Infinitesimals
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Recent papers in Infinitesimals
We develop the integral calculus for quasi-standard smooth functions defined on the ring of Fermat reals. The approach is by proving the existence and uniqueness of primitives. Besides the classical integral formulas, we show the... more
Resumo: Neste artigo analisamos as críticas apresentadas por George Berkeley, em The analyst (1734), ao método das fluxões e à inconsistência intrínseca à noção de infinitésimo do cálculo diferencial e integral, introduzido por Isaac... more
Intuition can be seen as the primordial and pre-verbal faculty by means of which the mind gains immediate epistemic access to the phenomena. The concept of structural intuition is premised on the basic principle that an infallible... more
C omo anunciamos en el primer número del volumen 46 de la Revista Latinoamericana de Filosofía, se publica aquí la segunda parte del "Dossier Leibniz", con trabajos que abordan los antecedentes escolásticos de la teodicea leibniziana y la... more
Resumen: En este artículo hago una nueva lectura de un escrito poco conocido de George Berkeley: Of Infinites. Hasta ahora se ha leído de manera parcial, ya sea destacando las aportaciones matemáticas, mientras se resta importancia a la... more
We review the theory of Fermat reals and Fermat extensions, a relatively new theory of nilpotent infinitesimals which does not need any background in Mathematical Logic. We focus on some differences from Nonstandard Analysis and Synthetic... more
This draft presents an outline of the application of the method of infinitesiamls and infinite quantities such as it is developed by Leibniz in his treatise De quadratura arithmetica circuli. Our treatment is based mainly on Konbloch's... more
In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow... more
Through Zeno of Elea, as a representant of an early ancient thinking, there is revealed the reception of zero in the ancient culture in the thesis. The main part is devoted to the conception of infinity, infinite division and nothingness... more
As Weyl was interested in infinitesimal analysis and for some years embraced Brouwer's intuitionism, which he continued to see as an ideal even after he had convinced himself that it is a practical necessity for science to go beyond... more
Can knowledge have a history? Plato's emphatic differentiation between episteme (ἐπιστήμη) and doxa (δόξα) clearly points to an ahistoricity of episteme, which, unlike doxa (an opinion or a belief) refers to eternal ideas or to pure being... more
Several ways used to rank countries with respect to medals won during Olympic Games are discussed. In particular, it is shown that the unofficial rank used by the Olympic Committee is the only rank that does not allow one to use a... more
ABSTRACT This is a proof to show that 0.9999... is not = 1 (where '...' means recurring)1. This is all that is necessary to refute the class of problems known generally as Zeno’s Paradox2. There are only two parts to this proof.... more
We introduce a ring of the so-called Fermat reals, which is an extension of the real field containing nilpotent infinitesimals. The construction is inspired by Smooth Infini-tesimal Analysis (SIA) and provides a powerful theory of actual... more
Numerous problems arising in engineering applications can have several objectives to be satisfied. An important class of problems of this kind is lexicographic multi-objective problems where the first objective is incomparably more... more
The original idea of aporia in the philosophy of Plato and Aristotle points to a limiting experience in thought, that is, to the hopelessness and uncertainty in the thought process, in which one desperately looks for a solution. However,... more
The Koch snowflake is one of the first fractals that were mathematically described. It is interesting because it has an infinite perimeter in the limit but its limit area is finite. In this paper, a recently proposed computational... more
The edition of a manuscript letter from Giuseppe Veronese to Giovanni Vailati is the occasion to discuss the relations between two alternative proposals on the didactics of geometry made at the beginning of the 20th century in Italy. The... more
A well-known drawback of algorithms based on Taylor series formulae is that the explicit calculation of higher order derivatives formally is an over-elaborate task. To avoid the analytical computation of the successive derivatives,... more
the basic idea of the method of indivisibles is the comparison of "indivisibles" that in some way makes up the figures whose size is compared. the quadrature of the parable with the method of mechanics by Archimedes prepares the geometry... more
Standard calculus is developed using standard real numbers R, usually plotted on the number line. In Leibniz’s vision of calculus, there are infinitesimally small numbers dx, that is |dx| < 1/n for all n = 1, 2, 3, . . ., yet dx ≠ 0. This... more
Draft 2. Versión mejorada del Draft 1. A publicarse como parte de los trabajos presentados en las XIII Jornadas de la Cátedra Leibniz, en el marco del XVIII Congreso Interamericano de filosofía, Bogotá, Colombia, 15-18 de octubre de 2019.... more
The necessity to find the global optimum of multiextremal functions arises in many applied problems where finding local solutions is insufficient. One of the desirable properties of global optimization methods is strong homogeneity... more
Many biological processes and objects can be described by fractals. The paper uses a new type of objects – blinking fractals – that are not covered by traditional theories considering dynamics of self-similarity processes. It is shown... more
To discover derivatives, Pierre de Fermat used to assume a non zero increment h in the incremental ratio and, after some calculations, to set h = 0 in the final result. This methods, which sounds as inconsistent, can be perfectly... more
In standard probability theory, probability zero is not the same as impossibility. If an experiment has infinitely many possible outcomes, all equally likely, then all the outcomes must have probability zero, but one of them must occur... more
Standard calculus is developed using standard real numbers R, usually plotted on the number line. In Leibniz's vision of calculus, there are infinitesimally small numbers dx, that is |dx| < for all n = 1, 2, 3, . . . , yet dx ≠ 0. This... more
In this paper we will try to explain how Leibniz justified the idea of an exact arithmetical quadrature. We will do this by comparing Leibniz’s exposition with that of John Wallis. In short, we will show that the idea of exactitude in... more