The Law of Infinitesimals

Three weeks ago, I mentioned a book I was reading, How Mumbo-Jumbo Conquered the World: A Short History of Modern Delusions, by Francis Wheen, which is about…exactly what it sounds like it’s about.  The essential premise of the book is that the Age of Reason ushered in by the Enlightenment is under assault, that the forces of ignorance, superstition, and philistinism are everywhere.

Three weeks ago, I was pretty glib about this assault on reason.  I was only about halfway through the book, and I think I basically concluded that, because of a spin class I once took, everything was just fine.

I was wrong!  I’ve finished the book, and I’m frankly terrified.

Here is but one scary and demoralizing example:

Have you heard of Hahnemann’s ‘Law of Infinitesimals’?  It is one of the three laws invented by Samuel Hahnemann in the late 1700’s, and it remains one of the foundational tenets of homeopathic medicine.

Hahnemann’s Law of Infinitesimals states that the more you dilute a medically active substance in a medically inert solution (like milk or water), the more potent it becomes.

I’m going to repeat that very slowly, just so we’re all clear:

The more you dilute a substance (so, the less of it there is per unit solvent), the more potent it becomes.

Let me put that another way:

Let’s suppose you have two cups of water, Cup A and Cup B, and into Cup A you put 100 molecules of Medicine X, and into Cup B you put 1 molecule of Medicine X.

The Law of Infinitesimals

Cup A                           Cup B

According to one of the cardinal principles of homeopathy, Cup B will be the more powerful medicine.  Not the correct dose, not the more medically advisable, the more powerful.

Some homeopathic remedies are sold at dilutions so high that they almost certainly contain no molecules of the original “medical” substance.

(Of course, it would be very difficult to say this for sure of any given container of diluent; however, Avogadro’s Limit is generally held to be the dilution at which no more original substance remains.  Avogadro’s Limit is around 13C (1 x 10²⁶) if 1 mole of the original substance were used in first dilution.  Hahnemann apparently advocated 30C as the best “usual” dilution of homeopathic remedies, and one remedy, Oscillococcinum, is famously sold at 200C.  Since every ‘C’, or “centesimal dilution” is one part substance diluted by 99 parts solvent, that would mean that a 200C solution (one part into 99 parts, then one part of that into 99, then one part of that into 99, 200 times) would have one molecule of active ingredient for approximately every 100²⁰⁰ molecules of diluting solution (that would be 10 with 400 zeroes after it).  Since that is way, way more than all the molecules in the universe, it is very unlikely that that one molecule is in your dose.)

No problem, say homeopaths!  Even if your bottle of Oscillococcinum has zero molecules of Oscillococcinum in it and is, in actuality, a bottle of water, no matter: the water “remembers” the Oscillococcinum, and that’s just as good.  No, scratch that – it’s even better.

This is so stupid I can’t believe that anyone believes it.

First of all, heaven help us all if water molecules “remember” other substances with which they’ve come into contact.  Worse still if, through that “memory”, water can impart the properties of those substances to the drinker.  Because, if that is the case, when you drink that Oscillcoccinum, or even when you down a bottle of Poland Spring, you’re also drinking all the things all the water molecules in that bottle have ever touched: feces, dirt, rotten flesh, every manner of poison and putrescence that has burped out of the surface of the earth in the gadzillion-odd years it’s been roiling around.

But failure to see that is merely failure of imagination, a failure of which we are all guilty.  Active belief in the Law of Infinitesimals, on the other hand, is an almost deliberately perverse misreading of actual principles of medicinal dosing.  It is a committed idiocy.

It also violates the sort of everyday experience that informs common sense:

If you’re making margaritas, and you make one with a single shot of tequila, and one with two shots of tequila, which will be the stronger drink?  Is there a single homeopathic consumer on the planet, no matter how credulous and stupid, who would accept the Law of Infinitesimals from their bartender?

I live in a country where a frightening percentage of the population refuses to believe in Evolution.  Where there is a movement against vaccines, the enormous benefit of which has been demonstrated well beyond reasonable doubt.  My countrymen are so skeptical that they literally pose a threat to themselves and to their own children.

But they believe in this, in Memory Water and the Law of Infinitesimals.

And the angels wept.

If the sight of serial dilutions makes you want to curl and weep, Wikipedia actually has surprisingly good and clear information on homeopathic doses.  Their basic article on homeopathy is also pretty informative.

Featured image is not of Hahnemann, who I’d love to kick in the shins right about now and so will not feature here and who looks like a tool (see the Wikipedia page on Homeopathy if you don’t believe me), but of Amedeo Avogadro, who might not have been a looker, but was right.  It is taken from his Wikipedia page.

“To See and Listen to the Wicked is Already the Beginning of Wickedness”

I used to work in the lab with a young woman, on her way to medical school, who didn’t “believe in schizophrenia”.  Everyone hears those voices, she explained to us completely in earnest; to succumb to them was simple weakness.  “Schizophrenics”, she would say with air-quotes, were just lazy people gaming the system, exploiting disability and trying to get out of working.

This woman was obviously a malignant idiot (and she is now in medical school, so be very afraid), but I was stunned and fascinated by her.  I have spent my entire adult life working in neuroscience research, where people don’t question the essential validity of psychiatric illness.  We study schizophrenia because we believe it exists, because, and I can’t stress this enough, it does.  It emphatically does: schizophrenics are ill – they are not “lazy”.  I’d never met anyone who’d admit to not believing in mental illness before.

Imputing moral failings to people who suffer from psychiatric disease is retrograde and stupid, and I mention that here because I am about to do it.

There’s been a fair amount of press this week, in the wake of the conviction of Lacey Spears of the second degree murder of her son Garnett, about Munchausen by Proxy.

Munchausen is a factitious disorder, a disorder characterized by the fabrication of symptoms of somatic illness.  People with Munchausen often go to extraordinary lengths to receive medical attention for their fake illnesses: they research their “condition” obsessively, run up huge medical bills, poison or mutilate themselves; they have even been known to undergo surgeries that they do not need.

They do not believe that they have these disorders – Munchausen is not a disorder of delusion or psychosis.  Munchausen patients are deceptive, they are seeking attention and sympathy.

Despite the fact that there is, at this time, no known neurological lesion associated with Munchausen, I won’t dispute that someone who is so desperate for attention that they will drink bleach or undergo unnecessary amputations is mentally ill.  However, Lacey Spears didn’t have Munchausen; she had Munchausen by Proxy.

Munchausen by Proxy is Munchausen Disorder where the medical charades are enacted not on the sufferer but on someone else, usually a dependent, a child or an elderly parent.  Instead of poisoning themselves, people with Munchausen by Proxy poison their dependents, over and over and over again.

I just don’t know that I have what it takes to consider someone with Munchausen by Proxy mentally ill, a victim of their condition in the same way that someone with schizophrenia or bipolar is a victim of their’s.

The argument is that Munchausen by Proxy is a compulsion, that the “need” for attention is like the need that someone with obsessive compulsive disorder might have to wash their hands repeatedly.  That may be, but I doubt it: Munchausen by Proxy deceptions are methodical and controlled, not frantic and compulsive, and well enough in-hand to be perpetrated on other, vulnerable people.

More than that, I’m not sure I care.  This may be my failing – certainly, I believe that it is my responsibility to try to see the world as it is, whether or not it accords with my wants or expectations.  I know I’m not the first person to see a diagnosis as an excuse, and I’m not at all sure that’s company I want to keep.

But I’m not sure it’s right or wise to let every category of human wickedness hide behind pathology.  If you are “compelled” to put feces in your child’s feeding tube because you’re so needy for the attention of others, are you sick or are you so supremely selfish as to be fairly called evil?  Or are you both?  And do your actions really deserve our moral parsing?  Do we owe you the time and energy it takes to figure out why you’ve done this, why you’ve killed your own child for attention?  Do you deserve the protection a diagnosis gives you?

I’ve spent a lot of time thinking about evil people and evil acts – I find them interesting and I want to understand them, maybe more than I should.  I care where the line between illness and evil lies, and I think that, perhaps, I want to draw that line between compulsion to self-harm and compulsion to harm another.  I think it behooves us to treat self-mutilators as sick: they hurt, primarily, themselves.  But we lose moral credibility if, as a society, we decide that anyone who hurts anyone else must be “crazy”, or ill – we should not rob all wrong-doers of volition.

For whatever reason, the coverage of Lacey Spears left me cold and angry.  Even I have my limit, and maybe this is it.  I have a difficult time seeing Munchausen by Proxy as an illness – I see it instead as vileness, as tremendous and terrible selfishness.  And I don’t want to look for moral or psychological complexity in that kind of darkness any more; I just want to look away.

Title quotation by Confucius.

A Brief Note in Defense of Georges Cuvier

Jean-Léopold-Nicholas-Frédéric Cuvier, known for reasons that are unclear to me (it is not as though he lacked for names) as Georges, is one of my intellectual heroes.  A French paleontologist when being French was trendy, but before being a paleontologist was, Cuvier is considered the foundational thinker of vertebrate paleontology.

He is also the person responsible for clearly formulating, and perhaps proving, the idea of catastrophic extinction.  Before Cuvier, the idea that animal species went extinct, that they simply ceased to exist, was considered something of a crackpot theory, more the province of poets than of scientists (Lucretius, for example, wrote about something very much like it in ‘On the Nature of Things’, in 50 B.C.).  Even Darwin, who understood that species must die, did not believe that that they went extinct all at once, in single events.

Cuvier saw that they might.  Cuvier saw an astonishing number of things: he discovered a number of species, made a number of correct family and order distinctions, and he saw it all from bones.  He had a remarkable ability to divine what once was from the little that remained.

Cuvier is not so well remembered as Darwin, and when he is remembered, it is usually remarked that he did not believe in evolution; in fact, he derided and punished his colleagues who did.

That is unfortunate and unattractive.  It is one thing to be wrong; it is quite another to be wrong and gloat about it.  More than that, we tend to hold it against intellectuals who did not then immediately see the good sense in evolutionary theory, in much the same way that we conclude that people who now persist in denying it are philistines and morons.

But there was, then, a great deal less accumulated evidence in support of evolution, and I would like to offer one small word in defense of Cuvier: in France, when he was living, the new theory of evolution of the species was called transformisme.

Transformisme is a fairly silly-sounding word, and not merely because it is French.  Evolution sounds like a process with impressive scope; natural selection is a machine whose gears might, over eons, grind out the diversity of life on earth.

Transformisme is a smaller word – it sounds like a journey of self-discovery, like something the ‘Eat, Pray, Love’ woman might have tattooed on her lower back.  When one confronts the expanse of geologic time, and examines the bones of the monsters that populated it, and grasps for a word to grapple with them, one finds transformisme insufficient.

(Interesting datum: Spouse, who is a scientist, when asked his opinion of the word transformisme, immediately sang, “More than meets the eye…”)

I’m kidding and I’m not: Cuvier had a penetrating analytic mind, and I don’t mean to imply that his scientific ideas were informed solely by how important-sounding the words for those ideas were.  There is no denying it: though his batting average was high, he completely whiffed it with evolution.  But there is a reason that scientists give their own projects weighty names: they signal to other people our seriousness, and the seriousness of our work.  Cuvier, who coined the names ‘mammoth’ and ‘mastodon’ and ‘pterodactyl’, understood that, and Darwin certainly did.  Scientists think very hard about what they call their discoveries – they understand that there is an element of marketing in the act of naming.  And transformisme is a right idea that starts off on the wrong foot.

Image taken from Wikipedia.

Have You Raped Today? Take Our Simple Survey!

I’ve been hearing a lot lately, not least in the wake of the UVA fraternity gang-rape allegations and the passing of California’s Yes Means Yes law, about a 2002 study by David Lisak and Paul H. Miller, ‘Repeat Rape and Multiple Offending Among Undetected Rapists’.

The authors of the study set up distribution tables at major pedestrian thoroughfares at “a mid-sized, urban commuter university”, and told students that they were conducting a study on “childhood experiences and adult functioning”.  Study participants were promised anonymity and allowed to complete the survey in private.

The survey asked a number of questions, including questions about sexual advances made towards children and prior violent conduct.  There were also four questions which, while not mentioning rape by name, if answered in the affirmative, triggered a follow-up interview:


  1. Have you ever been in a situation where you tried, but for various reasons did not succeed, in having sexual intercourse with an adult by using or threatening to use physical force (twisting their arm, holding them down, etc.) if they did not cooperate?
  2. Have you ever had sexual intercourse with someone, even though they no want to [sic], because they were too intoxicated (on alcohol or drugs) to resist your sexual advances (e.g. removing their clothes)?
  3. Have you ever had sexual intercourse with an adult when they didn’t want to because you used or threatened to use physical force (twisting their arm; holding them down, etc.) if they didn’t cooperate?
  4. Have you ever had oral sex with an adult when they didn’t want to because you used or threatened to use physical force (twisting their arm; holding them down, etc.) if they didn’t cooperate?


The authors gathered 1882 responses from men ranging in age from 18 to 71.  Of those 1882 men, a whopping 120 (or 6.4%) “met criteria for rape or attempted rape”, which means that they answered ‘Yes’ to at least one of those questions, and confirmed their assent in a follow-up interview.  Of those 120 men, 76 (or 63.3%) admitted to committing multiple rapes.  In total, the 120 admitted rapists, none of whom had been convicted or incarcerated, had committed an average of 4 rapes apiece.

There are many important and disturbing implications of this study, but my overwhelming impression was one of methodological confusion: who on earth answers ‘Yes’ to those questions?

When the authors of the study said that their questions would “never use words such as “rape”, “assault”, “abuse”, or “battery”’, I thought that the questions would be…subtler, craftier.  I’m not sure what I expected, but “Have you ever forced anyone to have sex with you?” is pretty blunt.

Here’s my question: let’s say you were a rapist (not a fun mental exercise, I know, but bear with me…).  You’re walking around, minding your own rapist-business, at school, and some stranger at a table offers you a few bucks to participate in a study about childhood and adulthood and stuff.  The study is apparently anonymous, but there are a couple of questions which bear directly on your rapey-er activities, and you don’t know the people administering it.  Now, do you, as an ambitious, as-yet-uncaught rapist-about-town, answer ‘Yes’ to those questions, to those strangers, and then, in the follow-up interviews, confirm that, yeah, you force people to have sex with you all the time?

Who does that?  Well, 6.4% of men on that college campus is who, but come on!  I suppose it’s a small mercy for the data-gatherers that they are so willing, but it does make one wonder: how many men read those questions and thought, “There is no way I’m going to admit that to a stranger”.  And that ambiguity is important, because it suggests that it is possible, maybe even probable, that Lisak and Miller’s study didn’t gather affirmative responses from all rapists in their sample populations, or even most of them: maybe it only gathered affirmative responses from the stupid ones.

Die, Dead, Deceased, Lifeless, and Suicide

Last week, I wrote about the Implicit-Association Test (IAT), and whether it has use as a real-world predictor of behavior.  I have been skeptical of the IAT, because I do not believe that a longer latency on a language-pairing task necessarily reveals a real-world social bias.  How, as I mentioned last week, the IAT has predicted differences in treatment between black and white cardiac patients by physicians who did not declare any explicit racial bias.

And while I was researching the IAT, I found another paper (on which Dr. Mahzarin Banaji was also senior author), by Nock et al., from 2010, ‘Measuring the Suicidal Mind: Implicit Cognition Predicts Suicidal Behavior’, in which the IAT was used to positively predict suicidality in an at-risk patient population.

The IAT, as I have explained, is a measure of the speed at which a patient can assign a word to one of four categories, arranged into two pairs.

IAT - Task 3

The assumption which underlies the IAT is that, when the paired categories are cognitively linked in the mind of the subject, he or she will perform the task more quickly than when the paired categories are cognitively dissonant.

The authors of this paper administered a version of the IAT to 157 patients in a psychiatric emergency room, none of whom showed cognitive impairment.  Some had a history of suicidal behavior; some did not.  Some had co-morbid psychiatric disorders; some did not.

The particular IAT administered to these patients measured the latencies between matching words which signified ‘life’ versus ‘death’ (i.e. ‘alive’, ‘live’, ‘thrive’, and ‘die’, ‘dead’, ‘deceased’, ‘lifeless’, ‘suicide’), and words which signified ‘me’ versus ‘not me’ (i.e. ‘I’, ‘myself’, ‘mine’ and ‘they’, ‘them’, ‘other’)

Nock et al. found that a strong association on the IAT between ‘death’ and ‘self’ significantly predicted a suicide attempt: “Specifically, the presence of an implicit association with death/suicide was associated with an approximately 6-fold increase in the odds of making a suicide attempt in the next 6 months.”  This means that the IAT was a better predictor of future suicidality than either the prediction of the hospital clinicians or of the patients themselves.

Importantly, the Nock study didn’t establish a causal relationship between cognitive affect bias and suicidality.  There is no way to tell, from their data, whether negative cognitive bias actually makes you more likely to to attempt suicide, or whether it is an effect of the same emotional, chemical, or cognitive disruption that will lead one psychiatric patient to suicide and not another.  Either way, the paper suggests another way in which we are governed by, or reveal ourselves through, cognitive processes of which we are not consciously aware.

Image taken from Wikipedia.

Racial Bias and Implicit Associations

One of the speakers I was excited to see at the Society for Neuroscience conference this year was Dr. Mahzarin Banaji.  I have been curious, and had strong feelings, about her work for a long time.

Banaji is one of the researchers who has developed the idea that memory retrieval can be affected by our unconscious bias, and she is part of the team that runs Project Implicit, which uses the Implicit-Association Test (IAT) to examine unconscious racial bias (among other things).

The IAT asks subjects to assign words to one of two categories: for example, is the name ‘Aaliyah’ the name of someone ‘Black’ or a ‘White’?  These categories are on either the left or right side of the screen; the subject signals choice by picking the side of the category.

IAT - Task 1

The test then introduces another category: for example, does the word ‘Suffering’ belong in the category ‘Pleasant’ or ‘Unpleasant’?

IAT - Task 2

Finally, the test layers the two category choices, and asks the subject to pick the side of the screen which contains the correct category, of the four.

IAT - Task 3

The testers hypothesized that subjects would perform this task more quickly when the two paired categories on either side of the screen were also cognitively paired for the subjects, i.e. ‘White’ + ‘Pleasant’, or ‘Black’ + ‘Unpleasant’, in the case of racial bias.  So far, there is a fair amount of data which supports their hypothesis.

However, the IAT is problematic.  First of all, it is very easy to manipulate the test – the mechanism is simple, and once you’ve grasped it, you can easily delay your responses along whatever axis for which you do not wish to express a bias.

Second, and more importantly, the IAT assumes that a slower sorting time in a cognitive retrieval task meaningfully reveals something as complicated as racial bias.  There are many reasons why this might not be the case: language sorting and recall are governed by different parts of the brain than those which govern social parsing.  The two processes might be unrelated.

Which is why studies showing that implicit biases as revealed by the IAT actually predict bias in behavior are valuable.  For example, Banaji and her team published a paper in 2007 (Green et al., 2007, ‘Implicit Bias among Physicians and its Prediction of Thrombolysis Decisions for Black and White Patients’) showing that physicians who scored on high on the IAT for implicit racial bias were less likely to prescribe thrombolysis for black patients than for white patients with identical case files, despite the fact that the physicians reported (and probably believed) that they had no explicit racial bias.

In fact, white patients are much more likely than black patients to receive thrombolysis treatment, and there are a number of well-documented racial disparities in the treatment of cardiac patients in the United States.  Papers like Green et al., are important not only because they support the usefulness of the IAT, but also because they help explain how subtle and pernicious racial bias can be, and because they propose a quantitative, though imperfect, mechanism for finding it out in ourselves.

Sample IAT images borrowed from Wikipedia.

Blood-Fluke Monogamy

You know how upsetting it is when a celebrity couple to whom you’ve become attached breaks up?  And you know how it’s the worst when they break up because of infidelity?

Then perhaps you can imagine how distressed I was this week when I learned that Schistosoma mansoni, the endoparasitic blood-fluke, wasn’t really monogamous.

Schistosomes are a genus of class Trematoda, or parasitic flatworms.  They are the cause, in humans, schistosomiasis, which kills tens or hundreds of thousands of people every year (depending on whom you ask), and which is the second-most impactful parasitic infection worldwide after malaria.

Unlike other flatworms, schistosomes are dioecious.  During mating, the male, which is much larger, essentially encloses the female in something called, unsexily, a “gynacophoric canal”, where reproduction will occur and where the female will spend the rest of her life.

Or so we thought!  Turns out, blood-fluke couples split up, or, as scientists kind of preposterously put it, “divorce”.  Schistosome divorce was apparently first reported in 2000 by Pica-Mattoccia et al., in the journal Parasitology, but I don’t read Parasitology (although clearly I need to), and so I missed it.

However, my attention fell this week on a 2009 paper by Beltran et al., ‘Adult sex ratio affects divorce rate in the monogamous endoparasite Schistosoma mansoni’.  The authors of this paper found that they could dramatically increase the blood-fluke divorce rate by flooding the bloodstream of infected animals with male schistosomes, thus increasing the number of unpaired, available males.  However, increasing the number of female schistosomes in the bloodstream, i.e. creating a population of unpaired female blood-flukes, had no effect on divorce rates.

There are two possible explanations for this gendered effect.  The first is that the female schistosome is the choosier sex, and so, when there are more available unpaired males from which to choose, some number of paired females will upgrade.  Supporting this view is the observation that there is greater genetic variety among male blood-flukes than among females; this means that a female schistosome may gain a significant advantage from switching mates, while a male probably will not.  For a male schistosome, one female is very like another, and so when you have one, best just to hold on to her; hence the lack of effect on divorce rate when the number of available females was increased.

The other possible explanation is that the free schistosome males act in a predatory manner, and may physically break-up happily joined couples.  Since the male schistosome is much larger than the female, it makes sense that this would only happen when the number of males was increased: the smaller females would be incapable of separating another blood-fluke couple, even if there were more of them.

Of course, these two hypotheses need not be mutually exclusive: as the authors of the study mention in a slightly victim-blaming aside, “the possibility exists that female schistosomes choose not to resist takeover attempts by rival males if they perceive their higher phenotypic and/or genetic value.”

None of this lightens the emotional blow to those of us who had pinned our hopes on blood-fluke monogamy, of course.  It’s over – it was all a lie.  Love is dead.

Image from National Geographic.