{"id":493,"date":"2007-08-16T21:41:47","date_gmt":"2007-08-16T21:41:47","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2007\/08\/16\/bad-homeopathic-differential-equations-yech\/"},"modified":"2007-08-16T21:41:47","modified_gmt":"2007-08-16T21:41:47","slug":"bad-homeopathic-differential-equations-yech","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2007\/08\/16\/bad-homeopathic-differential-equations-yech\/","title":{"rendered":"Bad Homeopathic Differential Equations. Yech."},"content":{"rendered":"

My friend and blog-father Orac<\/a> sent me a truly delectable piece of bad math today. It’s just
\nastonishing: a supposed mathematical model for why homeopathic dilution works, and for why the
\nstandard 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.<\/a> 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<\/em>. <\/p>\n

To give you an idea of what you’re in for, here’s the abstract:<\/p>\n

\n

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.<\/p>\n

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).<\/p>\n<\/blockquote>\n

What you find in this paper is both an astonishingly bad example of mathematical modeling, and
\na dreadful abuse of differential equations. It’s pathetic to realize that anyone<\/em> thought
\nthat this piece of dreck was not too embarrasingly bad to publish.<\/p>\n

<\/p>\n

As I’ve said before: mathematical modeling is a tricky thing. You need to abstract the
\nproperties of a real system that you want to model, and find a model that appears to
\nmatch observations, develop the model, and then validate it against tests.<\/p>\n

This paper manages to get all<\/em> of those steps – every single one – wrong.<\/p>\n

To start, they say that they don’t have a clue<\/em> about what they’re modeling, or how it works:<\/p>\n

\n

\tVarious hypotheses have been put forward to ‘explain’ homeopathy in terms of
\nconventional physics and chemistry. ‘Local’ hypotheses posit that remedies differ from
\nuntreated water in that they contain a population or concentration of an active ingredient.
\nFor some explanations, the active ingredient is a (hypothetical) persistent structural feature
\nin what is chemically pure water, such as a zwitterion,1 a clathrate,2 or nano-bubble.3 The
\n‘silica hypothesis’ posits that SiO2<\/sub> derived from the glass walls of the succussed vials is
\ncondensed into remedy-specific oligomers or nanocrystals, or else that silica nanoparticle
\nsurface is modified in patches to carry remedy-specific information.4 <\/p>\n

\tThe mathematical model developed here is compatible with any of these explanations. Let Q
\ndenote the concentration of ‘active ingredient’. Depending on the hypothesis, Q could be
\nthe concentration of a particular zwitterion, of a particular species of nano-bubble, of a
\nparticular silica oligomer (or family of oligomers), or of a specific silica nanoparticle
\nsurface feature. Note that the concentration of active ingredient in ordinary solvent is zero
\nor is assumed to be negligible. Right after dilution, the concentration will be Q
\ndil=Q\/H. <\/p>\n<\/blockquote>\n

Get rid of the babble, and what that comes down to “there are a bunch of completely different
\nexplanations 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
\nproduce dramatically different results.”<\/p>\n

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
\nthe fact that knocking silicate fragments off the walls of a glass jar is not going to behave the same
\nas creating tiny bubbles in water.<\/p>\n

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.<\/p>\n

\nThe fundamental assumption underlying our mathematical model is the following. Since a
\n1000c and 1001c are (essentially) identical, we assume that the effect of diluting a remedy
\nof concentration Q, followed by succussion, is to regenerate (approximately) the same
\nconcentration Q of the same active ingredient. The model will shortly be made more
\ncomplex by postulating multiple species of active ingredients, but let us start with the
\nassumption of a single active ingredient. Then succussion must raise the concentration from
\nQdil<\/sub> back up to Q=HQdil<\/sub>. If succussion did not raise the concentration
\nby a factor of (on average) H, then after repeated cycles the concentration would dwindle to zero.\n<\/p><\/blockquote>\n

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

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

\n

\tThis already tells us something interesting about the number of succussion strokes needed.
\nIf our growth rate reflects ‘perfect’ replication when very dilute, ie R=2, then to get RS>H
\nwe require a minimun of 7 succussion strokes per cycle for H=100 (since 27<\/sup>>100 but
\n26<\/sup><100), and a minimum of 16 strokes for the LM series. For a slower growth rate like
\nR=1.2, we need at least 38 strokes per cycle to bring the concentration u to 90% of the
\nmaximum when H=100, and 72 strokes per cycle for LM’s. (These stroke counts are
\nobtained by setting QS<\/sub>\/C=0.9 in Eq. (4) and solving for S).<\/p>\n

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

Gotta love that last bit, huh?<\/p>\n

Of course, this leaves an obvious problem: if the whole dilution\/succussion thing just reproduces
\nconcentrations of amorphous unknown thingies, then how can dilution accomplish anything? Homeopaths
\nclaim that something diluted 100 times is stronger<\/em> than something diluted 10 times; or at least, they claim that the 100 diluted solution is different<\/em> 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!<\/p>\n

\nIf there were just a single active ingredient, dilution would reduce and succussion would
\nrestore the concentration each cycle. Nothing would change with dilution-succussion
\ncycles and there would be no point in repeating dilution and succussion. But suppose there
\nare two active ingredients, each of which would, if it were alone, increase according to Eq.
\n(1). Approximate Eq. (1) by a continuous version, with the stroke count parameter ‘m’
\nbeing replaced by a ‘time’ parameter t.\n<\/p><\/blockquote>\n

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

\n

\tClearly, what happens with increasing potency is that the slower-growing species ‘X’ is
\ngradually replaced by the faster-growing species ‘Y’. Exponentiating Eq. (9) we see that the
\nconcentration ratio YP<\/sub>\/XP<\/sub> increases by a factor of between 10h(s-r)\/s<\/sup> and 10h(s-r)\/r<\/sup>, or
\nbetween H(s-r)\/s<\/sup> and H(s-r)\/r<\/sup>, with each dilution-succussion cycle.<\/p>\n<\/blockquote>\n

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

\n

Pre-transition the ratio
\nincrease is very close to H(s-r)\/r<\/sup>, while post-transition it is very close to H(s-r)\/s<\/sup>. Thus, the
\ntransition potency can be predicted fairly easily if one knows the growth rates and the initial
\nconcentration ratio at a low potency. <\/p>\n<\/blockquote>\n

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

\n

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

See, it all works!<\/p>\n

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
\ndominate the way they predict their unknown thingies in homeopathic solutions purportedly do.<\/p>\n

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
\nunobservable model of an unknown process, with no supporting data, and no validation. This pile
\nof nonsense is strung together with remarkably sloppy applications of irrelevant differential
\nequations in order to make it look credible. In other
\nwords, the whole thing comes down to a dreadful exercise in sloppy intellectual masturbation.<\/p>\n","protected":false},"excerpt":{"rendered":"

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 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[68],"tags":[],"class_list":["post-493","post","type-post","status-publish","format-standard","hentry","category-woo"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-7X","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/493","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=493"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/493\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=493"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=493"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=493"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}