Theory of B(X)-Module: Algebraic Module Structure of Generally Unbounded Infinitesimal Generators

The concept of logarithmic representation of infinitesimal generators is introduced, and it is applied to clarify the algebraic structure of bounded and unbounded infinitesimal generators. In particular, by means of the logarithmic representation, the bounded components can be extracted from generally unbounded infinitesimal generators. In conclusion, the concept of module over a Banach algebra is proposed as the generalization of the Banach algebra. As an application to mathematical physics, the rigorous formulation of a rotation group, which consists of unbounded operators being written by differential operators, is provided using the module over a Banach algebra.

The Great Patriotic War and the noosphere of V. I. Vernadsky

The article is devoted to the teachings of V. I. Vernadsky on the transition of the biosphere to the noosphere,which was developed by the creator of biogeochemistry during the Great Patriotic war. The o riginal phenomenon of Russian cosmism was revealed during the Gorbachev «perestroika» years. Some philosophers and scientists after the fall of Marxist ideology in our country began to consider the philosophy of Russian cosmism as «the last word of world philosophy». Russian cosmism gained the strength of the messianic myth largely thanks to scientists and philosophers who were participants in the Great Patriotic War. During the War, Vernadsky believed that after the victory over fascism, scientists would be involved in leadership in diff erent countries. Vernadsky’s teaching of the transition of the biosphere to the stage of the noosphere has not adopted a fi nal, rigorous formulation. This gives rise to numerous confl icting interpretations of the concept of the noosphere. According to Vernadsky, human activity on the planet qualitatively changes the biosphere, introducing excess energy. Vernadsky calls it cultural biochemical energy or the energy of human culture.

Church’s thesis

An algorithm or mechanical procedure A is said to ‘compute’ a function f if, for any n in the domain of f, when given n as input, A eventually produces fn as output. A function is ‘computable’ if there is an algorithm that computes it. A set S is ‘decidable’ if there is an algorithm that decides membership in S: if, given any appropriate n as input, the algorithm would output ‘yes’ if n∈S, and ‘no’ if n∉S. The notions of ‘algorithm’, ‘computable’ and ‘decidable’ are informal (or pre-formal) in that they have meaning independently of, and prior to, attempts at rigorous formulation. Church’s thesis, first proposed by Alonzo Church in a paper published in 1936, is the assertion that a function is computable if and only if it is recursive: ‘We now define the notion…of an effectively calculable function…by identifying it with the notion of a recursive function….’ Independently, Alan Turing argued that a function is computable if and only if there is a Turing machine that computes it; and he showed that a function is Turing-computable if and only if it is recursive. Church’s thesis is widely accepted today. Since an algorithm can be ‘read off’ a recursive derivation, every recursive function is computable. Three types of ‘evidence’ have been cited for the converse. First, every algorithm that has been examined has been shown to compute a recursive function. Second, Turing, Church and others provided analyses of the moves available to a person following a mechanical procedure, arguing that everything can be simulated by a Turing machine, a recursive derivation, and so on. The third consideration is ‘confluence’. Several different characterizations, developed more or less independently, have been shown to be coextensive, suggesting that all of them are on target. The list includes recursiveness, Turing computability, Herbrand–Gödel derivability, λ-definability and Markov algorithm computability.

The Regularity Account of Phenomenal Unity

This chapter presents the regularity account of phenomenal unity (RPU). The basic idea of RPU is that when the brain tracks a regularity that is predictive of different features (or of different objects or events), there will be an experienced connection between those features (or the respective objects or events). We can then say that the regularity connects those features (or objects or events). According to RPU, unity comes in degrees, and in ordinary conscious experience we find a hierarchy of experienced wholes. This chapter provides a preliminary taxonomy of experienced wholes, with many examples. Drawing on formal concepts of the predictive processing framework, a formal description of possible computational underpinnings of experienced wholeness is given. Finally, a rigorous formulation of the mélange model (first proposed in chapter 4) is provided.

Łukasiewicz and the Several Senses of Possibility

The most famous achievement of Jan Łukasiewicz (1878–1956) was to give the first rigorous formulation of many-valued logic. In this paper I discuss his motivations for this, which were philosophical, and the legacy of his work.