Quantum geometry, logic and probability

Quantum geometry on a discrete set means a directed graph with a weight associated to each arrow defining the quantum metric. However, these `lattice spacing' weights do not have to be independent of the direction of the arrow. We use this greater freedom to give a quantum geometric interpretation of discrete Markov processes with transition probabilities as arrow weights, namely taking the diffusion form $\partial_+ f=(-\Delta_\theta+ q-p)f$ for the graph Laplacian $\Delta_\theta$, potential functions $q,p$ built from the probabilities, and finite difference $\partial_+$ in the time direction. Motivated by this new point of view, we introduce a `discrete Schroedinger process' as $\partial_+\psi=\imath(-\Delta+V)\psi$ for the Laplacian associated to a bimodule connection such that the discrete evolution is unitary. We solve this explicitly for the 2-state graph, finding a 1-parameter family of such connections and an induced `generalised Markov process' for $f=|\psi|^2$ in which there is an additional source current built from $\psi$. We also discuss our recent work on the quantum geometry of logic in `digital' form over the field $\Bbb F_2=\{0,1\}$, including de Morgan duality and its possible generalisations.Comment: 23 pages, 5 figure

Logical Localism in the Context of Combining Logics

[eng] Logical localism is a claim in the philosophy of logic stating that different logics are correct in different domains. There are different ways in which this thesis can be motivated and I will explore the most important ones. However, localism has an obvious and major challenge which is known as ‘the problem of mixed inferences’. The main goal of this dissertation is to solve this challenge and to extend the solution to the related problem of mixed compounds for alethic pluralism. My approach in order to offer a solution is one that has not been considered in the literature as far as I am aware. I will study different methods for combining logics, concentrating on the method of juxtaposition, by Joshua Schechter, and I will try to solve the problem of mixed inferences by making a finer translation of the arguments and using combination mechanisms as the criterion of validity. One of the most intriguing aspects of the dissertation is the synergy that is created between the philosophical debate and the technical methods with the problem of mixed inferences at the center of that synergy. I hope to show that not only the philosophical debate benefits from the methods for combining logics, but also that these methods can be developed in new and interesting ways motivated by the philosophical problem of mixed inferences. The problem suggests that there are relevant interactions between connectives, justified by the philosophical considerations for conceptualising different logic systems, that the methods for combining logics should allow to emerge. The recognition of this fact is what drives the improvements on the method of juxtaposition that I develop. That is, in order to allow for the emergence of desirable interaction principles, I will propose alternative ways of combining logic systems -specifically classical and intuitionistic logics- that go beyond the standard for combinations, which is based on minimality conditions so as to avoid the so-called collapse theorems.[spa] El localismo lógico es una tesis en filosofía de la lógica según la cual diferentes sistemas lógicos son correctos en función del dominio en el que se aplican. Dicha tesis cuenta, prima facie, con cierta plausibilidad y con varios argumentos que la respaldan como mostraré. Sin embargo, el localismo se presta a un evidente y poderoso contraargumento conocido como ‘el problema de las inferencias mixtas’. El objetivo principal de esta disertación es dar respuesta a ese problema y extender la solución al problema afín de los compuestos mixtos que afecta al pluralismo alético. La manera de abordar el problema de las inferencias mixtas consistirá en analizar casos paradigmáticos en la literatura a la luz de los métodos de combinación de lógicas. En concreto, me centraré en el método de la yuxtaposición, desarrollado por Joshua Schechter. Así, ofreceré una solución al problema de las inferencias mixtas que pasará por realizar un análisis más sutil y una formalización más precisa de las mismas, para después aplicar los mecanismos de combinación como criterio de validez. Además, mostraré que el problema de las inferencias mixtas provee de multitud de ejemplos que invitan a desarrollar los métodos de combinación de lógicas de formas novedosas. Una de las aportaciones más relevantes de la disertación consistirá en modificar el método de la yuxtaposición para obtener mecanismos que van más allá del estándar de las extensiones mínimas conservativas. En concreto, propondré diferentes mecanismos para combinar la lógica clásica y la intuicionista, de manera que se permita la aparición de distintos principios puente para los que tenemos buenas razones que los justifican, sin que ello conduzca al colapso de las lógicas que se combinan

Logical Localism in the Context of Combining Logics

Programa de Doctorat en Ciència Cognitiva i Llenguatge[eng] Logical localism is a claim in the philosophy of logic stating that different logics are correct in different domains. There are different ways in which this thesis can be motivated and I will explore the most important ones. However, localism has an obvious and major challenge which is known as ‘the problem of mixed inferences’. The main goal of this dissertation is to solve this challenge and to extend the solution to the related problem of mixed compounds for alethic pluralism. My approach in order to offer a solution is one that has not been considered in the literature as far as I am aware. I will study different methods for combining logics, concentrating on the method of juxtaposition, by Joshua Schechter, and I will try to solve the problem of mixed inferences by making a finer translation of the arguments and using combination mechanisms as the criterion of validity. One of the most intriguing aspects of the dissertation is the synergy that is created between the philosophical debate and the technical methods with the problem of mixed inferences at the center of that synergy. I hope to show that not only the philosophical debate benefits from the methods for combining logics, but also that these methods can be developed in new and interesting ways motivated by the philosophical problem of mixed inferences. The problem suggests that there are relevant interactions between connectives, justified by the philosophical considerations for conceptualising different logic systems, that the methods for combining logics should allow to emerge. The recognition of this fact is what drives the improvements on the method of juxtaposition that I develop. That is, in order to allow for the emergence of desirable interaction principles, I will propose alternative ways of combining logic systems -specifically classical and intuitionistic logics- that go beyond the standard for combinations, which is based on minimality conditions so as to avoid the so-called collapse theorems.[spa] El localismo lógico es una tesis en filosofía de la lógica según la cual diferentes sistemas lógicos son correctos en función del dominio en el que se aplican. Dicha tesis cuenta, prima facie, con cierta plausibilidad y con varios argumentos que la respaldan como mostraré. Sin embargo, el localismo se presta a un evidente y poderoso contraargumento conocido como ‘el problema de las inferencias mixtas’. El objetivo principal de esta disertación es dar respuesta a ese problema y extender la solución al problema afín de los compuestos mixtos que afecta al pluralismo alético. La manera de abordar el problema de las inferencias mixtas consistirá en analizar casos paradigmáticos en la literatura a la luz de los métodos de combinación de lógicas. En concreto, me centraré en el método de la yuxtaposición, desarrollado por Joshua Schechter. Así, ofreceré una solución al problema de las inferencias mixtas que pasará por realizar un análisis más sutil y una formalización más precisa de las mismas, para después aplicar los mecanismos de combinación como criterio de validez. Además, mostraré que el problema de las inferencias mixtas provee de multitud de ejemplos que invitan a desarrollar los métodos de combinación de lógicas de formas novedosas. Una de las aportaciones más relevantes de la disertación consistirá en modificar el método de la yuxtaposición para obtener mecanismos que van más allá del estándar de las extensiones mínimas conservativas. En concreto, propondré diferentes mecanismos para combinar la lógica clásica y la intuicionista, de manera que se permita la aparición de distintos principios puente para los que tenemos buenas razones que los justifican, sin que ello conduzca al colapso de las lógicas que se combinan

To the Beat of Different Drumer....Freedom, Anarchy and Conformism in Research

In this paper I attempt to make a case for promoting the courage of rebels within the citadels of orthodoxy in academic research environments. Wicksell in Macroeconomics, Brouwer in the Foundations of Mathematics,Turing in Computability Theory, Sraffa in the Theories of Value and Distribution are, in my own fields of research, paradigmatic examples of rebels, adventurers and non-conformists of the highest calibre in scientific research within University environments. In what sense, and how, can such rebels, adventurers and nonconformists be fostered in the current University research environment dominated by the cult of picking winners? This is the motivational question lying behind the historical outlines of the work of Wicksell, Brouwer, Hilbert, Bishop, Veronese, Gödel, Turing and Sraffa that I describe in this paper. The debate between freedom in research and teaching and the naked imposition of correct thinking, on potential dissenters of the mind, is of serious concern in this age of austerity of material facilities. It is a debate that has occupied some the finest minds working at the deepest levels of foundational issues in mathematics, metamathematics and economic theory. By making some of the issues explicit, I hope it is possible to encourage dissenters to remain courageous in the face of current dogmas.Non-conformist research, macroeconomics, foundations of mathematics, intuitionism, constructivism, formalism, Hilbertís Dogma, Hilbertís Program, computability theory

Operational theories and Categorical quantum mechanics

A central theme in current work in quantum information and quantum foundations is to see quantum mechanics as occupying one point in a space of possible theories, and to use this perspective to understand the special features and properties which single it out, and the possibilities for alternative theories. Two formalisms which have been used in this context are operational theories, and categorical quantum mechanics. The aim of the present paper is to establish strong connections between these two formalisms. We show how models of categorical quantum mechanics have representations as operational theories. We then show how nonlocality can be formulated at this level of generality, and study a number of examples from this point of view, including Hilbert spaces, sets and relations, and stochastic maps. The local, quantum, and no-signalling models are characterized in these terms.Comment: 37 pages, updated bibliograph

Logical and algebraic structures from Quantum Computation

The main motivation for this thesis is given by the open problems regarding the axiomatisation of quantum computational logics. This thesis will be structured as follows: in Chapter 2 we will review some basics of universal algebra and functional analysis. In Chapters 3 through 6 the fundamentals of quantum gate theory will be produced. In Chapter 7 we will introduce quasi-MV algebras, a formal study of a suitable selection of algebraic operations associated with quantum gates. In Chapter 8 quasi-MV algebras will be expanded by a unary operation hereby dubbed square root of the inverse, formalising a quantum gate which allows to induce entanglement states. In Chapter 9 we will investigate some categorial dualities for the classes of algebras introduced in Chapters 7 and 8. In Chapter 10 the discriminator variety of linear Heyting quantum computational structures, an algebraic counterpart of the strong quantum computational logic, will be considered. In Chapter 11, we will list some open problems and, at the same time, draw some tentative conclusions. Lastly, in Chapter 12 we will provide a few examples of the previously investigated structures

Philosophy of mathematics education

PHILOSOPHY OF MATHEMATICS EDUCATION\ud This thesis supports the view that mathematics teachers should be aware of differing views of the nature of mathematics and of a range of teaching perspectives. The first part of the thesis discusses differing ways in which the subject 'mathematics' can be identified, by relying on existing philosophy of mathematics. The thesis describes three traditionally recognised philosophies of mathematics: logicism, formalism and intuitionism. A fourth philosophy is constructed, the hypothetical, bringing together the ideas of Peirce and of Lakatos, in particular. The second part of the thesis introduces differing ways of teaching mathematics, and identifies the logical and sometimes contingent connections that exist between the philosophies of mathematics discussed in part 1, and the philosophies of mathematics teaching that arise in part 2. Four teaching perspectives are outlined: the teaching of mathematics as aestheticallyorientated, the teaching of mathematics as a game, the teaching of mathematics as a member of the natural sciences, and the teaching of mathematics as technology-orientated. It is argued that a possible fifth perspective, the teaching of mathematics as a language, is not a distinctive approach. A further approach, the Inter-disciplinary perspective, is recognised as a valid alternative within previously identified philosophical constraints. Thus parts 1 and 2 clarify the range of interpretations found in both the philosophy of mathematics and of mathematics teaching and show that they present realistic choices for the mathematics teacher. The foundations are thereby laid for the arguments generated in part 3, that any mathematics teacher ought to appreciate the full range of teaching 4 perspectives which may be chosen and how these link to views of the nature of mathematics. This would hopefully reverse 'the trend at the moment... towards excessively narrow interpretation of the subject' as reported by Her Majesty's Inspectorate (Aspects of Secondary Education in England, 7.6.20, H. M. S. O., 1979). While the thesis does not contain infallible prescriptions it is concluded that the technology-orientated perspective supported by the hypothetical philosophy of mathematics facilitates the aims of those educators who show concern for the recognition of mathematics in the curriculum, both for its intrinsic and extrinsic value. But the main thrust of the thesis is that the training of future mathematics educators must include opportunities for gaining awareness of the diversity of teaching perspectives and the influence on them of philosophies of mathematics