Related papers: 19th century real analysis, forward and backward
This article presents a novel interpretation of quantum mechanics. It extends the meaning of ``measurement'' to include all property-indicating facts. Intrinsically space is undifferentiated: there are no points on which a world of locally…
Realism -- the idea that the concepts in physical theories refer to 'things' existing in the real world -- is introduced as a tool to analyze the status of the wave-function. Although the physical entities are recognized by the existence of…
This article is devoted to completing some aspects of the classical Cauchy-Lipschitz (or Picard-Lindel\"of) theory for general nonlinear systems posed on time scales, that are closed subsets of the set of real numbers. Partial results do…
The need to describe abrupt changes or response of nonlinear systems to impulsive stimuli is ubiquitous in applications. Also the informal use of infinitesimal and infinite quantities is still a method used to construct idealized but…
We study a real valued propositional logic with unbounded positive and negative truth values that we call R-valued logic. Such logic slightly extends continuous propositional logic which, in turn, builds on Lukasiewicz many-valued logic.…
Aristotelian assertoric syllogistic, which is currently of growing interest, has attracted the attention of the founders of modern logic, who approached it in several (semantical and syntactical) ways. Further approaches were introduced…
We study the asymptotic behavior of solutions of the Cauchy problem associated to a quantitative genetics model with a sexual mode of reproduction. It combines trait-dependent mortality and a nonlinear integral reproduction operator "the…
We give a number of formal proofs of theorems from the field of computable analysis. Many of our results specify executable algorithms that work on infinite inputs by means of operating on finite approximations and are proven correct in the…
The aim of this paper is to show Cauchy-Kowalevski and Holmgren type theorems with infinite number of variables. We adopt von Koch and Hilbert's definition of analyticity of functions as monomial expansions. Our Cauchy-Kowalevski type…
There is a problem with the foundations of classical mathematics, and potentially even with the foundations of computer science, that mathematicians have by-and-large ignored. This essay is a call for practicing mathematicians who have been…
The concept of determinism for a classical system is interpreted as the requirement that the solution to the Cauchy problem for the equations of motion governing this system be unique. This requirement is generally assumed to hold for all…
In this paper we offer a reconstruction of the evolution of Leibniz's thought concerning the problem of the infinite divisibility of bodies, the tension between actuality, unassignability and syncategorematicity, and the closely related…
We introduce the ring of Fermat reals, an extension of the real field containing nilpotent infinitesimals. The construction takes inspiration from Smooth Infinitesimal Analysis (SIA), but provides a powerful theory of actual infinitesimals…
We present a detailed and elementary construction of the real numbers from the rational numbers a la Bourbaki. The real numbers are defined to be the set of all minimal Cauchy filters in $\mathbb{Q}$ (where the Cauchy condition is defined…
Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past…
The stipulation that no measurable quantity could have an infinite value is indispensable in physics. At the same time, in mathematics, the possibility of considering an infinite procedure as a whole is usually taken for granted. However,…
This is the first volume of a textbook for a two-semester course in mathematical analysis. This first volume is about analysis of functions of a single variable. The topics covered include completeness axiom, Archimedean property,…
This commentary considers non-standard analysis and a recently introduced computational methodology based on the notion of \G1 (this symbol is called \emph{grossone}). The latter approach was developed with the intention to allow one to…
We consider a class of Fuchsian equations that, for instance, describes the evolution of compressible fluid flows on a cosmological spacetime. Using the method of lines, we introduce a numerical algorithm for the singular initial value…
We prove Cauchy's formula for repeated integration on time scales. The obtained relation gives rise to new notions of fractional integration and differentiation on arbitrary nonempty closed sets.