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A psychophysical investigation of quantum cognition:
An interdisciplinary synthesis
by Christopher B. Germann
A thesis submitted to the University of Plymouth
in partial fulfilment for the degree of
Doctor of Philosophy
January 2019
3
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Indagate Fingite Invenite (Explore, Dream, Discover)
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Copyright Statement
This copy of the thesis has been supplied on condition that anyone who consults it is
understood to recognise that its copyright rests with its author and that no quotation
from the thesis and no information derived from it may be published without the
author's prior consent.
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Author’s declaration
At no time during the registration for the degree of Doctor of Philosophy has the
author been registered for any other University award without prior agreement of the
Doctoral College Quality Sub-Committee. Work submitted for this research degree at
the University of Plymouth has not formed part of any other degree either at University
of Plymouth or at another establishment. This research was financed with the aid of the
Marie Curie Initial Training Network FP7-PEOPLE-2013-ITN-604764.
URL: https://ec.europa.eu/research/mariecurieactions/
Additional information can be found under the following URL:
https://www.cognovo.eu/christopher-germann
E-Mail: mail@christopher.germann.de
Word count of main body of thesis: 71,221
Signed: ___________________________
Date: ___________________________
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Prefix: Interpretation of the cover illustration
The cover of this thesis depicts a variation of Möbius band which has been
eponymously named after the German astronomer and mathematician August Ferdinand
Möbius. An animated version of the digital artwork and further information can be
found on the following custom-made website:
URL: http://moebius-band.ga
The Möbius band has very peculiar geometrical properties because the inner and the
outer surface create a single continuous surface, that is, it has only one boundary. A
Gedanken-experiment is illustrative: If one imagines walking along the Möbius band
starting from the seam down the middle, one would end back up at the seam, but at the
opposite side. One would thus traverse a single infinite path even though an outside
observer would think that we are following two diverging orbits. We suggest that the
Möbius band can be interpreted as a visual metaphor for dual-aspect monism
(Benovsky, 2016), a theory which postulates that the psychological and the physical are
two aspects of the same penultimate substance, i.e., they are different manifestations of
the same ontology. Gustav Fechner (the founding father of psychophysics) was a
proponent of this Weltanschauung, as were William James, Baruch de Spinoza, Arthur
Schopenhauer, and quantum physicists Wolfgang Pauli and David Bohm, inter alia.
The nondual perspective is incompatible with the reigning paradigm of reductionist
materialism which postulates that matter is ontologically primary and fundamental and
that the mental realm emerges out of the physical, e.g., epiphenomenalism/evolutionary
emergentism (cf. Bawden, 1906; Stephan, 1999)). The nondual perspective has been
concisely articulated by Nobel laureate Bertrand Russel:
The whole duality of mind and matter [...] is a mistake; there is only one kind of stuff
8
out of which the world is made, and this stuff is called mental in one arrangement,
physical in the other.” (Russell, 1913, p.15)
From a psychophysical perspective it is interesting to note that quantum physicist and
Nobel laureate Wolfgang Pauli and depth psychologist Carl Gustav Jung discussed
dual-aspect monism extensively in their long-lasting correspondence which spanned
many years. In particular, the “Pauli-Jung conjecture” (Atmanspacher, 2012) implies
that psychological and physical states exhibit complementarity in a quantum physical
sense (Atmanspacher, 2014b; Atmanspacher & Fuchs, 2014). We suggest that the
Möbius band provides a “traceable” visual representation of the conceptual basis of the
dual-aspect perspective. A prototypical Möbius band (or Möbius strip) can be
mathematically represented in three-dimensional Euclidean space. The following
equation provides a simple geometric parametrization schema:
(, ) = (3 +
2
cos
2
)cos
(, ) = (3 +
2
cos
2
)sin
(, ) =
2
sin
2
where 0 ≤ u < 2π and −1 ≤ v ≤ 1. This parametrization produces a single Möbius band
with a width of 1 and a middle circle with a radius of 3. The band is positioned in the xy
plane and is centred at coordinates (0, 0, 0). We plotted the Möbius band in R and the
associated code utilised to create the graphic is based on the packages “rgl(Murdoch,
2001) and “plot3D(Soetaert, 2014) and can be found in Appendix A1. The code
creates an interactive plot that allows to scale and rotate the Möbius band in three-
dimensional space.
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Figure 1. Möbius band as a visual metaphor for dual-aspect monism.
The cover image of this thesis is composed of seven parallel Möbius bands (to be
accurate these three-folded variations of the original Möbius band). It is easy to create a
Möbius band manually from a rectangular strip of paper. One simply needs to twist one
end of the strip by 180° and then join the two ends together (see Starostin & Van Der
Heijden, 2007). The graphic artist M.C. Escher (Crato, 2010; Hofstadter, 2013) was
mathematically inspired by the Möbius band and depicted it in several sophisticated
artworks,e.g., “Möbius Strip I” (1961) and “Möbius Strip II(1963).
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Figure 2. “Möbis Strip I” by M.C. Escher, 1961 (woodcut and wood engraving)
A recent math/visual-arts project digitally animated complex Möbius transformations in
a video entitled “Möbius Transformations Revealed” (Möbiustransformationen
beleuchtet). The computer-based animation demonstrates various multidimensional
Möbius transformation and shows that “moving to a higher dimension reveals their
essential unity”
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(Arnold & Rogness, 2008). The associated video
2
can be found under
the following URL:
http://www-users.math.umn.edu/~arnold/moebius/
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Interestingly, a similar notion forms the basis of “Brane cosmology(Brax, van de Bruck, & Davis,
2004; Papantonopoulos, 2002) and its conception of multidimensional hyperspace. Cosmologists have
posed the following question: “Do we live inside a domain wall?” (Rubakov & Shaposhnikov, 1983).
Specifically, it has been argued that(light) particles are confined in a potential well which is narrow
along N spatial directions and flat along three others.”
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The video is part of a DVD titled “MathFilm Festival 2008: a collection of mathematical videos
published by Springer (Apostol et al., 2008) which is available under the following URL:
http://www.springer.com/gb/book/9783540689027
Moreover, the computer animation was among the winners of the “Science and Engineering Visualization
Challenge” in 2007.
© 2018 The M.C. Escher Company
All rights reserved.
Used by permission.
www.mcescher.com
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Additionally, we integrated a high-resolution version of the video in our website,
together with supplementary background information:
http://irrational-decisions.com/?page_id=2599
Mathematics and particularly its subordinate branch geometry have always been
regarded as cognitive activities which enable access to transcendental/metaphysical
realms (e.g., for instance Pythagoras's theorem and Plato's transcendent forms) and there
is a longstanding interrelation between geometry, mathematics, and mysticism (e.g.,
sacred geometry, Fibonacci numbers, etc.) as has been pointed out by eminent
mathematicians who argue for the pivotal importance of mystical influences in the
history of mathematics (e.g., Abraham, 2015, 2017). For instance, it has been argued
that there is a close relation between geometry, space-time, and consciousness (Beutel,
2012), a perspective which can be found in many religions and ancient knowledge
traditions, e.g. Yantra (Sanskrit: ) and Mandala () in ancient Indian schools of
thought (also found in Buddhism, inter alia). Moreover, geometry was pivotal for the
progress of the exact sciences like cosmology and astronomy. For instance, when the
Lutheran astronomer Johannes Keppler’s published his “mysterium cosmographicum
at Tübingen in 1596, he based his theory on five Pythagorean polyhedra (Platonic
solids) which he conjectured form the basis of the structure of the universe and thus
realise God's ideas through geometry (Voelkel, 1999).
The geometry of the Möbius band has broad interdisciplinary pertinence. Besides its
contemporary relevance in the sciences like chemistry (e.g., “Möbius aromaticity” (Jux,
2008), “Möbius molecules” (Herges, 2006)), mathematics (Waterman, 1993), and
physics (Chang et al., 2010)) “the curious band between dimensions” has significance
for perceptual psychology. For instance, it has been argued that ”we can also use its
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dynamics to reveal the mechanisms of our perception (or rather, its deceptions as in the
case of optical illusions) in an augmented space-time.” (Petresin & Robert, 2002)
To sum up this annotation, the interpretation of the Möbius band has multifarious
semantic/hermeneutic layers and provides an apt visual primer for the concept of
psychophysical complementarity which will be discussed in greater detail in the
subsequent thesis, particularly in the context of nonduality and quantum cognition.
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Acknowledgements
First, I would like to acknowledge my supervisor Prof. Chris Harris who gave me the
“cognitive liberty” to engage in interdisciplinary, innovative, and unconventional
research. He kindly invited me to pursue this prestigious Ph.D. as a Marie-Curie Fellow.
Moreover, I would like to express my gratitude to Prof. Geetika Tankha who
proficiently supported my experimental studies during my secondment at Manipal
University Jaipur in India.
Furthermore, I would like to thank Dr. Christopher Berry and Prof. Harald Walach for
adopting the role of the internal and external examiner, respectively.
Finally, I would like to remember Dr. Martha Blassnigg (*1969; †2015) who was a
special and truly gifted scholar in many respects. She had a deep interest in holistic
approaches to the mind-body correlation, a theme which is of great pertinence for the
thesis at hand.
The primary impetus for the present interdisciplinary thesis is derived from a personal
initiatory nondual experience of “unity consciousness” (nondual consciousness). This
profound topic has recently received great attention in the pertinent contemporary
psychological and neuroscientific literature even though it has been discussed by
philosophers of mind for time immemorial. Hence, the topic of nonduality is of great
psychological importance and it intersects with various disciplines such as
neurochemistry, quantum physics, and various ancient eastern knowledge traditions,
inter alia. It is thus a truly interdisciplinary topic with great pragmatic importance for
the evolution of science and humanity as a species.
Special thanks are directed towards the Sivananda Yoga Vedānta Ashram in Kerala in
South India. I had the great privilege to take part in a knowledge tradition which dates
back several thousand years. My experiences in this centre for spiritual growth and
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learning further strengthened my conviction in the importance of ethics and morality
and specifically purity of thought, word, and action. Yoga is a truly psychologically
transformative practice and Swami Sivananda’s dictum “an ounce of practice is worth
tons of theory” illustrates the importance of first-person phenomenological experience
for which there is no substitute. One of the essential teachings of yoga is that the
individual must change before the world can change, viz., the microcosm and the
macrocosm are intimately interrelated. Consequently, self-reflection, self-actualisation,
and self-realisation (in the Maslowian sense) are of utmost significance. Moreover,
Advaita Vedānta emphasises “unity in diversity”, a philosophical perspective which has
great relevance for the thesis at hand due to its pertinence for a nondual
conceptualisation of reality.
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Table of contents
Figures
Tables
Equations
Code
Electronic supplementary materials
34
35
Abstract .......................................................................................................................... 38
Chapter 1. Introduction ................................................................................................ 40
1.1 Psychology: A Newtonian science of mind ......................................................... 48
1.2 Shifting paradigms: From Newtonian determinism to quantum indeterminism . 51
1.3 Quantum cognition: An emerging novel paradigm in psychology ...................... 55
1.4 Observer-effects, noncommutativity, and uncertainty in psychology ................. 56
1.5 Psychophysics: The interface between Psyche and Physis .................................. 60
1.6 A brief history of the evolution of the “complementarity” meme in physics ...... 77
1.7 Quantum cognitive science? ................................................................................ 83
1.8 Perceptual judgments under uncertainty .............................................................. 90
1.9 A real-word example of superposition and collapse ............................................ 97
1.10 Determinism vs. constructivism ........................................................................... 99
1.11 Quantum logic .................................................................................................... 101
1.12 Noncommutative decisions: QQ-equality in sequential measurements ............. 103
1.13 Quantum models of cognitive processes ............................................................ 108
1.14 Contextualism, borderline vagueness, and Sôritês paradox ............................... 109
1.15 Quantum-like constructivism in attitudinal and emotional judgements ............ 116
1.16 Current empirical research ................................................................................. 121
Chapter 2. Experiment #1: Noncommutativity in sequential visual perceptual
judgments ..................................................................................................................... 124
2.1 Experimental purpose ........................................................................................ 124
2.2 A priori hypotheses ............................................................................................ 127
2.3 Method ......................................................................................................... 128
2.3.1 Participants and Design .................................................................................................. 128
2.3.2 Apparatus and materials ................................................................................................. 129
2.3.3 Experimental application in PsychoPy ........................................................................... 131
2.3.4 Experimental Design ...................................................................................................... 133
2.3.5 Procedure ........................................................................................................................ 133
2.3.6 Sequential visual perception paradigm ........................................................................... 133
2.3.7 Statistical Analysis ......................................................................................................... 137
2.3.8 Data treatment and statistical software ........................................................................... 140
2.3.9 Frequentist NHST analysis ............................................................................................. 140
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2.3.10 Assumption Checks ....................................................................................................... 142
2.3.11 Parametric paired samples t-tests ................................................................................... 147
2.3.12 Bayes Factor analysis ..................................................................................................... 152
2.3.13 Bayesian a posteriori parameter estimation via Markov Chain Monte Carlo simulations ...
....................................................................................................................................... 165
2.4 Discussion ......................................................................................................... 196
Chapter 3. Experiment #2: Constructive measurement effects in sequential visual
perceptual judgments ................................................................................................. 199
3.1 Experimental purpose ........................................................................................ 199
3.2 A priori hypotheses ............................................................................................ 201
3.3 Method ......................................................................................................... 202
3.3.1 Participants and Design .................................................................................................. 202
3.3.2 Apparatus and materials ................................................................................................. 203
3.3.3 Experimental Design ...................................................................................................... 203
3.3.4 Experimental procedure ................................................................................................. 203
3.3.5 Sequential visual perception paradigm .......................................................................... 204
3.4 Statistical Analysis ............................................................................................. 207
3.4.1 Frequentist NHST analysis ............................................................................................ 208
3.4.2 Bayes Factor analysis ..................................................................................................... 213
3.4.3 Bayesian parameter estimation using Markov chain Monte Carlo methods .................. 222
3.5 Discussion ......................................................................................................... 230
Chapter 4. Experiment #3: Noncommutativity in sequential auditory perceptual
judgments ..................................................................................................................... 232
4.1 Experimental purpose ........................................................................................ 232
4.2 A priori hypotheses ............................................................................................ 234
4.3 Method ......................................................................................................... 235
4.3.1 Participants and Design .................................................................................................. 235
4.3.2 Apparatus and materials ................................................................................................. 235
4.3.3 Experimental Design ...................................................................................................... 236
4.3.4 Sequential auditory perception paradigm ....................................................................... 237
4.4 Statistical Analysis ............................................................................................. 237
4.4.1 Parametric paired samples t-tests ................................................................................... 240
4.4.2 Bayes Factor analysis ..................................................................................................... 244
4.4.3 Bayesian a posteriori parameter estimation using Markov chain Monte Carlo methods 248
4.5 Discussion ......................................................................................................... 253
Chapter 5. Experiment #4: Constructive measurement effects in sequential
auditory perceptual judgments .................................................................................. 254
5.1 Experimental purpose ........................................................................................ 254
5.2 A priori hypotheses ............................................................................................ 254
5.3 Method ......................................................................................................... 255
5.3.1 Participants and Design .................................................................................................. 255
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5.3.2 Apparatus and materials ................................................................................................. 256
5.3.3 Experimental Design ...................................................................................................... 256
5.3.4 Procedure ........................................................................................................................ 257
5.3.5 Sequential auditory perception paradigm ....................................................................... 257
5.4 Statistical Analysis ............................................................................................. 259
5.4.1 Frequentist analysis ........................................................................................................ 260
5.4.2 Bayes Factor analysis ..................................................................................................... 265
5.4.3 Bayesian a posteriori parameter estimation using Markov chain Monte Carlo methods 272
5.5 Discussion ......................................................................................................... 281
Chapter 6. General discussion ................................................................................... 282
6.1 Potential alternative explanatory accounts ......................................................... 288
6.2 The Duhem–Quine Thesis: The underdetermination of theory by data ............ 290
6.3 Experimental limitations and potential confounding factors ............................. 298
6.3.1 Sampling bias ................................................................................................................. 299
6.3.2 Operationalization of the term “measurement” .............................................................. 300
6.3.3 Response bias and the depletion of executive resources (ego-depletion) ....................... 301
6.4 Quantum logic .................................................................................................... 302
6.5 The interface theory of perception ..................................................................... 306
6.6 The Kochen-Specker theorem and the role of the observer ............................... 315
6.7 Consciousness and the collapse of the wave-function ....................................... 323
6.8 An embodied cognition perspective on quantum logic ...................................... 332
6.9 Advaita Vedānta, the art and science of yoga, introspection, and the hard
problem of consciousness ............................................................................................. 342
6.10 Dŗg-Dŗśya-Viveka: An inquiry into the nature of the seers and the seen .......... 349
6.11 Statistical considerations .................................................................................... 360
6.11.1 General remarks on NHST ............................................................................................. 360
6.11.2 The syllogistic logic of NHST ........................................................................................ 374
6.11.3 Implications of the ubiquity of misinterpretations of NHST results ............................... 376
6.11.4 P
rep
: A misguided proposal for a new metric of replicability ......................................... 377
6.11.5 Controlling experimentwise and familywise α-inflation in multiple hypothesis testing 381
6.11.6 α-correction for simultaneous statistical inference: familywise error rate vs. per-family
error rate ........................................................................................................................................ 396
6.11.7 Protected versus unprotected pairwise comparisons ...................................................... 397
6.11.8 Decentralised network systems of trust: Blockchain technology for scientific research 398
6.12 Potential future experiments .............................................................................. 402
6.12.1 Investigating quantum cognition principles across species and taxa: Conceptual cross-
validation and scientific consilience ............................................................................................... 402
6.12.2 Suggestions for future research: Mixed modality experiments ...................................... 404
6.13 Final remarks ..................................................................................................... 405
References .................................................................................................................... 407
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Appendices ................................................................................................................... 543
Appendix A Introduction ....................................................................................... 543
Möbius band .................................................................................... 543
Orchestrated objective reduction (Orch-OR): The quantum brain
hypothesis à la Penrose and Hameroff .......................................................................... 545
Algorithmic art to explore epistemological horizons ...................... 547
Psilocybin and the HT
2A
receptor ................................................... 552
Gustav Fechner on psychophysical complementarity ..................... 557
Belief bias in syllogistic reasoning ................................................. 560
Dual-process theories of cognition.................................................. 564
Bistability as a visual metaphor for paradigm shifts ....................... 572
CogNovo NHST survey: A brief synopsis ...................................... 573
Reanalysis of the NHST results reported by White et al. (2014) in a
Bayesian framework...................................................................................................... 586
Appendix B Experiment 1 ...................................................................................... 590
Embodied cognition and conceptual metaphor theory: The role of
brightness perception in affective and attitudinal judgments ........................................ 590
Custom made HTML/JavaScript/ActionScript multimedia website
for participant recruitment ............................................................................................ 599
PsychoPy benchmark report ............................................................ 602
Participant briefing .......................................................................... 608
Informed consent form .................................................................... 609
Verbatim instruction/screenshots .................................................... 610
Debriefing ....................................................................................... 625
Q-Q plots ......................................................................................... 625
The Cramér-von Mises criterion ..................................................... 627
Shapiro-Francia test ........................................................................ 627
Fisher’s multivariate skewness and kurtosis ................................... 628
Median-based boxplots ................................................................... 629
Tolerance intervals based on the Howe method ............................. 632
Alternative effect-size indices ......................................................... 637
Nonparametric bootstrapping .......................................................... 639
Bootstrapped effect sizes and 95% confidence intervals ................ 647
Bayesian bootstrap .......................................................................... 651
Probability Plot Correlation Coefficient (PPCC) ............................ 666
Ngrams for various statistical methodologies ................................. 671
Bayes Factor analysis (supplementary materials) ........................... 672
T-distribution with varying ν parametrisation ................................ 679
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Evaluation of null-hypotheses in a Bayesian framework: A ROPE
and HDI-based decision algorithm ............................................................................... 681
Bayesian parameter estimation via Markov Chain Monte Carlo
methods ......................................................................................................... 688
Markov Chain convergence diagnostics for condition V
00
and V
10
701
Markov Chain convergence diagnostics for condition V
00
and V
10
(correlational analysis) .................................................................................................. 706
Markov Chain convergence diagnostics for condition V
10
and V
11
(correlational analysis) .................................................................................................. 713
Correlational analysis ...................................................................... 720
Appendix C Experiment 2 ...................................................................................... 727
Skewness and kurtosis .................................................................... 727
Anscombe-Glynn kurtosis tests (Anscombe & Glynn, 1983)......... 728
Connected boxplots ......................................................................... 730
MCMC convergence diagnostics for experimental condition V
00
vs.
V
01
......................................................................................................... 733
MCMC convergence diagnostics for xperimental condition V
10
vs
V
11
......................................................................................................... 737
Visualisation of MCMC: 3-dimensional scatterplot with associated
concentration eclipse ..................................................................................................... 740
Correlational analysis ...................................................................... 744
Appendix C7.1 Hierarchical Bayesian model .......................................................... 744
Appendix C7.2 Convergence diagnostics for the Bayesian correlational analysis (V
10
vs. V
11
) 745
Appendix C7.3 Convergence diagnostics for the Bayesian correlational analysis (V
10
and V
11
) 748
Appendix C7.4 Pearson's product-moment correlation between experimental
condition V00 vs. V10 .................................................................................................. 751
Appendix C7.5 Pearson's product-moment correlations between experimental
conditions V
01
vs V
11
.................................................................................................... 755
JAGS model code for the correlational analysis ............................. 758
Tests of Gaussianity ........................................................................ 761
Symmetric beanplots for direct visual comparison between
experimental conditions ................................................................................................ 762
Descriptive statistics and various normality tests ........................... 763
χ2 Q-Q plot (Mahalanobis Distance) .............................................. 764
Connected boxplots (with Wilcoxon test) ....................................... 766
Correlational analysis ...................................................................... 769
Inferential Plots for Bayes Factor analysis ..................................... 774
Appendix D Experiment 3 ...................................................................................... 780
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Parametrisation of auditory stimuli ................................................. 780
Electronic supplementary materials: Auditory stimuli ................... 782
Bayesian parameter estimation ....................................................... 783
Correlational analysis ...................................................................... 785
Appendix E Experiment 4 ...................................................................................... 791
Markov chain Monte Carlo simulations .......................................... 791
Theoretical background of Bayesian inference ............................... 792
Mathematical foundations of Bayesian inference ........................... 803
Markov chain Monte Carlo (MCMC) methods .............................. 808
Software for Bayesian parameter estimation via MCMC methods 811
R code to find various dependencies of the “BEST” package. ....... 812
Hierarchical Bayesian model .......................................................... 813
Definition of the descriptive model and specification of priors ...... 814
Summary of the model for Bayesian parameter estimation ............ 822
MCMC computations of the posterior distributions ....................... 824
MCMC convergence diagnostics .................................................... 828
Diagnostics ...................................................................................... 829
Probability Plot Correlation Coefficient Test ................................. 829
P
rep
function in R ............................................................................. 831
MCMC convergence diagnostic ...................................................... 840
Appendix F Discussion ........................................................................................... 848
Extrapolation of methodological/statistical future trends based on
large data corpora ......................................................................................................... 848
Annex 1 N,N-Dimethyltryptamine: An endogenous neurotransmitter with
extraordinary effects. .................................................................................................. 851
Annex 2 5-methoxy-N,N-dimethyltryptamine: An ego-dissolving catalyst of
creativity? .................................................................................................................... 872
Vitæ auctoris ................................................................................................................ 912
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Figures
Figure 1. Möbius band as a visual metaphor for dual-aspect monism. ............................. 9
Figure 2. “Möbis Strip I” by M.C. Escher, 1961 (woodcut and wood engraving) ......... 10
Figure 3. Indra's net is a visual metaphor that illustrates the ontological concepts of
dependent origination and interpenetration (see Cook, 1977). ....................................... 69
Figure 4. Rubin’s Vase: A bistable percept as a visual example of complementarity-
coupling between foreground and background. .............................................................. 79
Figure 5. Photograph of Niels Bohr and Edgar Rubin as members of the club
“Ekliptika” (Royal Library of Denmark). ....................................................................... 81
Figure 6. Escutcheon worn by Niels Bohr during the award of the “Order of the
Elephant”. ........................................................................................................................ 82
Figure 7. Bloch sphere: a geometrical representation of a qubit. ................................... 86
Figure 8. Classical sequential model (Markov). ............................................................. 98
Figure 9. Quantum probability model (Schrödinger)...................................................... 99
Figure 10. Noncommutativity in attitudinal decisions. ................................................. 105
Figure 11. Sôritês paradox in visual brightness perception. ......................................... 111
Figure 12. Trustworthiness ratings as a function of experimental condition (White et al.,
2015). ............................................................................................................................ 118
Figure 13. Emotional valence as a function of experimental condition (White et al.,
2014b). .......................................................................................................................... 119
Figure 14. The HSV colour space lends itself to geometric modelling of perceptual
probabilities in the QP framework. ............................................................................... 131
Figure 15. Demographic data collected at the beginning of the experiment. ............... 134
Figure 16. Diagrammatic representation of the experimental paradigm. ..................... 136
Figure 17. Beanplots visualising distributional characteristics of experimental
conditions. ..................................................................................................................... 144
Figure 18. Asymmetric beanplots visualising pairwise contrasts and various
distributional characteristics. ........................................................................................ 145
Figure 19. Statistically significant differences between grand means of experimental
conditions and their associated 95% confidence intervals. ........................................... 150
Figure 20. Comparison of V
00
vs. V
10
(means per condition with associated 95%
Bayesian credible intervals). ......................................................................................... 157
23
Figure 21. Comparison of condition V
01
vs. V
11
(means per condition with associated
95% Bayesian credible intervals). ................................................................................. 157
Figure 22. Prior and posterior plot for the difference between V
00
vs. V
10
. ................. 158
Figure 23. Prior and posterior plot for the difference between V
01
vs. V
11
. ................. 159
Figure 24. Visual summary of the Bayes Factor robustness check for condition V
00
vs.
V
10
using various Cauchy priors. .................................................................................. 160
Figure 25. Visual summary of the Bayes Factor robustness check for condition V
01
vs.
V
11
using various Cauchy priors. .................................................................................. 161
Figure 26. Sequential analysis depicting the flow of evidence as n accumulates over
time (experimental condition V
00
vs. V
10
). .................................................................... 162
Figure 27. The visualisations thus show the evolution of the Bayes Factor (y-axis) as a
function of n (x-axis). In addition, the graphic depicts the accrual of evidence for
various Cauchy priors (experimental condition V
01
vs. V
11
). ....................................... 163
Figure 28. Hierarchically organised pictogram of the descriptive model for the Bayesian
parameter estimation (adaptd from Kruschke, 2013, p. 575). ....................................... 177
Figure 29. Visual comparison of the Gaussian versus Student distribution. ................ 179
Figure 30. Visual comparison of the distributional characteristics of the Gaussian versus
Student distribution. ...................................................................................................... 180
Figure 31. Visualisation of various MCMC convergence diagnostics for μ
1
(corresponding to experimental condition V
00
)............................................................. 182
Figure 32. Correlation matrix for the estimated parameters (μ
1
, μ
2
, σ
1
, σ
1
, ν) for
experimental condition V
00
and V
10
. ............................................................................. 187
Figure 33. Posterior distributions of μ
1
(condition V
00
, upper panel) and μ
2
(condition
V
10
, lower panel) with associated 95% posterior high density credible intervals. ........ 188
Figure 34. Randomly selected posterior predictive plots (n = 30) superimposed on the
histogram of the experimental data (upper panel: condition V
00
; lower panel condition
V
10
). ............................................................................................................................... 189
Figure 35. Posterior distributions of σ
1
(condition V
00
, upper panel), σ
2
(condition V
10
,
lower panel), and the Gaussianity parameter ν with associated 95% high density
intervals. ........................................................................................................................ 190
Figure 36. Visual summary of the Bayesian parameter estimation for the difference
between means for experimental condition V
00
vs. V
01
with associated 95% HDI and a
ROPE ranging from [-0.1, 0.1]. .................................................................................... 192
Figure 37. Posterior predictive plot (n=30) for the mean difference between
experimental condition V
00
vs. V
01
. .............................................................................. 193
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Figure 38. Visual summary of the Bayesian parameter estimation for the effect size of
the difference between means for experimental condition V
00
vs. V
01
with associated
95% HDI and a ROPE ranging from [-0.1, 0.1]. .......................................................... 194
Figure 39. Visual summary of the Bayesian parameter estimation for the standard
deviation of the difference between means for experimental condition V
00
vs. V
01
with
associated 95% HDI and a ROPE ranging from [-0.1, 0.1]. ......................................... 194
Figure 40. Visual summary of the Bayesian parameter estimation for the difference
between means for experimental condition V
10
vs. V
11
with associated 95% HDIs and a
ROPEs ranging from [-0.1, 0.1]. ................................................................................... 196
Figure 41. Schematic visualisation of the temporal sequence of events within two
successive experimental trials. ...................................................................................... 206
Figure 42. Visual summary of differences between means with associated 95%
confidence intervals. ..................................................................................................... 210
Figure 43. Asymmetric beanplots (Kampstra, 2008) depicting the differences in means
and various distributional characteristics of the dataset................................................ 211
Figure 44. Means per condition with associated 95% Bayesian credible intervals. ..... 215
Figure 45. Prior and posterior plot for the difference between V
00
vs. V
01
. ................. 216
Figure 46. Prior and posterior plot for the difference between V
10
vs. V
11
. ................. 217
Figure 47. Bayes Factor robustness check for condition V
00
vs. V
10
using various
Cauchy priors. ............................................................................................................... 218
Figure 48. Bayes Factor robustness check for condition V
01
vs. V
11
using various
Cauchy priors. ............................................................................................................... 219
Figure 49. Sequential analysis depicting the accumulation of evidence as n accumulates
over time (for experimental condition V
00
vs. V
10
). ...................................................... 219
Figure 50. Sequential analysis depicting the accumulation of evidence as n accumulates
over time (for experimental condition V
00
vs. V
10
). ...................................................... 220
Figure 51. Comprehensive summary of the Bayesian parameter estimation. ............... 226
Figure 52. Visual synopsis of the results of the Bayesian parameter estimation. ......... 229
Figure 53. Visualisation of differences in means between conditions with associated
95% confidence intervals. ............................................................................................. 242
Figure 54. Difference between means per condition with associated 95% Bayesian
credible intervals. .......................................................................................................... 246
Figure 55. Prior and posterior plot for the difference between V
00
vs. V
10
. ................. 246
Figure 56. Prior and posterior plot for the difference between V
01
vs. V
11
. ................. 247
25
Figure 57. Visual summary of the Bayesian parameter estimation for the difference
between means for experimental condition V
00
vs. V
01
with associated 95% HDIs and a
ROPEs ranging from [-0.1, 0.1]. ................................................................................... 250
Figure 58. Visual summary of the Bayesian parameter estimation for the difference
between means for experimental condition V
10
vs. V
11
................................................ 252
Figure 59. Diagrammatic representation of the temporal sequence of events within two
successive experimental trials in Experiment 4. ........................................................... 259
Figure 60. Visual summary of differences between means with associated 95%
confidence intervals. ..................................................................................................... 262
Figure 61. Beanplots depicting the differences in means and various distributional
characteristics of the dataset.......................................................................................... 263
Figure 62. Means per condition with associated 95% Bayesian credible intervals. ..... 266
Figure 63. Prior and posterior plot for the difference between V
00
vs. V
01
. ................. 267
Figure 64. Prior and posterior plot for the difference between V
10
vs. V
11
. ................. 268
Figure 65. Bayes Factor robustness check for condition V
00
vs. V
10
using various
Cauchy priors. ............................................................................................................... 269
Figure 66. Bayes Factor robustness check for condition V
01
vs. V
11
using various
Cauchy priors. ............................................................................................................... 270
Figure 67. Sequential analysis depicting the accumulation of evidence as n accumulates
over time (for experimental condition V
00
vs. V
10
). ...................................................... 271
Figure 68. Sequential analysis depicting the accumulation of evidence as n accumulates
over time (for experimental condition V
00
vs. V
10
). ...................................................... 272
Figure 69. Trace plot of the predicted difference between means for one of the three
Markov Chains. The patterns suggest convergence to the equilibrium distribution π. . 274
Figure 70. Density plot for the predicted difference between means. .......................... 275
Figure 71. Comprehensive summary of the Bayesian parameter estimation. ............... 278
Figure 72. Posterior distributions for the mean pairwise difference between
experimental conditions (V
10
vs. V
11
), the standard deviation of the pairwise difference,
and the associated effect size, calculated as (0)/. .......................................... 280
Figure 73. Classical (commutative) probability theory as special case within the more
general overarching/unifying (noncommutative) quantum probability framework. ..... 293
Figure 74. The Duhem-Quine Thesis: The underdetermination of theory by data. ...... 297
Figure 75. Supernormal stimuli: Seagull with a natural “normal” red dot on its beak. 310
Figure 76. Photograph of Albert Einstein and Ravīndranātha Ṭhākura in Berlin, 1930
(adapted from Gosling, 2007). ...................................................................................... 315
26
Figure 77. The attitudes of physicists concerning foundational issues of quantum
mechanics (adapted from Schlosshauer, Kofler, & Zeilinger, 2013; cf. Sivasundaram &
Nielsen, 2016). .............................................................................................................. 329
Figure 78. Graph indicating the continuously increasing popularity of p-values since
1950. .............................................................................................................................. 364
Figure 79. Questionable research practices that compromise the hypothetico-deductive
model which underpins scientific research (adapted from C. D. Chambers, Feredoes,
Muthukumaraswamy, & Etchells, 2014). ..................................................................... 371
Figure 80. Flowchart of preregistration procedure in scientific research. .................... 373
Figure 81. Graphical illustration of the iterative sequential Bonferroni–Holm procedure
weighted (adapted from Bretz, Maurer, Brannath, & Posch, 2009, p. 589). ................ 387
Figure 82. Neuronal microtubules are composed of tubulin. The motor protein kinesin
(powered by the hydrolysis of adenosine triphosphate, ATP) plays a central in vesicle
transport along the microtubule network (adapted from Stebbings, 2005). .................. 546
Figure 83. Space filling generative software art installed in Barclays Technology Center
Dallas Lobby (November 2014-15). ............................................................................. 548
Figure 84. Algorithmic art: An artistic visual representation of multidimensional Hilbert
space (© Don Relyea). .................................................................................................. 549
Figure 85. Average functional connectivity density Φ under the experimental vs. control
condition (adapted from Tagliazucchi et al., 2016, p. 1044) ........................................ 554
Figure 86. Flowchart depicting the default-interventionist model. ............................... 562
Figure 87. The Müller-Lyer illusion (Müller-Lyer, 1889). ........................................... 568
Figure 88. Neuroanatomical correlates of executive functions (DL-PFC, vmPFC, and
ACC) ............................................................................................................................. 570
Figure 89. Bistable visual stimulus used by Thomas Kuhn in order to illustrate the
concept of a paradigm-shift........................................................................................... 572
Figure 90. Results of CogNovo NHST survey ............................................................. 578
Figure 91. Logical consistency rates ............................................................................. 580
Figure 92. Bayesian reanalysis of the results NHST reported by White et al., 2014. ... 586
Figure 93. Q-Q plots identifying the 5 most extreme observation per experimental
condition (linearity indicates Gaussianity). .................................................................. 626
Figure 94. Boxplots visualising differences between experimental conditions (i.e.,
median, upper and lower quartile). ............................................................................... 629
Figure 95. Tolerance interval based on Howe method for experimental condition V
00
.
....................................................................................................................................... 633
27
Figure 96. Tolerance interval based on Howe method for experimental condition V
01
.
....................................................................................................................................... 634
Figure 97. Tolerance interval based on Howe method for experimental condition V
10
.
....................................................................................................................................... 635
Figure 98. Tolerance interval based on Howe method for experimental condition V
11
.
....................................................................................................................................... 636
Figure 99. Bootstrapped mean difference for experimental conditions V
00
vs. V
10
based
on 100000 replicas. ....................................................................................................... 640
Figure 100. Bootstrapped mean difference for experimental conditions V
10
vs. V
11
based
on 100000 replicas. ....................................................................................................... 642
Figure 101. Histogram of the bootstrapped mean difference between experimental
condition V
00
and V
10
based on 100000 replicates (bias-corrected & accelerated) with
associated 95% confidence intervals. ............................................................................ 644
Figure 102. Histogram of the bootstrapped mean difference between experimental
condition V
01
and V
11
based on 100000 replicates (bias-corrected & accelerated) with
associated 95% confidence intervals. ............................................................................ 645
Figure 103. Bootstrapped effect size (Cohen’s d) for condition V
00
vs V
01
based on
R=100000. ..................................................................................................................... 647
Figure 104. Bootstrapped effect size (Cohen’s d) for condition V
10
vs V
11
based on
R=100000. ..................................................................................................................... 649
Figure 105. Posterior distributions for experimental conditions V
00
and V
10
with
associated 95% high density intervals. ......................................................................... 652
Figure 106. Posterior distributions (based on 100000 posterior draws) for experimental
conditions V
01
and V
11
with associated 95% high density intervals. ............................ 656
Figure 107. Histogram of the Bayesian bootstrap (R=100000) for condition V
00
vs. V
10
with 95% HDI and prespecified ROPE ranging from [-0.1, 0.1]. ................................. 659
Figure 108. Posterior distribution (n=100000) of the mean difference between V
00
vs.
V
10.
................................................................................................................................ 660
Figure 109. Histogram of the Bayesian bootstrap (R=100000) for condition V
01
vs. V
11
with 95% HDI and prespecified ROPE ranging from [-0.1, 0.1]. ................................. 662
Figure 110. Posterior distribution (n=100000) of the mean difference between V
01
vs.
V
11.
................................................................................................................................ 663
Figure 111. Visual comparison of Cauchy versus Gaussian prior distributions
symmetrically centred around δ. The abscissa is standard deviation and ordinate is the
density. .......................................................................................................................... 673
Figure 112. Graphic of Gaussian versus (heavy tailed) Cauchy distribution. X axis is
standard deviation and y axis is the density .................................................................. 675
28
Figure 113. MCMC diagnostics for μ
1
(experimental condition V
00
). .......................... 701
Figure 114. MCMC diagnostics for μ
2
(experimental condition V
01
). .......................... 702
Figure 115. MCMC diagnostics for σ
1
(experimental condition V
00
). .......................... 703
Figure 116. MCMC diagnostics for σ
2
(experimental condition V
11
). .......................... 704
Figure 117. MCMC diagnostics for ν. .......................................................................... 705
Figure 118. Pictogram of the Bayesian hierarchical model for the correlational analysis
(Friendly et al., 2013). The underlying JAGS-model can be downloaded from the
following URL: http://irrational-decisions.com/?page_id=2370 .................................. 721
Figure 119. Visualisation of the results of the Bayesian correlational analysis for
experimental condition V
00
and V
01
with associated posterior high density credible
intervals and marginal posterior predictive plots. ......................................................... 724
Figure 120. Visualisation of the results of the Bayesian correlational analysis for
experimental condition V
10
and V
11
with associated posterior high density credible
intervals and marginal posterior predictive plots. ......................................................... 726
Figure 121. 3D scatterplot of the MCMC dataset with 50% concentration ellipsoid
visualising the relation between μ
1
(V
00
) and μ
2
(V
01
), and v in 3-dimensional parameter
space. ............................................................................................................................. 741
Figure 122. 3D scatterplot (with regression plane) of MCMC dataset with increased
zoom-factor in order to emphasize the concentration of the values of θ. ..................... 742
Figure 123. Visualisation of the results of the Bayesian correlational analysis for
experimental condition V
00
and V
01
with associated posterior high density credible
intervals and marginal posterior predictive plots. ......................................................... 753
Figure 124. Visualisation of the results of the Bayesian correlational analysis for
experimental condition V
10
and V
11
with associated posterior high density credible
intervals and marginal posterior predictive plots. ......................................................... 757
Figure 125. Q-Q plots for visual inspection of distribution characteristics. ................. 761
Figure 126. Symmetric beanplots for visual inspection of distribution characteristics.762
Figure 127. χ2 Q-Q plot (Mahalanobis Distance, D
2
)................................................... 764
Figure 128. Visualisation of the results of the Bayesian correlational analysis for
experimental condition V
00
and V
01
with associated posterior high density credible
intervals and marginal posterior predictive plots. ......................................................... 771
Figure 129. Graphic depicting the frequency of the terms “Bayesian inference” and
Bayesian statistics” through time (with least square regression lines). ........................ 793
Figure 130. Hierarchically organised pictogram of the descriptive model for the
Bayesian parameter estimation (adaptd from Kruschke, 2013, p. 575). ....................... 815
Figure 131. Visual comparison of the Gaussian versus Student distribution. .............. 817
29
Figure 132. Visual comparison of the distributional characteristics of the Gaussian
versus Student distribution. ........................................................................................... 819
Figure 133. Edaplot created with “StatDA” package in R. ........................................ 829
Figure 134. Connected boxplots for condition V
00
vs. V
01
. .......................................... 837
Figure 135. Connected boxplots for condition V
10
vs. V
11
. .......................................... 838
Figure 136. Connected boxplots for condition V
00
, V
01
, V
10
, V
11
. ............................... 839
Figure 137. Graph indicating the increasing popularity of MCMC methods since 1990.
Data was extracted from the Google Books Ngram Corpus (Lin et al., 2012) with the R
package “ngramr”. ...................................................................................................... 848
Figure 138. Discrete time series for the hypertext web search query “Markov chain
Monte Carlo” since the beginning of GoogleTrends in 2013/2014 for various countries
(DE=Germany, GB=Great Britain, US=United States). ............................................... 850
Figure 139. Color-coded geographical map for the query “Markov chain Monte Carlo”
(interest by region). ....................................................................................................... 850
Figure 140. Chemical structures of Serotonin, Psilocin, and N,N-Dimethyltryptamine in
comparison. ................................................................................................................... 853
Figure 142. Average functional connectivity density Φ under LSD vs. control condition
(adapted from Tagliazucchi et al., 2016, p. 1044) ........................................................ 884
30
Tables
Table 1 Descriptive statistics for experimental conditions. .......................................... 141
Table 2 Shapiro-Wilk’s W test of Gaussianity. ............................................................ 146
Table 3 Paired samples t-tests and nonparametric Wilcoxon signed-rank tests ......... 151
Table 4 Bayes Factors for the orthogonal contrasts.................................................... 154
Table 5 Qualitative heuristic interpretation schema for various Bayes Factor quantities
(adapted from Jeffreys, 1961). ...................................................................................... 155
Table 6 Descriptive statistics and associated Bayesian credible intervals. ................. 156
Table 7 Summary of selected convergence diagnostics for μ
1
, μ
2
, σ
1
, σ
2
, and ν. .......... 185
Table 8 Results of Bayesian MCMC parameter estimation for experimental conditions
V
00
and V
10
with associated 95% posterior high density credible intervals. ................ 186
Table 9 Numerical summary of the Bayesian parameter estimation for the difference
between means for experimental condition V
00
vs. V
01
with associated 95% posterior
high density credible intervals. ..................................................................................... 191
Table 10 Numerical summary of the Bayesian parameter estimation for the difference
between means for experimental condition V
10
vs. V
11
with associated 95% posterior
high density credible intervals. ..................................................................................... 195
Table 11 Shapiro-Wilk’s W test of Gaussianity. ........................................................... 208
Table 12 Descriptive statistics for experimental conditions. ........................................ 209
Table 13 Paired samples t-tests and nonparametric Wilcoxon signed-rank tests. ....... 212
Table 14 Bayes Factors for the orthogonal contrasts................................................... 214
Table 15 Descriptive statistics with associated 95% Bayesian credible intervals. ...... 214
Table 16 MCMC convergence diagnostics based on 100002 simulations for the
difference in means between experimental condition V
00
vs. V
10
. ................................. 223
Table 17 MCMC results for Bayesian parameter estimation analysis based on 100002
simulations for the difference in means between experimental condition V
00
vs. V
10
. .. 225
Table 18 MCMC convergence diagnostics based on 100002 simulations for the
difference in means between experimental condition V
00
vs. V
10
. ................................. 227
Table 19 MCMC results for Bayesian parameter estimation analysis based on 100002
simulations for the difference in means between experimental condition V
01
vs. V
11
. .. 228
Table 20 Descriptive statistic for experimental conditions. ........................................ 239
Table 21 Shapiro-Wilk’s W test of Gaussianity. ........................................................... 240
Table 22 Paired samples t-test and nonparametric Wilcoxon signed-rank tests .......... 243
31
Table 23 Bayes Factors for orthogonal contrasts. ....................................................... 245
Table 24 Descriptive statistics and associated Bayesian 95% credible intervals. ...... 245
Table 25 Numerical summary of the Bayesian parameter estimation for the difference
between means for experimental condition V
00
vs. V
10
with associated 95% posterior
high density credible intervals. ..................................................................................... 249
Table 26 MCMC convergence diagnostics based on 100002 simulations for the
difference in means between experimental condition V
01
vs. V
11
. ................................. 251
Table 27 Numerical summary of the Bayesian parameter estimation for the difference
between means for experimental condition V
01
vs. V
11
with associated 95% posterior
high density credible intervals. ..................................................................................... 251
Table 28 Descriptive statistics for experimental conditions. ........................................ 260
Table 29 Shapiro-Wilk’s W test of Gaussianity. ........................................................... 261
Table 30 Paired samples t-tests and nonparametric Wilcoxon signed-rank tests. ....... 264
Table 31 Bayes Factors for both orthogonal contrasts. ............................................... 265
Table 32 Descriptive statistics with associated 95% Bayesian credible intervals. ...... 266
Table 33 Summary of selected convergence diagnostics. ............................................. 276
Table 34 Results of Bayesian MCMC parameter estimation for experimental conditions
V
00
and V
10
with associated 95% posterior high density credible intervals. ................. 277
Table 35 Summary of selected convergence diagnostics. ............................................. 279
Table 36 Results of Bayesian MCMC parameter estimation for experimental conditions
V
10
and V
11
with associated 95% posterior high density credible intervals. ................. 279
Table 37 Potential criteria for the multifactorial diagnosis of “pathological
publishing” (adapted from Buela-Casal, 2014, pp. 92–93). ........................................ 368
Table 38 Hypothesis testing decision matrix in inferential statistics. ........................... 383
Table 39 Features attributed by various theorists to the hypothesized cognitive systems.
....................................................................................................................................... 565
Table 40. Comparison between international universities and between academic
groups. ........................................................................................................................... 580
Table 41 ......................................................................................................................... 581
Table 42 Results of Bca bootstrap analysis (experimental condition V
00
vs. V
10
). ...... 641
Table 43 Results of Bca bootstrap analysis (experimental condition V
10
vs. V
11
). ...... 643
Table 44 Numerical summary of Bayesian bootstrap for condition V
00
. ...................... 653
Table 45 Numerical summary of Bayesian bootstrap for condition V
10
. ...................... 654
Table 46 Numerical summary of Bayesian bootstrap for condition V
01
. ...................... 657
Table 47 Numerical summary of Bayesian bootstrap for condition V
11
. ...................... 657
32
Table 48 Numerical summary of Bayesian bootstrap for the mean difference between
V
00
vs. V
10
. ..................................................................................................................... 661
Table 49 Numerical summary of Bayesian bootstrap for the mean difference between
V
00
vs. V
10
. ..................................................................................................................... 664
Table 50......................................................................................................................... 667
Table 51 Summary of convergence diagnostics for ρ, μ
1
, μ
2
, σ
1
, σ
2
, ν, and the posterior
predictive distribution of V
00
and V
10
. ........................................................................... 722
Table 52 Numerical summary for all parameters associated with experimental condition
V
10
and V
01
and their corresponding 95% posterior high density credible intervals. .. 723
Table 53 Numerical summary for all parameters associated with experimental condition
V
01
and V
11
and their corresponding 95% posterior high density credible intervals. .. 725
Table 54 Numerical summary for all parameters associated with experimental condition
V
10
and V
01
and their corresponding 95% posterior high density credible intervals. .. 756
Table 55 Descriptive statistics and various normality tests. ........................................ 763
Table 56 Royston’s multivariate normality test. ........................................................... 765
Table 57 Numerical summary for all parameters associated with experimental condition
V
10
and V
01
and their corresponding 95% posterior high density credible intervals. .. 770
Table 58 Amplitude statistics for stimulus-0.6.wav. ................................................... 780
Table 59 Amplitude statistics for stimulus-0.8.wav. ................................................... 781
33
Equations
Equation 1. Weber’s law. ................................................................................................ 63
Equation 2. Fechner’s law. .............................................................................................. 65
Equation 3. Stevens's power law. .................................................................................... 75
Equation 4. Mathematical representation of a qubit in Dirac notation. .......................... 85
Equation 5. Kolmogorov’s probability axiom .............................................................. 103
Equation 6. Classical probability theory axiom (commutative).................................... 104
Equation 7. Quantum probability theory axiom (noncommutative). ............................ 104
Equation 8. Bayes’ theorem (Bayes & Price, 1763) as specified for the hierarchical
descriptive model utilised to estimate θ. ....................................................................... 171
Equation 9. Formula to calculate P
rep
(a proposed estimate of replicability). .............. 377
Equation 10: Holm's sequential Bonferroni procedure (Holm, 1979). ......................... 384
Equation 11: Dunn-Šidák correction (Šidák, 1967) ...................................................... 388
Equation 12: Tukey's honest significance test (Tukey, 1949) ...................................... 388
Equation 13. The inverse probability problem .............................................................. 578
Equation 14. The Cramér-von Mises criterion (Cramér, 1936) .................................... 627
Equation 15. The Shapiro-Francia test (S. S. Shapiro & Francia, 1972) ...................... 627
Equation 16. Fisher’s multivariate skewness and kurtosis............................................ 628
Equation 17: Cohen's d (Cohen, 1988) ......................................................................... 637
Equation 18: Glass' Δ (Glass, 1976) ............................................................................. 638
Equation 19: Hedges' g (Hedges, 1981) ........................................................................ 638
Equation 20. Probability Plot Correlation Coefficient (PPCC) .................................... 666
Equation 21. HDI and ROPE based decision algorithm for hypothesis testing. ........... 686
34
Code
Code 1. R code for plotting an iteractive 3-D visualisation of a Möbius band. ............ 544
Code 2. Algorithmic digital art: C++ algorithm to create a visual representation of
multidimensional Hilbert space (© Don Relyea). ......................................................... 551
Code 3. R code associated with the Bayesian reanalysis of the NHST results reported
by White et al. (2014). .................................................................................................. 589
Code 4. HTML code with Shockwave Flash® (ActionScript 2.0) embedded via
JavaScript. ..................................................................................................................... 601
Code 5. R code for symmetric and asymmetric “beanplots”. ................................. 631
Code 6. R code for plotting Cauchy versus Gaussian distribution (n=1000)
symmetrically centred around δ [-10,10]. ..................................................................... 674
Code 7. R code for plotting tails of Cauchy versus Gaussian distributions. ................. 676
Code 8. R code for plotting t-distributions with varying ν parametrisation. ................ 680
Code 9. R commander code for 3D scatterplot with concertation ellipsoid. ................ 743
Code 10. R code to download, save, and plot data from Google Ngram. Various R
packages are required (devtools, ngramr, ggplot2). ...................................................... 796
Code 11. R code to find various dependencies of the “BEST” package. ...................... 812
Code 12. R code for visualising a Gaussian versus Student distribution. ..................... 818
Code 13. R code for detailed comparison of differences between the Gaussian and the
superimposed t-distribution........................................................................................... 820
Code 14. R code for Bayesian analysis using the “BEST.R” function. ....................... 826
Code 15. “p.rep” function from the “psych” R package (after Killeen, 2005a) ......... 836
35
Electronic supplementary materials
Custom programmed meta-search tool for literature review:
http://irrational-decisions.com/?page_id=526
Animated version of the Möbius band which constitutes the cover image:
http://moebius-band.ga/
Online repository associated with this thesis containing all datasets:
http://irrational-decisions.com/phd-thesis/
Literature review on quantum cognition (HTML format):
http://irrational-decisions.com/?page_id=1440
Möbius band transformations:
http://irrational-decisions.com/?page_id=2599
Digital artworks depicting the Necker cube from a quantum cognition perspective
The “Quantum Necker cube”:
http://irrational-decisions.com/?page_id=420
Necker Qbism: Thinking outside the box – getting creative with the Necker cube:
http://irrational-decisions.com/?page_id=1354
The syllogistic logic of hypothesis testing – logical fallacies associated with NHST:
http://irrational-decisions.com/?page_id=441#nhst
Explanation of “rational intelligence” (IQ ≠ RQ):
http://irrational-decisions.com/?page_id=2448
BoseEinstein statistics: “Quatum dice” (included interactive Shockwave Flash
applet):
http://irrational-decisions.com/quantum_dice/
The Gott-Li self-creating fractal universe model (Vaas, 2004):
http://irrational-decisions.com/?page_id=2351
An interactive application of the HSV colour model programmed in Adobe® Flash:
http://irrational-decisions.com/?page_id=875
Visual stimuli as used in Experiment 1 and 2:
http://irrational-decisions.com/phd-thesis/visual-stimuli/low-luminance.jpg
http://irrational-decisions.com/phd-thesis/visual-stimuli/high-luminance.jpg
Python code for Experiment 1:
http://irrational-decisions.com/?page_id=618
36
High-resolution version of median-based connected boxplots:
http://irrational-decisions.com/phd-thesis/connected-boxplots-exp1-v00-v10.pdf
http://irrational-decisions.com/phd-thesis/connected-boxplots-exp1-v01-v11.pdf
Comprehensive summary NHST results if Experiment 1 including interactive
visualisation of the Vovk-Sellke maximum p-ratio (VS-MPR):
http://irrational-decisions.com/phd-thesis/results-exp1.html
JASP analysis script associated with the Bayes Factor analysis of Experiment 1:
http://irrational-decisions.com/phd-thesis/exp1.jasp
Open-source software for Markov chain Monte Carlo simulations and Bayesian
parameter estimation:
http://irrational-decisions.com/?page_id=1993
High-resolution version of the Bayesian parameter estimation correlation matrix of
Experiment 1:
http://irrational-decisions.com/phd-thesis/cor-matrix-exp1.pdf
High-resolution version of the posterior distributions associated with the Bayesian
parameter estimation analysis:
http://irrational-decisions.com/phd-thesis/summary-exp1-cond-v00-vs-v10.pdf
Comprehensive summary of the Bayes Factor analysis associated with Experiment
2:
http://irrational-decisions.com/phd-thesis/bayesfactor-analysis-exp2.html
JASP analysis script associated with Experiment 2:
http://irrational-decisions.com/phd-thesis/analysis-script-exp2.jasp
Auditory stimuli as utilised in Experiment 3 and 4 (*wav files)
http://irrational-decisions.com/phd-thesis/auditory-stimuli/stimulus-0.6.wav
http://irrational-decisions.com/phd-thesis/auditory-stimuli/stimulus-0.8.wav
Comprehensive summary of the NHST analysis associated with Experiment 3:
http://irrational-decisions.com/phd-thesis/exp3/results-exp3.html
Comprehensive summary of the NHST analysis associated with Experiment 4:
http://irrational-decisions.com/phd-thesis/frequentist-analysis-exp4.html
Comprehensive summary of the Bayes Factor analysis associated with Experiment
4:
http://irrational-decisions.com/phd-thesis/bayesfactor-analysis-exp4.html
JASP analysis script associated with Experiment 4:
37
http://irrational-decisions.com/phd-thesis/analysis-script-exp4.jasp
Interactive 3-dimensional scatterplot of the MCMC dataset associated with
Experiment 1 as a MP4 video file:
http://irrational-decisions.com/phd-thesis/scatterplot3d-openGL.mp4
Monte Carlo dataset associated with Experiment 1:
http://irrational-decisions.com/phd-thesis/mcmc-chain-exp2.txt
“BEST.R” script for MCMC based Bayesian parameter estimation:
http://irrational-decisions.com/?page_id=1996
High-resolution of “Google Trends” timeseries:
http://irrational-decisions.com/phd-thesis/gtrends-mcmc.pdf
Dataset underlying the “Google Trends” timeseries:
http://irrational-decisions.com/phd-thesis/gtrends-mcmc.txt
38
Author: Christopher B. Germann
Title: A psychophysical investigation of quantum cognition: An interdisciplinary
synthesis
Abstract
Quantum cognition is an interdisciplinary emerging field within the cognitive sciences
which applies various axioms of quantum mechanics to cognitive processes. This thesis
reports the results of several empirical investigations which focus on the applicability of
quantum cognition to psychophysical perceptual processes. Specifically, we
experimentally tested several a priori hypotheses concerning 1) constructive
measurement effects in sequential perceptual judgments and 2)noncommutativity in the
measurement of psychophysical observables . In order to establish the generalisability
of our findings, we evaluated our prediction across different sensory modalities (i.e.,
visual versus auditory perception) and in cross-cultural populations (United Kingdom
and India). Given the well-documented acute “statistical crisis” in science (Loken &
Gelman, 2017a) and the various paralogisms associated with Fisherian/Neyman-
Pearsonian null hypothesis significance testing, we contrasted various alternative
statistical approaches which are based on complementary inferential frameworks (i.e.,
classical null hypothesis significance testing, nonparametric bootstrapping, model
comparison based on Bayes Factors analysis, Bayesian bootstrapping, and Bayesian
parameter estimation via Markov chain Monte Carlo simulations). This multimethod
approach enabled us to analytically cross-validate our experimental results, thereby
increasing the robustness and reliability of our inferential conclusions. The findings are
discussed in an interdisciplinary context which synthesises knowledge from several
prima facie separate disciplines (i.e., psychology, quantum physics, neuroscience, and
philosophy). We propose a radical reconceptualization of various epistemological and
39
ontological assumptions which are ubiquitously taken for granted (e.g., naïve and local
realism/cognitive determinism). Our conclusions are motivated by recent cutting-edge
findings in experimental quantum physics which are incompatible with the
materialistic/deterministic metaphysical Weltanschauung internalised by the majority of
scientists. Consequently, we argue that scientists need to update their nonevidence-
based implicit beliefs in the light of this epistemologically challenging empirical
evidence.
40
CHAPTER 1. INTRODUCTION
We would like to set the stage for this thesis with a rather extensive
3
but highly apposite
prefatory quotation from the great polymath William James who can be regarded as the
founding father of American psychology. The following quote stems from the
introduction of his essay entitled “The hidden Self” which was published in 1890:
Round about the accredited and orderly facts of every science there ever floats a sort
of dust-cloud of exceptional observations, of occurrences minute and irregular, and
seldom met with, which it always proves less easy to attend to than to ignore. The ideal
of every science is that of a closed and completed system of truth. The charm of most
sciences to their more passive disciples consists in their appearing, in fact, to wear just
this ideal form. Each one of our various ‘ologies’ seems to offer a definite head of
classification for every possible phenomenon of the sort which it professes to cover;
and, so far from free is most men’s fancy, that when a consistent and organized scheme
of this sort has once been comprehended and assimilated, a different scheme is
unimaginable. No alternative, whether to whole or parts, can any longer be conceived
as possible. Phenomena unclassifiable within the system are therefore paradoxical
absurdities, and must be held untrue. When, moreover, as so often happens, the reports
of them are vague and indirect, when they come as mere marvels and oddities rather
than as things of serious moment, one neglects or denies them with the best of scientific
consciences. Only the born geniuses let themselves be worried and fascinated by these
outstanding exceptions, and get no peace till they are brought within the fold. Your
Galileos, Galvanis, Fresnels, Purkinjes, and Darwins are always getting confounded
3
It is easy to misinterpret a quote when it is taken out of its associated context. We tried to circumvent
this common scholarly fallacy by providing an exhaustive quotation, thereby significantly reducing the
odds of committing hermeneutic errors.
41
and troubled by insignificant things. Anyone will renovate his science who will steadily
look after the irregular phenomena. And when the science is renewed, its new formulas
often have more of the voice of the exceptions in them than of what were supposed to be
the rules. No part of the unclassed residuum has usually been treated with a more
contemptuous scientific disregard than the mass of phenomena generally called
mystical. Physiology will have nothing to do with them. Orthodox psychology turns its
back upon them. Medicine sweeps them out; or, at most, when in an anecdotal vein,
records a few of them as effects of the imagination’ a phrase of mere dismissal whose
meaning, in this connection, it is impossible to make precise. All the while, however, the
phenomena are there, lying broadcast over the surface of history. No matter where you
open its pages, you find things recorded under the name of divinations, inspirations,
demoniacal possessions, apparitions, trances, ecstasies, miraculous healings and
productions of disease, and occult powers possessed by peculiar individuals over
persons and things in their neighborhood. […] To no one type of mind is it given to
discern the totality of Truth. Something escapes the best of us, not accidentally, but
systematically, and because we have a twist. The scientific-academic mind and the
feminine-mystical mind shy from each other’s facts, just as they shy from each other’s
temper and spirit. Facts are there only for those who have a mental affinity with them.
When once they are indisputably ascertained and admitted, the academic and critical
minds are by far the best fitted ones to interpret and discuss them - for surely to pass
from mystical to scientific speculations is like passing from lunacy to sanity; but on the
other hand if there is anything which human history demonstrates, it is the extreme
slowness with which the ordinary academic and critical mind acknowledges facts to
exist which present themselves as wild facts with no stall or pigeon-hole, or as facts
which threaten to break up the accepted system. In psychology, physiology, and
42
medicine, wherever a debate between the Mystics and the Scientifics has been once for
all decided, it is the Mystics who have usually proved to be right about the facts, while
the Scientifics had the better of it in respect to the theories. (James, 1890a, pp. 361–362)
James is very explicit when he emphasises the irrational reluctance of the majority of
academic scientists to “face facts” when these are incongruent with the prevailing
internalised paradigm. Thomas Kuhn elaborates this point extensively in his seminal
book “The Structure of Scientific Revolutions” (T. Kuhn, 1970) in which he emphasises
the incommensurability of paradigms. Abraham Maslow discusses the “Psychology of
Science” in great detail in his eponymous book (Maslow, 1962). Maslow formulates a
quasi-Gödelian critique of orthodox science and its “unproved articles of faith, and
taken-for-granted definitions, axioms, and concepts”. Human beings (and therefore
scientists) are generally afraid of the unknown (Tart, 1972), even though the task of
science comprises the exploration of novel and uncharted territory. The history of
science clearly shows how difficult it is to revise deeply engrained theories. The
scientific mainstream community once believed in phrenology, preformationism,
telegony, phlogiston theory, luminiferous aether, contact electrification, the geocentric
universe, the flat earth theory, etc. pp, the errata is long... All these obsolete theories
have been superseded by novel scientific facts. The open question is: Which taken-for-
granted theory is up for revision next? Unfortunately, scientific training leads to
cognitive rigidity
4
, as opposed to cognitive flexibility which is needed for creative
ideation (ideoplasticity) and perspectival pluralism (Giere, 2006). From a
neuroscientific point of view, a possible explanation for this effect is based on a
4
Cognitive inflexibility has been investigated in obsessive-compulsive disorder and it has been correlated
with significantly decreased activation of the prefrontal cortices, specifically the dorsal frontal-striatal
regions (Britton et al., 2010; Gruner & Pittenger, 2017; Gu et al., 2008; Remijnse et al., 2013).
43
Hebbian neural consolidation account. That is, repeatedly utilised neural circuits are
strengthened (Hebb, 1949) and become dominant and rigid, e.g., via the neuronal
process of synaptic long-term potentiation
5
(Lomo, 2003). Interestingly, complex
system theory suggests a bipolar (orthogonal) continuum ranging from rigidity on one
end to chaos on the other. Integration lies interjacent between the extremes. Given that
the cognitive system can be regarded as a complex system, this generic account might
lend itself to conceptualise a “cognitive continuum of information processing states”
(Faust & Kenett, 2014) ranging from rigid cognition to chaotic cognition. In a rigid
neural network, nodes are only sparsely interconnected
6
(i.e., cognitive hyper-rigidity).
In a chaotic neural network topology, on the other hand, virtually all nodes are
interconnected (i.e., cognitive over-flexibility/chaos). In this schema, cognitive
integration (viz., the linkage of differentiated parts (Siegel, 2010)) is consequently
characterised by an intermediate neuronal network connectivity pattern which balances
and synchronizes the polar extremes (i.e., adaptive/dynamic cognitive coherence).
From a psychological point of view, scientist generally have great difficulties to revise
their (oftentimes implicit) theories and adjust their associated “degrees of belief
7
in the
light of new evidence (a desirable quasi-Bayesian epistemological approach),
particularly when they have vested personal/ideological interests in the predominant
5
Using human cerebral organoids and in silico analysis it has been demonstrated that 5-MeO-DMT has
modulatory effects on proteins associated with the formation of dendritic spines and neurite outgrowth
(Dakic et al., 2017) which may influence neuroplasticity and hence ideoplasticity. 5-MeO-DMT has been
found to match the σ
1
receptor. Because σ
1
R agonism regulates dendritic spine morphology and neurite
outgrowth it affects neuroplasticity which form the neural substrate for unconstrained cognition.
6
Network interconnectivity is often quantitatively specified by the rich-club coefficient Φ. This networks
metric quantifies the degree to which well-connected nodes (beyond a certain richness metric) also
connect to each other. Hence, the rich-club coefficient can be regarded as a notation which quantifies a
certain type of associativity.
7
The Quinan “Web of Beliefs” (Quine & Ullian, 1978) provides an applicable semantic analogy to
(Bayesian) neural network connectivity and the process of “belief updating” (i.e., modification of weights
between neuron nodes).
44
status quo.
8
This resistance towards new theories, change, and innovation (i.e.,
exnovation) is deeply rooted in various (primarily unconscious) psychological processes
which we will address later in more detail in the context of empirical findings which are
incompatible with the predominant mainstream paradigm in science (viz., reductionistic
materialism/physicalism/local realism) which has now been conclusively falsified by
recent empirical findings from experimental quantum physics (but see Gröblacher et al.,
2007; Hensen et al., 2015; Wiseman, 2015) — a milestone in the history of science.
Various dispositional factors play a role in this context. Dispositional factors may be
biological (e.g., genetic/epigenetic factors, specific receptor polymorphisms,
dissimilarities in neurotransmitter concentrations,
9
neuroanatomical idiosyncrasies,
variations in enteric microbiota composition/dysbiosis, etc.) and/or psychological in
nature (e.g., personality traits, individual differences in cognitive abilities, childhood
conditioning, cultural disparities, etc.). For instance, the psychological trait
“closedmindedness” (Kruglanski, 2014) appears to be relevant in this regard.
Closedmindedness is characterised as a general unreceptivity towards new ideas,
arguments, and empirical findings. It is anticorrelated with the personality trait
“openness to experience” which forms a major dimension in the widely applied five
8
This ego-driven modus operandi is unfortunately reinforced by an academic “climate of perverse
incentives and hypercompetition(Edwards & Roy, 2017) which does not foster sincere/genuine
scientific authenticity and integrity and is antagonistic towards altruistic behaviour (a selfless attitude is a
vital characteristic of an unbiased scientific ethos which transcends primitive personal interests). The
pressure to “publish or perish” (Fanelli, 2012; Rawat & Meena, 2014) leads to “publication-bias” (Franco
et al., 2014; J. D. Scargle, 2000) and promotes career-oriented behaviour which has been diagnosed as
“pathological publishing (Buela-Casal, 2014). Moreover, the quantitative (putatively “objective”)
evaluation of researchers based on bibliometric indices is causally related to an extrinsically motivated
“impact factor style of thinking(Fernández-Ríos & Rodríguez-Díaz, 2014) which is common among
researchers and compromised scientific values. These nontrivial systemic issues seriously impede the
scientific endeavour and have to be rectified for self-evident reasons. We are firmly convinced that
instead of “playing the game” serious scientific researchers have an obligation to try their best “to change
the rules” as it has recently been argued in an excellent AIMS
NEUROSCIENCE article (C. D. Chambers et
al., 2014). The ideals of science are fundamentally based on the quest for knowledge and truth and not on
egoic motives such as career aspirations, social status, and monetary rewards (Sassower, 2015).
9
See Appendix A3 for more details on the role of neurochemistry in the context of creativity and
“unconstrained cognition”.
45
factor model (FFM) of personality (McCrae, 1987). Openness
10
to experience (OTE) is
a rather complex psychological construct which is broadly related to interindividual
differences in information processing (Green-Hennessy & Reis, 1998), creative
cognition and behaviour (George & Zhou, 2001), aesthetic perception (McCrae, 2007),
absorption (Roche & McConkey, 1990), and cognitive style (Sadler-Smith, 2001), inter
alia. For example, higher OTE scores are related to novelty seeking, curiosity, and
intellectual achievement (Paunonen & Ashton, 2001). Moreover, OTE
11
is statistically
significantly correlated with fluid intelligence (Cattell, 1963), divergent thinking,
12
and
various facets of creativity (Silvia, Nusbaum, Berg, Martin, & O’Connor, 2009), to
mention just the most salient aspects of this multidimensional personality construct.
Hence, OTE is pivotal for the advancement of science into novel and unexplored
territory, for memetic evolution, and consequently, in sensu lato, for the evolution of
humanity as a species on this planet. Other important correlated psychological concepts
which are related to openness and a nondogmatic scientific attitude are intellectual
humility (Gregg, Mahadevan, & Sedikides, 2017; Krumrei-Mancuso & Rouse, 2016),
epistemic curiosity (Eigenberger, Critchley, & Sealander, 2007; Litman & Spielberger,
10
From a cognitive linguistic point of view, the usage of the concept “open” is interesting because it
indicative of a spatial metaphor (Lakoff, 1993, 2014; Lakoff & Nuñez, 2000). The psychological concepts
openness to experience” and “closedmindedness” are both based on primary conceptual metaphors (i.e.,
the spatial topology of containment (Lakoff & Johnson, 1980)). In other terms, the associated image
metaphor implies that the cognitive system tends to be open or closed to novel information (viz., the
diametrical psychological concepts can be represented as a gradual bipolar continuum: openness
closedness).
11
Recent neuropsychopharmacological work empirically demonstrated that the partial serotonin (5-
hydroxytryptamin) agonist Psilocybin (O-phosphoryl-4-hydroxy-N,N-dimethyltryptamine) (Hofmann et
al., 1958, 1959) enhances the personality trait openness to experience longitudinally (MacLean et al.,
2011).
12
Interestingly, it has been experimentally shown that psychotropic serotonergic compounds can enhance
divergent thinking while decreasing conventional convergent thinking (Kuypers et al., 2016), an empirical
finding of great importance which deserves much more detailed investigation. Moreover, it has been
noted that “plasticity and open-mindedness” are primarily 5-HT
2A
receptor mediated (as opposed to 5-
HT
1A
) and that “a key function of brain serotonin transmission is to engage in processes necessary for
change, when change is necessary(Carhart-Harris & Nutt, 2017, p. 1098). Moreover, cognitive
flexibility appears to be positively modulated by 5-HT
2A
agonists (Boulougouris, Glennon, & Robbins,
2008; Matias, Lottem, Dugué, & Mainen, 2017), thereby leading to enhancements in creative thinking
(Frecska, Móré, Vargha, & Luna, 2012).
46
2003), and rational intelligence (i.e., critical thinking) (K. Stanovich, 2014). Moreover,
group conformity and obedience to authority are important psychological constructs in
this context. To use Richard Feynman’s wise words which explicitly emphasise the
importance of a lack of respect for authority figures:
Science alone of all the subjects contains within itself the lesson of the danger of belief
in the infallibility of the greatest teachers in the preceding generation. […] Learn from
science that you must doubt the experts. As a matter of fact, I can also define science
another way: Science is the belief in the ignorance of experts.(Feynman, 1968)
The present thesis focuses on a novel emerging field within the cognitive science which
is referred to as “quantum cognition” (Aerts, 2009; Aerts & Sassoli de Bianchi, 2015;
Łukasik, 2018; Moreira & Wichert, 2016a; Z. Wang, Busemeyer, Atmanspacher, &
Pothos, 2013). Quantum cognition can be broadly defined as a combination of quantum
physics and cognitive psychology.
13
Recent empirical findings from quantum cognition
deeply challenge the prevailing academic modus operandi adopted by many experts in
cognitive psychology and the neurosciences. The counterintuitive theoretical and
epistemological implications of quantum physics require a great deal of OTE
(particularly divergent thinking), intellectual humility (as opposed to intellectual
arrogance), and epistemic curiosity (Echenique-Robba, 2013), because they challenge
some of our most fundamental beliefs about the nature of reality, specifically the widely
13
It should be emphasised at the outset that quantum cognition is independent from the Orch-OR
quantum brain hypothesis (Hameroff & Penrose, 2014b) which postulates that quantum processes within
the neuronal cytoskeleton (i.e., dendritic-somatic microtubules) form the basis for consciousness. Orch-
OR is an acronym for “orchestrated objective reduction” which has been popularised by Sir Roger
Penrose and Stuart Hameroff. We refer to Appendix A2for a brief synopsis of this integrative theory
which combines findings from neuroscience, molecular biology, quantum physics, pharmacology,
quantum information theory, and philosophy.
47
held notion oflocal realism
14
(Giustina et al., 2015; Hensen et al., 2015; Wiseman,
2015). Prima vista, many empirical findings from quantum physics seem
irrational/paradoxical, highly counterintuitive, and incompatible with our most
fundamental beliefs about reality. Consequently, they cause a significant amount of
“cognitive dissonance(i.e., mental discomfort/psychological stress due to
contradictory beliefs) (Festinger, 1957). Therefore, it is predictable that these
inconvenient empirical facts are ignored in order to circumvent psychological
tensions.
15
To appreciate the novel findings the deeply engrained “need for closure”
(Webster & Kruglanski, 1994) needs to be actively counteracted
16
in order to process
these novel seemingly irrational/paradoxical empirical facts in a less biased/prejudiced
manner. In other words, human beings generally strive for consistency and conflicting
information is likely to be disregarded. As William James pointed out, it is easy to
14
Local realism is the widely held belief that “the world is made up of real stuff, existing in space and
changing only through local interaction”, as Wiseman formulates it in a N
ATURE article entitled
Quantum physics: Death by experiment for local realism(Wiseman, 2015, p. 649). The widely held
belief in the veracity of this local-realism hypothesis has now been conclusively falsified., i.e., empirical
findings “rigorously reject local realism”. We urge the sceptical reader to verify this claim. The scientific
ramification of this cutting-edge evidence-based paradigm shift are extremely far-reaching and require a
substantial degree of open-mindedness, cognitive flexibility, and epistemological humility.
15
However, given that these inconvenient findings can be applied and economically exploited in the real-
world (e.g., quantum computation/communication/encryption/teleportation etc. pp.) it is no longer
feasible to just ignore them or dismiss them derogatively as “purely philosophical”. For instance, the
understanding and application of quantum principles like non-locality can be a decisive factor in cyber-
war and physical war (cf. Alan Touring and the enigma code (Hodges, 1995)). Google and NASA are
currently heavily investing in the technological application of quantum principles which were previously
thought to be “merely” of philosophical/theoretical relevance (e.g., quantum AI (Sgarbas, 2007; Ying,
2010)).
16
The default-interventionist account of thinking and reasoning (Evans, 2007) appears to be relevant in
this context. The need for closure is arguably an automatic and mainly unconscious process which needs
to be actively antagonised by more systematic higher-order cognitive processes which rely on executive
(prefrontal) cortical functions (Figner et al., 2010; Hare, Camerer, & Rangel, 2009). From a cognitive
economics perspective (Chater, 2015), these interventions upon frugal heuristic processes are costly in
energetic terms and therefore only used parsimoniously. Moreover, it should be noted that rational
intelligence is relatively independent from general intelligence, i.e., IQ ≠ RQ. As former APA president
Robert Sternberg formulated it “… IQ and rational thinking are two different constructs … The use of the
term ‘rational intelligence’ is virtually identical with the usual definition of critical thinking.” (Sternberg,
2018, p. 185). In other words, otherwise intelligent people frequently make irrational decisions and draw
logically invalid conclusions and are therefore perhaps not as smart as they are considered to be.
Stanovich labels the lack of rationality “disrationalia” in order to describe the inability to “think and
behave rationally despite adequate intelligence” (K. Stanovich, 2014, p. 18). We compiled additional
information and an RQ test under the following URL: http://irrational-decisions.com/?page_id=2448
48
prima facie reject ideas which are not readily classifiablein the prevailing scientific
framework. However, lateral and divergent “nonconformist” rational thinking is a much
harder task.
17
Immanuel Kant’s timeless advice which he formulated in his classic essay
“Was ist Aufklärung?” (What is enlightenment?) still reverberates with us today:
Sapere aude! (Kant, 1804)
1.1 Psychology: A Newtonian science of mind
Lateral thinkers interested in the mind have been inspired by the methods and results of
physics for a long time. For example, the British empiricist philosopher John Locke
(*1632; †1704) was imbued with the corpuscular theory of light (primarily formulated
by his friend Sir Isaac Newton) when he formulated his “corpuscular theory of ideas” in
his profoundly influential publication “An essay concerning human understanding”
which appeared in 1690. Locke transferred and generalised the axioms of Newtons
physical theory (which concerned the lawful behaviour of matter) to the psychological
(nonmaterial) domain. In other terms, Locke committed himself to a reductionist
Newtonian science of the mind (Ducheyne, 2009). Corpuscularianism is an ontological
theory which postulates that all matter is assembled of infinitesimally small particles
(Jacovides, 2002). This notion is similar to the theory of atomism, except that, in
contrast to atoms (from the Greek átomos, “that which is indivisible”)
18
, corpuscles can
17
At this place, a cautionary note should be cited: It has been convincingly argued that in the current
academic climate, critical “sincerely scientific” thinking is a dangerous activity which is associated with
various serious social risks which can have far-reaching consequences for the scientifically-minded
cogniser (Edwards & Roy, 2017). Divergent thinking can lead to ostracisms and various other detrimental
consequences, especially when central (oftentimes implicit) in-group norms are challenged, e.g.,
reductionist materialism/local realism. The extensive social psychology literature (e.g., group dynamics,
groupthink, conformity, consensus/dissent) is conclusive on this point (Bastian & Haslam, 2010; Postmes,
Spears, & Cihangir, 2001; K. D. Williams, 2007).
18
The idea behind the atom is that matter is composed of primordial material elements which are
fundamental to all of existence. Etymologically, the Greek term átomos (ἄτομος) is a composite lexeme
composed of the negating prefix á, meaning “not” and the word stem tomṓteros, “to cut”. Ergo, its literal
49
theoretically be further subdivided (ad infinitum). According to Newton, these
corpuscles are held together by a unifying force which he termed “gravitation”
(Rosenfeld, 1965). One of Locke’s primary concerns in this regard was: What are the
most elementary particles” of human understanding (i.e., what are the “atoms of
thought”), where do they come from, and how are they held together? Locke rejected
the Cartesian notion of innate (God-given) ideas, but he accepted some intuitive
principles of the mind (e.g., the law of contradiction) which he assumed must be in
place a priori in order for any knowledge to arise.
19
In addition to this kind of intuitive
knowledge about propositional logic, which he conceptualized as immediate,
indubitably knowable and certainly true, Locke also accepted some forms of
demonstrative knowledge to be certainly true. For example, the axioms of Euclidean
geometry. In contrast to intuitive knowledge, one has to perform a series of
mathematical proofs in order to reach a certain general conclusion which is true in all
contexts and circumstances.
20
Having defined these principles he pursued his initial
question: What are the most elementary “particles” of human cognition, where do they
come from, and how are they held together? Locke's answer is simple: Ideas come from
experience and are held together by associational forces (Halabi, 2005). That is,
empirical knowledge which is accumulated diachronically during the course of a
lifetime forms the basis of thought. Locke argues that the most elementary act is the
sensory act and the most elementary contents of the mind are sensations. He remarks:
meaning is “not cuttable”. In the memetic history of human thought, the term atom is ascribed to the
Greek philosophers Leucippus and Democritus (Pullman, 2001) even though similar atomistic concepts
were present in ancient Indian schools of thought (Rasmussen, 2006).
19
The Greek term “Epistemonicon” (i.e., the cognitive ability by which humans comprehend universal
propositions) provides an apposite semantic descriptor for this psychological faculty.
20
From a modern dual-systems perspective on cognitive processes, automatic (associative) and effortless
intuition is a System 1 process, whereas sequential and effortful logical reasoning is a System 2 process
(Kahneman, 2011) (but see Appendix A7). Hence, Locke’s theory can be regarded as a predecessor of
modern dual-process theories which are now ubiquitous in many fields of psychology and neuroscience
(Jonathan St B.T. Evans, 2003; Jonathan St B.T. Evans & Stanovich, 2013; Thompson, 2012).
50
For to imprint anything on the mind without the mind's perceiving it, seems to me
hardly intelligible” (Chapter 2 - On innate ideas). In other words, what enters the mind
comes through the sensorium and these elementary sensations must be connected
somehow. According to Newton, the corpuscular components of reality are held
together by gravitational forces, i.e., Newton's law of universal gravitation which
follows the inverse-square law.
21
Locke ingeniously applied this idea to elementary
sensations and proposes the principle of “associationas the mental counterpart to
physical gravitation.
22
Ex hypothesi, objects or events which are frequently experienced
together are connected by associative processes.
23
They thereby recombine to form
simple ideas. Out of simple ideas, increasingly complex ideas are hierarchically
assembled by the binding force of association – this the Lockean associative “logic of
ideas” (Yolton, 1955). The Lockean associationist memetic
24
account is still viable
today. e.g., associative (Bayesian) neural networks in artificial intelligence research.
21
The inverse-square law can be mathematically notated as follows: gravitational intensity
distance
In the context of Locke’s psychological theory, the term “gravitational intensity” can be replaced with
“associational intensity”. While gravitation is the attraction of two physical objects, association describes
the attraction between mental concepts (i.e., ideas). For instance, the “distance” between various concepts
can be indirectly quantified by variations in reaction-times in a semantic priming paradigm, for instance,
the implicit-association test (IAT) (Greenwald & Farnham, 2000; Sriram & Greenwald, 2009). The
concepts “tree” and “roots” are closer associated (i.e., the “associational intensity” is stronger) than the
concepts “university” and “beer” (perhaps this is an unfortunate example, but it illustrates the general
point).
22
Interestingly, it has been noted by historians of philosophy and science that “Locke's attitude towards
the nature of ideas in the Essay is reminiscent of Boyle's diffident attitude towards the nature of matter”
(Allen, 2010, p. 236).
23
This Lockean idea can be regarded as the predecessor of Hebbian engrams and assembly theory “cells
that fire together wire together” (Hebb, 1949). The formulaic description of Hebb's postulate is as
follows:

=
1

,
24
The science of memetics tries to (mathematically) understand the evolution of memes, analogous to the
way genetics aims to understand the evolution of genes (Kendal & Laland, 2000). Locke’s early
contributions are pivotal for the development of this discipline which is embedded in the general
framework of complex systems theory (Heylighen & Chielens, 2008). Memetics is of great importance
for our understanding of creativity and the longitudinal evolution of ideas in general. Memes reproduce,
recombine, mutate, compete and only the best adapted survive in a given fitness landscape. Similar to
genotypes, the degree of similarity/diversity between memes (and their associated fitness values)
determines the topology of the fitness landscape.
51
Locke was clearly far ahead of his time and the associative principles he formulated
where later partly experimentally confirmed by his scientific successors, e.g., Ivan
Pavlov (Mackintosh, 2003) and later by the behaviourists in the context of S-R
associations (Skinner, Watson, Thorndike, Tolman, etc. pp.). Furthermore, the
Newtonian/Lockean theory of how ideas are composed in the mind forms the basis of
the “British Associationist School” with its numerous eminent members (David Hartley,
Joseph Priestley, James Mill, John Stuart Mill, Alexander Bain, David Hume, inter
alia). In England, the Associationist School asserted an unique influence on science and
art alike and the principles of associationism and connectivism are still widely applied
in many scientific fields, for instance, in the psychology of associative learning and
memory (Rescorla, 1985) and in computer science (for instance, associative neural
networks like cutting-edge deep/multi-layered convolutional neural nets (Kivelä et al.,
2014; Lecun, Bengio, & Hinton, 2015)). To indicate Newton’s and Locke’s pervasive
influence on psychology it could for instance be noted that Pavlov’s classical and
Skinner’s operant conditioning can be classified as a form of associationism, as can
Hebbian learning which is ubiquitously utilised in science.
Until today, psychology and much of science operates on the basis of a materialistic,
mechanistic, and deterministic quasi-Newtonian paradigm.
1.2 Shifting paradigms: From Newtonian determinism
to quantum indeterminism
The crucial point is that Locke's associationist (Newtonian) theory of mind is
fundamentally deterministic (and consequently leaves no room for free will (cf. Conway
& Kochen, 2011)). Newton’s Philosophiæ Naturalis Principia Mathematica
(Mathematical Principles of Natural Philosophy) originally published in 1687 is among
52
the most influential works in the history of science and Newton’s materialistic
mechanistic determinism shaped and impacted scientific hypothesizing and theorising in
multifarious ways. In 1814, Pierre Simon Laplace famously wrote in his formative
Essai philosophique sur les probabilités” (A Philosophical Essay on Probabilities):
We may regard the present state of the universe as the effect of its past and the cause
of its future. An intellect which at a certain moment would know all forces that set
nature in motion, and all positions of all items of which nature is composed, if this
intellect were also vast enough to submit these data to analysis, it would embrace in a
single formula the movements of the greatest bodies of the universe and those of the
tiniest atom; for such an intellect nothing would be uncertain and the future just like the
past would be present before its eyes.” (Laplace, 1814, p. 4)
25
This deterministic view on reality was extremely influential until the late 18
th
century
and is still implicitly or explicitly the ideological modus operandi for the clear majority
of scientists today. However, in physics, unexplainable (anomalous) data and
inexplicable abnormalities kept accumulating (e.g.: the three-body-problem, the results
of Young’s double-slit experiment, etc.) and finally a non-deterministic (stochastic)
quantum perspective on physical reality evolved as exemplified by the following
concise quotation concerning the uncertainty principle by Werner Heisenberg from
Über die Grundprinzipien der Quantenmechanik(About the principles of quantum
mechanics):
25
The full essay is available on the Internet Archive under the following UR:
https://archive.org/details/essaiphilosophiq00lapluoft/page/n5
53
“In a stationary state of an atom its phase is in principle indeterminate,” (Heisenberg,
1927, p. 177)
26
One of the most eminent adversaries of this indeterministic theoretical approach, Albert
Einstein, vehemently disagreed with the stochastic uncertainty inherent to quantum
mechanics. For example, Einstein wrote in one of his letters to Max Born in 1944:
“We have become Antipodean in our scientific expectations. You believe in the God
who plays dice, and I in complete law and order in a world which objectively exists, and
which I, in a wildly speculative way, am trying to capture. I firmly believe, but I hope
that someone will discover a more realistic way, or rather a more tangible basis than it
has been my lot to find. Even the great initial success of the quantum theory does not
make me believe in the fundamental dice-game, although I am well aware that our
younger colleagues interpret this as a consequence of senility. No doubt the day will
come when we will see whose instinctive attitude was the correct one.” (Born, 1973,
p.149)
27
Einstein's general and special theory of relativity, radical though they were, explain
natural phenomena in a Newtonian deterministic fashion, thereby leaving the
established forms of reasoning, logic, and mathematics of the 19
th
century undisputed.
By comparison, quantum theory completely changed the conceptual framework of
science due to its fundamentally stochastic indeterminism. It has not just changed
26
The mathematical formulation of the Heisenbergian uncertainty principle is: 
,
where Δ signifies standard deviation (spread or uncertainty),
x and p signify the position and linear momentum of a given particle,
signifies a specific fraction of Planck's constant (Planck's constant divided by 2π).
That is, an accurate measurement of position disturbs momentum and vice versa (see Robertson, 1929).
For a discussion of the “inextricable” relation between non-locality and the uncertainty principle see
(Oppenheim & Wehner, 2010)
27
The Einstein-Born letter are available on the Internet Archive under the following URL:
https://archive.org/details/TheBornEinsteinLetters/
54
scientific concepts of physical reality but our understanding of the most essential
rationality principles in general, i.e., a new form of quantum logic was developed
(Beltrametti & Cassinelli, 1973). Quantum theory is now by a large margin the most
reliable theory science has ever developed because its quantitative predictions are
extremely accurate and have been tested in countless domains. Despite this unmatched
track record, contemporary psychology, the neurosciences, and the biomedical
sciences
28
(and their associated statistical methods) are still modelled after the
antiquated and de facto outdated Newtonian/Lockean deterministic worldview and these
scientific disciplines (and others) have not yet aligned themselves with the far-reaching
implications derived from quantum theory. In other words, the revolutionary
reformation of Newtonian mechanics has not yet reached psychology which is still
based on the hypothetical premise of local realism of classical physics. In fact, it could
be effectively argued that the classical probability framework (which is used in almost
exclusively in all cognitive modelling efforts) exhibits the defining characteristics of a
tenacious Kuhnian paradigm. As Thomas Kuhn articulates in his influential book “The
structure of scientific revolutions”:
“... ‘normal science’ means research firmly based upon one or more past scientific
achievements, achievements that some particular scientific community acknowledges
for a time as supplying the foundation for its further practice. Today such achievements
are recounted, though seldom in their original form, by science textbooks, elementary
and advanced. These textbooks expound the body of accepted theory, illustrate many or
28
It has been argued that the entire scientific endeavour has not yet come to terms with the radical
revolution which has been set in motion by quantum physics (Dowling & Milburn, 2003; Heisenberg,
1958). Science wants to define itself as objective, detached, and neutral. Several findings from quantum
physics challenge this identity. For instance, the observer effect questions the possibility of objective
measurements and the violations of Bell inequalities challenge the notion of local realism which forms the
basis of much of scientific theorising (Gröblacher et al., 2007).
55
all of its successful applications, and compare these applications with exemplary
observations and experiments. Before such books became popular early in the
nineteenth century (and until even more recently in the newly matured sciences), many
of the famous classics of science fulfilled a similar function. Aristotle’s Physica,
Ptolemy’s Almagest, Newton’s Principia and Opticks, Franklin’s Electricity,
Lavoisier’s Chemistry, and Lyell’s Geology—these and many other works served for a
time implicitly to define the legitimate problems and methods of a research field for
succeeding generations of practitioners.” (T. S. Kuhn, 1962, p. 10)
1.3 Quantum cognition: An emerging novel paradigm
in psychology
Psychology as a scientific discipline has primarily modelled its methods after the highly
successful achievements of classical physics, thereby longing for the acceptance as a
“hard” empirical science (this has been termed “physics envy” (Fish, 2000)). Hence, it
is not surprising that psychology almost always lags with regards to the evolution of
mathematical, methodological, and conceptual principles. Moreover, it follows that
physicists (who are generally aware of the paradigm shifts within their field) will be
among the first to accept a high degree of uncertainty and indeterminism in the methods
of psychology (e.g., Busemeyer, Pothos, Franco, & Trueblood, 2011b; Z. Wang et al.,
2013).
After John Locke’s quasi-Newtonian insights, the time is ripe that scholars of the mind
take a fresh look at the empirical findings physics provides in order to adapt their
epistemology and research methods. Especially quantum probability theory (herein after
referred to as QP theory) has very promising potential for the enrichment (and deep
revision) of many concepts that are widely and mainly unreflectively utilised in
56
psychology (and various other branches of science). Based on anecdotal data, we are
inclined to believe that the vast majority of psychologists and neuroscientists are utterly
unaware of the breakthroughs in quantum physics (let alone their ontological and
epistemological implications). This is presumably due to a lack of interdisciplinary
discourse (Lélé & Norgaard, 2005). Furthermore, QP theory has not yet been included
in any mainstream statistical textbook (let alone its integration into academic curricula).
However, the transdisciplinary ramifications of quantum physics are extremely far
reaching as Niels Bohr pointed out more than half a century ago:
“In atomic science, so far removed from ordinary experience, we have received a lesson
which points far beyond the domain of physics.” (Bohr, 1955, p. 171)
1.4 Observer-effects, noncommutativity, and
uncertainty in psychology
Based on accumulating converging empirical evidence (e.g., Aerts, Broekaert, &
Gabora, 2011; beim Graben, 2013; Moreira & Wichert, 2014; Z. Wang et al., 2013), it
seems plausible that measurements can affect not only physical processes (an empirical
fact that has been firmly established in quantum physics (e.g., Alsing & Fuentes, 2012;
Bell, 2004; Rosenblum & Kuttner, 2002)) but also cognitive and behavioural processes.
For example, the widely debated “unreliability” of introspection (Engelbert &
Carruthers, 2010), including all self-report measures (e.g., questionnaire studies), might
be partially due to interference effects caused by self-observation. That is, the mere act
of introspection (an internal self-measurement) interferes with the state of the cognitive
system to be evaluated, thereby confounding the introspective measurement outcome.
To be more explicit, introspection might distort the internal state in question because
this kind of self-observation focuses mental energy on the process in question
57
(analogous to a laser device focusing physical energy on a particle)
29
which causes the
state concerned to undergo a transformation, possibly via collapse of the “mental wave-
function” (A. Khrennikov, 2003, 2009, 2010). Moreover, the introspective process may
be influenced by idiosyncratic motives and intentions which makes the self-
measurement outcome even more unreliable due to a more systematically biased
distortion of the measurement of the psychological observable.
Apart from the observer-effect, the uncertainty-principle appears to be relevant to
cognitive processes, too (Busemeyer & Bruza, 2012). Uncertainty is ubiquitous in
multifarious decision-making scenarios (Kahneman & Tversky, 1974) and it has been
noted that “QP theory is potentially relevant in any behavioural situation that involves
uncertainty” (Pothos & Busemeyer, 2013, p.255). Moreover, QP has the potential to
parsimoniously account for empirical findings which appear paradoxical and irrational
in the classical probability framework (Z. Wang et al., 2013). Nobel Prize laureate
Daniel Kahneman, editor and co-author of the widely studied book “Judgement Under
Uncertainty: Heuristics and Biases” (inter alia), is momentarily presumably the most
eminent researcher in the field of reasoning and decision making. Therefore, his work is
the optimal starting point for an application of QP principles (but see Pothos and
Busemeyer, 2013). Kahneman can be categorized as a dual-process theorist (Jonathan St
B.T. Evans, 2003; Frankish, 2010). (The basic nexus of dual-process theories of
cognition is adumbrated in Appendix A7 and we recommend to the unfamiliar reader to
consult the addendum before continuing because a basic understanding of the dual-
process theory is required in order to appreciate the following argumentation.)
During his Nobel Prize lecture, Kahneman introduced his research agenda as an
29
A similar idea inspired by quantum physics has recently been published in a different context in a paper
published in the Philosophical Transactions of the Royal Society: “Social Laser: Action Amplification by
Stimulated Emission of Social Energy(A. Khrennikov, 2015).
58
“attempt to map departures from rational models and the mechanisms that explain
them”. Moreover, he formulated that one of the overarching features of his research
projects is to “introduce a general hypothesis about intuitive thinking, which accounts
for many systematic biases that have been observed in human beliefs and decisions”
(Kahneman, 2002). He advocates an evolutionary perspective on reasoning and his
reflections are based on the assumption that there is a kind of quasi biogenetic
progression in the evolution of cognitive processes starting from automatic processes
which form the fundamental basis for the evolution of more deliberate modes of
information processing. The postulated diachronic phylogenetic history of cognitive
processes can be adumbrated as follows:
PERCEPTION INTUITION REASONING
According to this sequential view on the Darwinian evolution of cognitive systems,
perception appears early on the time-line of history, whereas reasoning evolved
relatively recently. Intuition is intermediate between the automatic (System 1) processes
of perception and the deliberate, higher-order reasoning (System 2) processes that are
the hallmark of human intelligence (Kahneman, 2003). Furthermore, Kahneman
proposes that intuition is in many ways similar to perception and the analogy between
perception and intuition is the common denominator of much of his distinguished work.
Thus far, QP principles have primarily been tested in higher-order cognitive processes,
for instance, in political judgments and affective evaluations (e.g., Z. Wang &
Busemeyer, 2013; White, Barqué-Duran, & Pothos, 2015; White, Pothos, & Busemeyer,
2014b). Following Kahneman’s line of thought, one could ask the question: Do the
principles of QP also apply to more basic perceptual processes which evolved much
earlier in the phylogenetic evolutionary tree? That is, do the principles of quantum
59
cognition (for instance, the crucial noncommutativity axiom) also apply to the most
fundamental perceptual processes like visual perception? If so, this would provide
supporting evidence for the generalisability of QP principles. In addition, this kind of
evidence would have the potential to cross-validate recent findings concerning affective
(emotional) evaluations and attitudinal judgments (White et al., 2015, 2014b). However,
hitherto the literature on QP focuses primarily on judgments and decisions in higher-
order (System 2) cognitive processes
30
. Our experiments aim to bridge this empirical
gap. In this thesis, we report experimental evidence that extends this line of work into
the domain of basic perceptual (System 1) processes. We designed several experiments
in order to test various predictions derived from the QP model. Specifically, we
employed a reductionist psychophysics approach in order to address the question
whether QP principles are applicable to low-level perceptual processes. We argue, that
evidence which support the applicability of QP principles to perceptual processes would
cross-validate and corroborate the findings made in the domain of higher-order
cognitive processes (emotions, judgements, reasoning). The novelty of our approach is
thus to introduce principles from quantum probability to psychophysics. In the
following section, we will discuss why the marriage between psychophysics and
quantum cognition is fruitful.
30
There are some exceptions: For instance, the ingenious work by Atmanspacher et al. applied various
quantum principles (e.g., temporal nonlocality, superposition/complementarity, the quantum Zeno-effect)
to the perception of bistable ambiguous stimuli (Atmanspacher & Filk, 2010, 2013, Atmanspacher et al.,
2004, 2009). We will discuss these insightful findings in subsequent sections.
60
1.5 Psychophysics: The interface between Psyche and
Physis
In order to understand the relationship between psychophysics and quantum cognition it
is necessary to review the development of the discipline because the mainstream
accounts given in most textbooks on psychophysics is misleading and highly selective
(Boring, 1928, 1961; Scheerer, 1987), partly due to the fact that Fechner’s voluminous
work has only been partially translated from German into English. In the following
section, we will provide a brief account of the history of psychophysics with an
emphasis on Gustav Fechner’s formative contributions (Fechner has been regarded as
“inadvertent founder of psychophysics” (Boring, 1961)).
Contemporary psychology (the “nasty little subjectas William James labelled it) is an
amalgamation of science and philosophy. The scientific aspect of psychology is based
on the quantitative experimental scientific tradition and its focus on prediction,
experimental verification, and precision of measurement. The philosophical aspect of
psychology (which is complementary to the scientific aspect) is based on empiricisms
and its emphasis on observation as a means to acquire knowledge. Historically, precise
quantitative measurements became of great importance in the beginning of the 18
th
century and this development towards quantitative precision was primarily based on
pragmatic considerations. The ability to successfully navigate the oceans was of great
importance in this time period (not least for financial/economic reasons) and tools and
instruments were developed in order to enable accurate marine navigation. At the same
time, astronomy significantly gained in status due to Newtons and Kepplers theorizing.
Precise measurement instruments were required to empirically verify the novel
scientific theories. Especially in Great Britain (Wolfschmidt, 2009), for instance in
Greenwich (Howse, 1986), astronomical observatories were built. These observational
61
facilities systematically compared their findings in order to reach inter-observer
consensus, thereby increasing the accuracy and robustness of observations. At the same
time, the human sensory organs became a matter of great scientific interest, the reason
being that astronomy relied on the human observer (percipient) and on the precision of
the sensorium. Idiosyncratic observational differences could multiply and have large-
scale ramifications for the observational models which were formulated in this period.
Based on the philosophical school of empiricism, observational scientists developed a
keen interest in the optimal/ideal observer and the perceptual processes which undergird
signal detection. That is, a precise understanding of the perceptual system played a
pivotal role for very practical reasons. The key question was, how good are human
percipients in judging minute differences in the external world (for instance, the
brightness of visual stimuli, e.g., faint stars)
31
. That is, perceptual decision-making
became a topic of great interest because it had real-world implications and infinitesimal
perceptual deviations could incrementally amplify and have large scale real-world
implications. On the other hand, there was a philosophical interest in perception due to
the empiricist stance that the mind is a tabula rasa which is “furnished by experience”
(Locke, 1796), in accordance with the Peripatetic axiom: "Nihil est in intellectu quod
non prius fuerit in sensu" (nothing is in the intellect that was not first in the senses (but
see Kuksewicz, 1982)). According to the Aristotelian notion of the "intellectus agens
(active intellect) abstract universal meaning is inductively derivable from particular
empirical (sensory/perceptual) data. Consequently, how exactly the contents of the mind
are furnished by sensory inputs became a topic of great philosophical and psychological
importance (according to this perspective, incoming sensory data determines the
31
It has indeed been argued that Fechner’s law was anteceded by astronomers who investigated stellar
magnitudes, but that these early “astro-psychophysicists” are ignored in the historical discourse on
psychology (Pliskoff, 1977)
62
contents of the mind which was regarded as a “blank slate” which is imprinted by sense
data). From a purely pragmatic point of view, discriminatory acuity and the exact
quantification of perceptual measurement errors became subjects of particular interest
because they had far-reaching consequences in the real-world, for instance, navigation
on the sea relied on precise and accurate descriptions of various properties of the
external world. The refinement of exact measurement instruments was another closely
related research topic of utmost practical importance, primarily for political and
economic reasons (i.e., colonialism). Taken together, these historical developments
could be regarded as primary impetus for the development of western psychophysics.
However, it were German scientists in the beginning of the 19
th
century who started
psychophysics as a systematic experimental academic discipline. Particularly, Ernst
Heinrich Weber (1795 - 1878) who was a professor at the University of Leipzig (now
considered as one of the founding fathers of modern experimental psychology) started a
research program which focused meticulously on the precision of the human senses.
One of the textbook examples is Weber’s investigation of how accurate percpients are at
differentiating the intensity of two stimuli, for instance, between the brightness of two
lights. That is, what is the least perceptible difference
32
a human observer can detect
between two visual stimuli which differ only slightly in their brightness. In a
prototypical psychophysics experiment the subject would be presented with two lights
with varying brightness levels. One would be the standard light (modulus) and the other
the comparison light. Weber would then quantitatively determine at which point the
subject could detect a difference in brightness between the standard and the comparison
stimulus. On the basis of his experimental findings, he formulated the following law
32
The now widely used psychophysical concept is often acronymized as JND, i.e., just noticeable
difference (Gescheider, 1997).
63
known as Weber’s law or Weber’s ratio: The ratio of the value of difference between
the standard and the comparison stimulus ΔR divided by the value of the standard
stimulus R would produce a mathematical constant k. Weber’s law has been
systematically studied in many sensory modalities (e.g., audition, olfaction, gustation,
etc.). Weber published his findings in the 1830s. The main conclusion of his empirical
investigations was that perception can be quantified in a mathematical fashion and that
there is a systematic lawful relationship between the physical world and the mental
world of perception which can be precisely axiomatized.
Equation 1. Weber’s law.
=

Approximately 30 years later (at the same university in Leipzig) a physicist by the name
of Karl Gustav Fechner observed the sun to study visual negative afterimages. To his
great dismay he lost his eyesight due to photokeratitis (blindness caused by exposure of
insufficiently protection of the eyes from ultraviolet light). He already was a very
successful physicist and he received a professorial chair in his early 30s for his work on
electricity (one of the youngest professors of his time in Germany). However, his
blindness prevented him from pursuing his academic profession and ophthalmologists
predicted that his eyesight would not return. Fechner became seriously depressed and
lived a very melancholic life. Because he was unable to read, he spent most of his time
in contemplation in a dark room and began to become almost obsessively concerned
with the relationship between mind and matter.
However, after several months of “introspection” his ophthalmic condition reversed. At
this fortunate turning point in his life, he decided to dedicate his intellect to a new
endeavour. Inspired by his profound experiences, Fechner set out to prove that the same
64
divine force which is responsible for the creation of the external physical world is also
responsible for the creation of the internal psychological world. Fechner intended to
show that there is a set of connecting principles which connects the psychological realm
with the physical realm. That is, he intended to create a novel science which focuses on
the relationship between the psychological and the physically domain. He termed this
new scientific discipline “psychophysics”. Today psychophysics is a very well-
developed discipline within the arena of psychology and it can be said without any
doubt that it is the most quantitative and precise of all psychological schools of thought.
Modern psychophysics is in a position to produce highly reliable data with regards to
physical stimuli and the sensations and perceptions they produce in the percipient. To
be more exact, Bruce, Green, and Georgeson (1996) define psychophysics as "the
analysis of perceptual processes by studying the effect on a subject's experience or
behaviour by systematically varying the properties of a stimulus along one or more
physical dimensions."
According to historians of science, a solution to the problem of the relationship between
psyche and physis came to Fechner one morning in October 1850 in a sudden epiphany
(Meischner-Metge, 2010). This particular day is still yearly celebrated as “Fechner’s
day” which has even beenofficially celebrated in Asia (Mori, 2008). Fechner thought: If
he would be able to empirically establish quantitative relations between particular
physical stimuli and the accompanying sensation he would be able to proof the unity
(i.e., nonduality) of mind and matter (cf. Boring, 1928). In his meticulous experiments,
Fechner analysed countless judgments from his experimental subjects and he
65
logarithmically extended Weber’s law and developed what is now known as Fechner’s
law (Laming, 2011; Norwich & Wong, 1997)
33
:
Equation 2. Fechner’s law.
= ln
where k signifies a perceptual modality specific constant.
Fechner was keenly aware of the far-reaching implications of his idea, namely that an
element of human consciousness could be systematically quantified in mathematical
terms. Hence, Fechner played a pivotal role in the emergence of modern experimental
psychology and his achievements were later explicitly recognised by Wilhelm Wundt.
Fechner’s research methodology is widely emulated in countless psychology
laboratories until today. Contrary to mainstream belief, Fechner was antagonistic
towards materialism and the associated mechanistic paradigm which prevailed during
his lifetime until today (Scheerer, 1987). He rejected dualistic notions and became
convinced of the existence of a unitary reality which forms the foundation of the
material and the psychological reality (an ontological theory named “dual-aspect
monism” (Atmanspacher, 2012)). However, this fact is mainly neglected in the
psychophysics literature which focuses exclusively on his quantitative work and
neglects his deep philosophical motivation which provided the impetus for his
theorising, a well-known bias in the history of science which overemphasises the
nomological “context of justification” and neglects the idiosyncratic “context of
discovery” (Bowers, Regehr, Balthazard, & Parker, 1990). Fechner’s nondual
33
It should be noted that historians of science trace the antecedents of Fechner’s law to several British
astronomers, inter alia, the polymath Sir John Herschel. It has been argued that those early
psychophysicists have not been given their due (Pliskoff, 1977).
66
perspective on mind and matter is compatible with the monistic theory of Baruch de
Spinoza
34
, viz., dual-aspect monism (Charlton, 1981; Daniels, 1976; Della Rocca,
2002). A similar nondual conception was later discussed between the depth-
psychologist Karl Gustav Jung and quantum physicist and Nobel laureate Wolfgang
Pauli, i.e., the “Pauli-Jung conjecture(but see Atmanspacher, 2012)
35
. The British
quantum physicist David Bohm describes the mind-matter (psycho-physics) dichotomy
in terms of an ontological dimension he terms “implicit and explicit order”. The implicit
34
Albert Einstein was deeply influenced by Spinoza’s thoughts. In 1929, Einstein wrote (originally in
German): "I believe in Spinoza's God, who reveals himself in the harmony of all that exists, not in a God
who concerns himself with the fate and the doings of mankind.” Moreover, he stated in the Japanese
magazine “Kaizō” in 1923: “Scientific research can reduce superstition by encouraging people to think
and view things in terms of cause and effect. Certain it is that a conviction, akin to religious feeling, of
the rationality and intelligibility of the world lies behind all scientific work of a higher order. [...] This
firm belief, a belief bound up with a deep feeling, in a superior mind that reveals itself in the world of
experience, represents my conception of God. In common parlance this may be described as pantheistic”.
In a letter to a young girl named Phyllis he wrote in 1936 “… everyone who is seriously involved in the
pursuit of science becomes convinced that some spirit is manifest in the laws of the universe, one that is
vastly superior to that of man. In this way the pursuit of science leads to a religious feeling of a special
sort, which is surely quite different from the religiosity of someone more naive.” (Einstein & Alice
Calaprice (ed.), 2011)
35
This interdisciplinary discussion can be regarded as a first attempt to integrate quantum physics and
psychology into a unified theoretical “psychophysical” framework. We are convinced that many topics
which were addressed in the voluminous correspondence between Jung and Pauli will become of great
importance for future psychophysical theories which focus on the interplay between “mind and matter”
(note that dualistic terminology cannot be avoided). For instance, a fascinating topic Jung and Pauli
discussed in this context was the acausal connecting principle termed “synchronicity” (Donati, 2004; C.G.
Jung, 1975; Main, 2014). In his eponymous book Jung gives the following prototypical example of a
synchronistic event:
“My example concerns a young woman patient who, in spite of efforts made on both sides, proved to be
psychologically inaccessible. The difficulty lay in the fact that she always knew better about everything.
Her excellent education had provided her with a weapon ideally suited to this purpose, namely a highly
polished Cartesian rationalism with an impeccably "geometrical" idea of reality. After several fruitless
attempts to sweeten her rationalism with a somewhat more human understanding, I had to confine myself
to the hope that something unexpected and irrational would turn up, something that would burst the
intellectual retort into which she had sealed herself. Well, I was sitting opposite her one day, with my
back to the window, listening to her flow of rhetoric. She had an impressive dream the night before, in
which someone had given her a golden scarab a costly piece of jewellery. While she was still telling
me this dream, I heard something behind me gently tapping on the window. I turned round and saw that it
was a fairly large flying insect that was knocking against the window-pane from outside in the obvious
effort to get into the dark room. This seemed to me very strange. I opened the window immediately and
caught the insect in the air as it flew in. It was a scarabaeid beetle, or common rose-chafer (Cetonia
aurata), whose gold-green colour most nearly resembles that of a golden scarab. I handed the beetle to
my patient with the words, Here is your scarab.This experience punctured the desired hole in her
rationalism and broke the ice of her intellectual resistance. The treatment could now be continued with
satisfactory results.(C.G. Jung, 1975)
67
order is in principle epistemologically accessible whereas the implicit order is purely
ontological and epistemologically inaccessible:
At each level of subtlety there will be a “mental pole” and a “physical pole” . . . But
the deeper reality is something beyond either mind or matter, both of which are only
aspects that serve as terms for analysis.” (Bohm, 1990, p. 285)
Fechner also contributed significantly to the German psychology of unconscious.
cognition. However, his pioneering work on “unattended mental states” has not been
paid due attention in academic circles (Romand, 2012). Even though he was clearly
scientifically minded, he had spiritual ideas which were rather atypical even in the 19
th
century (and especially today in contemporary materialistic mainstream science)
36
.
Fechner could be classified as a panpsychist (or perhaps panentheist), i.e., he argued
that consciousness (or soul/psyche)
37
is a universal and primordial feature of all things.
According to Fechner, all things express the same anima mundi, or world soul, a
conception which is closely aligned with the Vedic concept of the “cosmic psyche” or
36
However, Fechner’s ideas resonated with William James’ thinking. For instance, "the compounding of
consciousness", a Jamesian idea which “postulates the theoretical possibility for individual entities within
a conscious system of thought to knowthe thoughts of others within the system(Hawkins, 2011, p.
68). Fechner and James both explicitly rejected materialist accounts of the relationship between mind and
brain (i.e., mind and matter). James experimented with the psychedelic Mescaline and nitrous-oxide and
he was very interested in spiritual ideas, as evidenced by his classic book “The varieties of religious
experience” (James 1842-1910, 1902). Moreover, James advocated a “radical empiricism” (James, 1976)
which is incongruent with the prevailing materialistic paradigm which disregards extraordinary (first-
person) qualitative experiences, for instance, those occasioned by naturally occurring “consciousness
expanding” (Metzner, 2010) psychedelics which have been utilised for spiritual purposes for millennia in
various cultures. That is, James was an advocate of a “science of subjective experience”, a stance which
become relevant in the subsequent discussion of complementarity (e.g., subjective vs. objective, the
observer vs. the observed).
37
The word psyche is etymologically derived from the ancient Greek ψυχή (psukhḗ, “mind, soul, spirit”).
Hence, psychology is the study of the “mind, soul, and spiriteven though most psychologists are utterly
unaware of this etymological definition. Moreover, they want to differentiate themselves from these
“metaphysical/philosophical” concepts in order to appear as “hard/materialistic” scientists. They thereby
neglect and extremely rich intellectual heritage which has deep historical roots which span many cultures
and epochs.
68
Ātman
38
(Orme-Johnson, Zimmerman, & Hawkins, 1997). The “rise of the world soul
theory in modern German philosophy” has been extensively discussed by historians of
science (Zachhuber, 2015). Fechner argued that all of existence is interconnected
through “spiritual nerves” or “long ropes” which constitute a unified web of existence
made of light, gravity, and yet unidentified forces.
39
This idea reverberates with the
ancient ontological concept of “dependent origination” or “dependent arising” (Sanskrit:

 Pratītyasamutpāda), which is a key concept, inter alia, in Hua-yen Buddhism
(Cook, 1977). Dependent origination is conceptually associated with the quantum
physical concept of entanglement
40
(e.g., violations of Bell inequalities, discussed later)
and quantum holism (Bohm, 1990). In eastern philosophy, the concept is often
illustrated with the visual metaphor of Indra’s net
41
(Sanskrit:  Indrajāla), a
concept which originated in early ancient Vedic cosmology (see Figure 3).
38
From a linguistic point of view the Sanskrit word Ātman forms the basis for the German word “Atmen”
which means “breathing”. Recall the etymology of the word psychology: The ancient Greek word psukhḗ
(ψυχή) or psyche means “life/soul/spirit” and also “breath. Likewise, the Chinese symbol for "spirit,
soul" is which also means breath”. Hence, the linkage between “soul/spirit” and breath was formed
independently by separate cultures. Thus defined, psychology is the study of “life/soul/spirit” and
“breath”, i.e., Ātman.
39
According to contemporary theorizing in physics and cosmology, ordinary atomic matter constitutes
only ≈ 5% of the observable Universe. The remaining 95% consist of dark matter (≈ 26%) and dark
energy (≈ 69%), which are hitherto completely mysterious to scientists. These values are in themselves
astonishing because they indicate numerically how limited our epistemic understanding regarding the
fundamental ontology of the Universe really is. Therefore, Fechner’s ideas about “yet unknown forces” is
not as absurd as it might seem prima facie (especially to scientists who were conditioned in a materialistic
worldview). As Sir Isaac Newton framed it: “What we know is a drop. What we don’t know is an ocean”.
Epistemological humility is a true virtue (Richards, 1988).
40
When quantum theory was approx.10 years old (around 1935) the concept of entanglement emerged
(quantum theory was invented/discovered around 1925-26). Entanglement is one of the most mind-
boggling concepts in quantum physics because it is so incongruent with our intuitions about reality and
specifically causality. Two particles that interacted at some point in time in the past are interconnected in
a “strange” way. That is, they remain interconnected even though there is no known physical medium
through which that interaction can be explained. This was discovered by Einstein and he believed that this
“wired” logical consequence of the mathematical formalism of quantum mechanics would proof its
invalidity. That is, if the mathematical axioms of quantum mechanics allow for such an absurd
phenomenon than it surely must be wrong. However, today we know that Einstein was wrong and this
nonlocal interaction between particles can be exploited for real world applications as, for instance,
quantum teleportation and quantum cryptography (discussed later).
41
In Hinduism, Indra is a Vedic deity (Flood, 2007) and is the most dominant deity in the ten anthological
books which comprise the Rigveda (the Sanskrit etymology of Rigveda is  ṛgvedapraise, shineand
69
"Imagine a multidimensional spider's web in the early morning covered with dew drops.
And every dew drop contains the reflection of all the other dew drops. And, in each
reflected dew drop, the reflections of all the other dew drops in that reflection. And so
ad infinitum. That is the Buddhist conception of the universe in an image." (A. Watts,
1969)
Figure 3. Indra's net is a visual metaphor that illustrates the ontological concepts of
dependent origination and interpenetration (see Cook, 1977).
The notion of interrelatedness has deep implications for morality and ethics and it has
been applied to social contexts, for instance, in a speech given by Martin Luther King
Jr.:
"It really boils down to this: that all life is interrelated. We are all caught in an
inescapable network of mutuality, tied into a single garment of destiny. Whatever affects
one destiny, affects all indirectly." (King, M.L., 1967)
 vedaknowledge”). In Buddhism, Indra is a guardian deity (Gethin, 1998). An artistic digital 3D
rendering of Indra’s net can be viewed under the following URL:
https://upload.wikimedia.org/wikipedia/commons/e/ea/Indrasnet.jpg
70
The fractal nature of reality, as metaphorically
42
symbolised by Indra’s net, was
conceived long before Benoît Mandelbrot invented fractal mathematics (Gomory,
2010). Interestingly, a recent paper published in SCIENTIFIC REPORTS investigated and
compared the scale-invariance of various network topologies using supercomputer-
simulations. Specifically, the paper discusses the significant structural similarity
between the network topology of galaxies in comparison to the neuronal network
architecture of brains (in line with the alchemical quasi-fractal principle "as above so
below)
43
. The authors suggest that “some universal laws might accurately describe the
dynamics of these networks, albeit the nature and common origin of such laws remain
elusive” (Krioukov et al., 2012). Interestingly in the context of interconnectivity and
relatedness, recent studies with the naturally alkaloid Psilocybin (a partial 5-
hydroxitryptamin agonist) indicate that insights into the interconnected nature of reality
can be neurochemically induced in controlled experimental settings (Lyons & Carhart-
Harris, 2018; MacLean, Johnson, & Griffiths, 2011; R. Watts, Day, Krzanowski, Nutt,
& Carhart-Harris, 2017), but see Appendix A3 for further information.
In the context of the “universal psyche”, Fechner was convinced that the psyche of
plants
44
is no more related to their physiology/phytochemistry than the human psyche is
linked to neurophysiology/neurochemistry (a notion which stands in sharp contrast with
42
The metaphoric nature of Indra’s net is in itself extremely interesting from a cognitive psychology
point of view, especially in the context of “contextual metaphor theory” (Gibbs, 2011; Lakoff, 1993).
However, a deeper linguistic analysis would go beyond the scope of this chapter and we refer the
interested reader to the seminal book “Metaphors we live by(Lakoff & Johnson, 1980).
43
Interestingly, the “Gott-Li self-creating universe model” (Vaas, 2004) postulates and eternal fractal
universe and thereby circumvents the antinomy associated with the infinite regress associated with causal
models of cosmology, e.g., Big Bang theory (Germann, 2015b).
For more information regarding the fractal universe theory visit:
http://irrational-decisions.com/?page_id=2351
44
Interestingly, “plant consciousness” (Barlow, 2015) has recently been discussed in the context of the
“orchestrated objective reduction(Orch-OR) theory of consciousness (Hameroff, 2013; Hameroff &
Penrose, 1996, 2004) which postulates that consciousness originates from quantum processes in neuronal
microtubule.
71
contemporary materialistic reductionism which predominates the neurosciences and
psychology which attempt to reduce qualia to physiological processes). Fechner wrote:
None of my limbs anticipates anything for itself … only I, the spirit of myself, sense
everything that happens to me(as cited in Falkowski, 2007). This perspective has
elements of Neo-Platonism
45
as well as of Spinoza and Leibniz. He published his
philosophical views, inter alia, in a book entitled “ZendAvesta: oder über die Dinge des
Himmels und des Jenseits” (ZendAvesta: or on the Things of Heaven and the
Hereafter)
46
. A detailed discussion of Fechner’s “inner psychophysics” goes beyond the
scope of this thesis and would lead to Hinduistic scriptures in which many Fechnerian
memes can be found back. For instance, Fechner wrote in “Die Tagesansicht” (cit., p.
243): At the bottom there is only one entity that appears different when observed from
different standpoints …” And in his classic work ”Elemente der Psychophysik” (cit.,
vol. I, p. 4.) he wrote similarly:
Neither do two causal chains unknown to each other interfere in disorderly fashion
with each other because there is only one causal chain that acts in one substance only
but can be perceived in two ways, that is, from two standpoints.”
As alluded to before, the notion of complementarity
47
and holism
48
can be found back
in interpretations of modern quantum physics, for instance, in the concept of “quantum
45
Plato stated the same idea a long time before Fechner: “Therefore, we may consequently state that: this
world is indeed a living being endowed with a soul and intelligence […] a single visible living entity
containing all other living entities, which by their nature are all related.” (J. C. Wilson, 1889)
46
Fechner’s book is in the public domain and available under the following URL:
https://archive.org/stream/zendavestaoderb01lassgoog#page/n17/mode/thumb
47
A broad quantum physical definition of complementarity is that physical objects have binary
(conjugate) pairs of (mutually exclusive) properties which can not be measured simultaneously. The
paradigmatic example is the wave-particle duality (cf. Young’s seminal double-slit experiment first
performed in 1801).
48
Similar concepts are currently revising our notions of evolution and biology. The “hologenome theory
of evolution” (Rosenberg et al., 2009) emphasises the interrelatedness of organisms, especially in
microbiology. Organism are no longer viewed as encapsulated entities but as mutually dependent
“holobionts” (Leggat, Ainsworth, Bythell, & Dove, 2007). The central concept of “symbiogenesis”
72
holism”, as advocated by the eminent British quantum physicists David Bohm (Bohm,
1990; Hiley & Peat, 2012; C. U. M. Smith, 2009) and Fritjof Capra (Capra &
Mansfield, 1976; McKinney, 1988), inter alia.
Fechner wanted to scientifically demonstrate the unity between the psychological and
the physical (i.e., the internal and the external, the observer and the observed, subject
and object). He thought if he could demonstrate lawful reliable relations between these
seemingly different realms this would prove his point. Fechner saw all living things as
having a psyche and this gave him a particularly animated perspective of nature. Even
though Fechner’s work had an extraordinary impact on the development of psychology
as a scientific discipline, his philosophical contemplations are largely left out of the
academic discourse and the majority of textbooks on psychophysics do not mention this
important aspect of his work. Ironically, his philosophical thoughts were the driving
motives behind the development of psychophysics. One reason for the selectivity bias is
that German is no longer understood by scientists outside of German-speaking countries
(Scheerer, 1987) and Fechner’s voluminous works have only been partially translated.
Another reason might be that Fechner’s ideas challenge the mainstream status quo of
science and are therefore disregarded. Fechner himself argued that his “inner
psychophysics” was much more important than his “outer psychophysics” even though
the former did not receive much attention in academic circles (D. K. Robinson, 2010)
and is not mentioned in most textbooks and those that mention it do not grasp its full
significance. While Fechner’s experimental work is widely acknowledged, his
philosophical views would be rejected by the vast majority of psychologists even
though they use Fechnerian methodologies in their own materialistic research agenda –
(Rosenberg & Zilber-Rosenberg, 2011) is reminiscent of the concept of interdependent arising discussed
earlier.
73
a paradigm which Fechner actually tried to invalidate with his work.
In the first chapter of his “Elements of Psychophysics” which was published in 1860,
Fechner explicates the motivation for his endeavour to connect psychology with
physics. After all, the external world is a chaotic conglomerate of multifarious
disordered physical processes and the human psyche is no more less chaotic in its
intricate workings. The obvious question is: Why would one assume that there is a
precisely quantifiable and reliable correlation between these external and internal
processes? Fechner refers to the work of Weber and in his review of Webers work, he is
the first to reference “Weber’s law” and Chapter 9 of his “Elements of Psychophysics
is even titled correspondingly (Das Weber’sche Gesetz), thereby emphasizing the lawful
relation between (physical) stimulus properties and (psychological) perception.
Fechner’s aim was to create laws of sensation, as opposed to Weber’s work on
discrimination. That is, Weber discovered the law of discrimination whereas Fechner
primarily wanted to develop a law of sensation (cf. Boring, 1928). Hence, Fechner’s
approach is much more ambitious because he wanted to find the laws that govern how
internal experience changes as a function of the physical properties of external physical
stimuli. That is, how does our conscious experience
49
change when the external world
changes. In other words, how does conscious perception vary as a function of the
physical stimuli that impinge on a specific sensory modality. As a good empirical
experimentalist, Fechner was keenly aware that one cannot investigate how physical
reality changes the psyche as a whole but that one has to isolate specific aspect of
physical reality in order to bring them under rigorous experimental control (i.e., the
science of psychophysics employs a reductionistic approach and progresses gradually in
49
Today, this first-person experience would be referred to as qualia (Jackson, 1982) due to its subjective
qualitative nature, as opposed to the postulated “objectively” quantifiable nature of the physical world (a
view which has been deeply challenged by quantum physics).
74
small increments). Hence, Fechner focused on the most elementary aspect of the psyche
and that is sensation. He reasoned: If one can develop the laws of elementary sensations,
then this is a first stepping stone in the hierarchy of understanding more complex
psychological phenomena which are more complex than simple sensations. One could
argue that the task of theoreticians is to look at the “bigger picture” whereas
experimentalists have to focus on isolated phenomena, viz., global vs. local levels of
analysis (even though both are mutually reciprocal). Fechner thus sought to develop a
way in which he could experimentally investigate how “sensation magnitude” varies as
a function of stimulus intensity. Fechner’s law formalises exactly this: it quantifies the
relationship between the magnitude of a physical stimulus and the consciously
perceived intensity of the sensory experience.
50
This relation between stimulus and
experience is logarithmic in nature, i.e., a stimulus varies as a logarithmic geometric
progression (i.e. multiplied by a fixed factor), the corresponding magnitude of
experience changes in a linear arithmetic progression (i.e. in additive fashion). Ergo, for
multiplications in stimulus intensity, the intensity of experience is only additive. For
example, if a given visual stimulus is increased by a factor of three (3 x 1), the
associated perception increases by a factor of two relative to its original value (i.e., 1 +
1). If the same stimulus is again increased by a factor of three (i.e., 3 x 3 x 1), the
associated perception is three times stronger relative to its original value (i.e., 1 + 1 +
50
The relation between stimuli and sensation is what Fechner called "outer psychophysics" and this forms
the main pillar of contemporary psychophysics. However, Fechner regarded "inner psychophysics" as
much more important. Inner psychophysics focuses on the relation between neuronal (physical) processes
and sensations. This topic has not received much attention in psychophysics (Murray, 1993) and it is
related to the mind-body problem in philosophy of mind which is much more complicated than the outer
psychophysics program. The question is, how does “objectively” quantifiable electrochemical
transduction of action potentials (a physical process) give rise to subjective first-person experiences
(quale). Currently, science cannot even begin to answer this central question even though it is crucial in
order to understand really understand sensation and perception in psychophysics (again the fundamental
question concerning the relation between the observer and the observed). Inner and outer psychophysics
can be regarded as complementary (J. C. Baird, 1997).
75
1). Fechner’s law and Weber’s law are two essential formulae in perceptual/sensory
psychology
51
(J. C. Baird, 1997). However, later, both have been revised and refined.
Weber’s law becomes imprecise when the absolute perceptual threshold is approached,
and the same imprecisions are encountered for very intense stimuli. Fechner’s law, on
the other hand is a good description of brightness perception but it does not hold for
loudness (i.e., loudness perception grows exponentially in proportion to stimulus
intensity as opposed to logarithmically). In the 1950s Harvard psychophysicist Stanley
Smith Stevens formulated a power law of the relation between the magnitude of a
physical stimulus and its perceived psychological experience which is more
generalizable across sensory modalities.
Equation 3. Stevens's power law.
() = 
where I denotes the magnitude of the stimulus, ψ(I) signifies the subjectively perceived
magnitude of the sensation evoked by the stimulus, and a is an exponent associated with
the type of sensory stimulation, and k is a constant that depends on the specific metric.
That is, the magnitude of perception increases as an exponent (i.e., power) of stimulus
intensity (the exponential factor can be >1). Hence by varying the exponent, Steven’s
power law can express exponential and logarithmic proportionality between stimulus
and perception. Hence, it can reproduce Weber’s and Fechner’s law and it can account
for situations which the former are unable to handle (i.e., it is more generalisable and
can be regarded as a “covering law”). Stevens law has also been a subject of extensive
criticism and revision. For instance, Robert Duncan Luce observed that "by introducing
51
Interestingly, there is a new branch in the literature on public finance which hypothesises that the
WeberFechner law can explain the increasing levels of public expenditures in mature democracies.
Election after election, voters demand more public goods to be effectively impressed; therefore,
politicians try to increase the „magnitude“ of this „signal of competence“ the size and composition of
public expenditures in order to collect more votes (Jorge Reis Mourao, 2012).
76
contexts such as background noise in loudness judgements, the shape of the magnitude
estimation functions certainly deviates sharply from a power function" (Luce, 1990, p.
73; cf. Luce, 2002). Furthermore, in order to utilise the scaling procedures in the
standard way as advocated by Stevens, several fundamental conditions that have to be
met empirically (Luce, 2002). One of these axioms is termed “commutativity” or
“threshold proportion commutativity(Narens, 1996, Axiom 4, p. 114). Specifically,
the commutativity axiom only holds true if the outcome of two successive adjustments
(e.g., 3x as loud and 4x as loud) is independent of the order in which these adjustments
are made. The concept of commutativity will be discussed in greater detail in the
context of quantum probability where it plays a crucial role. The fact that the same
target luminance can elicit different perceptions of brightness depending on the context
has puzzled psychophysicist ever since. More recently, it has been argued in a paper
published in the Proceedings of the National Academy of Sciences “that brightness is
determined by the statistics of natural light patterns implies that the relevant neural
circuitry is specifically organized to generate these probabilistic responses” (Yang &
Purves, 2004). However, the probabilistic framework which is utilised to account for
perceptual contextuality is Kolmogorovian in nature and therefore unable to account for
noncommutativity effects in a parsimonious fashion. Moreover, it is implicitly assumed
that the perceptual system itself is always in a discrete state, independent of the
probabilistic nature of natural light patterns (cf. Hoffman, 2016). We will subsequently
address this assumptions in the context of noncommutativity in visual judgments.
To conclude the brief discourse on the history and goals of psychophysics it should be
emphasised that this academic discipline is by far the most exact and reproducible area
of psychology. The data obtained in psychophysics experiments has usually such a high
degree of quantitative accuracy that it is more reliable and replicable than physiological
77
data associated with the same sensory modalities (e.g., neurophysiological
measurements). From a methodological point of view, it can oftentimes be reasonably
questioned whether the standard hypothetico deductive-nomological model (also known
as covering law model or Popper-Hempel model) is appropriate for many aspects of
psychological research. Psychophysics is an area of psychology were the application of
this nomological approach to hypothesis testing is most effectively justifiable because
the “explanans” are precisely defined. Psychophysics has demonstrated that the
sensitivity of the visual system is as low as five quanta at the photoreceptor level (D.
Robinson, 2001), and that the auditory system is able to detect acoustic signals at the
level of Brownian-motion. Hence, psychophysics is an exact, quantitative, and
nomological branch of psychology. Contemporary psychophysics focuses on “sensation
and perception” and this dichotomy has been fittingly described as “the
complementarity of psychophysics” (J. Baird, 1997). The psychophysical
complementarity also refers to what Fechner called “inner” and “outer psychophysics”
or as Stevens (1975, p. 7) put it, the “inner world of sensation” and the “outer world of
stimuli”. We will discuss this deep philosophical concept in more detail in the next
section because the complementarity principle is central to quantum physics and
quantum cognition.
1.6 A brief history of the evolution of the
“complementarity” meme in physics
It was a pivotal turning point for physics when Nils Bohr first introduced his
formulation of the idea of complementarity to his numerous colleagues. This historical
event took place at the International Congress of Physics in September 1927 in Como,
Italy and the world’s most eminent physicists were in the audience: Max Born, Enrico
78
Fermi, John von Neumann, Wolfgang Pauli, Max Planck, Werner Heisenberg, Eugene
Wigner, Louis de Broglie, to name just a few. However, Albert Einstein was noticeably
absent for some unbeknown reason (Holton, 1970).
The idea of complementarity fundamentally transformed physics. One of the crucial
points Bohr emphasised concerned “the impossibility of any sharp separation between
the behaviour of atomic objects and the interaction with the measuring instruments
which serve to define the conditions under which the phenomena appear” (Bohr, 1961).
In a theme issue of the journal DIALECTICA edited by Wolfgang Pauli and published in
1948 compiles various seminal papers on complementarity by eminent physicists. Bohr
also contributed an article to this special issue entitled “On the notions of causality and
complementarity” (Bohr, 1948) in which he discusses the dialectic complementarity
mode of description and the impossibility to objectively separate “between behaviour of
the atomic objects and their interaction with the measuring instruments defining the
conditions under which the phenomena appear” (Bohr, 1948, p.312).
Interestingly, Bohr was a cousin of the famous Danish psychologist Edgar Rubin who is
famous for his eponymous Rubin’s Vase (E. Rubin, 1915), see Figure 4. This
ambiguous visual stimulus is today still widely used in research on bistable perception
in psychology and neuroscience (e.g., Hasson, Hendler, Bashat, & Malach, 2001; Qiu et
al., 2009; X. Wang et al., 2017). Interestingly from a history of science point of view, it
was Rubin who introduced Bohr to the concept of complementarity. Both were
members of the club “Ekliptika” (see Figure 5). Rubin in turn adopted the idea from the
writings of the late William James who wrote about complementarity in Chapter 8 in his
timeless classic “Principles of Psychology” (James, 1890b). While Rubin focused on
perceptual complementarity, Bohr was primarily concerned with epistemological
complementarity (Pind, 2014) and much of his later writings were concerned with this
79
topic. Hence, from this historical vantage point, the quantum cognition paradigm is
bringing the meme of complementarity (which originated in psychology and spread to
change the fundamentals of physics) back to its roots.
Figure 4. Rubin’s Vase: A bistable percept as a visual example of complementarity-
coupling between foreground and background.
80
In an interview
52
with Thomas Kuhn
53
which took place in 1962, Bohr stated:
I was a close friend of Rubin, and, therefore, I read actually the work of William James.
William James is really wonderful in the way that he makes it clearI think I read the
book, or a paragraph, called —. No, what is that called?—It is called ‘‘The Stream of
Thoughts,’’ where he in a most clear manner shows that it is quite impossible to analyse
things in terms of—I don’t know what one calls them, not atoms. I mean simply, if you
have some things…they are so connected that if you try to separate them from each
other, it just has nothing to do with the actual situation. I think that we shall really go
into these things, and I know something about William James. That is coming first up
now. And that was because I spoke to people about other things, and then Rubin advised
me to read something of William James, and I thought he was most wonderful.”
The significance of complementarity beyond the domain of physics has been discussed
in greater detail by Atmanspacher, Römer, & Walach (2002). The complementarity
principle is closely related to the concepts of entanglement, superposition,
noncommutativity, and the stipulated collapse of the wave-function. In fact, “quantum
noncommutativity can be regarded as a mathematical expression of the complementarity
principle” (Plotnitsky, 2016).
52
The full transcript of the interview is available on the homepage of the American Institute of Physics
under the following URL: https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4517-5
53
Interestingly, Thomas Kuhn made use of ambiguous visual stimuli in his own work to demonstrate the
perceptual change that accompanies a paradigm-shift. He used the “duck-rabbit” (a bistable figure created
by the psychophysicist Joseph Jastrow and popularised by Ludwig Wittgenstein), as a visual metaphor to
illustrate that a paradigm-shift can cause the cogniser to perceive the same information in a completely
different way (see Appendix A7 for an example and a discussion). The complementarity principle was
thus utilised in the context of the perception of seemingly incompatible scientific paradigms. That is, it
illustrates the Kuhnian concept of incommensurability which is of great relevance for the discussion of
the perceived dichotomy between mind and matter. Moreover, the inability to entertain multiple
viewpoints simultaneously is of great pertinence for discussion of interdisciplinarity, e.g., psychology and
physics (mind/matter) can be regarded as complementary.
81
Figure 5. Photograph of Niels Bohr and Edgar Rubin as members of the club
“Ekliptika” (Royal Library of Denmark).
From left to right: Harald Bohr, Poul Nørlund, Edgar Rubin, Niels Bohr and Niels-Erik
Nørlund (Royal Library, Copenhagen
54
).
When Bohr received the prestigious Danish “Order of the Elephant” (a distinction
normally reserved for royalty) he emphasised the importance of the complementarity
principle. Bohr choose to wear the ancient Chinese Yin & Yang symbol on his coat
of arms together with the Latin slogan “Contraria sunt complementa” (opposites are
complementary), see Figure 6. The resemblance between the Yin and Yang symbol and
the ambiguous figures studied by Rubin is remarkable. Moreover, various
interdisciplinary scholars maintain that nonduality between mind and matter (psyche vs.
physis, percipient vs. perceived, observer vs. observed, inner vs. outer, etc. pp.) is a
54
Associated URL of the file in the digital Royal Library of Denmark:
http://www.kb.dk/images/billed/2010/okt/billeder/object73704/da/
82
fundamental pillar of Advaita Vedānta, Mahayana/Madhyamaka Buddhism, and Neo-
Platonism (e.g., Plotinus), inter alia.
Figure 6. Escutcheon worn by Niels Bohr during the award of the “Order of the
Elephant”.
In 1947 Bohr was awarded with the “Order of the Elephant” (Elefantordenen), Demarks
highest-ranked accolade. Bohr chose a “coat of arms” which was embroidered with the
Buddhistic Yin & Yang symbol in order to emphasise the centrality of nonduality and
complementarity
55
in his work on quantum physics. Chinese Buddhism is an offshoot of
early Hinduism, the womb of the ancient nondual philosophical school of Advaita
55
Interestingly from both a visual science and physics point of view, when light interacts with the eye the
wave-particle duality resolves, that is, observation collapses the superpositional state into a determinate
eigenvalue. In this context, Einstein wrote the following on the complementarity of physical descriptions:
It seems as though we must use sometimes the one theory and sometimes the other, while at times we
may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality;
separately neither of them fully explains the phenomena of light, but together they do.” (Einstein & Infeld,
1938, p. 278)
83
Vedānta
56
which is based on a highly sophisticated and extensive logic system (Gabbay
& Guenthner, 2014; Nicholson, 2007). Nils Bohr writes: “Altogether, the approach
towards the problem of explanation that is embodied in the notion of complementarity
suggests itself in our position as conscious beings and recalls forcefully the teaching of
ancient thinkers that, in the search for a harmonious attitude towards life, it must never
be forgotten that we ourselves are both actors and spectators in the drama of
existence” (Bohr, 1950, p.54).
Applied to the dichotomy between science and mysticism described by William James
(see introduction), the complementarity principle entails that science and mysticism are
not mutually exclusive but both necessary to complete the circle of human
understanding.
1.7 Quantum cognitive science?
All known information processing systems are physically embodied (i.e., they are
grounded in physical substrates). From a reductionist point of view, the underlying
physics of all information processing systems is consequently ultimately quantum-
mechanical in nature. It follows deductively
57
that science has to reconsider information
processing and computation in the light of recent evidence from quantum physics.
Information processing and computation play a major role in psychology, neuroscience,
and many other scientific disciplines (e.g., computational cognitive science (Sun, 1950),
56
According to Advaita Vedānta, consciousness and material reality do not exist in separation. This
schism, is an illusion or Māyā (Bhattacharji, 1970; Dabee, 2017). That is, the subject/object divide is also
part of Māyā or mere appearance”. Beyond the perceived duality is what quantum physicist John
Hagelin calls “the unified field” or “string field”pure abstract self-awareness which forms the nondual
basis for all of existence, material and immaterial (Hagelin & Hagelin, 1981).
57
One can construct a logically valid syllogistic argument in order to deduce the conclusion that quantum
physics is necessarily relevant for cognitive/computational processes and their neural correlates.
84
computational neuroscience (Sejnowski, Koch, & Churchland, 1988), computational
biology , etc. pp.). For instance, cognitive modelling is concerned with computational
models of cognition. These models assume “cognitive completeness” (Yearsley &
Pothos, 2014). Cognitive completeness implies that behaviour (e.g., perceptual
judgments) can be explained in purely cognitive terms without the invocation of neural
correlates. This is an implicit assumption of almost all cognitive models, otherwise
cognitive science would be forced to constantly integrate the complexities of
neurophysiology and neurochemistry into its modelling efforts (of course there are
exception). In sensu lato, cognitive completeness is embedded in the notion of “multiple
levels of description and explanation” (Coltheart, 2010; Perfors, 2012).
In the last century, quantum physics discovered extraordinary phenomena which shed
new light on the fundamental workings of reality. Among these phenomena are, for
instance, the concepts superposition, complementarity, and entanglement
(Atmanspacher et al., 2002). Besides their purely theoretical (and ontological)
relevance, these counterintuitive “strange” principles can be utilised for various
practical purposes. Real-world applications include, quantum encryption (Bernstein &
Lange, 2017), quantum communication (Zhang et al., 2017), and quantum computation
(Divincenzo, 1995), quantum teleportation (Ren et al., 2017), inter alia. For instance,
entanglement (see Bell’s theorem) can be utilised for extremely efficient transfer of
information (faster than the speed of light) and it has been convincingly argued that the
next generation of the internet (the “quantum internet” (Kimble, 2008; C. R. Monroe,
Schoelkopf, & Lukin, 2016; Pirandola & Braunstein, 2016)) will be based on the
principle of nonlocal entanglement, i.e., quantum nonlocality (Popescu & Rohrlich,
1992, 1994). However, the significance of these findings has not yet been realised by
the majority of cognitive and neuroscientists. Empirical research has clearly
85
demonstrated (i.e., beyond any reasonable doubt) that quantum computational resources
exists in nature and that they can be successfully employed for various pragmatic
purposes. However, hitherto these principles have not yet been given their due in
mainstream psychology (and neuroscience) and many researchers would argue that the
findings made by quantum physicists do not apply to cognitive processes (that is, they
are a priori assumed to be restricted to the physical microdomain). However, we argue
that the “burden of proof” (Hahn & Oaksford, 2007) rests on the side of those who
argue that QM principles do not apply to cognition: Why would the cognitive system
not make use of these extremely powerful computational resources?
An essential concept in this context is the qubit. The origination of the term qubit is
ascribed to Schumacher (1995) who proposes the term "quantum bits" or "qubits” in his
seminal paper entitled “quantum coding”. A qubit is a unit of quantum information (a
two-state quantum-mechanical system) and it can be regarded as an analogon to the
binary bit. By contrast to the classical bit, a qubit can be in a superpositional state. The
mathematical representation of a qubit is given in Equation 4, where α and β denote
probability amplitudes
|
=
|
0
+
|
1
Equation 4. Mathematical representation of a qubit in Dirac notation.
Conventionally quantum states are represented in Dirac notation (Equation 4) in which
computational basis states are enclosed in bra (|)–ket () notation, i.e., |0 and |1. A
geometrical (and more intuitive) representation of a qubit is provided in Figure 7.
86
Figure 7. Bloch sphere: a geometrical representation of a qubit.
Note: The qubit is a two-state system which can be in a superpositional state similar to
Youngs classical experiment (Østgård et al., 2014).
The qubit requires a completely new way of thinking about information and
computation. A qubit is a two-level quantum mechanical system and it can be in a
superpositional state, i.e., multiple states at the same time. Mathematically, a quantum
logical qubit state can be written as a linear combination (viz., superposition) of |0 and
|1. Moreover, a qubit can be visually represented as a Bloch sphere which is
eponymously named after its inventor (Bloch, 1946). Fascinatingly, a single qubit can in
principle carry the information of all libraries in the world (Muthukrishnan & Stroud,
2000), viz., continuous-valued quantum information in a linear superposition (the
problem is how to measure the information without destroying it via collapse of the
superposition caused by the measurement).
The primary difference between one- and two-qubit states is their dimensionality. While
a one-qubit state has two dimensions a two-qubit state has four dimensions. This is the
case because in mathematics the tensor product A B (where signifies the tensor
87
product, also known as Kronecker product
58
) of two vector spaces A and B forms a new
higher-dimensional vector space which has a dimensionality equal to the product of the
dimensions of the two factors. In linear algebraic notation this can be written as
follows:
59
00
1
0
1
0
=
1
0
0
0
, 01
1
0
0
1
=
0
1
0
0
,
10
0
1
1
0
=
0
0
1
0
, 11
0
1
0
1
=
0
0
0
1
.
However, the two-qubit states which cannot be simply reduced to the Kronecker
product of single-qubit states because they are in an entangled state and the contained
information is not reducible to the individual constituent qubits (i.e. the whole is more
than the sum of its parts). The contained information is rather stored in the correlation
between the two states, i.e., it is non-local quantum information (Nielsen & Chuang,
2010). This non-locality of information is a crucial criterion which distinguishes
quantum computation from classical computation. Moreover, this type of non-local
information storage is the basis of various quantum protocols, for instance quantum
teleportation (Gottesman & Chuang, 1999).
A qutrit is defined as a unit of quantum information that is composed of the
superposition of three orthogonal quantum states (Klimov, Guzmán, Retamal, &
Saavedra, 2003). While a qubit is analogous to a classical bit, a qutrit is analogous to the
58
Note that the Kronecker product is not identical to usual matrix multiplication which is a different
mathematical operation.
59
Matrix-notation adapted from Microsoft's Quantum Development Kit: https://docs.microsoft.com/en-
us/quantum/quantum-concepts-5-multiplequbits
88
classical trit (trinary digit), for instance as utilised by ternary
60
computers based on
ternary logic (aka. 3VL) (Putnam, 1957). Consequently, a multiqubit quantum computer
with 300 entangled qubits could instantaneously compute more calculations than there
are atoms in the known universe. However, due to decoherence, superpositions are
extremely delicate. The problem lies in measuring the contained information. As soon
as an invasive measurement on the system is performed, the superpositional states
collapse into an eigenstate (environmentally-induced decoherence) and the information
is lost.
61
In sum, superposition is an essential property which is utilised for quantum
computation and it also appears to be applicable to models of cognition (Busemeyer &
Bruza, 2012). Moreover, the future of the rapidly developing fields of machine learning
and artificial intelligence is likely based on these extremely powerful quantum
computational principles which require a radically new way to think about information
(Biamonte et al., 2017; Dunjko & Briegel, 2017; Prati, 2017). Therefore, cognitive
psychology is now carrying the burden of proof: Why should nature not make use of
these extremely effective quantum-principles in the domain of cognitive processes?
Most models of cognition are strongly influenced by the principles of digital binary
computation (Piccinini & Bahar, 2013), although some argue that “cognition is not
computation”
62
(Bringsjord & Zenzen, 1997). A classical bit can adopt two possible
states (i.e., binary states) usually symbolised as 0 and 1 (but more generally “true” or
60
For instance, in “The art of computer programming” Donal Knuth (creator of TeX which forms the
basis of LaTeX) explains that in balanced ternary, every digit takes on one of 3 values, i.e., [−1, 0, +1]
(which can be more parsimoniously notated as [− ,0, +]). In the context of ternary notation, he also writes
that “Positional number systems with negative digits have apparently been known for more than 1000
years in India in Vedic mathematics” (Knuth, 1973, p. 192).
61
First attempts have been made to create qudits which, in contrast to two-state qubits can have multiple
states simultaneously. A qudit based quantum computer with two 32-state qudits, could compute as many
calculations as 10 qubit quantum computer, thereby speeding-up computation and significantly reduce
problems associated with the delicate entanglement of multi-qubit systems (Neeley et al., 2009).
62
Specifically, the authors argue that “computation is reversible; cognition isn’t; ergo, cognition isn’t
computation(Bringsjord & Zenzen, 1997, p. 285). The irreversibility of cognitive processes might be
rooted in the stochastic nature of quantum processes (Aaronson, Grier, & Schaeffer, 2015; cf. Yearsley &
Pothos, 2014).
89
“false” or any other dichotomous notation, e.g., cats and dogs, as the physical substrate
in which the bit is realised is not important. This substrate independence is known as
multiple realizability, for a discussion of this fundamental concept see Shapiro (2000).
This implies that computation should be treated as logical abstraction what is
important is software (logic) not the physical substrate (hardware).
Alan Turing wrote:
“The [Babbage Engine's] storage was to be purely mechanical, using wheels and cards.
The fact that Babbage's Analytical Engine was to be entirely mechanical will help us rid
ourselves of a superstition. Importance is often attached to the fact that modern digital
computers are electrical, and the nervous system is also electrical. Since Babbage's
machine was not electrical, and since all digital computers are in a sense equivalent,
we see that this use of electricity cannot be of theoretical importance. ... If we wish to
find such similarities we should look rather for mathematical analogies of function.”
Richard Feynman argued in his lecture series on quantum computation that Turing’s
arguments were impeccable but that he did not consider substrates that behave
according to the “strange” laws of quantum logic. The crucial point is that it has become
very clear that classical notions of physics are no longer defendable on empirical
grounds (e.g., local realism) (Giustina et al., 2015; Hensen et al., 2015; Wiseman,
2015). All information processing systems are embodied in some form of physical
substrate. Given that those physical substrates ae governed by the laws of quantum
mechanics, it follows that classical (Newtonian) notions of computation have to be
revised (and in fact are currently being revised) in the light of insight derived from
quantum physics. For instance, Google and NASA are currently heavily investing into
quantum computation and quantum AI (both are grounded on quantum logic). In sum,
90
quantum computational principles will significantly speed up a large array of
computational processes (Rønnow et al., 2014) and might turn out to be a driving force
for the continuation of Moore’s law (Lundstrom, 2003; G. E. Moore, 1965).
Superposition and entanglement are pivotal concepts in quantum information and
quantum computing (Boyer, Liss, & Mor, 2017). Quantum information and
computation are closely related to quantum cognition, as cognition is understood to be
information processing. Many cognitive and neuroscientists believe that cognition is
essentially a form of computational, i.e., it can be modelled mathematically by utilising
various computational principles (i.e., Bayes’ rule). Therefore, it is obvious that
cognitive scientists should consider quantum computational principles which do not
obey Bayes’ rule (which is based on Kolmogorov’s probability axioms). The same
quantum computational principles are also important for neuroscience and particularly
(neuro)computational neuroscience and artificial intelligence. Currently, neurons are
almost exclusively modelled as binary states (firing vs. resting), even though several
researchers are now beginning to integrate quantum approaches into their efforts
(Schuld, Sinayskiy, & Petruccione, 2014). From a quantum perspective, neurons can be
modelled as superpositional states. Given that neurons are thought to underpin all of
cognition (at least in a reductionist materialism framework) this has implications for the
high-order cognitive processes and computational models of cognition which are based
on these neurocomputational processes.
1.8 Perceptual judgments under uncertainty
Random walk models (e.g., Ratcliff & Smith, 2004; Usher & McClelland, 2001) which
focus on reaction times in various decision scenarios assume that evidence
(information) is accumulated diachronically (over time) until a specific critical decision-
91
threshold (or criterion) is reached (Busemeyer & Bruza, 2012). In these models, the
weights associated with each option increases chronologically in a progressive manner.
However, at each discrete point in the temporal sequence the system is assumed to be in
a definite determinate state. This state can in principle be accessed by taking a
measurement. Moreover, it is assumed that the act of measuring does not influence the
state under investigation. That is, classical models presuppose that 1) a given system is
consistently in a specific state (even though the observers’ cognition of this state might
be uncertain) and 2) that this state is independent of the measurement operation which is
performed on the system. Prima facie, these postulates seem intuitive and logically
valid. How else could one build a model of a system if it is not in a definite (stable)
state at any point in time? And how else could one gain “objective” information about
the state of the system if not via independent (interference-free) measurements which
“read-out” the actual state of the system?
However, both assumptions stand in sharp contrast with one of the main ideas of
quantum probability (QP) theory which provides the axiomatic basis of quantum theory.
A fundamental insight derived from quantum theory is that taking a “physical
measurement” of a “physical system” actively creates rather than passively records the
property under investigation. By contrast, classical theories assume that taking a
measurement merely reads out an already pre-existing state of a system.
Moreover, QP is incompatible with the classical notion that a given system (be it
physical or psychological) is always in an a priori determinable state at any point in
time. By contrast, QP allows for the possibility that a system can be in a superpositional
state in which n possibilities can exist simultaneously. It is only when a measurement is
taken that these undetermined potentialities collapse into determinate actualities.
The collapse of the wave-function Ψ is caused by interactions with the environment, a
92
process known as decoherence, i.e., the destruction of interference (Zurek, 1994). This
environment-induced collapse causes a loss of information, i.e., entropy.
63
In other
words, decoherence is the transition from a quantum state to a classical state, a process
called “Einselection” (Zurek, 2003). Thus, Einselection imposes classicality via a
drastic reduction of the dimensionality of the Hilbert space, in other terms, it creates
coherence from decoherence (Zurek, Habib, & Paz, 1993). That is, “classical structure
of phase space emerges from the quantum Hilbert space” and “in measurements,
Einselection replaces quantum entanglement between the apparatus and the measured
system with the classical correlation” (Zurek, 2003, p. 715). Our foregone discussion of
the concept of complementarity in visual perception illustrates this point. The Rubin’s
vase can be regarded as a bistable superpositional quantum state. The visual percept is
in a superpositional state and it is only when a measurement is taken (i.e., an observer
observes the stimulus) that the superposition collapses into a mutually exclusive
“either/or” eigenstate (the dominance of either foreground or background) caused by the
process of Einselection. Similarly, the Necker cube (Necker, 1832) has been described
in terms of quantum superposition and temporal nonlocality
64
(Atmanspacher & Filk,
2010; Atmanspacher, Filk, & Römer, 2009; Conte, Khrennikov, Todarello, Federici,
Mendolicchio, et al., 2009) and the quantum Zeno effect (Asher Peres, 1980) has been
successfully applied to model the switching rates between bistable (ambiguous) visual
percepts (Atmanspacher & Filk, 2013; Atmanspacher, Filk, & Römer, 2004).
We created two websites with additional information. One contains a dynamic
63
Entropy is a function of t (time evolution of the system) and a functional of the systems initial
state. The entanglement between the system and the environment can be calculated by computing the
entropy using the following intuitive algorithm (but see Zurek et al., 1993):
() = Tr (
()log
())
64
Locality describes the notion that a given event X cannot cause a change in Y in less time than =
/, where T signifies time, D is the distance between X and Y, and c the (constant) speed of light, and.
93
“Adobe® Shockwave Flash” animation of the Necker cube from a quantum cognition
perspective
65
. The other contains several digital animated artworks we designed and is
entitled “Necker Qbism: Thinking outside the box – getting creative with the Necker
cube”, in analogy with the superpositional quantum qubit and the concept of
simultaneity in cubism (Fry, 1988).
66
Briefly, Atmanspacher et al. applied the concept of “temporal nonlocality” (Brunner,
Cavalcanti, Pironio, Scarani, & Wehner, 2014) to the perception of bistable stimuli (i.e.,
the Necker cube). Temporal nonlocality implies that “events cannot be uniquely fixed in
time” and it is based on temporal Bell inequalities (or Leggett–Garg inequalities). The
exact definition of TBI goes beyond the scope of this chapter (but see Eberly, 2002).
Temporal Bell inequalities are particularly important for quantum-like context effects,
quantum entanglement, and the Kochen-Specker theorem (Santos, 2016). Within the
context at hand, temporal Bell inequalities are most pertinent when multiple
measurements are performed at different points in time. According to physical realism
67
(A. J. Leggett, 2014), a given system with two or more possible states is at all times in a
definite (fixed) state. Such a realist system satisfies the temporal Bell inequality
(Yearsley & Pothos, 2014). In the history of science, the violation of TBI is one of the
most important findings of the 20
th
century. The violation of TBI has first been
empirically demonstrated by Aspect, Grangier, & Roger (1981) and various
independent labs replicated and extended this paradigm-changing experimental
65
URL associated with the “Quantum Necker cube”: http://irrational-decisions.com/?page_id=420
66
URL of the “Necker Qbism gallery”: http://irrational-decisions.com/?page_id=1354
67
Physical realism postulates a mind-independent reality that is composed of physical entities that are
located in space and time, and interact causally with each other (Ellis, 2005). The concept is crucial for an
understanding of quantum physics as it forms the basis for many discussions among scholars, e.g., the
prototypical Einstein vs. Bohr debate on epistemology and ontology (Mehra, 1987).
94
finding.
68
The wide ramifications of this scientific finding are staggering because the
violation of TBI negates the fundamental concept of “local causality” thereby ruling
out a large class of previously widely accepted physical models (Yearsley & Pothos,
2014), namely those which are based on local realism. However, in physics, local
realism has now been conclusively rejected (Giustina et al., 2015; Gröblacher et al.,
2007; Hensen et al., 2015). To those scientists who still persistently “believe” in an
objectively existing material reality
69
we recommend the concise NATURE article
entitled “The mental universe” authored by Richard Conn Henry, academy professor of
physics and astronomy at Johns Hopkins University (Henry, 2005) and the more recent
NATURE paper “Death by experiment for local realism” (Wiseman, 2015). In yet another
NATURE paper the following explicit statement has been formulated: ”Most working
68
Critics argue that the experiments might be confounded (e.g., by a loopholes/additional assumptions).
However, recent experiments successfully addressed these potentially confounding loopholes (e.g.,
Giustina et al., 2015) and provide strong empirical evidence of TBI violations, thereby paving the way for
implementing device-independent quantum-secure communication(Hensen et al., 2015). A even more
recent cutting edge experiment performed a “Cosmic Bell Test” by investigating violations of Bell
inequalities in polarization-entangled photons from distant Milky Way stars in real-time (Handsteiner et
al., 2017). The experiment confirmed the quantum theoretical prediction regarding statistical correlations
between measurements and provides further evidence against the classical local-realist theory. The
authors concluded that their experimental design rules out “any hidden-variable mechanism exploiting the
freedom-of-choice loophole” because it “would need to have been enacted prior to Gutenberg’s invention
of the printing press, which itself predates the publication of Newton’s Principia”. Interestingly, the
researchers report p-values < 1.8 x 10
-13
in support of their conclusion (Handsteiner et al., 2017, p. 4),
indicating that frequentist p-values are unfortunately still relevant in cutting-edge physics. The reporting
of such extremely small p-values (in the abstract) is misleading, as it capitalizes on the “replication
fallacy”, i.e., the widely shared fallacious belief that small p-values indicate reliable research (e.g.,
Amrhein, Korner-Nievergelt, & Roth, 2017).
69
Bishop Berkeley’s statement “esse est percipi (aut percipere)” — to be is to be perceived (or to
perceive) is relevant in this context. Samuel Johnson famously asserted in 1763 to have disproven
Berkeley's nonmaterialistic stance by kicking a rock and he is known to have said "I refute Berkeley
thus”, a non sequitur (cf. Priest, 1989; “Primary qualities are secondary qualities too”). This logical
fallacy goes by the name “argumentum ad lapidem(Latin for “appeal to the stone) (Winkler, 2005) as it
is no valid argument but merely superficially dismissing Berkley’s claim without providing any reasons
(Pirie, 2007). An example of this type of logical fallacy follows:
Person 1: Under the code name MK-Ultra, the CIA conducted illegal drug experiments on countless
nonconsenting human being to investigate mind control techniques which resulted in several deaths.
Person 2: This is just a conspiracy theory!
Person 1: Why do you think so?
Person 2: It is obviously just paranoia.
In this example of an “appeal to the stone” fallacy, Person 2 provides no logical reasons or facts. Person 2
merely asserts that the claim is absurd and therefore commits the same logical fallacy as Berkley’s
argumentative opponent..
95
scientists hold fast to the concept of 'realism' - a viewpoint according to which an
external reality exists independent of observation. But quantum physics has shattered
some of our cornerstone beliefs.” The authors go on and state that experimental
evidence (i.e., violation of Bell inequalities) has rendered “local realistic theories
untenable(Gröblacher et al., 2007).
70
Similarly to the breakthrough in quantum
physics, an experimental demonstration of a TBI violation in psychological observables
would herald a paradigm-shift in psychology because such a finding would rule out a
large class of cognitive models which assume that cognitive systems are always in a
deterministic state (Yearsley & Pothos, 2014). Experimental approaches that could
falsify TBI in the context of visual perception have already been formulated
(Atmanspacher & Filk, 2010). The implications such an empirical discovery would have
for psychology and neuroscience cannot be overemphasised as the violation of BI is one
of the most thought provoking finding physics has ever made. The rejection of local
realism is not only highly counterintuitive, it might also “feel” very uncomfortable
because our common-sense worldview is firmly anchored in this most constitutive
paradigm. Such a finding would certainly cause severe cognitive dissonance in the
minds of majority of scientists (Festinger, 1957, 1962). That is, if results from quantum
physics challenge our most fundamental beliefs and force us to rethink reality, this can
70
At this point it is important to differentiate between classical (spatial) Bell inqualities (BI) and temporal
Bell inequalities (TBI), i.e., Bell's theorem for temporal order (Paz & Mahler, 1993; Zych, Costa,
Pikovski, & Brukner, 2017) This difference is directly related to the Heisenberg uncertainty principle
(Heisenberg, 1927) which asserts a fundamental limit to the precision of measurements.
ΔxΔp≥h/, where h is Plancks constant.
Specifically, this principle describes a mathematical inequality which states that complementary variables
(i.e., complementary physical properties such as position x and momentum p) cannot be simultaneously
known (observed/measured) with an arbitrarily high degree of precision. It is important to emphasise that
this principle is completely independent of the inaccuracy of the measurement device or any other
experimental variables (e.g., noise, unknown experimental confounds, etc.). Rather, the uncertainty
principle is fundamental to the nature of the quantum mechanical description of reality. The
Heisenbergian uncertainty principle constitutes one of the defining difference between spatial and
temporal Bell inequalities as the constraint does not apply when two measurements are performed at the
same point in time on two different particles located in different space points. On the other hand, it does
constraint the ability to resolve the two states in a second measurement at a later time on the same particle
(Calarco, Cini, & Onofrio, 1999).
96
evoke strong emotional/affective responses and various cognitive defence mechanisms
might be activated to protect our conceptual schemata from the radical (Bayesian)
revision of beliefs which is necessary when these finding and their implications are
taken seriously. The well-studied phenomenon of belief-bias is relevant in this regard.
Belief-bias a phenomenon in the psychology of thinking and logical (syllogistic)
reasoning which demonstrates that reasoning is biased by a priori beliefs, even though
the logical argument might be syntactically valid (i.e., logically sound). This conflict
between semantic believability (a System 1 process) and syntactical logical validity (a
System 2 process) leads to large proportions of fallacious conclusions when these aspect
are incongruent, viz., the conclusion of a given argument is logically valid but
semantically unbelievable according to priors beliefs (J. St. B. T. Evans, Barston, &
Pollard, 1983; Kahneman, 2003; Tsujii & Watanabe, 2009). A more detailed description
of belief-bias can be found in Appendix A6. Hence, for proper scientific thinking it is
important to counteract this systematic belief-bias in order to deduce logically valid
conclusions.
There is general consensus
71
(i.e., a strong prior belief) in cognitive psychology and the
neurosciences that cognitive processes are ultimately reducible to neuronal processes, a
perspective which goes by the name of “materialistic reductionism” (however, this
71
Group-consensus (conformity) is another important factor which can dramatically distort the validity of
scientific judgments and reasoning (Asch, 1955). Social-identity theory (Tajfel & Turner, 1986) is yet
another powerful explanatory theoretical framework in this respect. If the social identity of a given
scientists (or a group of scientists, or a whole scientific discipline) is based on the (untested) assumption
of local realism, then any evidence which challenges this shared Weltanschauung is perceived as a threat
to the group norm. These group processes are in conflict with rational and “objective” scientific
reasoning. These well-documented effects are based on complex social dynamics which cannot be
ignored in the context of scientific reasoning. The “need to belong“ (Baumeister & Leary, 1995) is a
fundamental human motive which (implicitly) motivates much of human behaviour. Scientists (and the
groups they affiliate with) are no exception. Awareness of these confounding effects on reasoning and
decision-making is crucial but usually exclusively taught as part of a specialised social psychology
curriculum, which is (dis)regarded as a “soft” science even though it uses the same quantitative methods )
as other disciplines, e.g., the biomedical sciences (to be precise, a loically incoherent hybrid between
Fisherian and Neyman-Pearsonian hypothesis testing, but see Gigerenzer, 1993).
97
conceptual paradigm is not based on empirical evidenceit is merely hypothetical).
Therefore, the notion of realism (as used in physics) is an almost unquestioned
assumption of all mainstream cognitive (and neurological) models. An interesting
question is the following: If TBI is violated at the cognitive process level, but the brain
is assumed to be classical, then what exactly is the substrate of the quantum process
(Yearsley & Pothos, 2014)? And what role do quantum processes play in
neurophysiology/neurochemistry (Baars & Edelman, 2012; Koch & Hepp, 2006)?
Recently, several quantum models of the brain have been proposed. The most widely
known (and most controversial) theory is the “Orchestrated objective reduction(Orch-
OR) hypothesis formulated by Sir Roger Penrose and Stuart Hameroff which postulates
that quantum processes at the neuronal microtubular level are responsible for the
emergence of consciousness. Appendix A2 provides a synopsis of the conjectural Orch-
OR quantum-brain hypothesis.
1.9 A real-word example of superposition and collapse
The generic probability framework developed in quantum physics appears to be relevant
to multifarious psychological processes (Atmanspacher & Römer, 2012). Especially, the
concept of noncommutativity appears to be pertinent for cognitive operations.
Noncommutativity, in turn, is closely related to superposition and the collapse of the
wave-function. The following paragraph provides an intuitive simplistic illustration of
the principle of superposition applied to a real-world decision-making scenario.
Subsequently, we will discuss the concept in somewhat more technical terms in the
context of visual perception.
Here is the real-world example in the context of academic decision-making: Suppose an
examiner has to decide whether a Ph.D. thesis should be passed or failed. From a
98
classical information processing point of view the response format is binary, i.e., either
yes or no response (lets denote this with 1 or 0), a dichotomous decision. These values
might change dynamically over time as the examiner reads the thesis, but at any
moment in time, the associated cognitive variable is assumed to be in a definite fixed
state (see Figure 8). However, contrary to the classical notion, it seems plausible that the
examiners cognitive state does not jump from one discrete binary state to another (like a
flip-flop or an electron jumping from one orbit to another). Instead, the examiner might
experience ambiguity about both states simultaneously (see Figure 9). That is, until a
final decision is made, the cognitive system is in a superpositional state, i.e., an
indeterminate state. When the decision is finally reached (e.g., no corrections, i.e., 0),
the superpositional 1/0 state instantly transforms into a determinate state. This is the
simplified basic tenet of superposition and collapse in QP theory, explained in the form
of an intuitive analogy.
Observe state i at time t where p
i
= probability of state i
p(t | i) = [1,0,..,1,..0]'
p(t + s) = T (s) p(t | i)
Figure 8. Classical sequential model (Markov).
Observe state i at time t where ψ
i
= amplitude of state i
ψ (t | i) = [1,0,..,1,..0]'
ψ (t + s) =U(s) ψ(t | i)
1
0
1
0
1
1
0
0
(t)
99
Figure 9. Quantum probability model (Schrödinger).
This example illustrates the concept of “quantum indeterminacy” (Busch, 1985; cf.
Glick, 2017) which stands in direct contrast with deterministic physical theories which
predate quantum physics. Deterministic theories assumed that:
1) a given (physical) system always has a in principle determinable state that is
precisely defined by all its properties.
2) the state of the system is uniquely determined by the measurable properties of the
system (i.e., the inverse of point 1).
Thus, an adequate account of quantum indeterminacy needs to operationalise what
constitutes a measurement – an unresolved “hard” problem which we will address in
greater detail in the general discussion section.
1.10 Determinism vs. constructivism
“The procedure of measurement has an essential influence on the conditions on which
the very definition of the physical quantities in question rests.” (Bohr, 1935, p.1025).
According to the theoretical nexus of quantum cognition, superposition,
noncommutativity, and complementarity are closely interlinked phenomena. To reiterate
the basic principles of QP in more technical terms, superposition defines a state which
has a specific amplitude across >1 possibilities. QP postulates that taking a
1
0
0
0
0
1
1
1
(t)
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measurement causes a continuously distributed state to collapse into a discontinuous
discrete state (via wave function collapse as described by Schrödinger’s wave-
equation). That is, the quantity being measured changes from a superimposed state into
an Eigenstate.
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The crucial difference to sequential Markovian data models (e.g.,
Camastra & Vinciarelli, 2015) is the impossibility to create what Schrödinger called an
“Erwartungskatalog” (expectation catalogue), i.e., an index of the trajectory of states of
the system as a discrete time-series. Note that it is only when a measurement is taken
that a discrete value is created via collapse of Ψ. The trajectory of the state of a quantum
system is called a quantum trajectory” (Sanz & Borondo, 2007) and can be
conceptualised as stochastic random walk
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in a multidimensional Hilbert space.
However, in contrast to classical random walk models, the evolution of the quantum
system is conditioned upon measurements. In the current context of perceptual
judgments, we are particularly interested in the question whether the perceived
luminance level of a visual stimulus changes depending on whether there was an
antecedent psychophysical measurement or not. Let us assume that the perceptual
evaluation is developing over two stages. Each stage entails the presentation of a visual
stimulus (a grey rectangle with high or low luminance levels). From a classical
probability (CP) perspective, it should not make any difference if the percipient is
requested to provide a perceptual evaluation just after the second stage or after the first
stage as well. If an intermediate evaluation is required, this is assumed to merely read-
out an already pre-existing internal visual percept and therefore this should not have any
impact on the final perceptual judgment in the second stage. By contrast, from QP
perspective, a perceptual evaluation (an introspective measurement) can significantly
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The word “Eigenstateis derived from the German word “Eigen, meaning own, inherent, or
characteristic”.
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The term „random walk“ was first introduced by the English mathematician and biostatistician Karl
Pearson in a seminal N
ATURE article entitled „The Problem of the Random Walk“ (Pearson, 1905).
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change the state of the percipients’ cognitive system (the cognitive state vector is
realigned). Ergo, the intermittent perceptual judgment (i.e., cognitive measurement) can
causally interfere with the result of the subsequent judgement. Note that the CP model
does not predict any order effects due to an interjacent measurement whereas the QP
model predicts such effects a priori. Of course, it is possible to explain such a finding in
classical terms with auxiliary hypotheses (Leplin, 1982) which can be added a
posteriori to the CP model in order to provide a post hoc explanation for this kind of
carry-over effect. However, this can only be accomplished by adding additional
components to the model which are not inherent to CP theory and which have not been
predicted a priori. Consequently, according to the law of parsimony, i.e., Ockham's
razor (Rodríguez-Fernández, 1999), the QP model should be preferred over the CP
model.
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1.11 Quantum logic
The claim that logic should be subject to empirical research was first articulated by von
Neumann and Birkhoff in the Annals of Mathematics (Birkhoff & Neumann, 1936).
This position was later also advocated by Hilary Putnam (Cartwright, 2005; Maudlin,
2005). He argued that in the same way as non-Euclidean geometry revolutionised
geometry, quantum mechanics changed the fundamental assumptions of logic. In his
seminal paper entitled “Is logic empirical”, Putnam proposed the abandonment of the
algebraic principle of distributivity, a position which has been challenged on several
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Note that CP and QP theory are not necessarily mutually exclusive. Classical probability is a special
case within the more general overarching (unifying) quantum probability framework.
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grounds (Bacciagaluppi, 2009; Gardner, 1971). The distributivity principle has received
great attention in the context of irrational reasoning (Hampton, 2013; Sozzo, 2015), for
instance, in the context of the conjunction fallacy (e.g., the Linda paradox
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). However,
while violations of the distributivity principle are inconsistent with classical logic, they
are entirely consistent in the axiomatic framework and various prima vista seemingly
irrational reasoning fallacies have been successfully modelled using quantum logic
(Moreira & Wichert, 2016b). A pivotal difference from classical Boolean algebra is
described by the von Neumann’s concept of “simultaneous decidability” and extension
of simultaneous measurement. Birkhoff’s and von Neumann's interpretation of quantum
mechanics have been extensively discussed in philosophy of science, inter alia, by Karl
Popper (Popper, 1968).
In the psychological literature, classical probability theory dominates all modelling
efforts. That is, almost all cognitive and neuroscientific models implicitly assume the
validity of classical probability theory. The standard model of probability (known as
Boltzmann/Gibbs distribution in physics or Kolmogorov’s laws in classical probability
theory) is based on the set-theoretic assumption that probabilities always add up to 1.
This is formally axiomatized in the law of conditional probability. The Kolmogorov
formulation is as follows (Kolmogorov, 1956):
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A prototypical version of Linda paradox goes as follows (Tversky & Kahneman, 1983):
Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student,
she was deeply concerned with issues of discrimination and social justice, and also participated in
anti-nuclear demonstrations.
Which is more probable?
a) Linda is a bank teller.
b) Linda is a bank teller and is active in the feminist movement.
(We ask the reader to answer the question before reading the following paragraph.)
The majority of respondent “irrationally” choose option b) over option a). However, the conjunction of
both events occurring together is probabilistically less than or equal to either event occurring in isolation.
This inequality can be formalised as () () and () ().
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Equation 5. Kolmogorov’s probability axiom
(|) =
()
()
Current cognitive and decision models are almost exclusively derived from the
Kolmogorov axioms (Kolmogorov, 1933/1950). Quantum probability is based on
fundamentally different mathematical axioms and has the potential to provide a viable
alternative to the dominant Kolmogorovian paradigm
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.
1.12 Noncommutative decisions: QQ-equality in
sequential measurements
In the current experimental context, the most relevant difference between classical and
quantum probability models is the way in which they deal with violations of the
commutativity axiom (the quantum model allows for violations of symmetry, that is,
observables do not have to commute). In other terms, the defining difference between
classical probability theory and quantum probability theory is noncommutativity of
operators.
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If projectors do commute, classical probability theory applies, “iff” they do
not commute, quantum probability applies. Accordingly, quantum theory is only
applicable in cases of noncommutativity (Busemeyer & Bruza, 2012), otherwise it is
identical to the classical probability framework. Quantum stochastic calculus is the
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Bose-Einstein statistics are another counterintuitive instance of quantum probabilities which are
incongruent with classical notions of probability (quantum dice). The details go beyond the scope of this
chapter. However, for the curious reader, we created a website which contains additional information on
this topic: http://irrational-decisions.com/quantum_dice/
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In matrix algebra, the product of matrices does not necessarily commute, for instance:
[
0 2
0 1
] = [
1 1
0 1
] [
0 1
0 1
] [
0 1
0 1
] [
1 1
0 1
] = [
0 1
0 1
]
In matrix algebra, every subspace corresponds to a projector, i.e., the projector is an operator that takes a
vector and projects it onto the subspace (Busemeyer & Bruza, 2012) and projector A multiplied by
projector B does not always give the same result as projector B times projector A.
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mathematical framework which is used to model the random
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evolution of quantum
systems undergoing measurements. It is a generalization of stochastic calculus to
noncommuting observables (Hudson & Parthasarathy, 1984).
Equation 6. Classical probability theory axiom (commutative).
() = ()
Equation 7. Quantum probability theory axiom (noncommutative).
|
|
|
| |
|
|
|
How do we transfer this abstract mathematical formalism to actual real-world
phenomena? Let us consider a representative realistic example: In a Gallup poll
conducted in 1997, half of the sample (n = 1002) was asked, “Do you generally think
Bill Clinton is honest and trustworthy?” and subsequently they were asked the same
question about Al Gore (Moore, 2002). Using the standard (random) split sample
technique, the other 50% of respondents answered exactly the same questions but the
question order was reversed. When the question about Clinton was asked first, he
received a 53% agreement whereas Gore received 76% (Δ 23%). However, when the
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Werner Heisenberg differentiates between objective randomness and subjective randomness. While the
outcome of throwing two die is subjectively random, quantum randomness is objectively random. In
principle, the ooutcome of throwing a die could be determined however the Newtonian dynmics are just
to complex (viz., Laplace's omniscient demon could in principle predict the outcome). Quantum
randomness is by its very nature indeterminstic and therefore not dependent on the epistemological state
of the observer (e.g., unknown hidden variables). To twist Einsteins famous words: God does play
quantum dice, i.e., at its most fundamental level nature is indeterministic. This empirical fact poses a
serious problem for mechanistic causal models across the sciences. Specifically, because the demarcation
criterion between “quantum vs. not quantum” (i.e., micro vs. macro) appers to be arbitry (Arndt et al.,
1999; Van der Wal et al., 2000). That is quantum effects are observed in macro scale molecules and
eminent physicists argue that there is theoretically no upper limit to the size of object which obey
quantum laws (Zeilinger, 2012).
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question order (the order of sequential measurements) was inverted Clinton received
59% while Gore received only 67% (Δ 8%).
Figure 10. Noncommutativity in attitudinal decisions.
Classical probability theory cannot account for this kind of order effects because events
are represented as sets and are stipulated to be commutative, that is, P (A ∩ B) = P (B ∩
A). That is, the empirically observed order-effects clearly violate the Kolmogorovian
commutativity axiom. Quantum models of cognition can account for these prima facie
"irrational" judgment and decision-making phenomena and indeed predict them a
priori. In the pertinent literature, the effect of posing attitude questions successively in
different orders has been termed QQ-equality, i.e., quantum question equality (Z. Wang,
Solloway, Shiffrin, & Busemeyer, 2014). This measurement effect has been
investigated in a large scale meta-analytic study (based on 70 national representative
surveys each containing between 600-3000 participants). The results provided strong
support for the predicted QQ equality. Similar results in support of the broad
applicability of QQ-equality to cognitive processes have been obtained in various
unrelated domains, for instance, in dynamic semantics (beim Graben, 2013), thereby
Clinton
53%
Gore
67%
Gore
76%
Clinton
59%
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supporting the generalisability of QQ-equality across multiple domains of inquiry.
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Taken together, these findings suggest that QP, originally developed to explain
noncommutativity of measurements in quantum physics, provides a desirably
parsimonious explanation for measurement order effects in the social, behavioural, and
cognitive sciences (Z. Wang & Busemeyer, 2013). Classical Bayesian
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and Markov
models are unable to account for QQ-equality and are thus incapable of explaining the
empirical data. In the quantum probability framework events are subspaces in an n
dimensional Hilbert space and they may either be compatible or incompatible
(incompatible events are aligned orthogonal in respect to each other). In other words,
noncommutative order effects can be modelled in terms of incompatible projectors on a
Hilbert space (Z. Wang et al., 2014). If they are compatible, they can simultaneously
coexist without influencing each other. On the other hand, incompatible event, as
illustrated in the example above, interfere with each other, thereby causing order
interference effects. In quantum physics, these interference effects have been studied
extensively and the constructive role of measurements/observations is firmly established
even though the exact nature of what exactly defines a measurement/observation is a
wide-open question and is related to the measurement problem (Echenique-Robba,
2013). Several theorist argue that consciousness is crucial for the collapse of the wave-
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QQ-equality was initially developed to account for noncommutativity of measurements in quantum