Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later

No entailing laws, but enablement in the evolution of the biosphere

Biological evolution is a complex blend of ever changing structural stability, variability and emergence of new phenotypes, niches, ecosystems. We wish to argue that the evolution of life marks the end of a physics world view of law entailed dynamics. Our considerations depend upon discussing the variability of the very "contexts of life": the interactions between organisms, biological niches and ecosystems. These are ever changing, intrinsically indeterminate and even unprestatable: we do not know ahead of time the "niches" which constitute the boundary conditions on selection. More generally, by the mathematical unprestatability of the "phase space" (space of possibilities), no laws of motion can be formulated for evolution. We call this radical emergence, from life to life. The purpose of this paper is the integration of variation and diversity in a sound conceptual frame and situate unpredictability at a novel theoretical level, that of the very phase space. Our argument will be carried on in close comparisons with physics and the mathematical constructions of phase spaces in that discipline. The role of (theoretical) symmetries as invariant preserving transformations will allow us to understand the nature of physical phase spaces and to stress the differences required for a sound biological theoretizing. In this frame, we discuss the novel notion of "enablement". This will restrict causal analyses to differential cases (a difference that causes a difference). Mutations or other causal differences will allow us to stress that "non conservation principles" are at the core of evolution, in contrast to physical dynamics, largely based on conservation principles as symmetries. Critical transitions, the main locus of symmetry changes in physics, will be discussed, and lead to "extended criticality" as a conceptual frame for a better understanding of the living state of matter

Inflation, Large Branes, and the Shape of Space

Linde has recently argued that compact flat or negatively curved spatial sections should, in many circumstances, be considered typical in Inflationary cosmologies. We suggest that the "large brane instability" of Seiberg and Witten eliminates the negative candidates in the context of string theory. That leaves the flat, compact, three-dimensional manifolds -- Conway's *platycosms*. We show that deep theorems of Schoen, Yau, Gromov and Lawson imply that, even in this case, Seiberg-Witten instability can be avoided only with difficulty. Using a specific cosmological model of the Maldacena-Maoz type, we explain how to do this, and we also show how the list of platycosmic candidates can be reduced to three. This leads to an extension of the basic idea: the conformal compactification of the entire Euclidean spacetime also has the topology of a flat, compact, four-dimensional space.Comment: 29 pages, clarifications, typos fixed, references adde

Conceptual thinking in Hegel’s Science of logic

Filozofia analityczna po logicyzmie Fregego i atomizmie logicznym Russella odziedziczyła szereg założeń związanych z istnieniem rodzajowej dziedziny bytów indywidualnych, których tożsamość i elementarne określenia już mamy zdefiniowane. Te „indywidua” istnieją tylko w idealnych „światach możliwych” i nie są niczym innym jak zbiorami posiadającymi strukturę bądź czystymi zbiorami matematycznymi. W przeciwieństwie do takich czysto abstrakcyjnych modeli, Hegel analizuje rolę pojęciowych rozróżnień i odpowiednich brakujących inferencji w rzeczywistym świecie. Tutaj wszystkie obiekty są przestrzennie i czasowo skończone. Nawet jeśli rzeczywiste rzeczy poruszają się zgodnie z pewnymi formami, są tylko momentami w całościowym procesie. Wszelako, formy te nie są przedmiotami bezpośredniej, empirycznej obserwacji, lecz zakładają udane i powtarzalne działania i akty mowy. W rezultacie żadna semantyka odnoszącej się do świata referencji nie może obyć się bez kategorii Heglowskich, które wykraczają daleko poza narzędzia opartej wyłącznie na relacjach logiki matematycznej

Modeling High-Temperature Superconductivity: Correspondence at Bay?

How does a predecessor theory relate to its successor? According to Heinz Post's General Correspondence Principle, the successor theory has to account for the empirical success of its predecessor. After a critical discussion of this principle, I outline and discuss various kinds of correspondence relations that hold between successive scientific theories. I then look in some detail at a case study from contemporary physics: the various proposals for a theory of high-temperature superconductivity. The aim of this case study is to understand better the prospects and the place of a methodological principle such as the Generalized Correspondence Principle. Generalizing from the case study, I will then argue that some such principle has to be considered, at best, as one tool that might guide scientists in their theorizing. Finally I present a tentative account of why principles such as the Generalized Correspondence Principle work so often and why there is so much continuity in scientific theorizing.Articl