Abstract
One of the central goals of science is to find consistent and rational representations of observational data: a map of the world, if you will. How we do this depends on the specific tools, which are often mathematical. When dealing with real-world situations, where the data (or “territory”) has a random component, the mathematical tools most commonly used are those grounded in probability theory, defined in a precise way by the Russian mathematician Andrei Kolmogorov. In this paper we explore how experimental data (the “territory”) can be represented (or “mapped”) consistently in terms of probability theory, and present examples of situations, both in the physical and social sciences, where such representations are impossible. This suggests that some “territories” cannot be “mapped” in a way that is consistent with classical logic and probability theory.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
- 2.
Other values for certain and impossible events may be given, e.g. \(p(\varOmega )=100\) or \(p(\emptyset )=-1\), but the choice of 1 and 0 for them makes the overall expressions for the calculus of probabilities simpler (Jaynes 2003).
- 3.
To be precise, he used an algebra over \(\varOmega \) which did not necessarily include all the subsets. As importantly, later on it was discovered that for certain sets we need to require the algebra to be countable, i.e. a \(\sigma \)-algebra, but such nuances are not relevant for our discussions.
- 4.
i.e. if I have the belief that tomorrow it will rain with probability 0.5 and that a movie I want to watch has probability 0.3 of being interesting, and so on, I can rationally decide whether I will go to the beach or to the movies. Of course, probabilities only offer a measure of belief, and not whether I would extract more pleasure from one activity or another. To model this, economists use what is known as utility, a part of rational choice theory (Anand et al. 2009).
- 5.
Some authors, most notably Dzhafarov and Kujala (2017), label this type of contextuality as “direct influences,” to highlight the idea that the experimental context directly influences the variables in question. They reserve the label “contextuality” only for what we call here “hidden contextuality.”.
- 6.
The term “explicit” and “hidden” contextuality was, as far as the authors know, proposed by Kurzyński (2017).
- 7.
Another way is to use upper probabilities (Suppes and Zanotti 1991; de Barros and Suppes 2010). For example, Holik, Saenz, and Plastino showed that if we relaxed the requirement for a Boolean algebra and instead allowed for orthomodular lattices, one would get upper probabilities instead of standard probabiilty theory (Holik et al. 2014).
- 8.
Of course, here we mean negative numbers as counting of actual objects. As is well known, other useful interpretations of negative numbers such as placement on the number line were introduced that make sense.
References
S. Abramsky, A. Brandenburger, An operational interpretation of negative probabilities and no-signalling models, in Horizons of the Mind. A Tribute to Prakash Panangaden, ed. by F. van Breugel, E. Kashefi, C. Palamidessi, J. Rutten, number 8464 in Lecture Notes in Computer Science (Springer International Publishing, 2014), pp. 59–75
J.A. de Barros, Beyond the quantum formalism: consequences of a neural-oscillator model to quantum cognition, in Advances in Cognitive Neurodynamics (IV), ed. by H. Liljenström (Advances in Cognitive Neurodynamics (Springer, Netherlands, 2015), pp. 401–404
J.A. de Barros, Joint probabilities and quantum cognition, in AIP Conference Proceedings, vol. 1508, ed. by A. Khrennikov, A.L. Migdall, S. Polyakov, H. Atmanspacher (American Institute of Physics, Vaxjo, Sweden, 2012a), pp. 98–107
J.A. de Barros, Quantum-like model of behavioral response computation using neural oscillators. Biosystems 110(3), 171–182 (2012b)
J.A. de Barros, F. Holik, D. Krause, Contextuality and indistinguishability. Entropy 19(9), 435 (2017)
J.A. de Barros, J.V. Kujala, G. Oas, Negative probabilities and contextuality. J. Math. Psychol. 74, 34–45 (2016)
J.A. de Barros, P. Suppes, Quantum mechanics, interference, and the brain. J. Math. Psychol. 53(5), 306–313 (2009)
J.A. de Barros, P. Suppes, Probabilistic inequalities and upper probabilities in quantum mechanical entanglement. Manuscrito 33(1), 55–71 (2010)
J. A. de Barros, Decision Making for Inconsistent Expert Judgments Using Negative Probabilities. Lecture Notes in Computer Science (Springer, Berlin, Heidelberg, 2014), pp. 257–269
J. A. de Barros, E.N. Dzhafarov, J.V. Kujala, G. Oas, Measuring observable quantum contextuality, in Quantum Interaction ed. by H. Atmanspacher, T. Filk, E. Pothos, number 9535 in Lecture Notes in Computer Science, July 2015 (Springer International Publishing), pp. 36–47. https://doi.org/10.1007/978-3-319-28675-4_4
J. A. de Barros, G. Oas, Some examples of contextuality in Physics: implications to quantum cognition, in Contextuality, from Quantum Physics to Psychology, ed. by E.N. Dzhafarov, R. Zhang, S.M. Jordan (World Scientific, 2015), pp. 153–184
V.M. Alekseev, Quasirandom dynamical systems. I. Quasirandom diffeomorphisms. Sb.: Math. 5(1), 73–128 (1968)
V.M. Alekseev, Quasirandom dynamical systems. II. One-dimensional nonlinear oscillations in a field with periodic perturbation. Sb.: Math. 6(4), 505–560 (1968)
V.M. Alekseev, Quasirandom dynamical systems. III Quasirandom oscillations of one-dimensional oscillators. Sb.: Math. 7(1), 1–43 (1969)
P. Anand, P. Pattanaik, C. Puppe (eds.), The Handbook of Rational and Social Choice: An Overview of New Foundations and Applications (Oxford University Press, Oxford, England, 2009)
G. Boole, An Investigation of the Laws of Thought: On which are Founded the Mathematical Theories of Logic and Probabilities (Dover Publications, Mineola, New York, 1854)
M. Burgin, An introduction to symmetric inflated probabilities, in Quantum Interaction. Lecture Notes in Computer Science, July 2016 (Springer, Cham), pp. 206–223
M. Burgin, G. Meissner, Extended correlations in finance. J. Math. Finance 06(01), 178–188 (2016)
J.R. Busemeyer, P. Fakhari, P. Kvam, Neural implementation of operations used in quantum cognition, in Progress in Biophysics and Molecular Biology, May 2017
J.R. Busemeyer, P.D. Bruza, Quantum Models of Cognition and Decision (Cambridge University Press, Cambridge, 2012)
J.R. Busemeyer, Z. Wang, A. Lambert-Mogiliansky, Empirical comparison of Markov and quantum models of decision making. J. Math. Psychol. 53(5), 423–433 (2009)
V.H. Cervantes, E.N. Dzhafarov, Advanced analysis of quantum contextuality in a psychophysical double-detection experiment. J. Math. Psychol. 79, 77–84 (2017)
R.T. Cox, The Algebra of Probable Inference (The John Hopkins Press, Baltimore, 1961)
A. De Morgan, Formal Logic: Or (The Calculus of Inference, Necessary and Probable (Taylor and Walton, 1847)
A. De Morgan, On the Study and Difficulties of Mathematics (Open Court Publishing Company, 1910)
E.N. Dzhafarov, J.V. Kujala, Contextuality-by-default 2.0: systems with binary random variables, in Quantum Interaction: 10th International Conference, QI 2016 ed. by J. A. de Barros, B. Coecke, E. Pothos, volume 10106 of Lecture Notes in Computer Science (Springer International Publishing, 2017). arXiv:1604.04799
J. Ford, How random is a coin toss? Phys. Today 36, 40 (1983)
M.C. Galavotti, Philosophical Introduction to Probability, vol. 167 (CSLI Lecture Notes (CSLI Publications, Stanford, CA, 2005)
D.M. Greenberger, M.A. Horne, A. Zeilinger, Going beyond Bell’s theorem, in Bell’s Theorem, Quantum Theory, and Conceptions of the Universe, ed. by M. Kafatos, volume 37 of Fundamental Theories of Physics, (Kluwer, Dordrecht, Holland, 1989), pp. 69–72
E. Haven, A. Khrennikov, Quantum Social Science (Cambridge University Press, Cambridge, 2013)
F. Holik, M. Saenz, A. Plastino, A discussion on the origin of quantum probabilities. Ann. Phys. 340(1), 293–310 (2014)
E.T. Jaynes, Probability Theory: The Logic of Science (Cambridge University Press, Cambridge, Great Britain, 2003)
J.B. Keller, The probability of heads. Am. Math. Mon. 93(3), 191–197 (1986)
A. Khrennikov, Interpretations of Probability (Walter de Gruyter, 2009a)
A.Y. Khrennikov, Contextual Approach to Quantum Formalism (Springer Science & Business Media, 2009b)
A. Khrennikov, Contextual approach to quantum theory, in Information Dynamics in Cognitive, Psychological, Social and Anomalous Phenomena. Fundamental Theories of Physics (Springer, Dordrecht, 2004), pp. 153–185. https://doi.org/10.1007/978-94-017-0479-3_9
A. Khrennikov, Why so negative about negative probabilities?, in Derivatives: Models on Models (Wiley, West Sussex, England, 2007), pp. 323–334
A. Khrennikov, p-Adic probability theory and its applications. The principle of statistical stabilization of frequencies. Theor. Math. Phys. 97(3), 1340–1348 (1993)
A. Khrennikov, Quantum-like brain: "Interference of minds". Biosystems 84(3), 225–241 (2006)
A. Khrennikov, Ubiquitous Quantum Structure (Springer, Heidelberg, 2010)
AYu. Khrennikov, E. Haven, Quantum mechanics and violations of the sure-thing principle: the use of probability interference and other concepts. J. Math. Psychol. 53(5), 378–388 (2009)
A.N. Kolmogorov, Foundations of the Theory of Probability, 2nd edn. (Chelsea Publishing Company, Oxford, England, 1956)
P. Kurzyński, Contextuality of identical particles. Phys. Rev. A 95(1), 012133 (2017)
C. Moreira, A. Wichert, Exploring the relations between quantum-like Bayesian networks and decision-making tasks with regard to face stimuli. J. Math. Psychol. 78(Supplement C), 86–95 (2017)
C. Moreira, A. Wichert, Quantum-like Bayesian networks for modeling decision making. Front. Psychol. 7 (2016a)
C. Moreira, A. Wichert, Quantum probabilistic models revisited: the case of disjunction effects in cognition. Front. Phys. 4 (2016b)
G. Oas, J. A. de Barros, A survey of physical principles attempting to define quantum mechanics, in Contextuality From Quantum Physics to Psychology, ed. by E. Dzhafarov, R. Zhang, S.M. Jordan (World Scientific, 2015)
E.M. Pothos, J.R. Busemeyer, A quantum probability explanation for violations of ‘rational’ decision theory. Proc. Roy. Soc. B: Biol. Sci. 276(1665), 2171–2178 (2009)
I.Z. Ruzsa, G.J. Székely, Convolution quotients of nonnegative functions. Monatshefte für Mathematik 95(3), 235–239 (1983)
M.O. Scully, K. Druhl, Quantum eraser: a proposed photon correlation experiment concerning observation and "delayed choice" in quantum mechanics. Phys. Rev. A 25(4), 2208–2213 (1982)
P. Suppes, A linear learning model for a continuum of responses, in Studies in Mathematical Leaning Theory, ed. by R.R. Bush, W.K. Estes (Stanford University Press, Stanford, CA, 1959), pp. 400–414
P. Suppes, M. Zanotti, When are probabilistic explanations possible? Synthese 48(2), 191–199 (1981)
P. Suppes, M. Zanotti, Existence of hidden variables having only upper probabilities. Found. Phys. 21(12), 1479–1499 (1991)
G.J. Székely, Half of a coin: negative probabilities. Wilmott Mag. 50, 66–68 (2005)
V. Veitch, C. Ferrie, D. Gross, J. Emerson, Negative quasi-probability as a resource for quantum computation. New J. Phys. 14(11), 113011 (2012)
V.Z. Vulović, R.E. Prange, Randomness of a true coin toss. Phys. Rev. A 33(1), 576–582 (1986)
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Acacio de Barros, J., Oas, G. (2018). Mapping Quantum Reality: What to Do When the Territory Does Not Make Sense?. In: Wuppuluri, S., Doria, F. (eds) The Map and the Territory. The Frontiers Collection. Springer, Cham. https://doi.org/10.1007/978-3-319-72478-2_17
Download citation
DOI: https://doi.org/10.1007/978-3-319-72478-2_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-72477-5
Online ISBN: 978-3-319-72478-2
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)