Mathematics textbooks teach logical reasoning by example, a practice started by Euclid; while logic textbooks treat logic as a subject in its own right without practical application to mathematics. Stuck in the middle are students seeking mathematical proficiency and educators seeking to provide it. To assist them, the article explains in practical detail how to teach logic-based skills such as: making mathematical reasoning fully explicit; moving from step to step in a mathematical proof in logically correct ways; and checking to (...) make sure inferences are logically correct. The methodology can easily be extended beyond the four examples analyzed. (shrink)

We use tools of mathematical logic to analyse the notion of a path on a complex algebraic variety, and are led to formulate a “rigidity” property of fundamental groups specific to algebraic varieties, as well as to define a bona fide topology closely related to etale topology. These appear as criteria for ℵ1-categoricity, or rather stability and homogeneity, of the formal countable language we propose to describe homotopy classes of paths on a variety, or equivalently, its universal covering space.Technically, for (...) a variety A defined over a finite field extension of the field Q of rational numbers, we introduce a countable language L describing the universal covering space of A, or, equivalently, homotopy classes of paths in A. Under some assumptions on A source we show that the universal covering space of A is an analytic Zariski structure [26]), and present an Lω1ω)-sentence axiomatising the class containing the structure and that is stable and homogeneous over elementary submodels. The “rigidity” condition on fundamental groups says that projection of the fundamental group of a variety is the fundamental group of the projection, up to finite index and under some irreducibility assumptions, and is used to prove that the projection of an irreducible closed set is closed in the analytic Zariski structure.In particular, we define an analytic Zariski structure on the universal covering space of an Abelian variety defined over a finite extension of the field Q of rational numbers. (shrink)

Many different views on the problem of “war” and “peace” were substantively explored for several millennia. From our point of view, the theoretical and empirical basis of “war” and “peace” was made up principally from research in three areas: Religion, ethics, philosophy and psychology, as comprehension of the normative-evaluative categories of good and evil; Philosophy, political science, sociology and jurisprudence, in the theories of international relations; History, fiction as well as documentary, memoir, reference, technical literature on the subject of “war” (...) and “peace.” For such a significant amount of time, there was a wealth of factual and theoretical material accumulated in each of the three areas of research, which highlights the problem of “war” and “peace” in all variety of its meanings. To understand the scope of coverage and a range of research on the problem of “war” and “peace,” as well as the level of achievements in this field, let us briefly examine the history of “war” and “peace” in each of these three areas, proposed by us. (shrink)

Conceptual system developed in optimal control theory for technical purposes is used as a philosophical instrument applied to cyclic information processes, which are expected to be the basis of noosphere. Noosphere was perceived by the founding fathers of this concept, Vernadsky, Teilhard de Chardin, e.a. as an outgrowth of the evolutionary process, which begins with cosmogenesis and proceeds through geosphere and biosphere. We attempt to apply the optimal control concepts to all three levels – geospheric, biospheric, and noospheric – due (...) to their having a common structure of information processes. These processes include homeostasis, accumulation and expenditure of information, formation of hierarchical information structures, evolution involving the breaks of homeostasis etc. In noosphere, controlled system may have the same informational capabilities as controlling system, so that the term “dialog” is more adequate; in this case, we extend optimal control description to game theory. The cyclic, feedback logic of optimal control seems better adapted to noospheric processes than usual causeeffect logic. This second part of the paper proceeds from the geo- and biospheric levels discussed in the first part to the noospheric level. The basic structure at this level is the fusion of natural matter/energy cycles characteristic for geosphere with anthropogenic information cycles, which extend information accumulation and adaptation inherited from biospheric level into reflective realm. The basic type of informational interaction between these structures is construed in perspective of game theory between reflective players. Its essential feature is the interaction between reflective images that each player forms of other players and of oneself. We describe the nontrivial information flows that can arise in a distributed global system of such structures, including complex interactions between collective and individual levels and paradoxes. We analyze the role of science as the carrier of the “rational model”, on which this entire system is based, and the impact on this model of the sociobiological background of science inherited from the biospheric level. We also discuss the role of natural language as an alternative noospheric structure capable of supporting the irrational components of the noosphere. City is discussed as an example of an emerging structure integrating the basic noospheric components, albeit in an inchoate form. Finally, we consider the reflective identities, which may emerge in noosphere, and the relevant ethical issues. (shrink)

The high precision attained by cosmological data in the last few years has increased the interest in exact solutions. Analytic expressions for solutions in the Standard Model are presented here for all combinations of Λ = 0, Λ ≠ 0, κ = 0, and κ ≠ 0, in the presence and absence of radiation and nonrelativistic matter. The most complete case (here called the ΛγCDM Model) has Λ ≠ 0, κ ≠ 0, and supposes the presence of radiation and dust. (...) It exhibits clearly the recent onset of acceleration. The treatment includes particular models of interest such as the ΛCDM Model (which includes the cosmological constant plus cold dark matter as source constituents). (shrink)

У статті запропонований методологічний підхід до визначення сутності ефективності управління військово-правовим вихованням військовослужбовців Збройних Сил України, який покладений в основу математичного моделювання його оцінки. Особлива увага зосереджена на соціально-філософському аналізі оцінки ефекту управління, його кількісно-якісної визначеності, підкреслена її роль і місце у структурі управління.