Should one automatically object to the claim of resurrection as an answer to the origins of Christianity before any data is presented, simply because it would suppose a supernatural claim?" I can't imagine why today, of all days, he would think of this question (that was sarcasm, BTW). Anyways, after some discussion on his wall, I thought I'd blog a longer answer to those questions, framed as an implementation of Bernard Lonergan's thought about specialized knowledge.
A major aspect of Lonergan's thought is the need to cultivate specialized knowledge. His argument rests in part upon the recognition that methods must be commensurate with the questions they seek to answer. Historical questions must be answered by historical method. Metaphysical questions must be answered by metaphysical method. The difficulty with excluding the possibility of resurrection by excluding the supernatural a priori is that it potentially uses metaphysical method to answer a historical question. Now, one can in principle use the fruits of methodological investigation as data in historical investigation. If the supernatural can be excluded on metaphysical grounds, then that can serve as a datum in historical study. With that datum, we now know that any supernatural explanation is a non-starter. But the crucial point is to note that we've arrived there not via historical method but rather through metaphysical. It is not a historical conclusion, but rather a metaphysical conclusion arrived at prior to any historical investigation, and thus taken as a supposition for such work. This then answers Matt's questions: insofar as investigations have taken place antecedent to the work of historical investigation, and insofar as the fruits of those investigations reasonably exclude the supernatural, then it is legitimate to adopt that as a starting supposition in the work of doing history.
Practically, this brings us to the matter of expertise. Another part of why Lonergan encourages the cultivation of specialized knowledge is that no one can master everything that could be possibly mastered. If one is diligent, one can become an expert in a broad area of study, whether it is history, theology, physiology, etc. Within these broad areas however, one must further specialize. I would claim expertise in early Christian history, but not in the history of Reformation England, and even within early Christian history I would claim greater expertise in the first century than, say, the fourth. And this has important consequences for the matter under discussion. Historians do not tend to have expertise in metaphysics or, we might add, physiology. The time that they spent becoming historians precluded the development of such expertise. Conversely, experts in metaphysics or physiology tend not to have expertise in history, for much the same reason. What this means is that any investigation of, for instance, the physiological possibilities that could have followed from Jesus' crucifixion requires inter-disciplinary collaboration. The historian will tend not to have the expertise to pronounce on it alone, while the physiologist will tend not to have the expertise to get into the nitty-gritty details of what we know about Jesus' crucifixion and post-mortem appearances.
The above can be demonstrated by reference to the swoon theory, which is really a covering term for a range of hypotheses that hold that Jesus did not die on the cross but rather merely appeared to do so. He then, so goes the theory, regained consciousness post-burial and left the tomb. There are certain details of the crucifixion and resurrection narratives that could be taken to support this. The narratives emphasize that Jesus was only on the cross for a few hours, whereas crucifixions could ordinarily take days; we thus might think that he was taken off prematurely. There is the report that his legs were not broken, which could account for why he is described as walking in the days after his revival. There is the report that he still had visible wounds after his resurrection, which indicates that there was an awareness that the resurrection did not entail much healing: exactly as we'd probably expect from someone taken prematurely off a cross just a few days earlier. It could potentially account for why those who knew him had difficulty recognizing him after the crucifixion: no doubt he would have been quite marred by the ordeal. In fact, it could potentially explain virtually every detail of the narratives, apart from how the stone got rolled away, but for that one need merely posit that there were visitors to the tomb prior to those recorded in the gospels. But before a historian could affirm that this is what happened, she or he would need to consult with those with the requisite physiological knowledge, and inquire into the likelihood that Jesus could have been mistaken for dead on the cross, that he could have been capable of walking around a few days later, etc. The typical historian lacks this specialized knowledge.
Ultimately, it's not clear to me that the question of whether Jesus actually rose from the dead is really a question within the historian's ambit. The historian's task is not simply to establish what happened in the past, but to inquire into human operations in the past. An example that I often use is Hurricane Katrina. A meteorologist is keenly interested in how hurricanes work: what drives their destructive force, how do they emerge, etc. These are of very limited interest to the historian qua historian. The historian rather is interested in how the federal government responded, what might account for failures or successes in its response, etc. That's what the historian does. Likewise, the historian is interested in the question of whether Jesus was crucified. She or he is interested in the year in which said crucifixion took place (30 and 33 C.E. being the leading candidates). She or he is interested in the fact that the belief that he had risen from the dead took hold among his followers and had demonstrable consequences. She or he is interested in these consequences. She or he is less interested in the physiological or divine processes that generated that belief, just as the historian studying Hurricane Katrina has limited interests in how hurricanes operate.