Continuous Truth II: Reflections

Abstract. In the late 1960s, Dana Scott first showed how the Stone-Tarski topological interpretation of Heyting’s calculus could be extended to model intuitionistic analysis; in particular Brouwer’s continuity prin-ciple. In the early ’80s we and others outlined a general treatment of non-constructive objects, using sheaf models—constructions from topos theory—to model not only Brouwer’s non-classical conclusions, but also his creation of “new mathematical entities”. These categorical models are intimately related to, but more general than Scott’s topological model. The primary goal of this paper is to consider the question of iterated extensions. Can we derive new insights by repeating the second act? In Continuous Truth I, presented at Logic Colloquium ’82 in Florence, we showed that general principles of continuity, local choice and local com-pactness hold in the gros topos of sheaves over the category of separable locales equipped with the open cover topology. We touched on the question of iteration. Here we develop a more gen-eral analysis of iterated categorical extensions, that leads to a reflection schema for statements of predicative analysis. We also take the opportunity to revisit some aspects of both Continuous Truth I and Formal Spaces (Fourman & Grayson 1982), and correct two long-standing errors therein

Biosynthesis and emission of stress-induced volatile terpenes in roots and leaves of switchgrass (Panicum virgatum L.)

Switchgrass (Panicum virgatum L.), a perennial C4 grass, represents an important species in natural and anthropogenic grasslands of North America. Its resilience to abiotic and biotic stress has made switchgrass a preferred bioenergy crop. However, little is known about the mechanisms of resistance of switchgrass against pathogens and herbivores. Volatile compounds such as terpenes have important activities in plant direct and indirect defense. Here we show that switchgrass leaves emit blends of monoterpenes and sesquiterpenes upon feeding by the generalist insect herbivore Spodoptera frugiperda (fall armyworm) and in a systemic response to the treatment of roots with defense hormones. Belowground application of methyl jasmonate also induced the release of volatile terpenes from roots. To correlate the emission of terpenes with the expression and activity of their corresponding biosynthetic genes, we identified a gene family of 44 monoterpene and sesquiterpene synthases (mono- and sesqui-TPSs) of the type-a, b, g, and e subfamilies, of which 32 TPSs were found to be functionally active in vitro. The TPS genes are distributed over the K and N sub-genomes with clusters occurring on several chromosomes. Synteny analysis revealed syntenic networks for approximately 30-40% of the switchgrass TPS genes in the genomes of Panicum hallii, Setaria viridis, and Sorghum bicolor suggesting shared TPS ancestry in the common progenitor of these grass lineages. Eighteen switchgrass TPS genes were substantially induced upon insect and hormone treatment and the enzymatic products of nine of these genes correlated with compounds of the induced volatile blends. In accordance with the emission of volatiles, TPS gene expression was induced systemically in response to belowground treatment, whereas this response was not observed upon aboveground feeding of S. frugiperda. Our results demonstrate complex above and belowground responses of induced volatile terpene metabolism in switchgrass and provide a framework for more detailed investigations of the function of terpenes in stress resistance in this monocot crop

Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism

This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theory with choice (ZFC). The duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of $\Omega$-logical validity can then be countenanced within a coalgebraic logic, and $\Omega$-logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of $\Omega$-logical validity correspond to those of second-order logical consequence, $\Omega$-logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets

Techno-Ecological Synergy: A Framework for Sustainable Engineering

Even though the importance of ecosystems in sustaining all human activities is well-known, methods for sustainable engineering fail to fully account for this role of nature. Most methods account for the demand for ecosystem services, but almost none account for the supply. Incomplete accounting of the very foundation of human well-being can result in perverse outcomes from decisions meant to enhance sustainability and lost opportunities for benefiting from the ability of nature to satisfy human needs in an economically and environmentally superior manner. This paper develops a framework for understanding and designing synergies between technological and ecological systems to encourage greater harmony between human activities and nature. This framework considers technological systems ranging from individual processes to supply chains and life cycles, along with corresponding ecological systems at multiple spatial scales ranging from local to global. The demand for specific ecosystem services is determined from information about emissions and resource use, while the supply is obtained from information about the capacity of relevant ecosystems. Metrics calculate the sustainability of individual ecosystem services at multiple spatial scales and help define necessary but not sufficient conditions for local and global sustainability. Efforts to reduce ecological overshoot encourage enhancement of life cycle efficiency, development of industrial symbiosis, innovative designs and policies, and ecological restoration, thus combining the best features of many existing methods. Opportunities for theoretical and applied research to make this framework practical are also discussed

Photovoltaic power plants: a multicriteria approach to investment decisions and a case study in western Spain

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