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 flexibility of the Cartesian closed framework of Fermat spaces to deal with infinite dimensional integral operators. The total order relation between scalars permits to prove several classical order properties of these integrals and to study multiple integrals on Peano-Jordan-like integration domains.

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 Newton. Procuramos mostrar que as críticas de Berkeley não eram de todo infundadas, uma vez que foram necessários quase duzentos anos para que viesse a ser introduzida por Karl Weierstrass a definição rigorosa de limite, que propiciou uma solução para o problema dos infinitésimos. São mencionadas ainda duas outras teorias contemporâneas, com abordagens distintas para a solução da questão do infinitésimo: a análise não-standard de Abraham Robinson e o cálculo diferencial paraconsistente proposto por Newton da Costa. Apesar de serem citados alguns autores importantes para o desenvolvimento do cálculo, este artigo não se propõe a analisar suas obras e não pretende apresentar uma história do cálculo diferencial e integral. Palavras-chave: George Berkeley, método das fluxões, infinitésimo, cálculo diferencial paraconsistente.

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 resonance exists between internally intuitive and externally phenomenal structures; it generally proves to be compositional in that it unifies mathematical formality with me- chanical and material phenomenality. The present work is an attempt to extend the doctrine of structural intuition with regard to its epis- temological potential to an epistemic-structural intuition and, accord- ingly, to substantiate it as the characteristic trait of axiomatic intui- tions in the early modern mathematical and material sciences.


Intuition lässt sich als ein primordiales und vorsprachliches Ver- mögen betrachten, anhand dessen der Geist einen unmittelbaren epistemischen Zugang zu den Phänomenen erhält. Die strukturelle Intuition baut auf dem Grundprinzip auf, dass zwischen innerlich intuitiven und äußerlich phänomenalen Strukturen eine unfehlbare Resonanz besteht; sie erweist sich gemeinhin als kompositorisch, in- dem sie die mathematische Formhaftigkeit mit der mechanischen und materiellen Phänomenalität vereinheitlicht. Die vorliegende Abhand- lung ist ein Versuch, die Lehre der strukturellen Intuition hinsichtlich ihres erkenntnistheoretischen Potenzials auf eine epistemisch-struk- turelle Intuition zu erweitern und sie demnach als den Grundzug axiomatischer Intuitionen in den frühneuzeitlichen mathematischen und materiellen Wissenschaften zu begründen.

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 recepción de la que fueron objeto sus ideas en el pensamiento de Charles S. Peirce y Hermann Cohen. Al igual que los artículos de la primera parte, las versiones prelimi-nares de cada uno de ellos se discutieron en el simposio "Leibniz: ciencia, lógica y metafísica", celebrado el 12 y el 13 de diciembre de 2018 en la sede del Centro de Investigaciones Filosóficas (CIF). El primer artículo de esta segunda parte, "La metafísica de la creación de Duns Escoto: el entendi-miento divino como locus de los posibles", de Olga L. Larre, expone la doctrina escotista de la creación del mundo, con el objetivo de establecer las bases para un estudio comparativo con la teodicea leibniziana, que exhibe notables coincidencias con la metafísica de Duns Escoto. Por su parte, Javier Legris, en "La tradición del
http://rlfcif.org.ar/index.php/RLF/issue/current

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 parte filosófica o, lo más habitual, minimizando la parte matemática para resaltar sólo su incipiente inmaterialismo. Ambas lecturas han sido perniciosas para la comprensión cabal del texto, lo que ha traído como consecuencia que se siga minusvalorando su importancia y, por consiguiente, no se le estudie como debiera. Frente a las lecturas tradicionales realizo una que busca destacar, así como explicar, tanto los aspectos filosóficos como los matemáticos, con el fin de mostrar que su riqueza y complejidad le merecen un lugar importante dentro de la obra berkeleyana. /
Abstract: In this paper I propose a new reading of a little known George Berkeley´s work Of Infinites. Hitherto, the work has been studied partially, or emphasizing only the mathematical contributions, downplaying the philosophical aspects, or minimizing mathematical issues taking into account only the incipient immaterialism. Both readings have been pernicious for the correct comprehension of the work and that has brought as a result that will follow underestimated its importance, and therefore will not study as should be. Against traditional readings I make one that stand out both philosophical and mathematical aspects, with the purpose to show that richness and complexity of the work deserve that it has an special place within
Berkeley´s works.

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 Differential Geometry using the viewpoint of intuitive interpretation and applicability in Physics. Finally, we state some open problems.

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 investigations about the leibnizian treatise.

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 one to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor’s and non-standard analysis views and is based on the Euclid’s Common Notion no. 5 “The whole is greater than the part” applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses a computational device called the Infinity Computer (patented in USA and EU) working numerically (recall that traditional theories work with infinities and infinitesimals only symbolically) with infinite and infinitesimal numbers that can be written in a positional numeral system with an infinite radix. It is argued that numeral systems involved in computations limit our capabilities to compute and lead to ambiguities in theoretical assertions, as well. The introduced methodology gives the possibility to use the same numeral system for measuring infinite sets, working with divergent series, probability, fractals, optimization problems, numerical differentiation, ODEs, etc. (recall that traditionally different numerals 1; @0; !, etc. are used in different situations related to infinity). Numerous numerical examples and theoretical illustrations are given. The accuracy of the achieved results is continuously compared with those obtained by traditional tools used to work with infinities and infinitesimals. In particular, it is shown that the new approach allows one to observe mathematical objects involved in the Hypotheses of Continuum and the Riemann zeta function with a higher accuracy than it is done by traditional tools. It is stressed that the hardness of both problems is not related to their nature but is a consequence of the weakness of traditional numeral systems used to study them. It is shown that the introduced methodology and numeral system change our perception of the mathematical objects studied in the two problems.

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 in Zeno's paradoxes, including a detailed analysis and an attempt to interpret using a formal logical system. Subsequently Zeno´s approach is confronted with modern physics (especially with the theory of relativity) and there is also defined his position within the rationalist-empirical dichotomy and his uncompromisingness towards science in the thesis.

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 intuitionistic mathematics, this note presents some remarks on infinitesimals from a Brouwerian perspective. After an introduction and a look at Robinson's and Nelson's approaches to classical nonstandard analysis, three desiderata for an intuitionistic construction of infinitesimals are extracted from Brouwer's writings. These cannot be met, but in explicitly Brouwerian settings what might in different ways be called approximations to infinitesimals have been developed by early Brouwer, Vesley, and Reeb. I conclude that perhaps Reeb's approach, with its Brouwerian motivation for accepting Nelson's classical formalism, would have suited Weyl best.

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 which is not subjected to temporal or historical transformations. In my lecture, I try to show how the original ahistoricity of episteme, on which the philosophical and scientific epistemologies are based, was always presupposed by a secure epistemological referentiality – i.e. by a secure or safe epistemological access to pure ideas. Already in antiquity we can recognize many strategic preferences in terms of subject's epistemological reference to the object of knowledge. In comparison with Plato, Aristotle attempted to reverse the epistemological referentiality, so that scientific knowledge refers to natural phenomena, i.e. to sensible objects. With such an epistemological referentiality of knowledge, one clearly faces the inconsistencies of being that are inherent in natural phenomena (and which Plato strategically avoided through exclusive epistemological references to eternal ideas). The inconsistencies of objective being led to phenomenal aporias that characterized many of the Aristotelian discourses within the framework of his Metaphysics and Natural Philosophy. The medieval scholasticism inherited the ancient aporias that found various expressions in the aporetic discourses of the medieval natural philosophy (philosophia naturalis). The early modern age of Descartes and Newton tried to terminate the aporetics, which were characteristically represented in the medieval discourses on mechanical phenomena such as impetus, gravity, space, time, infinitesimal etc., the major tool for this termination being the strategic axiomatization of philosophical and scientific knowledge. Consequently, the axioms should establish themselves as stable and final cognitions and thus prove to be ahistorical. However, the Cartesian-epistemological turn had led to further problems of epistemological referentiality. The epistemological-referential access to objects seemed once again to be hampered by many new obstacles, as represented by mathematical formalism, pictorial representations of natural phenomena, and by historical contexts and paradigms of science. The problems of epistemological referentiality, which had emerged at the dawn of early modern age in mathematical and mechanical axiomatization of natural-philosophical knowledge, seemed to re-establish the historicity of scientific knowledge. In my lecture I try to explain – by means of a few examples from mechanics and optics – the emergence of the historicity of scientific knowledge in modern age.

Soundcloud Link: https://soundcloud.com/user-725946454/the-historicity-of-knowledge

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 numerical counter for ranking – this rank uses the lexicographic ordering to rank countries: one gold medal is more precious than any number of silver medals and one silver medal is more precious than any number of bronze medals. How can we quantify what do these words, more precious, mean? Can we introduce a counter that for any possible number of medals would allow us to compute a numerical rank of a country using the number of gold, silver, and bronze medals in such a way that the higher resulting number would put the country in the higher position in the rank? Here we show that it is impossible to solve this problem using the positional numeral system with any finite base. Then we demonstrate that this problem can be easily solved by applying numerical computations with recently developed actual infinite numbers. These computations can be done on a new kind of a computer – the recently patented Infinity Computer. Its working software prototype is described briefly and examples of computations are given. It is shown that the new way of counting can be used in all situations where the lexicographic ordering is required.

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.

Firstly, I represent 1 as 1.000...  and compare the finite decimal representation of 1.000...0  with its finite counterpart 0.999...9  (where the rightmost digit indicates a finite recurrence to an equivalent arbitrary accuracy) and show they can never be equal if the recurring digit recurs finitely.

Then I show that the number system breaks down if the recurring digit recurs infinitely because the interval between all numbers becomes zero and a Limit at infinity cannot and does not exist.

I conclude that recurring digits and asserting 0.999… = 1 (etc) are a mathematically incorrect non sequiturs arising from a failure to acknowledge a limitation of the decimal notation number system.

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 infinitesimals without any background in mathematical logic. In particular, in contrast to SIA, which admits models in intuitionistic logic only, the theory of Fermat reals is consistent with the classical logic. We face the problem of deciding whether or not a product of powers of nilpotent infinitesimals vanishes, study the identity principle for polynomials, and discuss the definition and properties of the total order relation. The construction is highly constructive, and every Fermat real admits a clear and order-preserving geometrical representation. Using nilpotent infinitesimals, every smooth function becomes a polynomial because the remainder in Taylor's formulas is now zero. Finally, we present several applications to informal classical calculations used in physics, and all these calculations now become rigorous, and at the same time, formally equal to the informal ones. In particular, an interesting rigorous deduction of the wave equation is given, which clarifies how to formalize the approximations tied with Hooke's law using the language of nilpotent infinitesimals.

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 important than the second one which, in its turn, is incomparably more important than the third one, etc. In this paper, Lexicographic Multi-Objective Linear Programming (LMOLP) problems are considered. To tackle them, traditional approaches either require solution of a series of linear programming problems or apply a scalarization of weighted multiple objectives into a single-objective function. The latter approach requires finding a set of weights that guarantees the equivalence of the original problem and the single-objective one and the search of correct weights can be very time consuming. In this work a new approach for solving LMOLP problems using a recently introduced computational methodology allowing one to work numerically with infinities and infinitesimals is proposed. It is shown that a smart application of infinitesimal weights allows one to construct a single-objective problem avoiding the necessity to determine finite weights. The equivalence between the original multi-objective problem and the new single-objective one is proved. A simplex-based algorithm working with finite and infinitesimal numbers is proposed, implemented, and discussed. Results of some numerical experiments are provided.

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, Aristotle points out that the hopelessness in thinking is still due to the aporetic nature of the object: “For those who wish to get clear of difficulties it is advantageous to state the difficulties well; for the subsequent free play of thought implies the solution of the previous difficulties, and it is not possible to untie a knot which one does not know. But the difficulty of our thinking points to a knot in the object.” (Aristotle, Metaphysics, B 1, 995 a24 – b4). This observation clearly suggests that Aristotle’s notion of aporetics was ultimately – in the final instance – premised on the aporia of the object. The Aristotelian emphasis on objective aporia was summarily rejected by the Cartesian modernity that restricted the aporetics within the strict framework of epistemology. The medieval-scholastic philosophia naturalis was characterised from the outset by the incessant discourses on natural phenomena such as gravity, void, impetus, infinitesimal, space, time etc. In fact, these aporetics or aporetic discourses owe the objective aporia, as represented in the above-mentioned natural phenomena, their characteristic unresolvability and perseverance. On the basis of the apodictic certainty of axioms, Cartesian modernity aspired to overcome once and for all the prevailing aporetics of the scholastic natural philosophy. Early modern mechanical philosophy was built on axioms, which in reality veiled the predominant aporetic discourses of late medieval natural philosophy or buried them beneath apodictic first principles. This resulted, as I would argue, from the strategic suppression of the objective-phenomenal aporia in favor of the apodicticity of subjective a priori knowledge. In my lecture I try to show how the historic- paradigmatic transition from aporetics to apodeictic scientific discourses – during the emergence of early modern mechanical philosophy – turns out to be a scientifically problematic or even failed undertaking. The rehabilitation and legitimization of the objective aporia could make it possible that the certainty or apodicticity of scientific knowledge can emerge from an aporetic discourse itself, or even be presupposed by it.

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 methodology allowing one to execute numerical computations with infinities and infinitesimals is applied to study the Koch snowflake at infinity. Numerical computations with actual infinite and infinitesimal numbers can be executed on the Infinity Computer being a new supercomputer patented in USA and EU. It is revealed in the paper that at infinity the snowflake is not unique, i.e., different snowflakes can be distinguished for different infinite numbers of steps executed during the process of their generation. It is then shown that for any given infinite number n of steps it becomes possible to calculate the exact infinite number, Nn, of sides of the snowflake, the exact infinitesimal length, Ln, of each side and the exact infinite perimeter, Pn, of the Koch snowflake as the result of multiplication of the infinite Nn by the infinitesimal Ln. It is established that for different infinite n and k the infinite perimeters Pn and Pk are also different and the difference can be infinite. It is shown that the finite areas An and Ak of the snowflakes can be also calculated exactly (up to infinitesimals) for different infinite n and k and the difference An − Ak results to be infinitesimal. Finally, snowflakes constructed starting from different initial conditions are also studied and their quantitative characteristics at infinity are computed.

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 analysis of the discussion between Vailati and Veronese shows that there was a constructive dialogue between some members of Peano’s ‘logical’ school and Veronese’s school based on ‘projective geometry’, notwithstanding the well-known and hostile criticism expressed by Peano on Veronese’s main work on geometry. The paper shows, on the basis of the edited letter, that certain traditional views on Italian mathematics at the beginning of the 20th century should be partially revised. The radical opposition between the two school is usually justified by reference to a previous controversy between Segre, Peano and Veronese that took place on Peano’s Revue des Mathématiques in 1891. Yet, the author claims that the friendly and scientifically interesting exchange between Veronese and Vailati is a counterexample to the idea that the two schools were so radically opposed as to avoid any communication.

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, numeric and automatic differentiation are usually used. A recent alternative to these techniques is based on the calculation of higher derivatives by using the Infinity Computer—a new computational device allowing one to work numerically with infinities and infinitesimals. Two variants of a one-step multi-point method closely related to the classical Taylor formula of order three are considered. It is shown that the new formula is order three accurate, though requiring only the first two derivatives of y(t) (rather than three if compared with the corresponding Taylor formula of order three). To get numerical evidence of the theoretical results, a few test problems are solved by means of the new methods and the obtained results are compared with the performance of Taylor methods of order up to four.

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 of the indivisibles of the t4th century..pag.22

This book is intended primary to serve as a study guide to the great con-cept of infinity that has tormented human thought for thousands of years. Infinitesimals are infinitely small quantities, which in the words of Ber-noulli, are so small that “if a quantity is increased or decreased by an infinitesimal, then the quantity is neither increased nor decreased. Mathematicians approached the concept of infinitesimals with various methods, the Greek method of exhaustion, the method of indivisibles of the early modern ages, the geometry of indivisibles, the differentials of Leibnitz (that are intuitive infinitesimals). Their history is a synthesis of three parts, of philosophical, mathematical and physical part. In this book we will see some basic elements of these three stories.

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 requires an extension of the standard reals R called the hyperreal numbers, denoted often by R*. Then the standard real numbers R form a proper subset of the hyperreal numbers R*, that is, R ⊂ R* with R ≠ R*. Leibniz himself could not formally establish the theory of hyperreal numbers, which was done much later in the 1960s by Abraham Robinson, using a more general theory of mathematical logic called Nonstandard Analysis, an advanced subject area.

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. ¿Qué es una ficción en matemáticas? Leibniz y los infinitesimales como ficciones. Oscar M. Esquisabel 1. Introducción El objeto de este trabajo es abordar la cuestión de los infinitesimales leibnizianos desde el punto de vista de las ficciones matemáticas. Para ello, el marco general que adoptaremos como hipótesis de trabajo son las concepciones de Leibniz acerca de la noción de conocimiento simbólico. De este modo, consideraremos que, para Leibniz, la ficción matemática es un tipo de noción o concepto simbólico 1 que tiene como características o notas fundamentales las siguientes: 2 1. En principio, se trata de un concepto vacuo, sin denotación o sin idea que le corresponda. En la clasificación leibniziana de nociones, le corresponde el estatus de...

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 meaning that a method produces the same sequences of points where the objective function is evaluated independently both of multiplication of the function by a scaling constant and of adding a shifting constant. In this paper, several aspects of global optimization using strongly homogeneous methods are considered. First, it is shown that even if a method possesses this property theoretically, numerically very small and large scaling constants can lead to illconditioning of the scaled problem. Second, a new class of global optimization problems where the objective function can have not only finite but also infinite or infinitesimal Lipschitz constants is introduced. Third, the strong homogeneity of several Lipschitz global optimization algorithms is studied in the framework of the Infinity Computing paradigm allowing one to work numerically with a variety of infinities and infinitesimals. Fourth, it is proved that a class of efficient univariate methods enjoys this property for finite, infinite and infinitesimal scaling and shifting constants. Finally, it is shown that in certain cases the usage of numerical infinities and infinitesimals can avoid ill-conditioning produced by scaling. Numerical experiments illustrating theoretical results are described.

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 that both traditional and blinking fractals can be successfully studied by a recent approach allowing one to work numerically with infinite and infinitesimal numbers. It is shown that blinking fractals can be applied for modeling complex processes of growth of biological systems including their season changes. The new approach allows one to give various quantitative characteristics of the obtained blinking fractals models of biological systems.

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 formalized with the Fermat-Reyes theorem about existence and uniqueness of a smooth incremental ratio. In the present work, we will introduce the cartesian closed category where to study and prove this theorem and describe in general the Fermat method. The framework is the theory of Fermat reals, an extension of the real field containing nilpotent infinitesimals which does not need any knowledge of mathematical logic. This key theorem will be essential in the development of differential and integral calculus for smooth functions defined on the ring of Fermat reals and also for infinite dimensional operators like derivatives and integrals.

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 nonetheless.  Many have suggested that this should not be so—that probabilities (ontic or epistemic, depending on the author) should be regular:  Only impossible events should have probability zero and only necessary or certain events should have probability one.  This can be arranged if we allow infinitesimal probabilities, but it turns out that infinitesimals do not solve all of the problems.  I will show that regular probabilities cannot be translation-invariant, even for bounded and disjoint events.  Hence, for various events confined to finite space and time (e.g., dart throws and vacuum fluctuations), regular chances cannot be determined by space-time invariant physical laws, and regular credences cannot satisfy seemingly reasonable symmetry principles.  Moreover, these examples are immune to the main objections against Timothy Williamson’s infinite coin flip examples.

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 requires an extension of the standard reals R called the hyperreal numbers, denoted often by R*. Then the standard real numbers R form a proper subset of the hyperreal numbers R*, that is, R ⊂ R* with R ≠ R*. Leibniz himself could not formally establish the theory of hyperreal numbers, which was done much later in the 1960s by Abraham Robinson, using a more general theory of mathematical logic called Nonstandard Analysis, an advanced subject area.

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 matters of quadratures relies on two fundamental requisites that, according to Leibniz, the infinite series have, namely, that of regularity and that of completeness. In the first part of this paper, we will go deeper into three main features of Leibniz’s method, that is: it is an infinitesimal method, it looks for an arithmetical quadrature and it proposes a result that is not approximate, but exact. After that, we will deal with the requisite of the regularity of the series, pointing out that, unlike the inductive method proposed by Wallis, Leibniz propounded some sort of intellectual recognition of what is invariant in the series. Finally, we will consider the requisite of completeness of the series. We will see that, although both Wallis and Leibniz introduced the supposition of completeness, the German thinker went beyond the English mathematician, since he recognized that it is not necessary to look for a number for the quadrature of the circle, given that we have a series that is equal to the area of that curvilinear figure.