The Logos Categorical Approach to Quantum Mechanics: I. Kochen-Specker Contextuality and Global Intensive Valuations
TL;DR: In this article, the authors present a new categorical approach which attempts to provide an original understanding of QM by considering the main features of the quantum formalism as the standpoint to develop a conceptual representation that explains what the theory is really talking about.
Abstract: In this paper we present a new categorical approach which attempts to provide an original understanding of QM. Our logos categorical approach attempts to consider the main features of the quantum formalism as the standpoint to develop a conceptual representation that explains what the theory is really talking about --rather than as problems that need to be bypassed in order to allow a restoration of a classical "common sense" understanding of what there is. In particular, we discuss a solution to Kochen-Specker contextuality through the generalization of the meaning of global valuation. This idea has been already addressed by the so called topos approach to QM --originally proposed by Isham, Butterfield and Doring-- in terms of sieve-valued valuations. The logos approach to QM presents a different solution in terms of the notion of intensive valuation. This new solution stresses an ontological (rather than epistemic) reading of the quantum formalism and the need to restore an objective (rather than classical) conceptual representation and understanding of quantum physical reality.
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TL;DR: In this paper, the authors argue that there exists a general confusion within the foundational literature arising from the improper scrambling of two different meanings of quantum contextuality: epistemic interpretation of contextuality and purely formal interpretation.
Abstract: In this paper we attempt to analyze the physical and philosophical meaning of quantum contextuality. We will argue that there exists a general confusion within the foundational literature arising from the improper "scrambling" of two different meanings of quantum contextuality. While the first one, introduced by Bohr, is related to an epistemic interpretation of contextuality which stresses the incompatibility (or complementarity) of certain measurement situations described in classical terms; the second meaning of contextuality is related to a purely formal understanding of contextuality as exposed by the Kochen-Specker (KS) theorem which focuses instead on the constraints of the orthodox quantum formalism in order to interpret projection operators as preexistent or actual (definite valued) properties. We will show how these two notions have been scrambled together creating an "omelette of contextuality" which has been fully widespread through a popularized "epistemic explanation" of the KS theorem according to which: The measurement outcome of the observable A when measured together with B or together with C will necessarily differ in case [A, B] = [A, C] = 0, and [B, C] /= 0. We will show why this statement is not only improperly scrambling epistemic and formal perspectives, but is also physically and philosophically meaningless. Finally, we analyze the consequences of such widespread epistemic reading of KS theorem as related to statistical statements of measurement outcomes.
20 citations
01 Jan 2016
TL;DR: In this paper, a translation of Schr6dinger's three-part 1935 paper in Die Naturzkissenschaften is presented, with a focus on the logical and physical content of the original, while at the same time trying to convey something of its semi-conversational, at times slightly sardonic flavor.
Abstract: This is a translation of Schr6dinger's three-part 1935 paper 1 in Die Naturzkissenschaften. Earlier that same year there had appeared the Einstein, Podolsky, Rosen paper 2 (also famous in "paradoxology") which, Schrodinger says, in a footnote, motivated his offering. Along with this article in German, Schrodinger had two closely related English-language publications.3 But the German, aside from its oneparagraph presentation of the famous cat, covers additional territory and gives many fascinating insights into Schrodinger's thought. The translator's goal has been to adhere to the logical and physical content of the original, while at the same time trying to convey something of its semi-conversational, at times slightly sardonic flavor.
19 citations
TL;DR: This paper argues that the logos categorical approach allows not only to better visualize the structural features of quantum superpositions providing an anschaulich content to all terms, but also to restore an objective representation of what QM is really talking about.
Abstract: Fil: de Ronde, Christian. Universidad de Buenos Aires. Facultad de Filosofia y Letras. Departamento de Filosofia; Argentina. Vrije Unviversiteit Brussel; Belgica. Universidad Nacional Arturo Jauretche; Argentina
17 citations
13 citations
Posted Content•
TL;DR: In this article, the authors analyze the deep link between the 20th Century positivist re-foundation of physics and the famous measurement problem of quantum mechanics, and show how through these same set of presuppositions it is easy to derive a completely analogous paradox for the case of classical mechanics.
Abstract: In this work we analyze the deep link between the 20th Century positivist re-foundation of physics and the famous measurement problem of quantum mechanics. We attempt to show why this is not an “obvious” nor “self evident” problem for the theory of quanta, but rather a direct consequence of the empirical-positivist understanding of physical theories when applied to the orthodox quantum formalism. In contraposition, we discuss a representational realist account of both physical ‘theories’ and ‘measurement’ which goes back to the works of Einstein, Heisenberg and Pauli. After presenting a critical analysis of Bohr’s definitions of ‘measurement’ we continue to discuss the way in which several contemporary approaches to QM —such as decoherence, modal interpretations and QBism— remain committed to Bohr’s general methodology. Finally, in order to expose the many inconsistencies present within the (empirical-positivist) presuppositions responsible for creating the quantum measurement problem, we show how through these same set of presuppositions it is easy to derive a completely analogous paradox for the case of classical mechanics.
11 citations
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TL;DR: Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that one is led to conclude that the description of reality as given by a wave function is not complete.
Abstract: In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false. One is thus led to conclude that the description of reality as given by a wave function is not complete.
13,778 citations
Book•
01 Jan 1971
TL;DR: In this article, the authors present a table of abstractions for categories, including Axioms for Categories, Functors, Natural Transformations, and Adjoints for Preorders.
Abstract: I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large Categories.- 8. Hom-sets.- II. Constructions on Categories.- 1. Duality.- 2. Contravariance and Opposites.- 3. Products of Categories.- 4. Functor Categories.- 5. The Category of All Categories.- 6. Comma Categories.- 7. Graphs and Free Categories.- 8. Quotient Categories.- III. Universals and Limits.- 1. Universal Arrows.- 2. The Yoneda Lemma.- 3. Coproducts and Colimits.- 4. Products and Limits.- 5. Categories with Finite Products.- 6. Groups in Categories.- IV. Adjoints.- 1. Adjunctions.- 2. Examples of Adjoints.- 3. Reflective Subcategories.- 4. Equivalence of Categories.- 5. Adjoints for Preorders.- 6. Cartesian Closed Categories.- 7. Transformations of Adjoints.- 8. Composition of Adjoints.- V. Limits.- 1. Creation of Limits.- 2. Limits by Products and Equalizers.- 3. Limits with Parameters.- 4. Preservation of Limits.- 5. Adjoints on Limits.- 6. Freyd's Adjoint Functor Theorem.- 7. Subobjects and Generators.- 8. The Special Adjoint Functor Theorem.- 9. Adjoints in Topology.- VI. Monads and Algebras.- 1. Monads in a Category.- 2. Algebras for a Monad.- 3. The Comparison with Algebras.- 4. Words and Free Semigroups.- 5. Free Algebras for a Monad.- 6. Split Coequalizers.- 7. Beck's Theorem.- 8. Algebras are T-algebras.- 9. Compact Hausdorff Spaces.- VII. Monoids.- 1. Monoidal Categories.- 2. Coherence.- 3. Monoids.- 4. Actions.- 5. The Simplicial Category.- 6. Monads and Homology.- 7. Closed Categories.- 8. Compactly Generated Spaces.- 9. Loops and Suspensions.- VIII. Abelian Categories.- 1. Kernels and Cokernels.- 2. Additive Categories.- 3. Abelian Categories.- 4. Diagram Lemmas.- IX. Special Limits.- 1. Filtered Limits.- 2. Interchange of Limits.- 3. Final Functors.- 4. Diagonal Naturality.- 5. Ends.- 6. Coends.- 7. Ends with Parameters.- 8. Iterated Ends and Limits.- X. Kan Extensions.- 1. Adjoints and Limits.- 2. Weak Universality.- 3. The Kan Extension.- 4. Kan Extensions as Coends.- 5. Pointwise Kan Extensions.- 6. Density.- 7. All Concepts are Kan Extensions.- Table of Terminology.
9,254 citations
Book•
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01 May 1997
TL;DR: Gaph Teory Fourth Edition is standard textbook of modern graph theory which covers the core material of the subject with concise yet reliably complete proofs, while offering glimpses of more advanced methods in each chapter by one or two deeper results.
Abstract: Gaph Teory Fourth Edition Th is standard textbook of modern graph theory, now in its fourth edition, combines the authority of a classic with the engaging freshness of style that is the hallmark of active mathematics. It covers the core material of the subject with concise yet reliably complete proofs, while offering glimpses of more advanced methods in each fi eld by one or two deeper results, again with proofs given in full detail.
6,255 citations
TL;DR: Koestler's book The Sleepwalkers as discussed by the authors is an account of the Copernican revolution, with Copernicus, Kepler, and Galilei as heroes, and he concluded that they were not really aware of what they were doing.
Abstract: ‘… the history of cosmic theories may without exaggeration be called a history of collective obsessions and controlled schizophrenias; and the manner in which some of the most important individual discoveries were arrived at reminds one of a sleepwalker's performance …’ This is a quotation from A. Koestler's book The Sleepwalkers . It is an account of the Copernican revolution, with Copernicus, Kepler, and Galilei as heroes. Koestler was of course impressed by the magnitude of the step made by these men. He was also fascinated by the manner in which they made it. He saw them as motivated by irrational prejudice, obstinately adhered to, making mistakes which they did not discover, which somehow cancelled at the important points, and unable to recognize what was important in their results, among the mass of details. He concluded that they were not really aware of what they were doing … sleepwalkers. I thought it would be interesting to keep Koestler's thesis in mind as we hear at this meeting about contemporary theories from contemporary theorists. For many decades now our fundamental theories have rested on the two great pillars to which this meeting is dedicated: quantum theory and relativity. We will see that the lines of research opened up by these theories remain splendidly vital. We will see that order is brought into a vast and expanding array of experimental data. We will see even a continuing ability to get ahead of the experimental data … as with the existence and masses of the W and Z mesons.
3,750 citations
TL;DR: The demonstrations of von Neumann and others, that quantum mechanics does not permit a hidden variable interpretation, are reconsidered in this article, and it is shown that their essential axioms are unreasonable.
Abstract: The demonstrations of von Neumann and others, that quantum mechanics does not permit a hidden variable interpretation, are reconsidered. It is shown that their essential axioms are unreasonable. It is urged that in further examination of this problem an interesting axiom would be that mutually distant systems are independent of one another.
3,230 citations