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A complex Hilbert space is a vector space over the field of complex numbers.
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A measurement is described by a self-adjoint operator, which is a linear function on the Hilubert space.
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For mathematical completeness, we mention that this is only the case when the operator has a point spectrum. Measurements described by operators without a point spectrum must be treated in a more sophisticated way. However, this is of no relevance to the points made here.
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Each state that is not an eigenstate is a superposition state.
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There are, however, quantum theories that do not involve collapse.
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Kolmogorov's axioms, and hence Kolmogorovian probability theories, are named after Andrei Nikolaevich Kolmogorov, who first formulated the axiomatic system for classical probability theory (see A. N. Kolmogorov: Grundbegriffe der Wahrscheinlichkeitrechnung; 1933, Berlin, Springer). The theory had in fact existed since the time of Laplace (Théorie Analytique des Probabilités, 1812) and had been worked out by various mathematicians, but in a rather informal way. It is well known that the probability calculus of quantum mechanics is non-classical, in the sense that it does not satisfy Kolmogorov's axioms (A. Wilce: 'Quantum logic and probability theory', in Stanford Encyclopedia of Philosophy, (ed. E. N. Zalta), Spring 2003 edn, plato.stanford.edu/archives/spr2003/entries/qt-quandog).
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Grundbegriffe der Wahrscheinlichkeitrechnung
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Kolmogorov's axioms, and hence Kolmogorovian probability theories, are named after Andrei Nikolaevich Kolmogorov, who first formulated the axiomatic system for classical probability theory (see A. N. Kolmogorov: Grundbegriffe der Wahrscheinlichkeitrechnung; 1933, Berlin, Springer). The theory had in fact existed since the time of Laplace (Théorie Analytique des Probabilités, 1812) and had been worked out by various mathematicians, but in a rather informal way. It is well known that the probability calculus of quantum mechanics is non-classical, in the sense that it does not satisfy Kolmogorov's axioms (A. Wilce: 'Quantum logic and probability theory', in Stanford Encyclopedia of Philosophy, (ed. E. N. Zalta), Spring 2003 edn, plato.stanford.edu/archives/spr2003/entries/qt-quandog).
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(ed. E. N. Zalta), Spring edn
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Kolmogorov's axioms, and hence Kolmogorovian probability theories, are named after Andrei Nikolaevich Kolmogorov, who first formulated the axiomatic system for classical probability theory (see A. N. Kolmogorov: Grundbegriffe der Wahrscheinlichkeitrechnung; 1933, Berlin, Springer). The theory had in fact existed since the time of Laplace (Théorie Analytique des Probabilités, 1812) and had been worked out by various mathematicians, but in a rather informal way. It is well known that the probability calculus of quantum mechanics is non-classical, in the sense that it does not satisfy Kolmogorov's axioms (A. Wilce: 'Quantum logic and probability theory', in Stanford Encyclopedia of Philosophy, (ed. E. N. Zalta), Spring 2003 edn, plato.stanford.edu/archives/spr2003/entries/qt-quandog).
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Yet another possibility is that the non-determinism is ontological, i.e. the universe is intrinsically non-deterministic at bottom. In this case, it can be shown that the mathematical structure necessary to model the situation would be equivalent to the mathematical structure needed to model the situation where the non-determinism arises through lack of knowledge of the context. (For details see D. Aerts: 'Quantum structures, separated physical entities and probability', Foundations of Physics, 1994, 24, 1227-1259.) Thus, ontological non-determinism can also be described in this framework.
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This story has a precedent. The same problem arose in physics in the last century, Classical mechanics could describe situations where the effect of the measurement was negligible, but not situations where the measurement intrinsically influenced the evolution of an entity. This is because it does not provide a way of coping with contextuality (except in the initial conditions or in an ad hoc way, by introducing an additional model of perturbation). Modern classical theories, such as chaos and complexity, though they provide a means of transcending reductionism, still have this limitation.
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In the SCOP formalism, an entity is described using a state space the algebraic structure of which is given by the set of atoms of a complete lattice (this plays the role of the rays of a complex Hilbert space in quantum mechanics). Measurements ate described by Boolean morphisms on the lattice (which play the role of the self-adjoint operators in quantum mechanics).
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D. Aerts: 'Quantum structures due to fluctuations of the measurement situations', International Journal of Theoretical Physics, 1993, 32, 2207-2220; D. Aerts and T. Durt: 'Quantum, classical, and intermediate, an illustrative example', Foundations of Physics, 1994, 24, 1353-1368; D. Foulis, C. Piron and C. Randall: 'Realism, operationalism and quantum mechanics', Foundations of Physics, 1983, 13, 813-841; D. Foulis and C. Randall: 'What are quantum logics and what ought they to be?', in Current Issues in Quantum Logic, (ed. E. Beltrametti and B. van Fraassen); 1981, New York, NY, Plenum;]. Jauch: Foundations of Quantum Mechanics-, 1968, Reading, MA, Addison-Wesley; G. Mackey: Mathematical Foundations of Quantum Mechanics-, 1963, Reading, MA, John Benjamin; C. Piron: 'Recent developments in quantum mechanics', Helvetica Physica Acta, 1989, 62, 439-468; C. Piron: Mecanique Quantique; Bases it Applications; 1990, Lausanne, Press Polytechnique de Lausanne; C. Randall and D. Foulis: 'The operational approach to quantum mechanics', in Physical Theories as Logico-Operational Structures, (ed. C. Hooker); 1978, Dordrecht, Reidel.
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D. Aerts: 'Quantum structures due to fluctuations of the measurement situations', International Journal of Theoretical Physics, 1993, 32, 2207-2220; D. Aerts and T. Durt: 'Quantum, classical, and intermediate, an illustrative example', Foundations of Physics, 1994, 24, 1353-1368; D. Foulis, C. Piron and C. Randall: 'Realism, operationalism and quantum mechanics', Foundations of Physics, 1983, 13, 813-841; D. Foulis and C. Randall: 'What are quantum logics and what ought they to be?', in Current Issues in Quantum Logic, (ed. E. Beltrametti and B. van Fraassen); 1981, New York, NY, Plenum;]. Jauch: Foundations of Quantum Mechanics-, 1968, Reading, MA, Addison-Wesley; G. Mackey: Mathematical Foundations of Quantum Mechanics-, 1963, Reading, MA, John Benjamin; C. Piron: 'Recent developments in quantum mechanics', Helvetica Physica Acta, 1989, 62, 439-468; C. Piron: Mecanique Quantique; Bases it Applications; 1990, Lausanne, Press Polytechnique de Lausanne; C. Randall and D. Foulis: 'The operational approach to quantum mechanics', in Physical Theories as Logico-Operational Structures, (ed. C. Hooker); 1978, Dordrecht, Reidel.
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D. Aerts: 'A mechanistic classical laboratory situation violating the Bell inequalities with 2√2, exactly "in the same way" as its violations by the EPR experiments', Helvetica Physica Acta, 1991, 64, 1-23; D. Aerts: 'Example of a macroscopical situation that violates Bell inequalities', Letters al Nuovo Cimento, 1982, 34, 107-111 (preprint at www.vub.ac.be/CLEA/ aerts/publications/1982ExMacViolBell.pdf); D. Aerts, S. Aerts, J. Broekaert and L. Gabora: 'The violation of Bell inequalities in the macroworld', Foundations of Physics, 2000, 30, 1387-1414.
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The reason this works is because when polymers interact, the number of different polymers increases exponentially, but the number of reactions by which they can interconvert increases faster than their total number. Thus, as their diversity increases, so does the probability that some subset of the total reaches a critical point where there is a catalytic pathway to every member.
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We have two copies of each gene, one from each parent. The two copies are sometimes different forms (or alleles) of the gene, in which case we are said to be 'heterozygous' for that gene. By contrast, when the two copies of a gene are of the same form (the same allele), we are said to be 'homozygous' for that gene.
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This is outlined in more elaborate detail in L. Gabora and D. Aerts: 'Distilling the essence of an evolutionary process, and implications for a formal description of culture', in Proceedings of Center for Human Evolution Workshop No. 5: Cultural Evolution, (ed. W. Kistler); 2005, Seattle, WA, Foundation for the Future.
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L. W. Barsalou: 'Context-independent and context-dependent information in concepts', Memory & Cognition, 1982, 10, 82-93; L. Gabora and D. Aerts: 'Contextualizing concepts using a mathematical generalization of the quantum formalism', Journal of Experimental and Theoretical Artificial Intelligence, 2002, 14, 327-358 (www.vub.ac.be/CLEA/liane/papers/jetai.pdf); D. Aerts and L. Gabora: 'A theory of concepts and their combinations I: The structure of the sets of contexts and properties', Kybernetes, 2005, 34, 151-175, uk.arxiv.org/abs/quant-ph/0402207; D. Aerts and L. Gabora: 'A theory of concepts and their combinations II: A Hilbert space representation', Kybernetes, 2005, 34, 176-205, uk.arxiv.org/abs/quant-ph/ 0402205.
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L. W. Barsalou: 'Context-independent and context-dependent information in concepts', Memory & Cognition, 1982, 10, 82-93; L. Gabora and D. Aerts: 'Contextualizing concepts using a mathematical generalization of the quantum formalism', Journal of Experimental and Theoretical Artificial Intelligence, 2002, 14, 327-358 (www.vub.ac.be/CLEA/liane/papers/jetai.pdf); D. Aerts and L. Gabora: 'A theory of concepts and their combinations I: The structure of the sets of contexts and properties', Kybernetes, 2005, 34, 151-175, uk.arxiv.org/abs/quant-ph/0402207; D. Aerts and L. Gabora: 'A theory of concepts and their combinations II: A Hilbert space representation', Kybernetes, 2005, 34, 176-205, uk.arxiv.org/abs/quant-ph/ 0402205.
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L. W. Barsalou: 'Context-independent and context-dependent information in concepts', Memory & Cognition, 1982, 10, 82-93; L. Gabora and D. Aerts: 'Contextualizing concepts using a mathematical generalization of the quantum formalism', Journal of Experimental and Theoretical Artificial Intelligence, 2002, 14, 327-358 (www.vub.ac.be/CLEA/liane/papers/jetai.pdf); D. Aerts and L. Gabora: 'A theory of concepts and their combinations I: The structure of the sets of contexts and properties', Kybernetes, 2005, 34, 151-175, uk.arxiv.org/abs/quant-ph/0402207; D. Aerts and L. Gabora: 'A theory of concepts and their combinations II: A Hilbert space representation', Kybernetes, 2005, 34, 176-205, uk.arxiv.org/abs/quant-ph/ 0402205.
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An affordance is a possibility offered by the environment. Thus the presence of food affords the possibil-. ity of eating.
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For example Bayes's formula for conditional probability is not satisfied
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For example Bayes's formula for conditional probability is not satisfied.
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