A mathematical explainantion why homeopathy is incompatible with our understanding of chemistry
Central to homeopathy is the concept that a substance becomes better at healing the more dilute it is. Homeopathy uses the notation Dn to describe dilutions, where one part of the substance is diluted in nine parts distilled water, n times.
Central to chemistry is the concept of the atom, the molecule, and the mole. Atoms are the smallest things that interact in chemical reactions. Atoms are composed of smaller particles, but their interaction is the realm of physics. Atoms bond to each other to form molecules, groups of atoms with definite chemical properties. When atoms change their groupings to form different molecules, a chemical reaction is said to have taken place. A mole is a precise number of molecules - avagadro's number. It is approximately equal to 6x10^23 .
If we consider homeopathic dilution to D25 (one of the lowest dilutions commonly used in homeopathy) of a mole of a substance that (for simplicity's sake) has the same number of molecules per litre as water does1. Before the first dilution (D0), there are 6x10^23 (600,000,000,000,000,000,000,000) molecules of the substance, and no molecules of water. We add 9 moles of water, mix well, and throw away 9 moles of the mixture. There are now (assuming the mixture was well mixed) 6x10^22 molecules of the substance to 5.4x10^23 molecules of water. We have thrown away an amount of the substance equal to the amount of water that is present, and are left with a D1 solution.
After ten dilutions (D10), we have 6x10^13 molecules of substance, and 5.9999999994x10^23 molecules of water.
Problems arise when we try to make a D24 solution. Our D22 solution had (approximately)60 molecules of substance in it. We added nine moles of water, mixed well, and threw away nine parts of the mixture. This leaves us with a D23 solution with only six molecules in it. Now we try to make a D24 solution: we add 9 moles of water, mix well, and tip out 9 moles of the mixture. There's a 10% chance that each of the 6 molecules of substance will be in our 1 mole of solution, rather than tipped down the drain. The odds of us having no molecules at all of our substance are 90%^62, or 53%.
There's some complicated maths involved in calculating the probability of having no molecules left if we dilute again from D24 to D25, as we can't say with much certainty how many molecules we're starting off with. Let's instead consider what would happen if we dilute straight from D23 to D25. We take our D23 solution (6 molecules of substance), and add 99 moles of water, mix well, and throw away 99 moles of the mixture. Each molecule of substance now has a 1% chance that it will be in the 1 mole of solution, rather than the 99 moles we threw away. The odds of us having no molecules at all of our substance are 99%^6, or 94%. At D30, the odds of us having no molecules at all are 99.99994%, at D35 99.9999999994%
Every time the solution is diluted past D23, the odds of there being any molecules at all of the substance decrease by a factor of ten. At D30, you're more likely to win the lottery than find a molecule of active ingredient, at D37 you're more likely to win the lottery twice in consecutive weeks. And yet we see homeopathic remedies with dilutions of D50 and over.
If homeopathic remedies do have an effect (something I don't presume to prove one way or the other), then it means that either our understanding of the behaviour (or even existence) of atoms is fundamentally wrong, that dilution works differently to how we'd expect, or that water has some previously unknown property that's causing pills made from sugar and pure water to heal people.
1 - For it to have any effect on the calculation, we would have to be wrong by at least a 10 times. For the pedants and chemists amongst us, let's assume we're doing this at standard temperature and pressure.
2 - See probability for why this is the case.