Платоническое учение о Едином, многом и космосе в контексте современной философии математики

The principle of continuity in evolution is often violated by discontinuous saltations leading to "punctuations" in evolutionary history. Highly accurate cellular replication fidelity is a requirement for biological evolution. In previous work, I have used a statistical theoretical model to demonstrate discontinuity in the evolution of high replication fidelity. Depending on the granularity of approach, both a continuous and a saltational view of evolutionary history are consistent with a scientific worldview of creation, and with the concept of simplification in biology as articulated by Emily Boring et al. The apparent contradiction between the complexity of biological systems with the idea of evolutionary simplification can be resolved by considering the globally simplifying selection of single systems and the local evolution of increasing system complexity. Explanations of thresholds and discontinuities during evolution might require the inclusion of paradigms such as teleology and agency in biological science, with theological implications.

Introduction. The change in general paradigm of education, its transition to a competent model and the permanent change in federal state standards of higher education have created the problem associated with selecting the content of course programmes studied by university students. In the field of mathematical knowledge, the problem of strengthening students’ mathematical training is particularly acute in connection with the declared task, in which mathematical analysis is central. One of the ways to solve this problem is to distinguish the invariant and variable components in the content of the university course.The aimof the present research is to describe the content of variable components developed by the authors for the university course of mathematical analysis and to present the results of their introduction into the practice of teaching.Methodology and research methods. The conducted research is based on the principles of continuity and systemacity of modern education, its current concepts (fundamentalisation, humanisation, humanitarisation, individualization and differentiation) and the provisions of competency-based, activity-based, personality-oriented and interdisciplinary approaches to teaching. The theoretical analysis and experiment were used as the main methods, the results of which were evaluated through empirical and praximetric methods.Results and scientific novelty. The present study, carried out for many years at Vyatka State University, has shown that the system-forming factor of variable education, determining the means and forms of its implementation, is the variable content of education. Firstly, this particular content provides additional information on key concepts, theories and mathematical analysis, taking into account the specifics of students’ specialties, which facilitated their successful professionalism. Secondly, the variable content of education offers the possibility to systematically rethink and rapidly revise educational material, taking into account new scientific facts and discoveries. Finally, it can develop cognitive autonomy of junior students, encouraging them to carry out regular and informal research activities. The coverage of students of several mathematical directions of education, their obligatory involvement in independent research activities and support for mechanisms of interdisciplinarity and transprofessionalism ensured the scientific novelty of the research undertaken. The results of the formation of professional competencies of future graduates obtained during pedagogical measurements (questionnaires, surveys of students, observation of their educational and research achievements) confirmed the effectiveness of using the designed variable components of the discipline “Mathematical Analysis” in the learning process.Practical significance. The research material and the authors’ conclusions described in the present article can be useful for methodologists of higher school and teachers of mathematics interested in improving the quality of mathematical training in universities.

This chapter puts forward the thesis that the different concepts of Wisdom in the book of Proverbs are not accidental. Rather, they can be connected to a theological discourse about the relationship between Wisdom and Torah in the post-exilic period. Against a careful examination of the texts it can be seen that this discourse shaped the composition and redaction of the book of Proverbs, leading up to a final position which reduces wisdom to an everyday concept without any theological implications. This thesis is developed in three parts. First, the influence of Deuteronomy on Proverbs 1-9 is analyzed. Secondly and following this analysis the different positions on Wisdom and Torah in late-exilic literature are presented. Finally it is shown how this discourse influences the composition of the book of Proverbs up to the final layer in Proverbs 30. Keywords: book of Proverbs; influence of Deuteronomy; late-exilic literature; theological discourse; Torah; Wisdom

In The Principles of Mechanics, physicist Heinrich Hertz argues that instead of replying to the question “what is force?” like physicists and philosophers had been doing unsuccessfully, Newtonian physics should be reformulated without considering “force” a basic concept. Decades after Hertz’s book, Ludwig Wittgenstein considered the physicist’s proposal a perfect model for how philosophical problems should be solved, to the point that he made it the foundation of his way of doing philosophy. This article addresses Wittgenstein’s way of doing philosophy, while it also proposes the reason why he failed in solving the philosophical problems — as did Hertz in his project on reformulating Newtonian physics without considering the concept “force”. And to illustrate Wittgenstein’s failure, it examines his disputes with mathematicians Kurt Gödel and Alan Turing on the foundations of mathematics.

The aim of this article is to analyze the format of a two-leveled training – bachelor and master – future teachers of mathematics from the point of view of the content of mathematical material, which is to develop prospective teachers of mathematics at those two levels, shaping their professional competence. Methods. The study involves the theoretical methods: the analysis of pedagogical and methodical literature, normative documents; historical, comparative and logical analysis of the content of pedagogical mathematical education; forecasting, planning and designing of two-leveled methodical system of training of future teachers of mathematics. Results and scientific novelty. The level differentiation of the higher education system requires developing the appropriate curricula for undergraduate and graduate programs. The fundamental principle must be the principle of continuity – the magister must continue to deepen and broaden knowledge and skills, along with competences acquired, developed and formed on the undergraduate level. From these positions, this paper examines the course «Number Systems» – the most important in terms of methodology course for future mathematics teachers, and shows what content should be filled with this course at the undergraduate level and the graduate level. At the undergraduate level it is proposed to study classical number systems – natural, integer, rational, real and complex. Further extensions of the number systems are studied at the graduate level. The theory of numeric systems is presented as a theory of algebraic systems, arising at the intersection of algebra and mathematical logic. Here we study algebras over a field, division algebra over a field, an alternative algebra with division over the field, Jordan algebra, Lie algebra. Comprehension of bases of the theory of algebras by the master of the «mathematical education» profile will promote more conscious understanding of an axiomatic method, a structure of axiomatic theories in mathematics, development mechanisms of mathematical science; at the same time it will help to develop to complete vision of mathematics as a single science. As a result, the educational level of the master will be above the educational level of the bachelor of pedagogical mathematical education. Practical significance. The article can be useful to heads of departments and graduate programs, faculties of classical and pedagogical universities, carrying out preparation of masters in the direction «Pedagogical Education (Mathematics)».

Extending the classical side-channel analysis framework to access-driven cache attacks

The foundations of mathematics cover mathematical as well as philosophical problems. At the turn of the 20th century logicism, formalism and intuitionism, main foundational schools were developed. A natural problem arose, namely of how much the foundations of mathematics influence the real practice of mathematicians. Although mathematics was and still is declared to be independent of philosophy, various foundational controversies concerned some mathematical axioms, e.g. the axiom of choice, or methods of proof (particularly, non-constructive ones) and sometimes qualified them as admissible (or not) in mathematical practice, relatively to their philosophical (and foundational) content. Polish Mathematical School was established in the years 1915–1920. Its research program was outlined by Zygmunt Janiszewski (the Janiszewski program) and suggested that Polish mathematicians should concentrate on special branches of studies, including set theory, topology and mathematical logic. In this way, the foundations of mathematics became a legitimate part of mathematics. In particular, the foundational investigations should be conducted independently of philosophical assumptions and apply all mathematically accepted methods, finitary or not, and the same concerns other branches of mathematics. This scientific ideology contributed essentially to the remarkable development of logic, set theory and topology in Poland.

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the fifteenth publication in the Lecture Notes in Logic series, collects papers presented at the symposium 'Reflections on the Foundations of Mathematics' held in celebration of Solomon Feferman's 70th birthday (The 'Feferfest') at Stanford University, California in 1988. Feferman has shaped the field of foundational research for nearly half a century. These papers reflect his broad interests as well as his approach to foundational research, which emphasizes the solution of mathematical and philosophical problems. There are four sections, covering proof theoretic analysis, logic and computation, applicative and self-applicative theories, and philosophy of modern mathematical and logic thought.

This chapter offers a comprehensive account of Levinas’s relation to Heidegger’s thought during the formative years of his philosophical development through to Totality and Infinity (1961). Heidegger’s thought informed the central problem that preoccupied Levinas during these years, namely, the rise of Nazism, Hitlerism, and the prospect of a radical collapse of civilization into barbarism and evil. It afforded Levinas a way of understanding these historical events as deep philosophical problems rather than cognitive or historical aberrations. The chapter pinpoints two very different critiques Levinas makes, a critique of Heidegger’s account of facticity and the more famous ethical critique of Heidegger’s thought. The critique of Heidegger’s account of facticity structures Levinas’s alternative account of the phenomenology of being-riveted that shapes his thinking from 1934 to Totality and Infinity and gives the ethical critique of being its philosophical traction.

The traditional treatment of the divine attributes in theological discourse has been criticised for what some scholars regard as the influence of Greek philosophy, which they argue may result in distorted concepts of the divine. A further development in the doctrine of God is the renewed consciousness of the importance of the doctrine of the Trinity. The purpose of this article is to consider how these developments may impact the doctrine of the divine attributes. Can the doctrine of the Trinity enhance an articulation of the divine attributes? To illustrate the difference that a trinitarian approach to the divine attributes could make, divine omnipotence, as well as the possibility of discovering new attributes will be considered.Intradisciplinary and/or interdisciplinary implications: This article is an intra-disciplinary study with implications for dogmatics or systematic theology. It addresses the doctrine of the divine attributes from a trinitarian perspective. At stake is the impact of different approaches within the same discipline. In this case, the engagement is between a trinitarian versus a classical approach to the study of the divine attributes.

This study investigates the development of creation narratives within Patristic literature, focusing on key theologians from the Apostolic to Medieval periods, such as Augustine, Origen, and Gregory of Nyssa. The primary issue explored is the theological theme of chaos and cosmos, which illustrates how God transforms chaos into order through His creative and redemptive actions. The research traces how early Christian thinkers understood and articulated this theme, particularly about God's kingship and the overarching narrative of creation, fall, redemption, and new creation. The study employs a qualitative textual analysis of prominent Patristic writings, systematically reviewing key texts that discuss creation narratives. This method allows for a detailed examination of the theological nuances present in these works, highlighting significant trends and variations in thought among different theologians. The iterative approach of this analysis uncovers the dynamic nature of early Christian cosmology and its implications for Trinitarian theology. The findings reveal that Patristic thinkers were deeply concerned with establishing a solid theological foundation for understanding creation about the Trinity. Their reflections continue to enrich contemporary Christian thought on creation, offering insights relevant to modern spiritual and theological discussions. This study recommends further engagement with Patristic literature to foster interdisciplinary dialogue, historical contextualisation, and meaningful theological reflection on creation in the context of contemporary issues.

ANALYSIS OF THE PRIMARY AND SECONDARY STRUCTURE OF GLYCOSAMINOGLYCANS FROM ALTERNATIVE SOURCES (NATURAL OR SYNTHETIC)ANALISI DELLA STRUTTURA PRIMARIA E SECONDARIA DI LICOSAMMINOGLICANI DA FONTE ALTERNATIVA(NATURALE O SINTETICA)

Reviewed by: Creatures of Possibility: The Theological Basis of Human Freedom by Ingolf U. Dalferth Adam Morton Creatures of Possibility: The Theological Basis of Human Freedom. By Ingolf U. Dalferth. Translated by Jo Bennett. Grand Rapids: Baker Academic, 2016. 216 pp. Despite the subtitle, this slim book is not a direct treatment of the theme of human freedom. It is instead a series of studies developing what Dalferth takes to be the surprising basis of such freedom: humanity's fundamental passivity coram Deo. Far from conceiving this passivity negatively, as if it were simply identifiable with deficiency, it stands as "the center of creativity from which all humans live" (6). Humans are thus conceived as beings of possibility, becoming passively what is strictly impossible under their own power. Proceeding from this thematic center, Dalferth engages a series of philosophical and theological positions. Chapter one gives a very brief philosophical introduction to the topic. This discussion is deepened in chapter two as Kant and Luther, especially the latter's 1536 Disputatio de homine, are employed to distinguish between philosophical and theological accounts. Here Luther's definition of the human as justified by faith is given central place in Dalferth's project: humans "become what they are through what happens to them. And they understand what happens to them as what God does to them and for them" (42). Subsequent chapters can be read as stand-alone essays, though they have significant thematic links. Chapter three engages further with Luther, ranging over many of his works to examine the theme of passivity in relation to the concept of gift, and touching on the difference between the old and new creatures, the nature of the will, and ontology. While distinguishing carefully between creation, justification, and sanctification, Dalferth sets them in a common [End Page 244] creation-theological framework: "Being and being Christian are thus understood … as divine acts of creation ex nihilo" (53–54). The fourth chapter takes up the concept of gift again, this time beginning with the phenomenological analyses of Derrida, that nothing is gift, and Marion, that all is gift. Against these, Dalferth shifts from a phenomenological to a hermeneutical analysis, focusing on the necessity of the gift as "for me," and so on the gift as passively determining its recipient. This allows him to take up the character of Christian faith as the interruption of the unknown and unlooked-for gift that, echoing thesis 28 of the Heidelberg Disputation, "creates its own mode of reception" (114) in excess of any humanly perceived need. Chapters five through seven follow a similar structure, each setting out from a philosophical problem toward a theological discussion. Thus, in chapter five, Derrida on sacrifice is a springboard toward provocative analyses of Abraham and Isaac, and of the crucifixion of Jesus, as divine love. Chapter six engages Hans Blumenberg on the question of the incarnation, from there returning to the relation between philosophy and theology. Chapter seven takes up a variety of voices, principally Nietzsche, to discuss human creativity and finally the image of God. The book has some limitations. A reader unfamiliar with the framing philosophical conversations may struggle to make use of certain chapters, and the author's style often requires inferring connections well beyond his explicit citations. Furthermore, there remains risk that theology is determined negatively by that philosophical framing, as when the crucifixion is described in terms calibrated to evade Derrida's critique of sacrifice. Moreover, the author tends to articulate the passivity of faith in abstract terms, neglecting to draw a more concrete connection to those divine words by which faith is bestowed, and so which constitute that passivity. Even so, this volume offers creative and insightful reformulations of Lutheran theology in close relation to its central theme. Scholars interested in theological anthropology or in the relation between philosophy and theology have much to gain from this book. [End Page 245] Adam Morton Holy Trinity Lutheran Church Lancaster, Pennsylvania Copyright © 2018 Johns Hopkins University Press and Lutheran Quarterly, Inc.

In his later works, the great logician and mathematician Kurt Gödel concentrates his focus on the philosophical problems such as the implications of set theory, the grammar and philosophy of language, objectivity and relativity, the ontological proof of God’s existence, and phenomenology as an exact method. This essay explores how Gödel reads the philosophy (of logic and mathematics) of his time and why he turns his attention to Husserl’s phenomenology for describing the foundations of mathematics. To begin with, Gödel employs Husserl’s significant distinction between Weltanschauung (worldview) philosophy and philosophy as rigorous science: According to the Weltanschauung philosophy, the spirit of time constantly changes so that the ideas discussed and goals attempted are meant to be temporal, and not for the sake of eternal truths, but for that of their own perfection; philosophy as rigorous science, on the other hand, is supratemporal so that its aim is to discover absolute and timeless values. As for the worldview of his time, Gödel sees the development of philosophy and mathematics leaned toward skepticism, pessimism, and positivism. The antinomies of set theory, for instance shaked the grounds on which mathematics and logic are founded. Gödel, too, uses these paradoxes in his incompleteness theorems in order to prove that there are some statements which can neither be proved nor disproved within a system. That also means that arithmetic is not eligible to prove its own consistency. From this, however, Gödel does not come to a conclusion for a nihilism in mathematics and logic: These mere antinomies of set theory do not “necessarily” lead us to logical positivism, and neither to such a materialism, nor to any kind of pessimistic theory of knowledge. The incompleteness theorems assert that there are arithmetical propositions that are true but neither provable nor unprovable within its own calculus, so that arithmetic is intrinsically incomplete. However, instead of Alfred Tarski’s pathological view of examining the detections within the faulty system and then reforming the system all together, Gödel holds that we need to change our methods to find new patterns that describe the antinomies pointing to the unrecoverable reality of the mathematical world. Thus, Gödel does not follow any variation of the Weltanschauung philosophy of his time, either attempting to reduce mathematical realities to mathematical proofs in order to get rid of antinomies, or endeavoring to rescue a complete system of truths by a closed formal system, both Weltanschauung philosophies fail to set forth a realistic method. In this context, Gödel finds the task of phenomenology analogous to what he pursues in terms of a systematic framework for the foundations of mathematics. Husserl’s phenomenology, in Gödel’s account, proliferates the intuition of (mathematical) essences and provides a clarification of meaning of undefinable concepts, such as the antinomies of set theory. Applying the phenomenological reduction to the objective reality of the mathematical world, Gödel believes one obtains a clear experiential reality of the essential characteristics of (mathematical and logical) concepts. Briefly put, what Gödel finds in Husserl’s phenomenology that corresponds to his way of mathematical realism is a thoroughly designated method giving us mathematical essences back again.

This publication is a non-academic discussion of the philosophical problems ofcontemporary Orthodox theology. Participants in the discussion were asked to expresstheir views on the most important challenges facing contemporary Orthodoxthinking, focusing on the following questions posed by the editors of the journalOtechestvennaya Philosophiya:1) What is Russian Orthodox theology at the end of the first quarter of the 21stcentury? What do you see as the possible and most important impulses for its development,but also as challenges, problems and obstacles?2) What are the philosophical problems facing Orthodox thought in the 21st century?3) Does modern Orthodox theological language correspond to the contemporaryintellectual context? Should Orthodox theology interact with contemporary philosophicalcurrents and adapt their conceptual and categorical apparatus for its ownpurposes, or is this an unacceptable modernisation of the theological heritage?4) Can we talk about the special nature of the relationship between philosophy andtheology in the local intellectual tradition?5) How do you assess the experience of interaction between philosophical andtheological thinking in the Russian religious-philosophical tradition of the 19th andfirst half of the 20th centuries? Can it beused in modern Orthodox theology, and if so,how?