# Bad Homeopathic Differential Equations. Yech.

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
Qdil back up to Q=HQdil. 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 QS/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 YP/XP increases by a factor of between 10h(s-r)/s and 10h(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.

## 0 thoughts on “Bad Homeopathic Differential Equations. Yech.”

1. PalMD

Thanks so much for the analysis. Next we need a physical chemist to go after a few of the articles in this issue of Homeopathy, which is entirely devoted to water memory.

2. Walker

So if two homeopathic remedies affect exactly the same symptoms, are they homeomorphic?

3. Shawn Wilkinson

Wow…I didn’t know such a journal existed from Elsevier…PalMD, I’ll give it my best shot…

4. spudbeach

I’ve heard that any competent theoretical physicist can match any set of data points, but these guys are even better than physicists: They can fit a theory to no data at all! Gotta love it! Makes me want to go down some homeopathically diluted poisons.

Hmm. Obviously this is bogus on a bogus field, but I must disagree with the post on some minor points.

If you want to put together a mathematical model, you need to know something about the thing you’re modeling

Agreed. It is very common to put up a first order model when analyzing a new phenomena, sketchy on the details. And that is what the author does.
The problem here is of course that there is no observed phenomena. It isn’t even a gedanken experiment without any evidence what so ever.

The only argument for it is what you read above: that if it wasn’t, then homeopathy couldn’t work.

There isn’t much of finetuning in the model, it has a threshold. Assuming they have observations, it can be tested. But again, no observations.

Why exponentiate the differential equation?

Um, it is an inequality. To translate back from the temporary used convenient log scale, perhaps?

Their idea of validating this model is showing that if you seed a culture dish …

That is testing the model equations, which is good to do. However, it is not testing its specific applicability. For the same reason we keep pounding here, there are no observations of homeopathy whatsoever.
This is a work that shouldn’t have been done. Nobody models the needed population of tooth fairies. And why should science lend credence to homeopathy?

6. Wayne McCoy

Reminds me of a paper I once did for fun. I gathered together various bits of data from random places and experiments, examined them for patterns that seemed to fit what I was looking for and came up with a theory called Hyperarcuate Whorls. This resulted in the paper “Hyperarcuate Whorls in Multiply-transient Levitron Dynamics.” Sounds impressive, hey?

David Anick was a professor of mathematics at MIT (though not an analyst!), before leaving the field to study medicine in the early 1990’s.
As a mathematician, I am sad to see my subject so abused and misused.

8. Andrew Briscoe

Buwh?
Had never heard of this before now, I’m so confused. Some questions: 1. If the substance they are using helps, why dilute it at all? 2. If no molecules are left, then the substance used wouldn’t matter, right? so could they just ‘dilute’ water a bunch of times if they think ‘air bubbles’ or something causes the effect? 3. A harvard medical professor supports this? have they been fired yet? or are they too busy working on their crystal healing paper?

Andrew Briscoe:

If the substance they are using helps, why dilute it at all?

I think it is the reverse of “like cures like”. IIRC they start with a poison that gives similar effects as a disease. Wouldn’t want to give too much to an already sick patient.
Besides, water is less expensive. 😛

Duh! Not the “reverse” of old superstition – a development of it.

11. Anonymous

If the idea is ‘like cures like’, why havn’t all the people doing this converted to vaccines? thats identical not just like, and it has proven effectiveness.

12. dave tweed

One small criticism: Often in purely empirical modelling a model is chosen “because we know how to work with it” which is quite close to “It’s a model pulled out of nowhere for no particular reason”. The key difference is that in those cases the model must be capable of, to reasonable accuracy, both fitting your training data and predicting new data. What’s really bizarre about the numerical model that it seems to have been chosen to fit two “explanations” which aren’t independently observable: (1) there is a key “dose” giving the curative response (conventional medicine)and (2) dilution followed by succussion is curative.

13. Greg

This is a very sad story, because David J. Anick was an excellent mathematician before he left the field to go into medicine. He has written some very good papers on algebraic topology in the 80’s. I can’t imagine what happened to him twelve years ago, but something very bad.

14. Mikael Johansson

Operadic, Greg: It really is that Anick? Whoa! I remember basing some of my M.Sc. on research he had done with the people at my department once upon a time. I knew he had gone into medicine, but not that he ended up a crackpot.

15. Greg

Mikael, I’m afraid yes:
David J. Anick, MD
Education:
BS, Massachusetts Institute of Technology, Cambridge, MA, 1976;
Ph.D., Massachusetts Institute of Technology, Cambridge, MA, 1980;
MD with Thesis, University of California, San Francisco, CA, 1995
Residency:
Psychiatry, McLean Hospital, Belmont, MA, 1995-99
Professional Associations:
American Medical Association, American Psychiatric Association, Massachusetts Medical Society

16. Dr Aust

Andrew Briscoe:
That’s the point of homeopathy – there IS no substance. Repeat, no substance. None. Not a single solitary molecule.
This is why homeopathy proponents have to claim the substance (diluted away) has somehow left a “memory imprint” on the water. Hence the shorthand often used for what homeopaths believe in: “the memory of water”.
The weird thing is one would think a person like Dr Anick, who had done a psychiatry residency AS WELL as being a proper mathematician, would clearly be able to see how the technique “works”.
In its classical form, which is involving at least one 1 hr consultation about your ailment(s)and “life issues” with a homeopathic “therapist” – the answer is that homeopaths offer a kind of stealth talking therapy. You talk to a nice empathetic and concerned person about yourself and your problems for an hour in a soothing office with lots of plants and artwork. S/he tells you they understand and that they will give you a special tailored remedy to “aid your body’s natural healing power”. You go away with your sugar pils, take them for a couple of weeks, and feel better. The power of placebo, with an added side order of what we used to call “medical paternalism” (authority therapist figure / man of wisdom tells me I will feel better, ergo…).
So it’s all about the power of belief. Or, less snidely, a demonstration of the role that the mind plays in how people perceive their health / illness / symptoms.
Since homeopaths deny this perfectly plausible explanation, which fits with everything known and requires no suspension of the laws of physics and chemistry, they have to come up with craziness like Dr Anick’s work, and the other papers in the journal it comes from.

17. Anonymous

PS Anyone else who can face it – please do go through the “math” in Anick’s article, and point out where any logical flaws / fallacies lie.
One of the weapons homeopaths use against us sceptics in bioscience and medicine is to point to stuff like this paper and say:
“You see! There are possible explanations for how it works which you guys don’t understand and aren’t qualified to comment on
(Another example of this is claiming there is some hidden explanation of how nothing can be something in homeopathy based on quantum theory)
This approach doesn’t cut much ice with us, but it is clear that it does work on many non-scientific people. This is why critiques like Mark’s are so important – hence the concept of the BadScience journal club.

18. Keifus

On one hand, it seems forgivable to employ phenomenology (meaning here, “ignoring the molecular mechanisms” and not “based on observation”) to make a case like this. To the extent I can understand by skimming the selections, they’re basically saying that the amount of phase X is reduced by dilution, but upon shaking (succussion), it re-precipitates as either phase X or Y. Presuming this is feasible (and I wouldn’t!), it should be straighforward to model the dynamics of how the amounts of X and Y evolve over time. You know, applying arbitrary rates and stuff for the magical processes.
On the other hand, I trust you that the math in this paper is as loopy as the physics and English. I don’t intend to read it, that’s for sure.

19. Nathan

> 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.
Wearing my Bayes-hat, I don’t think this is bad reasoning (locally!), just wildly stupid priors. If they were really sure (that is, P(homeopathy working) ~ 1), then they’d take the low P(homeopathy working|!succesion raising concentration) as evidence for succession raising concentration, because they’re obviously so sure of homeopathy.