These days, when you hear "faith versus science" the first thing you probably think is: evolution versus creationism. In Billy Budd, though, the tension between faith and science is not so simple or straightforward as Religion versus Science, or Believers versus Atheists. On some level, you might say that the real tension is not faith versus science so much as faith versus rationality. In the central moral debate of the story, Captain Vere tries to put aside his gut feeling that Billy is innocent; he tries to act rationally, according to the word of the law, in convicting him. If Captain Vere permitted himself to act on faith, then it is possible that Billy never would have been executed.
We'll focus in on one particular scene that more or less captures the flavor of a debate that runs throughout the book. This is the scene in Chapter 26, where the Purser and the Surgeon get into an argument over how to explain the fact that Billy did not twitch after being hanged. The Purser interprets it as "testimony to the force lodged in will power" (26.2). The Surgeon, dryly refuting him, says, "Such movement indicates mechanical spasm in the muscular system. Hence the absence of that is no more attributable to will power, as you call it, than to horsepower – begging your pardon" (26.1). As is typical in the portrayal of such debates, the Purser is portrayed as relatively simple and uneducated in contrast to the worldly and sophisticated philosophizing Surgeon.
Yet what is key is that the Purser admires Billy Budd. He thinks of him as something of a hero. He wants reality to align with his desire to glamorize Billy Budd, to portray him as stronger than the average man. The Surgeon is cut and dried; at least openly, he shares no such desire. Just as he dismisses the idea of Billy's willpower, so he dismisses the entire role of human desire in interpreting and explaining events.
As he says, in his scientific way, "It was phenomenal, Mr. Purser, in the sense that it was an appearance the cause of which is not immediately to be assigned" (26.7). Billy's failure to twitch is not of interest to the Surgeon because it cannot be explained. But doesn't this betray the Surgeon's own desire – his desire that the world operates rationally and that the only things that matter are those that can readily be explained?