This study follows up on a recent discussion about interaction information [1,2,3,4]. In that discussion, I argued that the concurrent generation of probabilistic entropy (with “the arrow of time” ) and redundancy (against the arrow of time) in complex systems could be made measurable by considering the Q-measure as an imprint of meaning processing on information processing . If a modeling system, for example, feeds back from the perspective of hindsight on the modeled one by providing the latter with meaning, a complex interaction between information and redundancy generating mechanisms can be expected. Might it be possible to measure the outcome of this balance as a difference?
Krippendorff  responded to this question for another interpretation of Q by elaborating the formula R = I – Q (or equivalently Q = I – R). In this formula, R indicates redundancy which, in Krippendorff’s formulation, is generated by a (Boolean) observer who assumes independence among three (or more) probability distributions, whereas IABC→AB:AC:BC measures the (Shannon-type) interaction information generated in the system in addition to the lower-level (i.e., bilateral) interactions. In other words, R is generated at the level of the model entertained by an observer because of the assumption that the probability distributions are independent. Given this assumption of independence, the algebraic derivation of Q is valid because circular relations among the variables cannot occur under this condition and all probabilities add up to unity. However, R measures the error caused by this assumption if the distributions are not independent.
In this contribution to the discussion, I shall argue that this possibility of using I and Q to measure R enables us to address the question of how to measure the impact of the communication of meaning in a system which provides meaning to the events by entertaining a model. The Boolean observer who makes an error can be considered as the simplest case of a model maker. Using Q, the error in the expectations can be quantified (in terms of bits of information) in addition to interaction information IABC→AB:AC:BC in the modeled system. In a final section, I provide empirical examples of how one can measure, for example, the effects of intellectual organization as an order of expectations of textual organization in interactions among (three or more) attributes of documents.
2. Dually Layered Systems Entertaining a Model of Themselves
Anticipatory systems were defined by Rosen  as systems which entertain a model of themselves. Since the modeling module is part of the system to be modeled, this reasoning would lead to an infinite regress. However, Dubois  showed that this problem can be solved by introducing incursive and hyper-incursive equations into the computation of anticipatory systems. Unlike a recursive equation—which refers for the computation of its current state to a previous state (t – 1)—an incursive equation refers also to states in the present among its independent variables, and a hyper-incursive one refers to future states as co-constructors of present states.
xt+1 = ½ ± ½ √[1 – (4/a) xt]
This equation has two real roots for a ≥ 4. As is well known, the (recursive) logistic equation is defined only for a < 4; the solutions become increasingly chaotic as a approaches four. In other words, a = 4 functions as a divide between the domain of the recursive version of the equation (along the arrow of time) and the hyper-incursive version (against the arrow of time). Furthermore, the incursive version of this equation generates a steady state for all values of a as follows: x = (a – 1)/a . This continuous curve crosses the noted divide between the two domains [10,11].
In general, a model advances in time with reference to the present state of the modeled system. In other words, the model contains a prediction of a next state of the system. Hyper-incursive equations show how models may additionally feed back on the present state of the modeled system. The total system—that is, the system including its modeling subroutines—develops over time and thus generates entropy . The model, however, reverses the time axis and thus can be expected to generate redundancy.
In other words, a model provides meaning to the modeled system. Meaning can be communicated in a reflexive discourse entertaining different models: a discursive model enables us to feed back on the system under study. This hyper-incursive meaning processing—formulated in Equation 1 exclusively in terms of possible future states—can be instantiated incursively by a reflexive observer who herself operates with an expectation in the present and historically with reference to a previous state. The model is thus instantiated .
The communication of meaning cannot be expected to be directly measurable because, unlike the communication of information, hyper-incursive communication of meaning generates only redundancy. When the communication of meaning occurs within the system, however, this communication at a next-order level may make a difference for the communication of information by being instantiated locally within a historical process of information exchanges. The latter process necessarily generates probabilistic entropy, while the next-order level of meaning exchange can from this perspective also be considered as an exogenous network variable potentially restricting the options . The model constrains the number of possible states in the system and can thus be considered as a filter reducing the uncertainty that prevails.
3. The Measurement of Redundancy and Interaction Information
Might one be able to measure the reduction of uncertainty in an information processing system which entertains a model of itself? The generation of redundancy in information exchanges has been studied as a possible outcome of mutual information among three or more dimensions. Ulanowicz  for example, suggested using this local reduction of uncertainty as an indicator of “ascendency.”
Depending on how the different variations disturb and condition one another, the outcome of this measure can be positive, negative, or zero. In other words, the interactions among three sources of variance may reduce the uncertainty which prevails at the systems level.
Krippendorff [1,2] convincingly showed that multivariate Q-measures of interaction information cannot be considered as Shannon-type information. Watanabe  had made the same argument, but this was forgotten in the blossoming literature using Q-measures for the measurement of “configurational information” . Yeung  used the deviant symbol μ* to indicate that mutual information in three or more dimensions is not Shannon-type information. However, Krippendorff specified in detail why the reasoning among the proponents of this measure as an information statistics (about probability distributions) is mistaken: when more than two probability distributions are multiplied, loops can be expected to disturb what seems algebraically a perfect derivation. The probabilities may no longer add up to unity, and in that case a major assumption of probability theory (Σipi = 1) is violated.
Building on earlier work, Krippendorff [1,23,24] proposed another algorithm which approximates interaction information in three or more dimensions iteratively by using maximum entropies. This interaction information (denoted in the case of three dimensions as IABC→AB:AC:BC) is Shannon-type information: it is the surplus information potentially generated in the three-way interactions which cannot be accounted for by the two-way interactions.
As noted, Krippendorff  provided an interpretation of Q expressed as Q = I – R. In this formula, R indicates redundancy which is generated at the level of the model entertained by an observer on the basis of the assumption that the probability distributions are independent and therefore multiplication is allowed. Given this assumption of independence, the algebraic derivation of Q is valid because circular relations among the variables cannot occur and all probabilities add up to unity. However, when this assumption is not true, a Shannon-type information IABC→AB:AC:BC is necessarily generated. Q can be positive or negative (or zero) depending on the difference between R and I.
Figure 1. Example of positive ternary interaction with Q = 0 .
Figure 1. Example of positive ternary interaction with Q = 0 .
In other words, Q = 0 should not be interpreted as the absence of interaction information, but only as an indicator of R = I.Figure 1 shows that the presence of (Shannon-type) interaction information I is compatible with the absence of a value for Q.Q cannot be interpreted as a unique property of the interactions. In other words, not Q, but R is urgently to be provided with an interpretation. Q can only be considered as the difference between the positive interaction information in the modeled system and redundancy generated in the model.
4. An Empirical Interpretation of R
R cannot be measured directly, but one can retrieve both I and Q from the data. For the measurement of IABC→AB:AC:BC, Krippendorff’s iterative algorithm is available, and Q can be computed using Equation 4. Because IABC→AB:AC:BC is a Shannon-type information it cannot be negative. μ* (= –Q) has to be added to (or subtracted from) I in order to find R.
Krippendorff  noted that in Shannon’s information theory processes can flow only in a single direction, that is, with the arrow of time: “Accordingly, a message received could have no effect on the message sent”. Meaning, however, is provided to the message from the perspective of hindsight, and meaning processing can be expected to operate orthogonally to—albeit potentially in interaction with—information processing. The modeling aims to reduce complexity and thus may add to redundancy without necessarily affecting the information processing. In the case of this independence, the model’s expected information content remains “hot air” only specifying other possible states. However, if the meaning processing is made relevant to the information processing system, an imprint is generated. This imprint makes a difference and can be measured as Q, that is, the difference between interaction information in the modeled subsystem and redundancy in the modeling subroutine.
What type of (meta-)model might model this relation between information processing and meaning processing in numerical terms? Let me propose a rotated three-factor model as an example of a model of constructs that can be provided with meaning in terms of observable variables. The factor model assumes orthogonality among the (three or more) dimensions and one can measure the relations among these dimensions in terms of the factor loadings of the variables . In other words, the data matrix represents a first-order network in which a second-order structure among the variables can be hypothesized using factor analysis.
The dimensions of the second-order structure are latent at the first-order level; they remain expectations about the main dimensions that span the network. By using orthogonal rotation among the factors, one can assure that the dimensions are analytically independent. Let us assume a three-factor model because the interactions among three dimensions provide us with the most parsimonious example for analyzing the problem of how redundancy because of data reduction operates in relation to interaction information .
The factor model spans a three-dimensional space in which the variables can be positioned as vectors. The variables are associated with the eigenvectors in terms of the factor loadings. (Factor loadings are by definition equal to Pearson correlation coefficients between variables and factors.) In other words, the (orthogonally rotated) factor matrix provides us with a representation of the variables in three main dimensions. Among these three dimensions one can compute an IABC→AB:AC:BC, μ*, and therefore R. From this perspective, the difference between R and I can be considered as the remaining redundancy of the model that is not consumed by the Shannon information contained in the empirical distributions. If R < I, this difference can be considered as remaining uncertainty.
While R is a property of the model and cannot be measured directly, the remaining redundancy or uncertainty (R – I) can be provided with historical meaning because it is manifest in the instantiation as μ* (= –Q). Remaining redundancy can be considered as redundancy that could have been filled by the events. Remaining uncertainty would indicate that the model is not sufficiently complex to reduce all uncertainty in the data. Note that our meta-model assumes that the model generating R
Johannes Diderik van der Waals (Dutch: [joːˈɦɑnəz ˈdidəˌrɪk fɑn dɛr ˈʋaːls] ( listen); 23 November 1837 – 8 March 1923) was a Dutch theoretical physicist and thermodynamicist famous for his work on an equation of state for gases and liquids.
His name is primarily associated with the van der Waals equation of state that describes the behavior of gases and their condensation to the liquid phase. His name is also associated with van der Waals forces (forces between stable molecules), with van der Waals molecules (small molecular clusters bound by van der Waals forces), and with van der Waals radii (sizes of molecules). As James Clerk Maxwell said about Van der Waals, "there can be no doubt that the name of Van der Waals will soon be among the foremost in molecular science."
In his 1873 thesis, van der Waals noted the non-ideality of real gases and attributed it to the existence of intermolecular interactions. He introduced the first equation of state derived by the assumption of a finite volume occupied by the constituent molecules. Spearheaded by Ernst Mach and Wilhelm Ostwald, a strong philosophical current that denied the existence of molecules arose towards the end of the 19th century. The molecular existence was considered unproven and the molecular hypothesis unnecessary. At the time van der Waals' thesis was written (1873), the molecular structure of fluids had not been accepted by most physicists, and liquid and vapor were often considered as chemically distinct. But van der Waals's work affirmed the reality of molecules and allowed an assessment of their size and attractive strength. His new formula revolutionized the study of equations of state. By comparing his equation of state with experimental data, Van der Waals was able to obtain estimates for the actual size of molecules and the strength of their mutual attraction. The effect of Van der Waals's work on molecular physics in the 20th century was direct and fundamental. By introducing parameters characterizing molecular size and attraction in constructing his equation of state, Van der Waals set the tone for modern molecular science. That molecular aspects such as size, shape, attraction, and multipolar interactions should form the basis for mathematical formulations of the thermodynamic and transport properties of fluids is presently considered an axiom. With the help of the van der Waals's equation of state, the critical-point parameters of gases could be accurately predicted from thermodynamic measurements made at much higher temperatures. Nitrogen, oxygen, hydrogen, and helium subsequently succumbed to liquefaction. Heike Kamerlingh Onnes was significantly influenced by the pioneer work of van der Waals. In 1908, Onnes became the winner of the race to make liquid helium and because of this, he was also to be the discoverer of superconductivity in 1911.
Van der Waals started his career as a school teacher. He became the first physics professor of the University of Amsterdam when in 1877 the old Athenaeum was upgraded to Municipal University. Van der Waals won the 1910 Nobel Prize in physics for his work on the equation of state for gases and liquids.
Early years and education
Johannes Diderik van der Waals was born on 23 November 1837 in Leiden in the Netherlands. He was the eldest of ten children born to Jacobus van der Waals and Elisabeth van den Berg. His father was a carpenter in Leiden. As was usual for working-class children in the 19th century, he did not go to the kind of secondary school that would have given him the right to enter university. Instead he went to a school of “advanced primary education”, which he finished at the age of fifteen. He then became a teacher's apprentice in an elementary school. Between 1856 and 1861 he followed courses and gained the necessary qualifications to become a primary school teacher and head teacher.
In 1862, he began to attend lectures in mathematics, physics and astronomy at the University in his city of birth, although he was not qualified to be enrolled as a regular student in part because of his lack of education in classical languages. However, the University of Leiden had a provision that enabled outside students to take up to four courses a year. In 1863 the Dutch government started a new kind of secondary school (HBS, a school aiming at the children of the higher middle classes). Van der Waals—at that time head of an elementary school—wanted to become a HBS teacher in mathematics and physics and spent two years studying in his spare time for the required examinations.
In 1865, he was appointed as a physics teacher at the HBS in Deventer and in 1866, he received such a position in The Hague, which was close enough to Leiden to allow van der Waals to resume his courses at the University there. In September 1865, just before moving to Deventer, van der Waals married the eighteen-year-old Anna Magdalena Smit.
Van der Waals still lacked the knowledge of the classical languages that would have given him the right to enter university as a regular student and to take examinations. However, it so happened that the law regulating the university entrance was changed and dispensation from the study of classical languages could be given by the minister of education. Van der Waals was given this dispensation and passed the qualification exams in physics and mathematics for doctoral studies.
At Leiden University, on June 14, 1873, he defended his doctoral thesis Over de Continuïteit van den Gas- en Vloeistoftoestand (on the continuity of the gaseous and liquid state) under Pieter Rijke. In the thesis, he introduced the concepts of molecular volume and molecular attraction.
In September 1877 van der Waals was appointed the first professor of physics at the newly founded Municipal University of Amsterdam. Two of his notable colleagues were the physical chemist Jacobus Henricus van 't Hoff and the biologist Hugo de Vries. Until his retirement at the age of 70 van der Waals remained at the Amsterdam University. He was succeeded by his son Johannes Diderik van der Waals, Jr., who also was a theoretical physicist. In 1910, at the age of 72, van der Waals was awarded the Nobel Prize in physics. He died at the age of 85 on March 8, 1923.
The main interest of van der Waals was in the field of thermodynamics. He was influenced by Rudolf Clausius' 1857 treatise entitled Über die Art der Bewegung, welche wir Wärme nennen (On the Kind of Motion which we Call Heat). Van der Waals was later greatly influenced by the writings of James Clerk Maxwell, Ludwig Boltzmann, and Willard Gibbs. Clausius' work led him to look for an explanation of Thomas Andrews' experiments that had revealed, in 1869, the existence of critical temperatures in fluids. He managed to give a semi-quantitative description of the phenomena of condensation and critical temperatures in his 1873 thesis, entitled Over de Continuïteit van den Gas- en Vloeistoftoestand (On the continuity of the gas and liquid state). This dissertation represented a hallmark in physics and was immediately recognized as such, e.g. by James Clerk Maxwell who reviewed it in Nature in a laudatory manner.
In this thesis he derived the equation of state bearing his name. This work gave a model in which the liquid and the gas phase of a substance merge into each other in a continuous manner. It shows that the two phases are of the same nature. In deriving his equation of state van der Waals assumed not only the existence of molecules (the existence of atoms was disputed at the time), but also that they are of finite size and attract each other. Since he was one of the first to postulate an intermolecular force, however rudimentary, such a force is now sometimes called a van der Waals force.
A second great discovery was published in 1880, when he formulated the Law of Corresponding States. This showed that the van der Waals equation of state can be expressed as a simple function of the critical pressure, critical volume, and critical temperature. This general form is applicable to all substances (see van der Waals equation.) The compound-specific constants a and b in the original equation are replaced by universal (compound-independent) quantities. It was this law which served as a guide during experiments which ultimately led to the liquefaction of hydrogen by James Dewar in 1898 and of helium by Heike Kamerlingh Onnes in 1908.
In 1890, van der Waals published a treatise on the Theory of Binary Solutions in the Archives Néerlandaises. By relating his equation of state with the Second Law of Thermodynamics, in the form first proposed by Willard Gibbs, he was able to arrive at a graphical representation of his mathematical formulations in the form of a surface which he called Ψ (Psi) surface following Gibbs, who used the Greek letter Ψ for the free energy of a system with different phases in equilibrium.
Mention should also be made of van der Waals' theory of capillarity which in its basic form first appeared in 1893. In contrast to the mechanical perspective on the subject provided earlier by Pierre-Simon Laplace, van der Waals took a thermodynamic approach. This was controversial at the time, since the existence of molecules and their permanent, rapid motion were not universally accepted before Jean Baptiste Perrin's experimental verification of Albert Einstein's theoretical explanation of Brownian motion.
He married Anna Magdalena Smit in 1865, and the couple had three daughters (Anne Madeleine, Jacqueline E. van der Waals (nl), Johanna Diderica) and one son, the physicist Johannes Diderik van der Waals, Jr. (nl) Jacqueline was a poet of some note. Van der Waals' nephew Peter van der Waals was a cabinet maker and a leading figure in the Sapperton, Gloucestershire school of the Arts and Crafts movement. The wife of Johannes van der Waals died of tuberculosis at 34 years old in 1881. After becoming a widower Van der Waals never remarried and was so shaken by the death of his wife that he did not publish anything for about a decade. He died in Amsterdam on March 8, 1923, one year after his daughter Jacqueline had died.
His grandson, Christopher D. Vanderwal is a distinguished professor of Chemistry at the University of California, Irvine.
Van der Waals received numerous honors and distinctions, besides winning the 1910 Nobel Prize in Physics. He was awarded an honorary doctorate of the University of Cambridge; was made honorary member of the Imperial Society of Naturalists of Moscow, the Royal Irish Academy and the American Philosophical Society; corresponding member of the Institut de France and the Royal Academy of Sciences of Berlin; associate member of the Royal Academy of Sciences of Belgium; and foreign member of the Chemical Society of London, the National Academy of Sciences of the U.S., and of the Accademia dei Lincei of Rome. Van der Waals was a member of the Koninklijke Nederlandse Akademie van Wetenschappen (Royal Netherlands Academy of Sciences) since 1875. From 1896 until 1912, he was secretary of this society.
|“||...There can be no doubt that the name of Van der Waals will soon be among the foremost in molecular science,||”|
|— James Clerk Maxwell's remarks in Nature magazine (1873).|
|“||...It will be perfectly clear that in all my studies I was quite convinced of the real existence of molecules, that I never regarded them as a figment of my imagination, nor even as mere centres of force effects. I considered them to be the actual bodies, thus what we term "body" in daily speech ought better to be called "pseudo body". It is an aggregate of bodies and empty space. We do not know the nature of a molecule consisting of a single chemical atom. It would be premature to seek to answer this question but to admit this ignorance in no way impairs the belief in its real existence. When I began my studies I had the feeling that I was almost alone in holding that view. And when, as occurred already in my 1873 treatise, I determined their number in one gram-mol, their size and the nature of their action, I was strengthened in my opinion, yet still there often arose within me the question whether in the final analysis a molecule is a figment of the imagination and the entire molecular theory too. And now I do not think it any exaggeration to state that the real existence of molecules is universally assumed by physicists. Many of those who opposed it most have ultimately been won over, and my theory may have been a contributory factor. And precisely this, I feel, is a step forward. Anyone acquainted with the writings of Boltzmann and Willard Gibbs will admit that physicists carrying great authority believe that the complex phenomena of the heat theory can only be interpreted in this way. It is a great pleasure for me that an increasing number of younger physicists find the inspiration for their work in studies and contemplations of the molecular theory...||”|
|— Johannes D. van der Waals's notes in Nobel Lecture, The equation of state for gases and liquids (12 December 1910).|
- This article incorporates material from the Citizendium article "Johannes Diderik van der Waals", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL.
- ^Every word in isolation: [joːˈɦɑnəs ˈdidəˌrɪk vɑn dɛr ˈʋaːls].
- ^Parsegian, V. Adrian (2005). Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists. (Cambridge University Press), p. 2. “The first clear evidence of forces between what were soon to be called molecules came from Johannes Diderik van der Waals' 1873 Ph.D. thesis formulation of the pressure p, volume V, and temperature T of dense gases.”
- ^ abJohannes Diderik van der Waals - Biographical - Nobelprize.org
- ^van der Waals; J. D. (1873). On the Continuity of the Gaseous and Liquid States (doctoral dissertation). Universiteit Leiden.
- ^Sengers, Johanna Levelt (2002), p. 16
- ^Kipnis, A. Ya.; Yavelov, B. E.; Rowlinson, J. S.: Van der Waals and Molecular Science. (Oxford: Clarendon Press, 1996)
- ^Sengers, Johanna Levelt (2002), p. 255-256
- ^Blundell, Stephen: Superconductivity: A Very Short Introduction. (Oxford University Press, 1st edition, 2009, p. 20)
- ^"The Nobel Prize in Physics 1910". Nobel Foundation. Retrieved 2008-10-09.
- ^see the article on the van der Waals equation for the technical background
- ^J.D. van der Waals, 1910, "The equation of state for gases and liquids," Nobel Lectures in Physics, pp. 254-265 (December 12, 1910), see , accessed 25 June 2015.
- ^Clausius, R. (1857). "Über die Art der Bewegung, welche wir Wärme nennen". Annalen der Physik. 176 (3): 353–380. Bibcode:1857AnP...176..353C. doi:10.1002/andp.18571760302.
- ^Andrews, T. (1869). "The Bakerian Lecture: On the Gaseous State of Matter". Philosophical Transactions of the Royal Society of London. 159: 575–590. doi:10.1098/rstl.1869.0021.
- ^van der Waals, JD (1873) Over de Continuiteit van den Gas- en Vloeistoftoestand (on the continuity of the gas and liquid state). PhD thesis, Leiden, The Netherlands.
- ^Maxwell, J.C. (1874). "Van der Waals on the Continuity of Gaseous and Liquid States". Nature. 10 (259): 477–480. Bibcode:1874Natur..10..477C. doi:10.1038/010477a0.
- ^Tang, K.-T.; Toennies, J. P. (2010). "Johannes Diderik van der Waals: A Pioneer in the Molecular Sciences and Nobel Prize Winner in 1910". Angewandte Chemie International Edition. 49: 9574–9579. doi:10.1002/anie.201002332. PMID 21077069.
- ^van der Waals, J.D. (1893). "Thermodynamische theorie der capillariteit in de onderstelling van continue dichtheidsverandering". Verhand. Kon. Akad. V Wetensch. Amst. Sect. 1 (Dutch; English translation in J. Stat. Phys., 1979, 20:197).
- ^Laplace, P.S. (1806). Sur l'action capillaire (Suppl. au livre X, Traité de Mécanique Céleste). Crapelet; Courcier; Bachelier, Paris.
- ^"Johannes Diderik van der Waals Senior (1837 - 1923)". Royal Netherlands Academy of Arts and Sciences. Retrieved 17 July 2015.
- Kipnis, A. Ya.; Yavelov, B. E.; Rowlinson, J. S.: Van der Waals and Molecular Science. (Oxford: Clarendon Press, 1996, 313pp) ISBN 0-19-855210-6
- Sengers, Johanna Levelt: How Fluids Unmix: Discoveries by the School of Van der Waals and Kamerlingh Onnes (Edita - History of Science and Scholarship in the Netherlands). (Edita-the Publishing House of the Royal, 2002, 318pp)
- Shachtman, Tom: Absolute Zero and the Conquest of Cold. (Boston: Houghton Mifflin, 1999)
- Van Delft, Dirk: Freezing Physics: Heike Kamerlingh Onnes and the Quest for Cold. (Edita-the Publishing House of the Royal, 2008, 592pp)
- Scientists of the Dutch School Van der Waals, Royal Netherlands Academy of Arts and Sciences
- Albert van Helden Johannes Diderik van der Waals 1837 – 1923 In: K. van Berkel, A. van Helden and L. Palm ed., A History of Science in the Netherlands. Survey, Themes and Reference (Leiden: Brill, 1999) 596 – 598.
- Johannes Diderik van der Waals – Biography at Nobelprize.org.
- Museum Boerhaave "Negen Nederlandse Nobelprijswinnaars"(PDF). Archived from the original(PDF) on June 7, 2011. (2.32 MiB)
- H.A.M. Snelders, Waals Sr., Johannes Diderik van der (1837–1923), in Biografisch Woordenboek van Nederland.
- Biography of Johannes Diderik van der Waals (1837–1923) at the National Library of the Netherlands.