Supplementary material
A mole is more than an unwelcome visitor to our garden: it is also the number 6.02214129 × 1023. That power 23 up there means we could write it 602 214 129 000 000 000 000 000 if we had a big enough page.
Why is that number worth remembering? Because it is Avogadro's number, that's why, and Avogadro is one mean bastard who will break our legs if we show any disrespect to his number!!! No, I kid. It is worth remembering because it allows us to make a connection between the mass of a substance and the number of atoms it contains.
The periodic table of the elements classifies all known atoms according to the number of protons their nucleus contains, and it gives us the atomic mass of each. You'll notice that the mass given does not carry a unit: Oxygen, for example, has a mass of 15,9994. Sometimes we'll say it has a mass of 15,9994 Daltons. To connect those numbers to real life, we'll use Avogadro's number: in the case of oxygen, 6.02214129 × 1023 atoms weigh 15,9994 grams. Hydrogen has an atomic mass of about 1, meaning that 6.02214129 × 1023 atoms of hydrogen weigh 1 gram.
This concept explains what is the biggest problem with the homeopathic principle, which relies on great dilutions of certain substances to treat patients. Homeopathy is a therapeutic method invented by the German doctor Christian Friedrich Samuel Hahnemann in 1796, which by a funny coincidence is also the year Edward Jenner proceeded with the first vaccination. I find it amusing because both vaccination and homeopathy share at least one basic principle, in a sense: that of using like to fight like.
Past that one similarity, the big difference between them is that after two centuries, one has eliminated smallpox and pretty much made polio a forgotten nightmare, and the other works no better than any old placebo.
Vaccination relies on our adaptive immune system. Whenever we are exposed to a foreign substance of a certain size, one which does not belong in our body, certain specialized cells recognize it and try to get rid of it. This includes the molecules found on the surface of pathogens. How is this done? Partly through cells that specialize in identifying and swallowing foreign bodies; partly through cells that specialize in identifying other infected cells and destroying them, and partly through specialized cells that, once appraised to the presence of a specific foreign component in the body, produce proteins called antibodies that act as magic bullets that hunt it down and mark it for destruction. Oh, and once we've generated an immune reaction against something, there are cells that retain that information... if perchance we are exposed once again to the same pathogen, they'll allow us to generate such a quick response that we won't get ill again. We will be immune.
This system is actually quite powerful and can deal with pretty much any pathogen one can imagine, as long as we give it time to function. The reason we still get ill is that many pathogens evolved ways to circumvent our immune system's strategy (by hiding in certain places, for example, or by changing their molecular appearance very quickly)... and some pathogens are so damn efficient that by the time we raise a good immune response, we're dead.
Vaccination is based on the pre-emptive preparation of our immune system. It mimics an infection by using (a) more-or-less harmless pathogens that look so much like really bad ones that our system makes antibodies against both, as was the case with the vaccinia virus, far less dangerous than the smallpox virus; (b) attenuated pathogens, which have been modified in the lab so as not to cause disease (although immunosuppressed people can not use them); (c) subunits of pathogens, which are not dangerous on their own but teach our system to recognize the pathogens usually carrying them; and variations on these themes. The general idea is to expose the immune system to some part of a pathogen to give it a chance to recognize it later, without actually making us ill.
Homeopathy, meanwhile, relies on the philosophical principle of similarity; the same idea that gave us the expression "hair of the dog". Nowadays it mostly refers to the idea of drinking some alcohol to treat a hangover, but it comes from the suggestion to put some hair of the dog that bit you on the wound it left to help make it better. The idea is that when it is known that a substance can cause some disease in an otherwise healthy person, that substance (properly diluted) can be used to treat said disease in a patient.
Of course, presented like that, it could very well be that homeopathy is just vaccination under another guise: exposing the body to a small amount of a harmful agent, like a virus or a bacterium, or some other toxin. However, for the immune system to do its thing, it has to be exposed to at least a few molecules of a foreign origin... and homeopathic remedies are rather dilute. No, make that very dilute. Better yet, make that extremely dilute.
These remedies generally give you an idea of how dilute they are. A "1C" dilution, for example, means that whatever's considered has been diluted 100 times. A 6C dilution has been diluted 1012 times, or 1 part in 1 000 000 000 000. Where it gets silly is when we get to dilutions 12C and 14C, respectively one part in 1024 and in 1026... Since we exceed Avogadro's number, we don't even have one molecule of the original substance left.
Let's do a virtual experiment. Let's make a sucrose solution, which we can also call sugared water (although that sounds far less sciency)! Just so I don't seem to start with already diluted material, I'll put 40 packets of sugar in 1 litre of pure water. At 5 g per pack, it comes up to 200 g of sugar. It's not quite twice as sweet as Coca-cola, which contains 108 grams of sugar per litre, but I suppose we'll all agree that it is plenty sweet. Since we know that sucrose has the formula C12H22O11, we can look at the periodic table and determine that a molecule of sucrose weighs 342 Daltons, or 342 grams per mole. Our 200 grams here represent a fraction of that, or 0,585 mole; in terms of molecules, it comes up to 0,585 mole times 6.02214129 × 1023 molecules per mole, or 3,5× 1023 of sugar in our litre of water.
To make a 1C dilution of our sugared water, we'd need to take one part of it and mix it with 99 parts pure water. (It is said that homeopaths must mix stuff in a certain way for the remedy to work, and someone once amusingly said in the journal Nature that this must be how James Bond can tell that a vodka-martini has been shaken and not stirred). Such a 1:100 dilution would contain 3,5× 1023 / 100 molecules, or 3,5× 1021. That's still plenty sweet. But now let's look at the 12C dilution: 3,5× 1023 / 1024 leaves us with... 0,35 molecule! Less than one! Since we can not cut a molecule in two by diluting it, we either have one molecule left or none at all, with a 35% chance that we do. At the 13C dilution, 1 in 1026, we're down to 0,0035 molecule (or 3,5 chances in a thousand) that we still have a molecule in our container.
The dilution recommended by the good doctor Hahnemann is 30C, or 1 : 1060. In these conditions we'd have less than one chance in 100 000 000 000 000 000 000 000 000 000 000 000 to have even one molecule of sugar in our water. No, it won't taste very sweet. This is pure water. And that's not even the most dilute of homeopathic recipes! A popular remedy against flu called oscillococcinum is apparently prepared at a dilution of about 1 in 10400; as the initial material is duck liver, perhaps that's not such a bad thing.
Some fans of homeopathy refer to something called water memory to explain how infinitely diluted material might still work. In this model, water molecules could arrange in space so as to retain some structural information about past solutes. This hypothesis got a lot of momentum thanks to a controversial paper published in Nature in 1988 (with an odd advisory note by the publisher, saying in essence "I can find no fault with the technical aspects of this paper but I think it's wrong"); the experiments described therein could not be reproduced independently, nor when under the scrutiny of stage magician and pseudoscience debunker James Randi.
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