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The concepts of infinity and infinitesimal in mathematics date back to anciens Greek and have always attracted great attention. Very recently, a new methodology has been proposed by Sergeyev for performing calculations with infinite and…

The aim of this paper is to highlight a hitherto unknown computational aspect of Nonstandard Analysis. Recently, a number of nonstandard versions of Goedel's system T have been introduced ([2,9,12]), and it was shown in [26] that the…

It is well known that the set of algebraic numbers (let us call it $A$) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using \G1-based…

A recently developed computational methodology for executing numerical calculations with infinities and infinitesimals is described in this paper. The developed approach has a pronounced applied character and is based on the principle `The…

We develop some nonstandard techniques for bornological and coarse spaces. We first generalise the notion of bornology to prebornology, which better fits to coarse spaces. We then give nonstandard characterisations of some basic large-scale…

This is a Research and Instructional Development Project from the U. S. Naval Academy. In this monograph, the basic methods of nonstandard analysis for n-dimensional Euclidean spaces are presented. Specific rules are deveoped and these…

It is often claimed that analysis with infinitesimals requires more substantial use of the Axiom of Choice than traditional elementary analysis. The claim is based on the observation that the hyperreals entail the existence of nonprincipal…

In this article, we will introduce methods of non-standard analysis into projective geometry. Especially, we will analyze the properties of a projective space over a non-Archimedean field. Non-Archimedean fields contain numbers that are…

Nonstandard analysis is very complex, so finding a simple description of infinitesimal points will be useful. In this paper, ultrafilters as infinitesimal points in a topological space will be proposed, and some topological concepts is…

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…

This is a research announcement on what is best termed `nonlocal' methods in mathematics. (This is not to be confused with global analysis.) The nonlocal formulation of physics in \cite{principia} points to a fresh viewpoint in mathematics:…

Non-Archimedean mathematics (in particular, nonstandard analysis) allows to construct some useful models to study certain phenomena arising in PDE's; for example, it allows to construct generalized solutions of differential equations and…

Extending the classical result that the roots of a polynomial with coefficients in $\mathbf{C}$ are continuous functions of the coefficients of the polynomial, nonstandard analysis is used to prove that if $\mathcal{F} = \{f_{\lambda}…

Non-linear gravitational collapse introduces non-Gaussian statistics into the matter fields of the late Universe. As the large-scale structure is the target of current and future observational campaigns, one would ideally like to have the…

In his remarkable paper Formalism64, Robinson defends his philsophocal position as follows: (i) Any mention of infinite totalities is literally meaningless. (ii) We should act as if infinite totalities really existed. Being the originator…

In the present paper, we propose a new axiomatic approach to nonstandard analysis and its application to the general theory of spatial structures in terms of category theory. Our framework is based on the idea of internal set theory, while…

In La G\'eom\'etrie, Descartes proposed a balance between geometric constructions and symbolic manipulation with the introduction of suitable ideal machines. In modern terms, that is a balance between analog and symbolic computation.…

We determine sufficient structure for an elementary topos to emulate E. Nelson's Internal Set Theory in its internal language, and show that any topos satisfying the internal axiom of choice occurs as a universe of standard objects and…

Non standard analysis is an area of Mathematics dealing with notions of infinitesimal and infinitely large numbers, in which many statements from classical analysis can be expressed very naturally. Cheap non-standard analysis introduced by…

I present a novel mathematical technique for dealing with the infinities arising from divergent sums and integrals. It assigns them fine-grained infinite values from the set of hyperreal numbers in a manner that refines the standard…