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Consequences of Theoretically Modeling the Mind As a Computer PONTIFICIA UNIVERSIDAD CATOLICA DE CHILE ESCUELA DE PSICOLOGIA´ CONSEQUENCES OF THEORETICALLY MODELING THE MIND AS A COMPUTER ESTEBAN HURTADO LEON´ Thesis submitted to the Office of Research and Graduate Studies in partial fulfillment of the requirements for the degree of Doctor in Psychology Advisor: CARLOS CORNEJO ALARCON´ Santiago de Chile, August 2017 c MMXVII, ESTEBAN HURTADO PONTIFICIA UNIVERSIDAD CATOLICA DE CHILE ESCUELA DE PSICOLOGIA´ CONSEQUENCES OF THEORETICALLY MODELING THE MIND AS A COMPUTER ESTEBAN HURTADO LEON´ Members of the Committee: CARLOS CORNEJO ALARCON´ DIEGO COSMELLI SANCHEZ LUIS DISSETT VELEZ JAAN VALSINER ......... Thesis submitted to the Office of Research and Graduate Studies in partial fulfillment of the requirements for the degree of Doctor in Psychology Santiago de Chile, August 2017 c MMXVII, ESTEBAN HURTADO To Carmen and Fulvio ACKNOWLEDGEMENTS I would like to thank the School of Psychology at Pontificia Universidad Catolica´ de Chile for taking me in and walking me through the diversity of the study of the mind. I am in debt to all the teachers who kindly and passionately shared their knowledge with me, and very specially to the kind and helpful work of the administrative staff. I took my first steps in theoretical computer science at the School of Engineering of the same university, with Dr. Alvaro´ Campos, who is no longer with us. His passion for knowledge, dedication and warmth continue to inspire those of us who where lucky enough to cross paths with him. The generous and theoretically profound support of the committee members has been fundamental to the production of this text. I am deeply thankful to all of them. I would like to specially thank Dr. Luis Dissett, from whom I learnt many of the issues discussed in this work. From him I also learnt to see logic as a beautiful form of art. His love for maths reverbs in the hearts of many of his students, myself included. I thank my dear friend Dr. Agust´ın Ibanez,˜ for introducing me to cognitive science, stimulating my development of an individual critical position in the area, and encouraging me to pursue an academic career. This work would have not been possible without his inspiration and support. I also thank Roberto Musa and David Carre,´ for the stimulating conversations we have had have been a real contribution to this thesis. I want to thank my thesis advisor, Dr. Carlos Cornejo, in a very special way. This work, its approach, and my whole interest in the questions embedded in it, would have not been possible without his humble and theoretically profound guidance. With humility and true admiration, I hope this is only the beginning of a long collaboration. I would like to thank CONICYT for funding my doctoral studies. I also thank the generous spiritual and administrative support I received from Jan Krebs, Hector´ Bernales, David Huepe, and Gorka Naverrete, all people I have had the privilege to work for. iv I dedicate this work to my parents, Carmen and Fulvio. I thank them deeply for their encouragement, dedication and support. v TABLE OF CONTENTS ACKNOWLEDGEMENTS ::::::::::::::::::::::::::::: iv LIST OF FIGURES ::::::::::::::::::::::::::::::::: ix LIST OF TABLES :::::::::::::::::::::::::::::::::: xi ABSTRACT ::::::::::::::::::::::::::::::::::::: xii RESUMEN ::::::::::::::::::::::::::::::::::::: xiii 1. Introduction ::::::::::::::::::::::::::::::::::: 1 2. What is a computer system? ::::::::::::::::::::::::::: 5 2.1. Multiple definition, multiple implementation, and computational power :: 7 2.2. Some well known computer models :::::::::::::::::::: 9 2.2.1. Finite-state machines :::::::::::::::::::::::::: 9 2.2.2. Pushdown automata :::::::::::::::::::::::::: 12 2.2.3. The Turing machine :::::::::::::::::::::::::: 14 2.2.4. Universal Turing machines ::::::::::::::::::::::: 17 2.3. Actual implementation of computing models :::::::::::::::: 17 3. Countable and uncountable sets ::::::::::::::::::::::::: 21 3.1. How to define a mathematical set :::::::::::::::::::::: 21 3.2. The cardinality of infinite sets ::::::::::::::::::::::: 23 3.2.1. Natural numbers :::::::::::::::::::::::::::: 23 3.2.2. Some subsets of natural numbers :::::::::::::::::::: 24 3.2.3. Integer numbers :::::::::::::::::::::::::::: 25 3.2.4. Rational numbers :::::::::::::::::::::::::::: 26 3.2.5. Real numbers :::::::::::::::::::::::::::::: 27 3.3. Countability of computational procedures ::::::::::::::::: 29 vi 4. Turing halting :::::::::::::::::::::::::::::::::: 32 4.1. The Turing halting problem ::::::::::::::::::::::::: 33 4.1.1. Undecidability proof :::::::::::::::::::::::::: 33 4.2. Computable numbers :::::::::::::::::::::::::::: 35 4.3. Consequences :::::::::::::::::::::::::::::::: 36 4.3.1. A few undecidable problems :::::::::::::::::::::: 36 4.3.2. Undecidability of Turing machine equivalence ::::::::::::: 37 4.3.3. Not all definable functions are computable ::::::::::::::: 38 5. Godel’s¨ incompleteness ::::::::::::::::::::::::::::: 40 5.1. Axiomatic systems and formal languages :::::::::::::::::: 41 5.1.1. Theories and proofs :::::::::::::::::::::::::: 41 5.1.2. The natural number system ::::::::::::::::::::::: 43 5.2. Hilbert’s program :::::::::::::::::::::::::::::: 46 5.3. Godel’s¨ incompleteness ::::::::::::::::::::::::::: 47 6. Consequences of Godel’s¨ incompleteness :::::::::::::::::::: 49 6.1. Consequences for axiomatic systems :::::::::::::::::::: 49 6.1.1. There is no proof of arithmetic consistency in arithmetic :::::::: 49 6.1.2. There is no definition of arithmetic truth in arithmetic ::::::::: 49 6.2. Relationship with Turing halting :::::::::::::::::::::: 51 6.3. Implications for logic language ::::::::::::::::::::::: 53 7. Psychological interpretations of incompleteness :::::::::::::::: 56 7.1. Piaget’s reaction to Godel’s¨ theorems :::::::::::::::::::: 56 7.2. Did Wittgenstein misunderstand Godel?¨ :::::::::::::::::: 57 7.3. Lucas-Penrose interpretation of Godel¨ ::::::::::::::::::: 59 7.4. The “truth” of Godel’s¨ sentence :::::::::::::::::::::: 60 8. Knowledge in an incomplete system :::::::::::::::::::::: 63 8.1. Truth values and world facts :::::::::::::::::::::::: 63 8.2. Dichotomous truth ::::::::::::::::::::::::::::: 66 vii 8.3. Trichotomous truth ::::::::::::::::::::::::::::: 66 9. Computers and non-determinism :::::::::::::::::::::::: 70 9.1. Non-deterministic Turing machines and quantum computing :::::::: 70 9.2. Bayesian modeling ::::::::::::::::::::::::::::: 73 9.3. Markovian modeling :::::::::::::::::::::::::::: 75 9.3.1. Bayes’ theorem in time: an example :::::::::::::::::: 76 9.3.2. Acting under uncertainty :::::::::::::::::::::::: 80 9.4. Non-determinism and computational modeling of the mind :::::::: 82 10. Some controversies around a computational mind ::::::::::::::: 84 10.1. Being versus being like a computer :::::::::::::::::::: 84 10.2. Brain and mind :::::::::::::::::::::::::::::: 85 10.3. Is there anything beyond the physical? :::::::::::::::::: 86 10.4. Representation of knowledge ::::::::::::::::::::::: 87 10.5. Architecture of the mind :::::::::::::::::::::::::: 89 10.6. Body and environment :::::::::::::::::::::::::: 91 11. Incompleteness and epiphenomenal qualia ::::::::::::::::::: 93 11.1. The incompleteness of Mary’s knowledge ::::::::::::::::: 94 12. A few limitations of the computational theory of mind :::::::::::: 98 12.1. Some basic limitations ::::::::::::::::::::::::::: 99 12.2. Underdetermination of theoretical models by evidence :::::::::: 100 12.3. Metaphysical orthogonality of the computational theory of the mind ::: 101 13. Conclusions :::::::::::::::::::::::::::::::::: 104 References :::::::::::::::::::::::::::::::::::::: 110 viii LIST OF FIGURES 2.1 Example of equivalent definitions. The defining property of p-quadrilaterals is having parallel opposite sides, while that of a-quadrilaterals is having same value opposite angles. Despite having differently worded definitions, both define the exact same class of polygons, because one property implies the other. ::::: 8 2.3 Simplified finite state machine for two floor elevator. It models how position changes as a result of pressing up or down buttons. :::::::::::::: 10 2.4 Finite state machine for a three question quiz show. States circled in double lines are final. Prize can be up to $3,000 if all three questions are answered correctly. Retiring without an answer yields an amount that depends on the number of previous right answers. A wrong answer finishes the game with no prize. ::: 11 2.5 Finite state machine for telling whether a sequence of characters contains an even amount of ‘a’ symbols. It has essentially the same form as the elevator model from Figure 2.3. For instance, processing characters “aabaaz” goes through several state transitions (arrows) each one resulting in a new output. The last output indicates that the input has an even number of ‘a’. ::::::::::: 11 2.6 Both pushdown automata and Turing machines are the combination of a finite- state machine and some additions. In the case of a PDA the addition is a stack for piling up symbols. Instead of a stack, a Turing machine has a tape and a read/write head that transfers symbols between the tape and the FSM. The tape is infinite, in the sense that there is a beginning to it but no end. In constrast to a PDA, a TM does not need separate input and output, because the input can be written on the tape at the begining, and the output can be read from the same tape after processing has finished. ::::::::::::::::::::::::: 12 3.1 Matching of integer numbers and natural numbers on the number line. :::: 25 ix 3.2 Countability
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