Related papers: Bernard Bolzano: from Topological to Arithmetical …
In his Foundations of a General Theory of Manifolds, Georg Cantor praised Bernard Bolzano as a clear defender of actual infinity who had the courage to work with infinite numbers. At the same time, he sharply criticized the way Bolzano…
During the early 1830's Bernard Bolzano, working in Prague, wrote a manuscript giving a foundational account of numbers and their properties. In the final section of his work he described what he called `infinite number expressions' and…
Bolzano and Cantor were the first mathematicians to make significant attempts to measure the size (numerosity) of different infinite collections. They differed in their methodological approaches, with Cantor's prevailing. This led to the…
The embracing of actual infinity in mathematics leads naturally to the question of comparing the sizes of infinite collections. The basic dilemma is that the Cantor Principle (CP), according to which two sets have the same size if there is…
The paper introduces the notion of the size of countable sets that preserves the Part-Whole Principle and generalizes the notion of the cardinality of finite sets. The sizes of natural numbers, integers, rational numbers, and all their…
A description of physical reality in which wholeness is the foundation is discussed along with the motivation for such an attempt. As a possible mathematical framework within which a physical theory based on wholeness may be expressed,…
The concept of measurement is discussed. It is argued that counting process in mathematics is also measurement which requires a basic unit. The idea of scale is put forward. The basic unit itself, which are composed of the infinitesimal of…
This article critically reappraises arguments in support of Cantor's theory of transfinite numbers. The following results are reported: i) Cantor's proofs of nondenumerability are refuted by analyzing the logical inconsistencies in…
We present the model theoretic concepts that allow mathematics to be developed with the notion of the potential infinite instead of the actual infinite. The potential infinite is understood as a dynamic notion, being an indefinitely…
Mathematical conception of infinite quantities forms a cornerstone of many disciplines of modern mathematics --- from differential calculus to set theory. In fact, it could be argued that the most significant revolutions in mathematics in…
In a recent paper as an alternative to models based on the notion of ideal mathematical point, characterized by a property of separatedness, we considered a viewpoint based on the notion of continuous change, making use of elements of a…
Measurement theory is the cornerstone of science, but no equivalent theory underpins the huge volumes of non-numerical data now being generated. In this study, we show that replacing numbers with alternative mathematical models, such as…
An alternative mathematics based on qualitative plurality of finiteness is developed to make non-standard mathematics independent of infinite set theory. The vague concept "accessibility" is used coherently within finite set theory whose…
Zeno's paradoxes are explained as being the result of inappropriate combination of discrete and continuous mathematical systems. It is proposed that the source of this confusion lies in the course of development of the number system, which…
Set-theoretical, physical, and intuitive notions of continuum are compared. It is shown that the independence of the continuum hypothesis determines status and properties of the set of intermediate cardinality. The intermediate set is a…
The goal of this paper consists of developing a new (more physical and numerical in comparison with standard and non-standard analysis approaches) point of view on Calculus with functions assuming infinite and infinitesimal values. It uses…
Projections of finite dimensional sets and their measures are investigated in infinite-dimensional power measure spaces. The starting point is the known algebraic formula, expressing \ the $y$-projection of a finite-dimensional set $a$ as a…
Throughout the course of mathematical history, generalizations of previously understood concepts and structures have led to the fruitful development of the hierarchy of number systems, non-euclidean geometry, and many other epochal phases…
A subset of a topological space is said to be \emph{universally measurable} if it is measured by the completion of each countably additive $\sigma$-finite Borel measure on the space, and \emph{universally null} if it has measure zero for…
Since the theory developed by Georg Cantor, mathematicians have taken a sharp interest in the sizes of infinite sets. We know that the set of integers is infinitely countable and that its cardinality is Aleph0. Cantor proved in 1891 with…