Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity

Previous work on applications of Abstract Differential Geometry (ADG) to discrete Lorentzian quantum gravity is brought to its categorical climax by organizing the curved finitary spacetime sheaves of quantum causal sets involved therein, on which a finitary (:locally finite), singularity-free, background manifold independent and geometrically prequantized version of the gravitational vacuum Einstein field equations were seen to hold, into a topos structure. This topos is seen to be a finitary instance of both an elementary and a Grothendieck topos, generalizing in a differential geometric setting, as befits ADG, Sorkin's finitary substitutes of continuous spacetime topologies. The paper closes with a thorough discussion of four future routes we could take in order to further develop our topos-theoretic perspective on ADG-gravity along certain categorical trends in current quantum gravity research.Comment: 49 pages, latest updated version (errata corrected, references polished) Submitted to the International Journal of Theoretical Physic

On the connection between Nonstandard Analysis and Constructive Analysis

Constructive Analysis and Nonstandard Analysis are often characterized as completely antipodal approaches to analysis. We discuss the possibility of capturing the central notion of Constructive Analysis (i.e. algorithm, finite procedure or explicit construction) by a simple concept inside Nonstandard Analysis. To this end, we introduce Omega-invariance and argue that it partially satisfies our goal. Our results provide a dual approach to Erik Palmgren's development of Nonstandard Analysis inside constructive mathematics

Hyperlogic: A System for Talking about Logics

Sentences about logic are often used to show that certain embedding expressions, including attitude verbs, conditionals, and epistemic modals, are hyperintensional. Yet it not clear how to regiment “logic talk” in the object language so that it can be compositionally embedded under such expressions. This paper does two things. First, it argues against a standard account of logic talk, viz., the impossible worlds semantics. It is shown that this semantics does not easily extend to a language with propositional quantifiers, which are necessary for regimenting some logic talk. Second, it develops an alternative framework based on logical expressivism, which explains logic talk using shifting conventions. When combined with the standard S5π+ semantics for propositional quantifiers, this framework results in a well-behaved system that does not face the problems of the impossible worlds semantics. It can also be naturally extended with hybrid operators to regiment a broader range of logic talk, e.g., claims about what laws hold according to other logics. The resulting system, called hyperlogic, is therefore a better framework for modeling logic talk than previous accounts

Freedom, Anarchy and Conformism in Academic 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 caliber in scientific research within University environments. In what sense, and how, can such rebels, adventurers and non-conformists 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 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 of 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 dogmasNon-conformist research, economic theory, mathematical economics, 'Hilbert's Dogma', Hilbert's Program, computability theory

Sheaf Logic, Quantum Set Theory and the Interpretation of Quantum Mechanics

Based on the Sheaf Logic approach to set theoretic forcing, a hierarchy of Quantum Variable Sets is constructed which generalizes and simplifies the analogous construction developed by Takeuti on boolean valued models of set theory. Over this model two alternative proofs of Takeuti's correspondence, between self adjoint operators and the real numbers of the model, are given. This approach results to be more constructive showing a direct relation with the Gelfand representation theorem, revealing also the importance of these results with respect to the interpretation of Quantum Mechanics in close connection with the Deutsch-Everett multiversal interpretation. Finally, it is shown how in this context the notion of genericity and the corresponding generic model theorem can help to explain the emergence of classicality also in connection with the Deutsch- Everett perspective.Comment: 34 pages, 2 figure