than in natural science, however, even the representation that one has of, say, a function (be it a graph, a process, etc.) will have an influence on how research is conducted. Of course, in the case of maths, it will not have any influence on the validity of the result, but it will explain how concepts come to tie in together, and may justify new definitions.
Then, there's the influence on the work itself. Not only which questions are asked, but also their phrasing has an influence, especially in social studies. The effect of the researcher on education has been studied at length - between others, the Hawthorne effect may play a significant role. In quantum physics, researchers have to deal with the influence of observation on the observed phenomena.
Saying "I" allows to take in, and acknowledge all those side-effects of research, improving both internal and external validity of the piece of work.
This is why I believe that researchers in education place randomisation far too high in their esteem. Choosing subjects at random goes in the direction of negating the researcher's role, for very little gain. Indeed, choosing samples completely at random allows for very disproportionate (in terms of ethnicity, gender, achievement, ...) groups to be compared. This is why researchers have come up with the most half-arsed form of randomisation ever: stratified randomisation. The idea is to divide the population into groups, see in which proportion each comes, and then draw at random within those groups accordingly. The problem this poses is that it hides the researcher's choice of criteria, or how far the division into subgroups goes. It does not provide more external validity either, in so far as having a disproportionate group is just as likely as having any other group. The law of large numbers can only be applied if the study is repeated a number of times, on different cases (considered individually), not if you consider one case of a group (treated as such), no matter how randomly people in the group were chosen.


The advantage to the reader
I am placing this discussion within the framework of legitemate peripheral participation (don't go to the wikipedia entry), introduced by Jean Lave. The basic idea is that learning is only one facet of a social process. It happens when the learner can legitimately participate in an activity that is accessible to him (a refined, social-oriented version of Vygotsky's zone of proximal development, as it were).
Now, if research is presented as distanced from any situation of enunciation, and particularly from the researcher, it is far from being addressable by the learner. There is no room for challenging what is being read, and there is no two-way social process. Participation on the learner's part is not legitimate, and learning becomes more difficult.
It also prevents critical thinking on the part of the reader, as - again - there is noone to challenge, and the researcher is not accountable. Finally, the process leading up to the results becomes hidden - preventing any identification to the researcher, or researcher training.
I'd like to thank Dr Candia Morgan (from London's Institute of Education), as her seminar yesterday completed my previous thoughts on the matter.