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Benveniste’s Experiments and the So-Called “Water Memory” Phenomenon: an Example of Serendipity?

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Benveniste’s experiments – known in the lay press as the “water memory” phenomenon – are generally considered to be a closed case. However, the amount of data generated by twenty years of well-conducted experiments prevents closing the file so simply. An issue, which has been little highlighted so far, merits to be emphasized. Indeed, if Benveniste failed to persuade his peers of the value of his experiments, it was mainly because of a stumbling block, namely the difficulty of convincingly proving the causal relationship between the supposed cause (“informed water”) and the experimental outcomes in different biological models. To progress in the understanding of this phenomenon, we abandon the idea of any role of water in these experiments (“water memory” and its avatars). In other words, we assume that control and test conditions that were evaluated were all physically identical; only their respective designations (labels) differentiated them. As a consequence, labels (“controls” vs. “tests”) and the corresponding states of the biological system (no change vs. change) are independent variables. We show in this article how simple considerations based on probability theory allow to build a probability model where the order of measurements matters. This model provides an alternative explanation to Benveniste’s experiments where water plays no role and where the place of the experimenter is central.
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Benveniste’s Experiments and the So-Called Water Memory”
Phenomenon: an Example of Serendipity?
Francis Beauvais 1, *
1 Scientific and Medical Writing, Viroflay, France
* Correspondence:
Benveniste’s experiments known in the lay press as the “water memory” phenomenon are
generally considered to be a closed case. However, the amount of data generated by twenty years
of well-conducted experiments prevents closing the file so simply. An issue, which has been little
highlighted so far, merits to be emphasized. Indeed, if Benveniste failed to persuade his peers of
the value of his experiments, it was mainly because of a stumbling block, namely the difficulty of
convincingly proving the causal relationship between the supposed cause (“informed water”) and
the experimental outcomes in different biological models. To progress in the understanding of this
phenomenon, we abandon the idea of any role of water in these experiments (“water memory” and
its avatars). In other words, we assume that control and test conditions that were evaluated were
all physically identical; only their respective designations (labels) differentiated them. As a
consequence, labels (“controls” vs. “tests”) and the corresponding states of the biological system
(no change vs. change) are independent variables. We show in this article how simple
considerations based on probability theory allow to build a probability model where the order of
measurements matters. This model provides an alternative explanation to Benveniste’s experiments
where water plays no role and where the place of the experimenter is central.
Keywords: Water memory; High dilutions; Scientific controversy; Serendipity; Experimenter effect.
1. Introduction
Serendipity has been defined as the faculty or phenomenon of finding valuable or agreeable things not sought
for according to Merriam-Webster Dictionary [1]. This word was coined in 1754 by Horace
Walpole as a reference to the fairy tale The three princes of Serendip” from Cristoforo Armeno. The
heroes of this story discovered by chance and sagacitythings they were not looking for. The history
of science is full of such chance discoveries as X-rays, radioactivity or penicillin. In this article, we
will see why Benveniste’s experiments also known in the lay press as the water memory
phenomenon could be a new example of serendipity.
The purpose of this article is not to tell again the entire story with all the details of the scientific
debate and controversies that can be found elsewhere. A complete and systematic account of this
case has been published by the author in a freely available text, both in a French version [2] and in
its English translation [3]. Other points of views can be found in several books or articles [4-12].
The present article is hoped to be an element to reach a successful conclusion and, more
importantly, to open new horizons.
It is important to remember that the experiments we are talking about were spread over almost 20
years (from 1983 to Benveniste’s death in 2003). This saga is too often reduced to the famous
controversy with the journal Nature in 1988 [13, 14]. Moreover, before this controversy, Benveniste
had a position in the scientific community that was far from marginal. His research unit was
affiliated with INSERM, the French national medical research institution, and Benveniste himself
was recognized internationally. The original idea to test extreme dilutions was not from Benveniste
himself who, as a good Cartesian, was rather reluctant initially to test homeopathic medicines, but
was nevertheless open to countercurrent ideas. Indeed, homeopathy is an alternative medicine
which, to put it mildly, is based on ideas from another time and has never proven its efficacy.
B. Poitevin who was a thesis student at that time was also involved in the practice of homeopathy
and proposed to Benveniste collaborations with the homeopathy industry [3, 7]. Thus, the famous
“Benveniste’s experiments” were initially nothing more than research contracts with homeopathy
firms aimed to evaluate some homeopathic medications. It is important to notice that Benveniste’s
team was not the first to test homeopathic medications. What was probably new was that a
renowned laboratory from a public research institution managed such a research topic and that
Benveniste did not hesitate to promote the high-dilution issue into the academic debate.
In the former experiments (years 19831991), high dilutions of various preparations were tested
on human basophils which are cells involved in allergic disorders. Some of the diluted compounds
were used in homeopathic practice, but others were not homeopathic medications (antibodies, for
example). The hypothesis behind these first experiments performed in the 1980’ was that a drug
or a biologically-active compound could apparently continue to manifest some activity after being
highly diluted. Obtaining a specific effect on a biological model with high dilutions was theoretically
impossible given physico-chemical laws. To get an idea, less one molecule of antibody was
theoretically present in the assay after about fifteen serial ten-fold dilutions of the initial sample.
Therefore, the first positive results were received with surprise and skepticism in the laboratory,
including Benveniste himself.
However, the basophil model needed fastidious and time-consuming counting under a microscope
by trained experimenters and required a number of tricks to avoid pitfalls. In addition, the
controversy with the journal Nature prompted Benveniste to consider other biological models. A
physiological model already in use in the laboratory, namely the isolated rodent heart, appeared to
be promising (years 19921998). Changes in the state of this biological model (coronary flow
variations) could be followed live by observers and this new model was therefore more convincing
and demonstrative than the basophil model. The results with high dilutions of various compounds
were confirmed using the isolated rodent heart model [3].
After high dilutions, Benveniste developed from the year 1992 different devices based on
electromagnetism and made of electric coils and electronic amplifiers which were supposed to
“transfer the activity” of biologically-active molecules directly to water samples without the dilution
process [3, 15-20]. The transmission experiments were also supposed to avoid contaminations
that could be responsible for the observed effects. In a further refinement, Benveniste obtained in
1995 experimental data suggesting that the “activity” issued from a biologically-active solution
could be captured and stored in a computer memory. The electric current that passed through a
coil surrounding a biologically-active sample was supposed to be modulated by an electromagnetic
emission from the sample and was digitized before being stored in a computer file. Then the
biological activity” recorded in the file could be diffused via an electric coil to a water sample
initially devoid of any “information”. In another variation of this device, the diffusion of the
“biological activity” was done directly to the biological system via the electric coil without
intermediate water sample. Benveniste coined at this occasion the term of “digital biology”. These
new developments and his new ideas about a future breakthrough in biology and medicine thanks
to digital biology further isolated Benveniste from the scientific community and caused him to
lose his early supports.
The isolated heart rodent model, however, appeared to be difficult to export into other laboratories
because few teams had the experience to use it routinely. Benveniste then developed experiments
based on plasma coagulation that could be easily performed in most laboratories (years 19992001).
This experimental model offered also the possibility of being completely automated. A robot that
performed all steps of an experiment, including the random selection of experimental conditions
and processing of biological samples, was thus developed. Again, the successful results obtained
with this new device convinced Benveniste that he was on the right track. This robot attracted the
attention of the US Defense Advanced Research Projects Agency (DARPA), which commissioned
an expertise [21]. We will talk of the results of this expertise later in the text.
However, if these astonishing results were so obvious, why did Benveniste fail to convince his
peers? The main reason was that he could not get rid of a strange phenomenon that literally
poisoned his experiments, in particularly when he tried to carry out proof-of-concept experiments
involving other researchers he wanted to convince. In a next section, we shall see what this
stumbling block was and why it systematically and constantly stopped Benveniste in his race to the
“decisive experiment”. But first, we need some conventions and definitions to describe these
2. Causal relationships in experimental biology
Suppose that we are interested in a parameter of a biological experimental system that we study in
the laboratory in various experimental conditions. A change of this parameter during an experiment
is noted Δ+ while no change is noted Δ0. Experimental conditions are either control condition
(noted C) or test condition (noted T). If we notice that C is always associated to Δ0 and that T is
always associated to Δ+, we conclude that a (strong) relationship exists between the experimental
conditions (designated by the labels C and T) and the system states 0 and Δ+ obtained after
measurement). By convention, we name this relationship as “direct”; the “reverse” relationship
associates C with Δ+ and T with Δ0.
The purpose of experimental biology is precisely to reveal such relationships. Indeed, modern
biology is based on causal relationships where X induces Y which in turn induces Z, etc. For
example, if we inject a solution of histamine (test T) into the skin, we obtain a local edema at the
place of the injection surrounded by a red area indicating that the system state has changed +). If
we inject only the carrier solution without the active compound (control C), there is no change of
the system state 0). Since the experimenter can choose at will the experimental conditions, we
conclude that histamine induces a skin response in a causal relationship. Describing the complete
chain of events that occurs locally from the initial cause (histamine) to the final effect (edema/skin
redness) is a typical research program. In this description, the principle of locality, which states that
a physical object is only influenced by its immediate environment, is respected. As a matter of fact,
most biologists do not even imagine that other types of relationships could be involved or simply
envisioned. In Benveniste’s experiments, the principle of locality was also implicit since the
supposed causes (“structured water” or electromagnetic fields) were thought to exert their effects
locally. But are local cause-and-effect relationships the only ones possible? A clue is provided by
the results of Benveniste’s experiments themselves as we will see in the next section.
3. When the scientific interest is not where it was supposed to be
We must now explain in detail why the field of research based on “water memory” was stopped in
its development in spite of well-conducted experiments and seemingly convincing results. In short,
the central problem was the impossibility of proving the causal nature of the relationship observed
between experimental conditions and states of the biological system. Another reason which is
related to it, as we shall see was the difficulty of having these experiments replicated by
independent teams.
First, we will explain why Benveniste was convinced with sound arguments that he had
discovered “something”. For this purpose, we will describe three experimental settings: open-label
experiments, internal blinding and external blinding (Figure 1).
Figure 1. Configurations for (A) open-label, (B) “internally” blinded and (C) “externally” blinded
experiments. The “internal” or “external” supervisor provides the experimenter with “control” and
“test” samples whose labels have been masked (blind controls/tests). After the experiment is done,
the experimenter returns to the supervisor the outcomes which have been obtained (change or no
change of system state noted Δ0 and Δ+, respectively) for each control or test. The supervisor
assesses the rate of success” of the experiment (i.e., how many direct relationships have been
obtained with the masked samples: “control” associated with “no change” or “test” associated with
Open-label experiments. Figure 2 reports results on two devices that were used in parallel in 1992
1996. There was no blinding between the two measurements; these duplicate experiments were
done in order to consolidate the data. To analyze these results, we do not care of the nature of the
samples tested (C or T, high dilutions, electronic transmission, etc.) We only consider the
concordance of the outcomes obtained with the two devices. We observe a high correlation of
these measurement (the most probable pairs of outcomes are Δ0/Δ0 and Δ+/Δ+). The important
point is for the pairs Δ++. Indeed, these correlated pairs are a strong argument in favor of
Benveniste’s theses. They indicate that there is “something” that occurs in these experiments.
Whether one agrees or not with Benveniste’s views on the interpretation of these experiments, the
source of correlations needs to be explored. This is precisely the purpose of the present article.
Internal blinding. The data of Table 1 indicate that these correlations persist after an internal blinding.
Internal blinding means a blinding of labels of the samples to be tested (controls and tests) by a
colleague who interacts with the experimenter in the laboratory (Figure 1). These data show that
the pairs C/Δ0 and T/Δ+ are the most probable, thus strongly suggesting a direct relationship
between labels and system states. Again, we understand why Benveniste defended his research with
determination and why he could not resign himself to forgetting his results in a drawer as many
suggested he should do [3].
External blinding. To share his experimental results and to convince his peers, Benveniste organized
“public demonstrations” carried out in the presence of an audience of colleagues and observers
who did not belong to the laboratory. These demonstrations were designed in order to give a
definitive confirmation on the reality of these experiments and to establish their scientific
interest. For this purpose, an experimental protocol was written and then a study report with all
raw data was shared with participants [3]. During a typical session of demonstration, controls and
tests were prepared (either high dilutions for the early demonstrations or computer files later) in
another laboratory than Benveniste’s laboratory. These samples were prepared under supervision
of all participants. Then, the labels of the control and test samples were replaced by a code by
participants not belonging to Benveniste’s laboratory. Some samples were also kept unblinded to
check that everything was fine in non-blinded conditions. Then, Benveniste’s team recovered all
samples and tested them in its laboratory within the next days. After all measurements had been
done, the outcomes 0 and Δ+) corresponding to all blind samples were sent to the external
supervisor who detained the code (Figure 1). This supervisor compared the two lists: list of labels
(C vs. T) under a code name and list of outcomes (Δ0 vs. Δ+) and assessed whether outcomes were
as expected (C associated to Δ0 and T associated to Δ+).
Figure 2. Open-label experiments made in duplicate on two experimental systems (dates: 19921996;
experimental model: changes of coronary flow in isolated rodent heart; N=574 pairs of
measurements) [22]. (a) In this figure we do not care about the label (control or test), only the
concordance of the system states (Δ0 or Δ+) on the two devices matters. We also do not care of the
method used by Benveniste’s team to produce controls and tests (high dilutions, electronic
transmission, digital biology, etc.) The data obtained in devices A and B appear to be highly correlated.
(b) If the value measured on the device A is < 10% (no change; Δ0), then the probability of a value
< 10% on device B is 0.93. (c) If the value measured on the device A is 10% (change; Δ+), then the
probability of a value 10% on device B is 0.89. Note that scales are logarithmic.
110 100
Measure with device A
Measure with device B
110 100
Measure dev ice B
(for measures dev ice A < 10%)
93% 7%
110 100
Measure dev ice B
(for measures dev ice A > 10%)
11% 89%
Measurement with device A
Measurement with device B
Measurement with device B
(for measurements with device A < 10%)
Measurement with device B
(for measurements with device A 10%)
Table 1. Experiments with internal blinding (dates: 19931997; experimental model: changes of coronary
flow in isolated rodent heart) [22].
Experimental situations
System states after internal blinding
No change (Δ0)
Change (Δ+)
Interim blinding between 1st and 2nd measurement of the same sample
No change (Δ0) for 1st measurement
(2nd measurement)
(2nd measurement)
Change (Δ+) for 1st measurement
(2nd measurement)
(2nd measurement)
Blind labels
Label “Control”
Label “Test”
Table 2. Digital biology” experiments with external blinding (dates: 1996-1997; experimental model:
changes of coronary flow in isolated rodent heart) [3]. In these experiments (public demonstrations with
external supervisor), system states are distributed at random in contrast with open-label or internal blinding
experiments presented in Figure 2 and Table 1). Therefore, no relationship can be established between labels
and system states. If a causal relationship existed, one would expect square labels to be superimposed on
round labels of the same color. These “mismatches” were considered by Benveniste as experimental failures.
In the present analysis, we consider that “successfuland “failedexperiments are the two faces of the same
phenomenon. Note that the observation of changes of state (noted ●), whatever their place, requires an
explanation. Indeed, according to current scientific knowledge, we would expect to observe only no change of
system state (------○, etc.) The scientific context and the experimental details of these experiments
can be found in [3].
February 27, 1996
Labels (= C, = T)
States (= Δ0, = Δ+)
May 7, 1996
Labels (□ = C, = T)
States (○ = Δ0, = Δ+)
June 12, 1996* (two parts)
Labels (□ = C, = T)
-------■ / ■-------
States (○ = Δ0, = Δ+)
-------● / ●-------
September 30, 1996*
Labels (□ = C, = T)
States (○ = Δ0, = Δ+)
November 4, 1996
Labels (□ = C, = T)
States (○ = Δ0, = Δ+)
December 4, 1996*
Labels (□ = C, = T)
States (○ = Δ0, = Δ+)
September 27, 1997*
Labels (□ = C, = T)
States (○ = Δ0, = Δ+)
* The number of controls and tests to be evaluated was not known from the experimenter.
There was an internal blinding between the series of measurements for all these experiments (except
November 4, 1996) in order to verify the consistency of the outcomes. Note that experiments done in
1996 were performed on two apparatus to confirm results as described in Figure 2.
A series of experiments with external blinding is presented in Table 2. It is easy to see that the
relationship between labels and system states was lost in the experiments with external blinding.
Indeed, in this table, square labels are not superimposed on round labels of the same color, but
distributed at random. There was still an effect, i.e., the observation of changes of system state
+), but not at the place where they were supposed to be.
To explain these oddities, Benveniste put forward various hypotheses (human errors for label
allocation, water contamination, electromagnetic pollution, “remnant activity” in the apparatus,
spontaneous “jumps of activity” from one sample to another, etc.) These mismatches prompted a
technological race in which Benveniste engaged to rule out these supposedly disturbing external
events. In contrast, in the present analysis, we consider that “successes” with open-label/internal
blinding experiments (A and B in Figure 1) and “failures” with external blinding experiments (C
in Figure 1) are the two faces of the same phenomenon. Actually, we think that these disturbing
events are the scientific fact of this story. One important point must again be stressed. If
Benveniste’s experiments were of no scientific interest, no change of the system state 0) should be
observed under either control or test conditions.
In order to take into account all these aspects of Benveniste’s experiments i.e., both “successes”
and “failures” we propose to abandon the idea of any role of water (“water memory” and its
avatars) and to begin by revisiting the notion of relationship in experimental biology.
4. Probabilities of events vs. probabilities of their relationships
To progress in the understanding of the phenomena reported by Benvenistes team, we consider
that all various procedures for diluting solutions or electronic stuff aimed to “transfer biological
activity” make no sense. We state that they have no more value than a ritual or meaningless
We assume that the control and test conditions are physically identical and that their difference lies
only in the respective labels (“control or test) that they randomly or subjectively receive.
Consequently, we consider that labels (“control” vs. “test”) and corresponding states of the
biological system (no change vs. change) are independent variables. Two events are
independent if the occurrence of one does not affect the probability of occurrence of the other.
Yet a relationship between labels and system states was observed in Benveniste’s experiments. This
is precisely the problem we have to solve.
Figure 3. Law of total probability for two sets of independent variables: experimental conditions
(control C vs. test T) and system states (no change of system state, Δ0 vs. change of system state, Δ+).
According to the law of total probability, the sum of the probabilities of the four possible branches
is equal to one. Note that P (C) + P (T) = 1 and P 0) + P +) = 1.
For this purpose, we suppose a simple situation where labels (control, C; test, T) and system states
(no change, Δ0; change of system state, Δ+) are independent variables with corresponding probabilities
to be observed (Figure 3).
In this case, the law of total probability is:
P (C) × P (Δ0) + P (C) × P (Δ+) + P (T) × P (Δ0) + P (T) × P (Δ+) = 1
(Eq. 1)
with P (C) + P (T) = 1 and P (Δ0) + P (Δ+) = 1
For reasons that will appear later, we write:
i. P (C) = a2 and P (T) = b2 where a and b are real numbers
ii. P (Δ0) = cos2 and P (Δ+) = sin2
With these conventions, the law of total probability described in Eq. 1 becomes:
(a.cos)2 + (b.sin)2 + (b.cos)2 + (a.sin)2 = 1 (Eq. 2)
We add δ = 2ab.cossin 2ab.cossin = 0 to Eq. 2:
[(a.cos)2 + (b.sin)2 + 2ab.cossin] + [(b.cos)2 + (a.sin)2 2ab.cossin] = 1 (Eq. 3)
δ is equal to zero, but its addition allows to reorganize the equation with two remarkable identities:
(a.cos + b.sin)2 + (b.cos a.sin)2 = (a.cos)2 + (b.sin)2 + (b.cos)2 + (a.sin)2 = 1 (Eq. 4)
Eq. 4 shows that there are two possible writings of the law of total probability for two independent
dichotomous random variables. We can also represent this equation graphically (Figure 4). We
recognize a logical structure similar to an experiment where a photon “interferes with itself” such
as in a two-slit Young’s experiment or in a Mach-Zehnder interferometer.
Figure 4. Law of total probability for two independent dichotomous random variables (labels and
system states). Probabilities of direct and reverse relationships are calculated as the square of the sum
of the probability amplitudes of the two possible paths: P (direct)I = (a.cos + b.sin)2 and P (reverse)I
= (b.cos a.sin)2. In contrast, when a path measurement is performed, probabilities are calculated
as the sum of the squares of the probability amplitudes of paths: P (direct)II = a2cos2 + b2sin2 and
P (reverse)II = b2cos2 + a2sin2.
Note that P (C) and P (T) are based on the a priori knowledge of the experimenter about the number of
controls and tests to be evaluated (this point is important for blind experiments).
In the same logic as an interference experiment, the left side of Eq. 4 corresponds to the absence
of path measurement (superposition with interference pattern), while the right side corresponds to
path measurement (no interference pattern and labels associated randomly to system states). The
real number a (resp. b) can be assimilated to a “probability amplitude” while a2 (resp. b2) is the
corresponding probability.
Therefore, there are two possible definitions of P (direct) according to path measurement or not:
P (direct)I = (a.cos + b.sin)2 if no path measurement (Eq. 5)
P (direct)II = (a.cos)2 + (b.sin)2 if path measurement (Eq. 6)
Similarly, there are two possible definitions of P (reverse):
P (reverse)I = (b.cos a.sin)2 if no path measurement (Eq. 7)
P (reverse)II = (b.cos)2 + (a.sin)2 if path measurement (Eq. 8)
P (direct)II can be also written by using conditional probabilities where P (XY) means probability
of the event X knowing that the event Y has occurred:
P (direct)II = a2cos2 + b2sin2 = P (C) × P (Δ0C) + P (T) × P (Δ+T) (Eq. 9)
We see clearly with Eq. 9 how P (direct)II is related to the measurement of the path (from a point of
view outside the laboratory). It could be tempting to use conditional probabilities to rewrite Eq. 3.
However, the difference δ of the “interference terms” is not egal to zero in all cases if conditional
probabilities are considered:
))2))2 (((( 00 CPTPabTPCPab ++ =
(Eq. 10)
Indeed, δ = 0 only if P (Δ0C) = P (Δ0T) = P (Δ0) and if P (Δ+T) = P (Δ+C) = P (Δ+). In other
terms, the two “interference terms” cancel each other only if labels (C or T) and system states (Δ0
or Δ+) are independent variables as initially postulated in Eq. 1. The “same randomness” must
operate for system states, regardless of the paths C and T (see Figure 4).
Another important remark is for a particular case of Eq. 5 and Eq. 7:
If a = cos and b = sin, then P (direct)I = 1 and P (reverse)I = 0 (Eq. 11)
P (direct)I = 1 means that each label C is associated with Δ0 and each label T is associated with Δ+.
Note that there is no constraint on the order of the pairs (C, Δ0) and (T, Δ+); there is only a
constraint on probabilities.
We can also represent Eq. 4 geometrically in a Cartesian coordinate system (Figure 5). In this
representation, two bases, one for the labels (C and T) and the other for the relationships (direct
and reverse), are necessary. Cos and sin are the probability amplitudes of Δ0 and Δ+,
respectively. The two bases are related by a rotation matrix (with >0 in the clockwise direction):
(Eq. 12)
Each point can be projected (i.e., expressed) in one basis or the other (C/T or direct/reverse). The
projection of a point of the unit circle leads to different results if projected directly to the
direct/reverse basis or if projected first on the C/T basis and then on the direct/reverse basis. This
last case (projection first on the C/T basis) is equivalent to a which-path measurement. For
0, direct/reverse relationships and control/test labels are said “non-commutative variables
because the final outcomes depend on the order of the measurements/observations.
Figure 5. Representation of Equation 4 in a Cartesian coordinate system. The point Ω with
coordinates a and b in C/T basis has coordinate a.cos + b.sin on the direct relationship axis; the
associated probability is therefore (a.cos + b.sin)2. If the relationship is assessed separately for C
and T (path measurement), the coordinates on the direct relationship axis are a.cos for C and b.sin
for T, with the corresponding probability for direct relationship equal to (a.cos)2 + (b.sin)2. For
clarity, projections for reverse relationship are not presented on the figure.
In summary of this section, Eq. 4 (or equivalently Eq. 12) describes the probabilities of events vs.
the probabilities of their relationships. In this equation, each relationship (direct or reverse) is
considered in its entirety, not as the sum of its parts.
5. Are all random systems suitable?
Until now, we did not impose any condition for the experimental system. However, one may ask
whether it would be possible to replace the biological system with any other system that produces
randomness such as a random draw of black and white balls from a bag or a dice roll.
All experimental systems used in Benveniste’s experiments were biological systems involving cells,
isolated organs or enzymatic reactions. Such systems are submitted even at rest to tiny random
fluctuations. Therefore, in Figure 5, we can assume that the angle slightly fluctuates around zero.
As a consequence, the probability to observe Δ+, which is P (Δ+) = (sin)2, is not equal to zero,
even though this probability is low. The transition of the system state from no change to change
(Δ0 Δ+) is therefore a possible event.
In contrast, suppose a bag full of many white balls (Δ0) and only a few black balls (Δ+). The
probability to draw a black ball is low and white balls cannot transform into black balls. In other
words, the transition of the system state from no change to change (Δ0 → Δ+) is not a possible event.
Therefore, such a random system would not allow results comparable to those observed in
Benveniste’s experiments.
6. The emergence of non-local correlations
If we want to describe Benveniste’s experiments using Eq. 4 or its geometrical equivalent, we
need to explain how this equation can be in a non-trivial form (i.e., with a non-zero value of ).
In other words, how is it possible that each test label (T) is matched with a change (Δ+) of the
experimental system? Indeed, in the previous section, we have seen that Δ+ is a possible event,
but with a low probability. We propose that non-trivial values of have their origin in prior
observations of repeated experiments where labels and system states exhibit a direct
relationship. This can be done by the experimenter observing, for example, a “classical”
experiment. By classical, we mean an experiment with a classical cause” (e.g., molecules of a
biologically-active compound added to the biological system at the usual micromolar
concentration). Note that this was the case in Benveniste's laboratory; the experimental systems
were routinely used for “classical” experiments and biologically-active compounds were also
used to verify that the systems were functioning correctly under “classical” conditions.
Suppose an experiment for which labels and system states are involved in a direct relationship,
i.e., the only possible pairs of labels and system states are (C=, Δ0=○) and (T=■, Δ+=●). We
name n1 the number of trials where the experimenter observes (□○) and n2 the number of trials
where the experimenter observes (■●) (with n1 + n2 = N). The series of outcomes that the
experimenter records are, for example: (□○), (■●), (■●), (□○), (□○), (■●), (■●), (□○), (■●), etc.
where the probability of C= is a2 and the probability of T= is b2.
When the number N of trials increases, n1/Na2 and n2/N b2. Since a2 + b2 = 1, the point
Ω with coordinates
Nna 1
Nnb 2
can be represented on the unit circle of Figure 6. Ω
allows defining the position of the axis for direct relationship and the perpendicular axis for
reverse relationship. It defines also the value of since sin = b. The abscissa of Ω on the
direct relationship” axis is equal to a.cos + b.sin = a.a + b.b = 1.
Figure 6. Defining the position of the direct/reverse basis. (A) After repeated experiments (e.g. with
a “classical” biologically-active compound), the label C (probability a2) is systematically associated to
Δ0 and T (probability b2) is systematically associated to Δ+. This relationship can be represented on
the unit circle by the point Ω with coordinates a and b. (B) Ω is not an experimental result, but is a
limit (
) which integrates results of repeated correlation measurements between
labels and system states in “classical experiments”. This point allows defining the axis of direct
relationship and a new basis (direct/reverse). In this new basis, the direct relationship is recognized
as such, independently of the elements (labels and system states) that permitted to define it.
Therefore, the model shows how a non-classical relationship can emerge after repeated
observations of correlated variables. This step can be described as learning or conditioning and we
use the term conditioning relationship to describe these repeated observations of correlations
which “condition” the experimenter’s cognitive state. Note that the underlying nature of the
conditioning relationship does not matter (it can be related to a local cause or be any type of
7. Self-sustained non-local correlations
The question arises now how the non-local correlations persist after the conditioning relationship
is no longer observed. In Benveniste’s experiments, the removal of the local cause could be
performed, for example, by extremely diluting biologically-active molecules.
First, we need to define the physical structures that underlie the logic we propose. Obviously, the
physical support for Δ0 and Δ+ is the experimental biological system. Concerning control and test
conditions (for example, samples of “informed water”), we have seen that we consider that they
are physically identical and the experimental differences are founded only on their denominations
(labels). The latter have the meaning that the observer attributes to them and consequently the difference
between a control experiment (C) and a test experiment (T) is based on differences in the cognitive
structures of the experimenter. Therefore, labels are subjective concepts which are embodied in the
experimenter’s cognitive structures. In the first part of Eq. 4, the experimenter’s cognitive structures and
the experimental system are entangled in a relationship and form an inseparable whole.
During the conditioning (cond.) phase, a direct relationship is observed which obeys to:
P (direct)cond. = P (C) × P (Δ0C) + P (T) × P (Δ+T) = a2 × 1 + b2 × 1
Since a2 + b2 = 1, we can write a = cos and b = sin.
Consequently, P (direct)cond. = a2 + b2 = cos.cos + sin.sin (with =).
After the conditioning phase, everything happens as if an imprint or a ghost of the conditioning
relationship persists in the universe of all possible probability amplitudes of system states and
experimenter’s cognitive states (represented by and , respectively). This imprint takes the form
of a probability amplitude equal to cos.cos + sin.sin (with =) and is nothing more than the
abscissa of the point Ω on the “direct relationship” axis in Figure 6. Therefore, the conditioning
relationship and its imprint (“ghost relationship”) have the same mathematical form (cos.cos +
sin.sin with =), but the first is a classical probability while the second is a probability amplitude.
Even once the initial conditioning relationship is no longer observed, a direct relationship between
system states and experimenter’s cognitive structures persists through self-sustained non-local
correlations (Figure 9). Indeed, the local or non-local nature of the conditioning relationship does
not matter. In other words, the ghost relationship that has been produced during the
conditioning phase is self-sustained thanks to non-local correlations.
Figure 9. Self-sustained non-local correlations. Labels C () and T () which are embodied in the
experimenter’s cognitive structures are characterized by the angle of their probability amplitudes
(cos=a and sin=b) and states Δ0 () and Δ+ () of the experimental system by the angle of their
probability amplitudes (cos and sin). With = , non-local correlations are established between
labels and system states (these latter can be said to be “in phase”). The establishment of non-local
correlations needs first a “conditioning relationship. Then, the relationship is self-sustained:
repeated observations of the direct relationship by the experimenter reinforce (via conditioning) the
mathematical equality between and , which in turn guarantees the direct relationship (via non-
local correlations).
P (reverse) = 0
P (direct) = 1
(cos.cos + sin.sin)2 + (sin.cos cos.sin)2 = 1
8. Open-label vs. blind experiments explained
We will now examine how the formalism we describe could explain all aspects of Benveniste’s
Open-label experiments/internal blinding. Both the labels (C or T) and the system states (Δ0 or Δ+) are
random events that are governed by their respective probabilities. The records of these events by
the experimenter are measurements: during open-label experiments, the experimenter measures
first the label (C or T) and then the system state (Δ0 or Δ+); during internal blinding, the
experimenter measures first the system state and then the label (Figure 1). Internal blinding of
labels can be performed by an automatic device or by a colleague present in the neighborhood of
the experimenter. In this case, the blinding device (machine or human) is nothing more than a part
of the experimental system.
We see in Eq. 4 that the order of probability amplitudes referring to labels and states of the system
does not matter (labels and system states could be said “entangled”). Therefore, in our modelling,
there is no difference in nature between open-label experiments and experiments with internal
In addition, we have seen that P (direct)I = 1 and P (reverse)I = 0 for sin = b and cos = a, thus
showing that the probability amplitudes of labels and outcomes are merging when the correlation
is maximal. C and Δ0 are combined in a single path (Path 1) and T and Δ+ are combined in another
single path (Path 2) (Figure 10).
Figure 10. Consequences of self-sustained non-local correlations in open-label, “internally” blinded
and “externally” blinded experiments. (A) The probability of a direct relationship is maximal with
a = cos and b = sin. Any couple (a, b) defines an angle and P (direct) I = 1 if = ; labels and
system states can be said in phase. In this case, C and Δ0 are combined in a single path (Path 1) and T
and Δ+ are combined in another single path (Path 2) with P (direct)I = (a.a + b.b)2 = 1 (no path
measurement; open-label experiments or “internal” blinding) and P (direct)II = (a.a)2 + (b.b)2 (path
measurement; external blinding). (B) Geometrical representation of the two bases (Direct/Reverse
and Path 1/Path 2) on a unit circle.
The internal blind experiments made always a deep impression on Benvenistes team because they
strongly suggested that the effects of high dilutions or digital biology were a tangible reality (“it
works”). It was these experiments that convinced Benveniste that he had to persevere and that one
day his theories and discoveries would be recognized. However, we have seen in this article that a
logic other than a classical causal relationship is possible. This does not mean that causality is absent
of this description, but causality operates at a higher level, i.e. at the level of the relationship. Internal
blinding in Benveniste’s experiments should be no more surprising than internal blinding for a
classical causal relationship. We must simply admit that what is at stake is not the physical
composition of the sample to be tested they are all identical but its designation (label C or T).
This label can be “measured” like any other parameter and a probability can be attributed to it.
External blinding. The external blinding which constituted the “stunning block” described above
is simply explained by a which-path measurement with the laboratory in its ensemble being
considered as an “interferometer” (Figure 9). Everything happens as if the external supervisor is
making a measurement of the path from the outside by exchanging data on labels and system states
(Figure 1). It is the external supervisor who provides the experimenter with coded samples to be
Finally, the external supervisor receives the records of the corresponding system states and,
after unblinding, establishes whether the experiment is “successful”. Therefore, we can understand
why these experiments were so disturbing for Benveniste's team. Indeed, even if these experiments
were considered as “failures”, they nevertheless indicated that “something” happened because
changes of the system were recorded (but randomly associated with C and T). The experiments
seemed to be going “crazy”. In the “classical” and local approach of Benveniste’s team, there was
no place for such outcomes. Therefore, external causes of disturbances were searched leading to a
technological race for the “perfect” and indisputable experiment.
The experimental outcomes and underlying mathematics for open-label and blind experiments are
summarized in Table 3.
Table 3. Summary of experimental results according to the conditioning of the experimenter and path
measurement. Non-commutativity of the variables (labels and systems states) is the underlying logic
explaining the outcomes of the different experimental situations.
No conditioned experimenter
Conditioned experimenter
No path measurement
by external supervisor
Path measurement
by external supervisor
Order of the experiments
Labels ( = C, = T)
States ( = Δ0, = Δ+)
1 2 3 4 5 6 7 8 9 10 etc.
□ ■ ■ □ ■ □ ■ □ ■ □
○ ○ ○ ○ ○ ○ ○ ○ ()
1 2 3 4 5 6 7 8 9 10 etc.
□ ■ ■ □ ■ □ ■ □ ■ □
○ ● ● ○ ● ○ ● ○ ● ○
1 2 3 4 5 6 7 8 9 10 etc.
□ ■ ■ □ ■ □ ■ □ ■ □
● ○ ● ● ● ○ ○ ● ○ ○
is a rare but possible event
is no more a rare event
at random places
(interpreted as “jumps of activity”
by Benveniste’s team)
P (direct)
P(□) = a2
= (a.cos + b.sin)2 = 1 a
(probability amplitudes add up)
= (a.cos)2 + (b.sin)2
= a4 + b4
P (reverse)
P(■) = b2
= (b.cos a.sin)2 = 0 a
(probability amplitudes cancel)
= (b.cos)2 + (a.sin)2
= 2a2b2
a For cos = a and sin = b
9. Discussion
In this analysis of Benveniste’s experiments, details on the biological models and the different
methods used to “inform” water are left aside. Only the logical structure of the outcomes is
analyzed. In order to review these experiments with a renewed perspective, a radical choice has
been done by considering that control and test conditions that were evaluated in Benveniste’s
experiments differed only by their specific names (labels). Indeed, we previously noted that
whatever the process used to inform water (high dilutions, direct electromagnetic transmission,
record on a computer memory, etc.), the effect size whatever the experimental system was of the
same order of magnitude [3, 22]. This suggested that an alternative unique explanation
By doing so, the experimenter prevents the random selection of each label (C or T) and conditional probabilities
apply (See Eq. 9). This action is equivalent to closing one of the two slits in Young’s experiment.
independent of the different methods and devices used might be at work. This radical choice
seems to make any explanation of Benveniste's experiments even more problematic. Indeed, how
could a relationship be observed if the samples tested are all physically comparable to simple
For this purpose, the mathematical formalism starts from scratch with the definition of the total
probability law for two dichotomous independent variables, namely labels (which designate the
experimental conditions, either control or test) and the corresponding system states. We easily
obtain Eq. 4 where the left and right parts can be considered as complementary. They are the two
opposite poles of the same reality, but both are necessary to understand the logic of Benveniste’s
experiments. The formalism requires a description from the outside where the laboratory and its
content behave with the same logic as an “interferometer”. The “causality” of the relationship
described by the present formalism is only apparent since measurement of the “path” destroys the
correlations: labels (C vs. T) are then randomly associated with system states (no change vs.
change). This is precisely the description of the Benveniste’s stumbling block.
The model we propose adopts some mathematical concepts from quantum physics (e.g., non-
commutativity, probability amplitudes, superposition, “which-path” measurement, entanglement).
Nevertheless, there is no quantum physics in this model, only quantum-like logic. In the field of
experimental psychology, quantum-like logic has been used with success in a new approach named
quantum cognition. Cognitive processes such as decision making, judgment, memory, reasoning,
language or perception, not adequately described by “classical” probabilities, are modelled with
mathematical quantum-like tools, thus better fitting experimental data [23, 24]. In quantum
cognition, combining abstract concepts is associated to quantum-like phenomena such as
superposition and interferences, as it is the case in our modelling. However, in quantum cognition,
all processes are supposed to be limited to brain, while our modelling involves not only the
experimenter’s cognitive structures, but also an experimental biological system. Nevertheless, there
is nothing magical since there is no “action” of the mental structures on the experimental system
(labels and system can be said as “entangled”). If one tries to demonstrate a causal relationship or
to use the correlations in a causal way to perform an action (e.g., giving an order or sending a
message), the direct relationship between the experimenter and the system breaks down.
The source of the non-classical correlations between the experimenter and the experimental system
is to be found in the complementarity of labels and system states as separated objects and their
relationship perceived as a unique entity. A parallel can be drawn with the wave-particle duality
where either waves or particles are detected depending on the measuring device. Particles and
waves are classical concepts that we can apprehend with our measurement tools, our senses, our
language and our usual concepts to describe reality. Depending on the experimental context, either
wave or particle descriptions are appropriate (they are said complementary). Thus, light behaves as
waves when it does not interact with matter and behaves as particles (i.e., photons) when it interacts
with matter (for example, with a measurement apparatus). These two descriptions particles or
waves are incompatible pictures of reality (we cannot think about them simultaneously).
Nevertheless, both descriptions are necessary to account for quantum or quantum-like phenomena
when we use our usual concepts about reality. In the formalism of this article, experimental
objects (C, T, Δ0 and Δ+) are described as “particles” while the relationship between them is
described as “wave”. Thus, P (direct)II = (a.cos)2 + (b.sin)2 is a particle description of the
experimental situation; P (direct)I = (a.cos + b.sin)2 is a “wave” description which differs from
P (direct)II by the interference term 2ab.cossin. Interferences are characteristic of wave interactions
that can be constructive or destructive according to the values of the amplitudes of the waves that
are superposed at a given point. Thus, to calculate P (direct)I, the two amplitudes a.cos and b.sin
are added (amplitudes can be positive or negative). In quantum or quantum-like formalism,
amplitudes are probability amplitudes which have no correspondence in the reality. Indeed, these
waves are not waves of matterbut waves of probability. The connection with reality is done
via probabilities, which are obtained after squaring the sum of all probability amplitudes that concur
to the outcome (direct or reverse relationship in the present case).
The crucial element of our description of Benveniste’s experiments is how to explain the emergence
of the direct/reverse basis. For this purpose, we propose that repeated observations of a
“classical” relationship, for example structure the system formed by the cognitive structures of
the experimenter “extended” with the experimental system. This can be described as learning or
conditioning. It is important to underscore that this combination of a label with the corresponding
system state is not a simple addition or juxtaposition. Indeed, during this process, the relationship
is recognized as such (i.e., independently of its constitutive elements). These considerations are
reminiscent of Gestalt theory. According to this theory, objects are perceived as a whole or as a
form (Gestalt) by the mind and not as the sum of their parts; the whole has therefore its own
independent existence [25, 26].
The automation of the experimental process is not necessarily the solution to avoid an
“experimenter effect”. As previously said, Benveniste’s team set up a robot analyzer (based on
fibrinogen coagulation) that required only to push one switch to launch an experiment. The choice
of control and test conditions was randomized by a computer and the experimenter was informed
of the results when the experiment was finished. An expertise of this apparatus was mandated by
the DARPA and was performed in Summer 2001 in a US laboratory at Bethesda [21]. In their
report, the multidisciplinary team concluded that they did not observe anything abnormal about
the robot analyzer and that the experiments they supervised in the presence of Benveniste's team
seemed to confirm the concepts of “digital biology”. They noted, however, that the presence of
the experimenter dedicated to these experiments seemed to be necessary for the expected results
to be observed. They also reported that after the departure of Benveniste’s team, no effect at all
(i.e., no change of system state whatever the label) could be observed using this robot. The existence
of unknown “experimenter factors” was suggested by some members of the supervisory team [21].
The final conclusion was that the alleged phenomenon based on “digital biology” was not
replicable. Note that this failure was different from the failures reported above with external
blinding. Here, what has been raised as a concern in the DARPA expertise was the requirement of
a trained/conditioned experimenter to record a change of the system state. The relationship
between conditioned vs. non-conditioned experimenter, on the one hand, and “success” vs.
“failure” of the experiment, on the other, appears to be the actual causal relationship that was at
work during Benveniste’s experiments.
With this latest episode of DARPA expertise, which concludes the “water memory” saga, it is
unreasonable to persist in considering as Benveniste and his supporters did that high dilutions,
“water memory” or “digital biology” constituted a major breakthrough in biology and medicine. If
plain causal relationships were really at work, the existence of “water memory” and its avatars
would certainly have been demonstrated without the difficulties we described. It is even possible
that such non-classical correlations between labels and biological systems occur in other
laboratories unwittingly to the experimenters who think that they have evidenced a “real” (causal)
relationship. We can also hypothesize that the lack of replicability of certain experiments could be
explained by such a non-classical phenomenon. If this type of phenomenon is suspected,
experiments with an external supervisor can help clarify. This modelling could also be useful in the
description of the placebo effect or in some alternative medicines.
In conclusion, we have seen in this article how simple considerations based on probability theory
led to describe non-classical correlations involving the experimenter. This probabilistic modelling
allows to propose an alternative explanation to Benveniste’s experiments where water plays no role
and where the place of the experimenter is central. All aspects of Benveniste’s experiments are
taken into account in this modelling, including the weird stumbling block. This obstacle prevented
proving the causal relationship between the information supposedly stored in water and the
corresponding effect observed on the biological system. Nevertheless, because Benveniste
persisted in his quest for the definitive experiment that would convince everyone, he has
transmitted to us a corpus of well-conducted experiments. For this reason, we must express our
gratitude to him. He was pursuing what seems today a chimera, but his experimental data allows
us to suggest the possible emergence of non-classical correlations between an experimenter and a
biological system. These correlations mimic classical relationships, but are not causal because no
local effect is involved. Causality is not absent, however, but the cause of these correlations is at
the level of the experimenter’s cognitive states and not at the local level of the experimental
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Declaration of conflicting interests
The author declares that there is no conflict of interest.
The author received no financial support.
ResearchGate has not been able to resolve any citations for this publication.
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