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In the paper we consider homogeneous systems of linear equations and classify coordinate monoids over the additive monoid of natural numbers which are defined by such systems. Further, we apply our results to the wide class of commutative monoids. [ABSTRACT FROM AUTHOR]

The Phenomenological Movement was one of the dominant philosophical fashions on the continent of Europe from the early 1920s until the late 1970s. Hundreds of philosophy professors in Europe, Japan, and in the United States conceived of themselves as phenomenologists in at least some stage of their careers, and countless articles and books have been published under the phenomenological flag. Among the many causes that explain the popularity of phenomenology, three may be mentioned. First, whereas the scientific revolutions of the beginning of the twentieth century threatened the traditional position of philosophy as a separate and foundational discipline, seminal phenomenologists such as Husserl and Heidegger, at least in his early works, reasserted the foundational role of philosophy, claiming that they had discovered a philosophical method distinct from the methods of empirical science. Second, this method seemed to enable phenomenologists to widen the scope of philosophical research in unprecedented ways, thereby attracting those who felt suffocated by philosophy as a limited academic discipline. Finally, phenomenology allowed philosophers to discuss problems of life which became pressing during and after the world wars, in particular because of the way in which Heidegger had integrated the existential philosophy of Kierkegaard and the philosophy of life of Wilhelm Dilthey into his phenomenological masterpiece Sein und Zeit (Being and Time, 1927). The popularity of phenomenology cannot be understood without taking such cultural factors into account. [ABSTRACT FROM AUTHOR]

Introduction It is uncontroversial to say that the period in question saw more important changes in the philosophy of mathematics than any previous period of similar length in the history of philosophy. Above all, it is in this period that the study of the foundations of mathematics became partly a mathematical investigation itself. So rich a period is it, that this survey article is only the merest sketch; inevitably, some subjects and figures will be inadequately treated (the most notable omission being discussion of Peano and the Italian schools of geometry and logic). Of prime importance in understanding the period are the changes in mathematics itself that the nineteenth century brought, for much foundational work is a reaction to these, resulting either in an expansion of the philosophical horizon to incorporate and systematise these changes, or in articulated opposition. What, in broad outline, were the changes? First, traditional subjects were treated in entirely new ways. This applies to arithmetic, the theory of real and complex numbers and functions, algebra, and geometry. (a) Some central concepts were characterised differently, or properly characterised for the first time, for example, from analysis, those of continuity (Weierstrass, Cantor, Dedekind) and integrability (Jordan, Lebesgue, Young), from geometry, that of congruence (Pasch, Hilbert), and geometry itself was recast as a purely synthetic theory (von Staudt, Pasch, Hilbert). (b) Theories were treated in entirely new ways, for example, as axiomatic systems (Pasch, Peano and the Italian School, Hilbert), as structures (Dedekind, Hilbert), or with entirely different primitives (Riemann, Cantor, Frege, Russell). [ABSTRACT FROM AUTHOR]

Chapter three of the book "The Human Being As a Logical Thinker" by Noel Balzer is presented. It discusses the natural numbers and the logic of arithmetic. It notes the universal application of numbers which are related to objects like trees, houses, and people. They are also often subjected to restrictions. It also outlines the development of the theory of classes and the theory of sets by mathematician Georg Cantor.

The article discusses natural multiples that are ordinals and transitivity's recurrence. Topics discussed include multiple of multiples and how concept of multiple of multiples is related to Ontology, views of mathematicians on ordinals, and Idea of multiple-presentation that belongs to nature and it belongs to itself.

The article presents the poem "Parnassus," and offers the author's views on the poem. He explores the philosophical and religious aspects of the numeral value of ten. He narrates his personal experiences of the power and importance of this number and holds that it always fascinated him to narrate his feelings in poetry. According to the author, number ten is a powerful value whose impact runs through the world.

We give an existence-uniqueness result of periodic motion for discrete dynamical system x(n + 1) = f(n, x(n)), (n,x) ∈ Z x Rm. [ABSTRACT FROM AUTHOR]

A chapter from the book "Think of a Number: A Fascinating Look at the World of Numbers" is presented. It provides an analysis of prime numbers. It states that a prime number is defined as a whole number that cannot be divided into other whole numbers except for 1. Some mathematicians reportedly call prime numbers the building blocks of math since other whole numbers can be created if prime numbers are multiplied together. There is also no pattern to determine prime numbers.

An introduction to various chapters of a section on Magic numbers featured in the book "Think of a Number: A Fascinating Look at the World of Numbers" is presented.