Amulets and health
To a modern reader of Shabbat 53b, the Rabbis are dealing with a problem of correlation: Do amulets protect the health of animals? If amulets truly correlate with health, then they may be carried on Shabbat. If there is merely a spurious correlation between amulets and health, then they do not work and carrying them is (or ought to be) a form of prohibited sabbatical work.
Lacking biochemical (etc.) knowledge of sickness, the Rabbis judged the efficacy of amulets by two critera (bShab 61): medical efficacy and expertise. First, amulets are deemed effective if they cure or prevent sickness three times (3x); Second, healers are deemed experts if their amulets cure three times (3x). Wisely, the rabbis did not assume that amulets that cured humans would necessarily work with animals.* For those of us scientifically inclined, we would not expect these criteria to adequately identify effective cures. Fortunately, Jewish law here is judging the amulets only in terms of Shabbat. So the downside to an error is rather limited (esp for a shogeg). In Quicksilver כספית, this daf inspired me to blog more about the problem of mercury in vaccines, where the stakes are higher.
* Starting with the gemara at bShab 61a-b, the Talmudic liberature applies the 3x criteria with permutations involving 3 different amulets, diseases, patients, or healers.