Related papers: Zooming in on infinitesimal 1-.9.. in a post-trium…
Is .999... equal to 1? Lightstone's decimal expansions yield an infinity of numbers in [0,1] whose expansion starts with an unbounded number of digits "9". We present some non-standard thoughts on the ambiguity of an ellipsis, modeling the…
We examine alternative interpretations of the symbol described as nought, point, nine recurring. Is "an infinite number of 9s" merely a figure of speech? How are such alternative interpretations related to infinite cardinalities? How are…
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…
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…
The notions of potential infinity (understood as expressing a direction) and actual infinity (expressing a quantity) are investigated. It is shown that the notion of actual infinity is inconsistent, because the set of all (finite) natural…
We define the indefinite logarithm [log x] of a real number x>0 to be a mathematical object representing the abstract concept of the logarithm of x with an indeterminate base (i.e., not specifically e, 10, 2, or any fixed number). The…
In this article we consider alternative definitions-descriptions of a set being Infinite within the primitive Axiomatic System of Zermelo.
A refinement of the classic equivalence relation among Cauchy sequences yields a useful infinitesimal-enriched number system. Such an approach can be seen as formalizing Cauchy's sentiment that a null sequence "becomes" an infinitesimal. We…
The mathematical study of infinity seems to have the ability to transport the mind to lofty and unusual realms. Decades ago, I was transported in this way by Rudy Rucker's book Infinity and the Mind. Despite much subsequent learning and…
Philosopher Benardete challenged both the conventional wisdom and the received mathematical treatment of zero, dot, nine recurring. An initially puzzling passage in Benardete on the intelligibility of the continuum reveals challenging…
Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not…
A new computational methodology for executing calculations with infinite and infinitesimal quantities is described in this paper. It is based on the principle `The part is less than the whole' introduced by Ancient Greeks and applied to all…
In this paper, we present a comprehensive system for the treatment of the topic of limits--conceptually, computationally, and formally. The system addresses fundamental linguistic flaws in the standard presentation of limits, which attempts…
This book is the final version of a course on algorithmic information theory and the epistemology of mathematics and physics. This is camera-ready copy prepared for publication as a book, but at the last minute I decided to publish it…
We discuss the repercussions of the development of infinitesimal calculus into modern analysis, beginning with viewpoints expressed in the nineteenth and twentieth centuries and relating them to the natural cognitive development of…
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…
We contribute to the lively debate in current scholarship on the Leibnizian calculus. In a recent text, Arthur and Rabouin argue that non-Archimedean continua are incompatible with Leibniz's concepts of number, quantity and magnitude. They…
We explore the issue of providing a foundational framework for Leibnizian infinitesimals in the light of modern standard and nonstandard approaches. We outline a trichotomy of ordinals, cardinals and ringinals as a historiographic tool. A…
The cognitive framework of conceptual spaces [3] provides geometric means for representing knowledge. A conceptual space is a high-dimensional space whose dimensions are partitioned into so-called domains. Within each domain, the Euclidean…
We reconsider the classical equality 0.999. .. = 1 with the tool of circular words, that is: finite words whose last letter is assumed to be followed by the first one. Such circular words are naturally embedded with algebraic structures…