Homeopathic medications are used by millions, and hundreds of millions of dollars are spent on these remedies in the USA alone. In the UK, the NHS covers homeopathic treatments. Nonetheless, homeopathy is held in considerable disrepute by much of the medical and scientific community. Many proponents of homeopathy are well aware of these criticisms but remain unimpressed. The differences of opinion run deep, and the debate seems deadlocked. We aim to shed some light on this situation. We briefly recap some of the major arguments on each side, but we try to go further by making explicit an underlying philosophical presupposition. In particular, we will claim that there is an important principle, which has ancient roots going back at least to Occam, some version of which constrains all empirical reasoning. We call this constraint the simplicity principle. We argue that this is not something specific to a scientific paradigm, but that, all of us, including proponents of homeopathy, are themselves deeply committed to the simplicity principle. However, once the simplicity principle is made explicit and applied to homeopathy, allegiance to homeopathy is clearly seen as irrational. The point is not merely the lack of clinical trials supporting homeopathy; rather, belief in the efficacy of homeopathy leaves a mountain of unexplained mysteries, and thereby flies in the face of the simplicity rule that guides the homeopaths' own reasoning and arguments. If nothing else, we hope that defenders of homeopathy will gain a greater understanding of why critics are so deeply reluctant to accept the efficacy of homeopathic interventions - and that this reluctance is not mere stubbornness or artificial allegiance to western medicine. Finally, we also hope thereby to illustrate the usefulness of philosophy in unearthing presuppositions in seemingly deadlocked debates.