This is a really useful paper – I read it a while back, but realised I hadn’t put it up here just yet. It’s majorly influenced my methodological approach. I met someone at a conference who’d worked with this guy.
Section 1 justifies the existence of a sociology of science, not just of scientists.
The most influential book in changing the sociology of science was Thomas KUHN’s (1962/1996) The Structure of Scientific Revolutions, in which KUHN challenged the traditional image of (natural) science as being continuously cumulative or straightforwardly progressive.2) This image, KUHN claimed, was based on a distorted reading of the history of science, which acknowledged past achievements only in relation to present concerns (i.e., earlier scientific theories were valued only according to the way they contributed to current scientific theories).
BLOOR’s (1976, p.1) “strong programme”:
“Can the sociology of knowledge investigate and explain the very content and nature of scientific knowledge? Many sociologists believe that it cannot. They say that knowledge as such, as distinct from the circumstances surrounding its production, is beyond their grasp. They voluntarily limit the scope of their own enquiries. I shall argue that this is a betrayal of their disciplinary standpoint. All knowledge, whether it be in the empirical sciences or even in mathematics, should be treated, through and through, as material for investigation.” 
This stuff might be useful for writing about Kant.
A variety of researchers conducted observational studies of the day-to-day work practices of laboratory scientists. These studies are often referred to as “laboratory studies” (e.g., LATOUR & WOOLGAR, 1979; KNORR-CETINA, 1981; LYNCH, 1985) and frequently contrasted idealized portrayals of science (as, e.g., operating through the hypothetico-deductive method) with how science actually gets done in the laboratory.
2. What About Mathematics?
“Although mathematics was central to David Bloor’s early discussion of the ‘strong programme of the sociology of knowledge’, most subsequent strong-programme work has been on the natural sciences. The deductive proofs at the centre of mathematics and logic have seldom been addressed in any detail.” (MACKENZIE, 1999, p.7) 
Here’s a useful thing: “Since it is a purely contingent matter that we have adopted the decimal system, “2 + 2 = 4″ can be said to be a social fact”
He references the low visibility of mathematical work, and goes on to assess some solutions:
LIVINGSTON’s “demonstrative sociology”(1986; 1987, pp.86-141; 1999; 2006) – he approaches mathematics from the analytic perspective of ethnomethodology, and so learns by doing, and inviting others to do, mathematics. He references the practical, social and retrospective-prospective nature of proving.
LIVINGSTON is able to dispel misconceptions about mathematics, in particular, the belief that mathematics is a simple application of logic and therefore a form of “cold” deductive reasoning, where each next step is dictated by its predecessor. LIVINGSTON’s demonstrations reveal that a lot of practical, situated work is involved in figuring out how to move from one step to the next
However, he’s limited by focusing on mathematics that an outsider can understand.
2. MERZ and KNORR-CETINA’s “e-mail ethnography”7)
MERZ and KNORR-CETINA adapted the anthropological approach to laboratory science for their study of theoretical physics:
“It also has to be admitted that the laboratory approach had to be adapted to the obdurateness of the field: the study is based rather less on the observation of physicists’ activities than on one analyst’s capability to exploit her physics training and interact with participants as a member of their culture. It is also anchored in a close ‘reading’ of physicists’ personal-professional communications (their e-mail correspondence […]), their calculation protocols, and their explanations to us, which invariably involved paper and pencil. The close ‘reading’ was adopted to gain access to the ethnomethods implicated in doing theoretical physics work.” (MERZ & KNORR-CETINA, 1997, p.74) 
MERZ and KNORR-CETINA do not take the e-mail exchange as their data (in order to analyze it as though it were the phenomena), but rather use it as a conduit for getting the scientists to reflect on, in more vernacular terms, what was going on. The danger of this strategy is that the resulting metaphorical descriptions (e.g., theoretical physics as “deconstruction”) might suggest that this practice is, “really,” not very different to other practices (e.g., literary criticism).
Unlike Livingston, they do engage with current, high-level work, but therefore encounter the problems of mediated translation.
3. Greiffenhagen then talks about his own work; his differs from mine significantly in one aspect:
As for LIVINGSTON and MERZ and KNORR-CETINA, my own mathematical competence (I studied mathematics for four years) is a crucial prerequisite for this kind of research, trying to follow GARFINKEL’s (2002, p.212) recommendation that as a work maxim the analyst should be competent in the practice that he or she is investigating.
He focuses on graduate lectures and PhD supervisor meetings, giving a good justification.
Like LIVINGSTON, my work has been informed by ethnomethodology (GARFINKEL, 1967, 2002; LYNCH, 1993) and conversation analysis (SACKS, 1992; SCHEGLOFF, 2007), which aim to exhibit the endogenous orderliness of activities.
He goes through some observations and conclusions – more on that later.