My friend and blog-father Orac sent me a truly delectable piece of bad math today. It’s just
astonishing: a supposed mathematical model for why homeopathic dilution works, and for why the
standard dilutions are correct. It’s called “The octave potencies convention: a mathematical model of dilution and succussion”, and I got a copy of it via the Bad Science blog. The only part of it that’s depressing is the location of the authors: this piece of dreck was published by someone from the Harvard medical school.

To give you an idea of what you’re in for, here’s the abstract:

Several hypothesized explanations for homeopathy posit that remedies contain a concentration of discrete information-carrying units, such as water clusters, nano-bubbles, or silicates. For any such explanation to be sustainable, dilution must reduce and succussion must restore the concentration of these units. Succussion can be modeled by a logistic equation, which leads to mathematical relationships involving the maximum concentration, the average growth of information-carrying units rate per succussion stroke, the number of succussion strokes, and the dilution factor (x, c, or LM). When multiple species of information-carrying units are present, the fastest-growing species will eventually come to dominate, as the potency is increased.

An analogy is explored between iterated cycles dilution and succussion, in making homeopathic remedies, and iterated cycles of reseeding and growth, in bacterial cultures. Drawing on this analogy, the active ingredients in low and medium potency remedies may be present at early dilutions but only gradually come to ‘dominate’, while high potencies may develop from the occurrence of low-probability but faster-growing ‘mutations.’ Conclusions from this model include: ‘x’ and ‘c’ potencies are best compared by the amount of dilution, not the amount of succussion; the minimum number of succussion strokes needed per cycle is proportional to the logarithm of the dilution factor; and a plausible interpretation of why potencies at approximately regular ratios are traditionally used (the octave potencies convention).

What you find in this paper is both an astonishingly bad example of mathematical modeling, and
a dreadful abuse of differential equations. It’s pathetic to realize that anyone thought
that this piece of dreck was not too embarrasingly bad to publish.

As I’ve said before: mathematical modeling is a tricky thing. You need to abstract the
properties of a real system that you want to model, and find a model that appears to
match observations, develop the model, and then validate it against tests.

This paper manages to get all of those steps – every single one – wrong.

To start, they say that they don’t have a clue about what they’re modeling, or how it works:

Various hypotheses have been put forward to ‘explain’ homeopathy in terms of
conventional physics and chemistry. ‘Local’ hypotheses posit that remedies differ from
untreated water in that they contain a population or concentration of an active ingredient.
For some explanations, the active ingredient is a (hypothetical) persistent structural feature
in what is chemically pure water, such as a zwitterion,1 a clathrate,2 or nano-bubble.3 The
‘silica hypothesis’ posits that SiO₂ derived from the glass walls of the succussed vials is
condensed into remedy-specific oligomers or nanocrystals, or else that silica nanoparticle
surface is modified in patches to carry remedy-specific information.4

The mathematical model developed here is compatible with any of these explanations. Let Q
denote the concentration of ‘active ingredient’. Depending on the hypothesis, Q could be
the concentration of a particular zwitterion, of a particular species of nano-bubble, of a
particular silica oligomer (or family of oligomers), or of a specific silica nanoparticle
surface feature. Note that the concentration of active ingredient in ordinary solvent is zero
or is assumed to be negligible. Right after dilution, the concentration will be Q
dil=Q/H.

Get rid of the babble, and what that comes down to “there are a bunch of completely different
explanations for phenomena that might cause homeopathy to work. But that doesn’t matter – because we’re just going to invent a model that completely ignores the fact that these different explanations would
produce dramatically different results.”

If you want to put together a mathematical model, you need to know something about the thing you’re modeling: you need to know enough about it to be able to state what you’re modeling mathematically, and to be able to show whether or not your model conforms to reality. “Nano-bubbles”, silicate crystals, clathrates, etc., are all different phenomena (assuming they exist at all): you can’t just blindly ignore
the fact that knocking silicate fragments off the walls of a glass jar is not going to behave the same
as creating tiny bubbles in water.

But they don’t let that stop them. They know nothing about the mechanism they’re purportedly modeling; they have no data about how it behaves – but they’re going to forge ahead anyway.

The fundamental assumption underlying our mathematical model is the following. Since a
1000c and 1001c are (essentially) identical, we assume that the effect of diluting a remedy
of concentration Q, followed by succussion, is to regenerate (approximately) the same
concentration Q of the same active ingredient. The model will shortly be made more
complex by postulating multiple species of active ingredients, but let us start with the
assumption of a single active ingredient. Then succussion must raise the concentration from
Q_dil back up to Q=HQ_dil. If succussion did not raise the concentration
by a factor of (on average) H, then after repeated cycles the concentration would dwindle to zero.

Well, gosh, isn’t that clever? Since just repeatedly diluting things reduces concentration to zero, we
can just arbitrarily assume that succussion (the homeopathic dilution process) just happens to arbitrarily
exactly reproduce concentrations! Wow! It’s just like a homeopath would hope. Oh wait, it is just what a
homeopath would hope, with absolutely no reason for it except that it’s just what a homeopath would hope.
There’s no reason to believe that replacement rate: they give no evidence for it – not a single shred. The
only argument for it is what you read above: that if it wasn’t, then homeopathy couldn’t work.
What would a reasonable person assume from that? Probably that homeopathy can’t work. But not these
authors: they know it must work, so they create a totally arbitrary model, with no evidence,
that just happens to show exactly what they want it to show.

Then, they hash together some sloppy differential equations based on this purported replication rate
of unknown thingies by unknown mechanisms, and use it to estimate the number of whacks needed per
succession – once again by totally bogus handwaving:

This already tells us something interesting about the number of succussion strokes needed.
If our growth rate reflects ‘perfect’ replication when very dilute, ie R=2, then to get RS>H
we require a minimun of 7 succussion strokes per cycle for H=100 (since 2⁷>100 but
2⁶<100), and a minimum of 16 strokes for the LM series. For a slower growth rate like
R=1.2, we need at least 38 strokes per cycle to bring the concentration u to 90% of the
maximum when H=100, and 72 strokes per cycle for LM’s. (These stroke counts are
obtained by setting Q_S/C=0.9 in Eq. (4) and solving for S).

Although we have no experimental evidence to give us a range for R, Eq. (4) suggests that
we should not skimp on succussion, with 40 strokes as a reasonable minimum when
making ‘c’ potencies. Hahnemann himself held changing views about the optimum value
for S. In the 5th edition of the Organon he recommended S=2 but revised the figure upward
to S=100 in the 6th edition [5, p. 270].

Gotta love that last bit, huh?

Of course, this leaves an obvious problem: if the whole dilution/succussion thing just reproduces
concentrations of amorphous unknown thingies, then how can dilution accomplish anything? Homeopaths
claim that something diluted 100 times is stronger than something diluted 10 times; or at least, they claim that the 100 diluted solution is different from the 10 diluted. But dey, no problem! We’ve got a crappy mathematical model totally unpolluted by data: we can make it say anything we want!

If there were just a single active ingredient, dilution would reduce and succussion would
restore the concentration each cycle. Nothing would change with dilution-succussion
cycles and there would be no point in repeating dilution and succussion. But suppose there
are two active ingredients, each of which would, if it were alone, increase according to Eq.
(1). Approximate Eq. (1) by a continuous version, with the stroke count parameter ‘m’
being replaced by a ‘time’ parameter t.

So they set up a two-ingredient “approximation” of their equations, and use it to argue
that differences in replication rates of different solvents during succussion are the key: the
more dilution/succussion cycles you go through, the more the faster replicator comes to dominate
the solution. And this, you see, is why 100 dilutions aren’t noticably different from 101, but
totally different than 1000 dilutions.

Clearly, what happens with increasing potency is that the slower-growing species ‘X’ is
gradually replaced by the faster-growing species ‘Y’. Exponentiating Eq. (9) we see that the
concentration ratio Y_P/X_P increases by a factor of between 10^h(s-r)/s and 10^h(s-r)/r, or
between H^(s-r)/s and H^(s-r)/r, with each dilution-succussion cycle.

I just have to inject here that they can’t even be bothered to do the basic math stuff right. All
of the differential equation stuff in this paper is pretty much like that: just arbitrary stuff
done for no particular reason, with no explanation other than that it produces the results that
they want it to. Why exponentiate the differential equation? Why not? It gives the answers they
want.

Pre-transition the ratio
increase is very close to H^(s-r)/r, while post-transition it is very close to H^(s-r)/s. Thus, the
transition potency can be predicted fairly easily if one knows the growth rates and the initial
concentration ratio at a low potency.

If only they had any way of measuring the stuff they’re talking about, and they had any actual
data, and they actually managed to validate any of it, then you could easily predict stuff
with this! See, it’s science, they’re describing an experiment where you could use their math!
Uhh… Except that they can’t measure it, and this whole model is a pile of rubbish that they
pulled out of their asses because it said what they wanted it to say.

If s/r is only slightly bigger than 1, it takes more
cycles to reach the transition and the transition occurs gradually over several cycles. If s/r is
substantially bigger than 1, the transition is reached quickly and occurs abruptly. Of course,
there is no transition at all if the initial concentration of ‘Y’ exceeds that of ‘X’: in this case
the slower growing ‘X’ just declines, out-competed by ‘Y’.

See, it all works!

It just keeps going like that. Their idea of validating this model is showing that if you seed a culture dish with two organisms that reproduce at different rates, you’ll see one of them come to
dominate the way they predict their unknown thingies in homeopathic solutions purportedly do.

So… Putting this all together: we have a great example of really horrific mathematical modeling. It’s a model pulled out of nowhere for no particular reason, supposedly modeling an unobserved and
unobservable model of an unknown process, with no supporting data, and no validation. This pile
of nonsense is strung together with remarkably sloppy applications of irrelevant differential
equations in order to make it look credible. In other
words, the whole thing comes down to a dreadful exercise in sloppy intellectual masturbation.