History of Continuity and Infinitesimals

The distinction between the discrete and the continuous lies at the heart of mathematics. Discrete mathematics (arithmetic, algebra, combinatorics, graph theory, cryptography, logic) has a set of concepts, techniques, and application areas largely distinct from continuous mathematics (traditional geometry, calculus, most of functional analysis, differential equations, topology). The interaction between the two – for example in computer models of continuous systems such as fluid flow – is a central issue in the applicable mathematics of the last hundred years. This article explains the distinction and why it has proved to be one of the great organizing themes of mathematics.

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.

Reality Through Fiction
: Leibniz on the nature of infinitesimals

Shinji IKEDA

Focusing on the “fictionalistic” interpretations of infinitesimals, we study in this article Leibniz’s conception of the nature of infinitesimals. First, we check his use and understanding of infinitesimals in the Nova Methodus of 1684. We find here that Leibniz lacks the definition of infinitesimals and several other fundamental ideas to justify his use of infinitesimals in his differential calculus. Secondly, we summarize recent interpretations of Leibniz’s infinitesimals. Surveying recent articles on this topic, and analyzing Leibniz’s letters to Varignon, we assert that fictionalism is the standard and natural view on the nature of Leibniz’s infinitesimals. Since fictionalism is not intended to give the justification and the rigorous foundation of the infinitesimal calculus, we then verify the way Leibniz justifies his use of infinitesimals, firstly in his largest mathematical monograph De Quaduratura arithmetica of 1675 and secondly in the “Justification du Calcul des infinitesimales par celuy de l’Algebre ordinaire” of 1702. To justify his calculus, he also needs to appeal to the law of continuity, which is derived from his considerations on the collision problem of bodies. We pay attention that the law of continuity needs more mathematical tools such as the convergence condition and more metaphysical foundations such as the Principle of Sufficient Reason. Finally, we discuss the meaning of the “syncategorematic actual infinity”, which is used to characterize Leibniz’s thought that “matter is actually infinitely divided”. Through this article, we conclude that Leibniz’s fictionalistic view of infinitesimals is not only coherent with the rigorous demonstration by the method of exhaustion, but also coherent with his understanding of the material world.

LEIBNIZ: THE INFINITE IN THE ORGANIC BODY
Abstract: The organic body is conceived as a unit and can be thought as a corporeal substance. The relationship between soul and body is not just a pre-established harmony relationship between a dominant monad and endless monads, but also a substantial bond between monads.
The substantial union and the pre-established harmony are two possible and compatible explanations about the same. Although the "bond" explanation only appears at the Leibniz— Des Bosses correspondence, the major theses about the body were already present in the philosophy of Leibniz since 1686. 
Keywords: substance, body, organic body, matter, monad 

Resumo: O corpo orgânico é concebido como uma unidade e pode ser pensado como substância corpórea. A relação entre alma e corpo não é apenas uma relação de harmonia preestabelecida entre uma mônada dominante e infinitas mônadas, mas também um vínculo substancial entre mônadas. A união substancial e a harmonia preestabelecida são duas explicações possíveis e compatíveis a respeito do mesmo. Embora a explicação do " vínculo " só compareça na correspondência entre Leibniz e Des Bosses, as grandes teses a respeito do corpo já estavam dadas na filosofia de Leibniz desde 1686.

“O infinito! Nenhuma outra questão tem tocado tão profundamente o espírito do homem; nenhuma outra idéia tem estimulado de forma tão frutífera seu intelecto; no entanto, nenhum outro conceito permanece com tanta necessidade de esclarecimento que o de infinito” (Hilbert 1926, 163)

As noções de infinito muitas vezes nos levam a colher frutos do paradoxo. Sobre isto, este trabalho pretende mostrar as várias características, assumidas e construídas pelos pensadores da filosofia e da matemática, ao longo da história, a respeito dos infinitésimos e infinitos.
Não há como separar absolutamente as noções de infinitamente grande e infinitamente pequeno, chamadas, respectivamente, de Infinito e Infinitésimo. Os dois conceitos estão atrelados por dois outros: o infinito não-pensável – definido pela impossibilidade de se pensar infinitos elementos - e o conceito de medida ou tamanho. Por esse caminho confuso e, ao mesmo tempo, iluminador, veremos que, mesmo depois da aritmetização da Análise Matemática e do Cálculo Diferencial e Integral no século XIX, que adotou majoritariamente a Teoria dos Limites deixando de lado a interpretação infinitesimal do cálculo, Abraham Robinson introduz pela Teoria dos Modelos os números infinitesimais e infinitos na Análise, seguindo, segundo suas próprias palavras, os passos deixados por Leibniz e surgindo, então, uma nova área de pesquisa em matemática pela Teoria dos Infinitesimais e outras abstrações como os números Hiperreais e os Surreais, por exemplo.
Resta, no entanto, se perguntar aonde essas abstrações formais irão nos levar enquanto não se entende plenamente seus significados, ou melhor, até que nível poderemos interpretar uma construção abstrata na obtenção de outra. Por exemplo, a questão ainda reminiscente disso sobre o problema lógico-simbólico na definição formal dos infinitos: por mais que a intuição nos pareça evoluir para a elaboração de uma formalização de uma dada quantidade infinitamente grande ou pequena – por exemplo, pelos axiomas de Peano, números Hiperreais ou números Surreais -, ainda estamos limitados pela quantidade finita de símbolos e ideias empregados e submetidos a uma interpretação mental, da onde vem a dificuldade de se entender seus significados. Nesse ponto, voltamos ao infinito não-pensável mencionado acima e ficamos tentados a questionar o título deste trabalho: não só um problema matemático, mas também, cognitivo, epistemológico e filosófico.
Uma explicação bem definida desse nosso ranço lógico do discreto e do finito, até mesmo numa construção formal incluindo infinitos e infinitésimos, se estendendo, consequentemente, por exemplo, aos conceitos da física, ainda é pendente.

At the turn of the seventeenth century, Bruno and Cavalieri independently developed two theories, central to which was the concept of the geometrical indivisible. The introduction of indivisibles had significant implications for geometryespecially

The aim of this article is to shed light on an understudied aspect of Giordano Bruno's intellectual biography, namely, his career as a mathematical practitioner. Early interpreters, especially, have criticized Bruno's mathematics for being “outdated” or too “concrete”. However, thanks to developments in the study of early modern mathematics and the rediscovery of Bruno's first mathematical writings (four dialogues on Fabrizio's Mordente proportional compass), we are in a position to better understand Bruno's mathematics. In particular, this article aims to reopen the question of whether Bruno anticipated the concept of infinitesimal quantity. It does so by providing an analysis of the dialogues on Mordente's compass and of the historical circumstances under which those dialogues were written.

The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the status of infinite divisibility in nature, rather than in mathematics. In line with this distinction, we offer a reading of the fictionality of infinitesimals. The letter has been claimed to support a reading of infinitesimals according to which they are logical fictions, contradictory in their definition, and thus absolutely impossible. The advocates of such a reading have lumped infinitesimals with infinite wholes, which are rejected by Leibniz as contradicting the part-whole principle. Far from supporting this reading, the letter is arguably consistent with the view that infinitesimals, as inassignable quantities, are mentis fictiones, i.e., (well-founded) fictions usable in mathematics, but possibly contrary to the Leibnizian principle of the harmony of things and not necessarily idealizing anything in rerum natura. Unlike infinite wholes, infinitesimals-as well as imaginary roots and other well-founded fictions-may involve accidental (as opposed to absolute) impossibilities, in accordance with the Leibnizian theories of knowledge and modality.

The purpose of this monograph is to show that Aristotle's revolutionary ideas are pertinent and current even in the twenty-first century and have not yet been fully understood. We shall start by explaining the concept of phenomenon, as understood in his time, but in the light of today's technologies. Next, we will show how Aristotle presents and develops the discussion about the Being, and how things arise from non-being, a discussion then ongoing for 400 years already.  And we shall finally see, how he, with a genius' insight, engendered a totally original conception of the Cosmos.  Analyzing the theories proposed by the philosophers who preceded him, Aristotle constructs a new relation between hylé and eidos, and modifies the idea of telos, introducing the revolutionary concepts of entelecheia (actuality) and dynamis (potential).  To this end, using mathematical concepts, he analyzes the differences between real numbers and natural numbers, and develops, from this discussion, the application of apeiron to the physical world of actual bodies.  Moreover, he rejects the concept of place and void as being a rigid space, a "little box" that remains after a body leaves it.  Finally, he analyzes what is time and asks in what way it is infinite.

The book seeks to resolve the so-called ‘problem of mass nouns’ — a problem which cannot be resolved on the basis of a conventional system of logic. It is not, for instance, possible to explicate assertions of the existence of air, oil, or water through the use of quantifiers and variables which take objectual values. The difficulty is attributable to the semantically distinctive status of non-count nouns — nouns which, although not plural, are nonetheless akin to plural nouns in being semantically non-singular. Such are the semantics of a non-singular noun, that there can be no such single thing or object as the thing of which the noun is true. However, standard approaches to understanding non-singular nouns tend to be reductive, construing them as singular expressions — expressions which, in the case of non-count nouns, are true of ‘parcels’ or ‘quantities’ of stuff, and in the case of plural nouns, are true of ‘plural entities’ or ‘sets’. It is argued that both approaches are equally misguided, that there are no distinctive objects in the extensions of non-singular nouns. With plural nouns, their extensions are identical with those of the corresponding singular expressions. With non-count nouns, because they are not plural, there can be no corresponding singular expressions. In consequence, there are no objects in the extensions of non-count nouns at all. In short, there are no such things as instances of stuff: the world of space and time contains not merely large numbers of discrete concrete things or individuals of diverse kinds, but also large amounts of sheer undifferentiated concrete stuff. Metaphysically, non-singular reference in general is an arbitrary modality of reference, ungrounded in the realities to which it is non-ideally or intransparently correlated.

Wiener in his mathematics and in his development of prosthetic devices often emphasized continuous processes, contrasting with Shannon's total emphasis on discrete bits .The connection between atomism and individualism since ancient Greece and China through the seventeenth century is surveyed, as well as the libertarian emphasis in Silicon Valley. Wiener's communalism contrasts with this social atomism and parallels his emphasis on the continuous as opposed to the discrete.

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.

Ce travail est une étude sur la méthode de Fermat pour le maximum et minimum. Fermat en parlait comme s' il s' agissait d'une méthode, mais il n'expliqua que quelques procédures. Nous reprenons les différentes procédures de Fermat et nous essayons de déterminer leur statut en tant que méthode. Celle-ci a, d'après Fermat, deux fondements: l'observation de Pappus à propos des extrêma et la syncrisis, ou comparaison entre équations quadratiques: la technique qui en découle fait usage de l'accroissement et de Yadaequatio (en termes modernes, f (a+ e) adégal à f (a)). Autrement dit, Fermat élabore l'expression algébrique de la comparaison de figures autour de l'extrêmum. La définition de la méthode de Fermat dépend égalememnt de sa diffusion parmi les contemporains, ainsi que de sa tradition parmi les savants. Nous parcourons donc les étapes de la transformation de la méthode, en privilégiant le Cursus mathematicus de Pierre Hérigone et la tradition de Van Schooten et Huygens, dont le rôle dans la diffusion de la méthode est mis en évidence par les nouveaux tomes de la Correspondance de Mersenne. En guise de conclusion, nous comparons la méthode de Fermat avec l'approche aux questions d'extrêma d'une théorie contemporaine, la géométrie différentielle synthétique, qui se réclame de Fermat.

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.

Tretji del trilogije Mnogo hrupa se odmika od samega problema staroslovanskega svetišča in se dotakne širšega konteksta Ptuja v času od pozne antike do visokega srednjega veka. Rdeča nit članka je kontinuiteta, specifično pa se posveti trem perečim problemom. Prvi je kontinuiteta med ljudstvi in kulturami na prehodu iz romanizirane antike v slovanski srednji vek. Drugi je kontinuiteta vere oziroma »trk« stare »poganske« vere z »novo« krščansko vero v zgodnjem srednjem veku. Tretji je družbenopolitična kontinuiteta oziroma uveljavitev fevdalnega reda pred madžarskimi vpadi in po njih. Pri raziskavi primerja različne vire, od arheoloških in zgodovinskih do filoloških in geografskih, pri čemer ponudi pri nekaterih virih novo branje in interpretacijo.

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.

In this paper I defend Aristotle on Finitism criticising in
a new way the notion of instantaneous velocity so closely associated with the
development of the calculus. Finitism, as it is here taken, is a cluster of views
centring on the claim that there are no completed infinities, only potential
ones. Amongst other things, as we shall see in the final section of this paper,
it is precisely through defining irrational numbers in terms of the potential
infinities involved in open intervals on the geometric line that we avoid the
usual conclusions about irrational ‘points’ completing the rational ones on
that line. In getting to this conclusion I look at some of the traditional philosophical
questions about derivatives, from Berkeley’s arguments against the
emerging calculus of Newton, through the ideas in ‘Smooth Infinitesimal
Analysis’ to another recent line of argument, which Robinson formulated, to make rigorous Leibniz’ ideas about infinitesimals. That leads me to consider
the nineteenth century thinkers, Dedekind, Cauchy and Weierstrass, who in
stages formalised the notion of a limit, using the ‘epsilon-delta’ method, and
established the belief in real numbers. The ‘epsilon-delta’ method is retained
in a modified form, but twentieth century Finitists, such as Tiles, and Van
Bendegem, in the tradition of Aristotle, have questioned whether real numbers
provide an appropriate foundation for anything. Constructivists first
tried improving on the mainline tradition in Analysis, but their doubts about
real numbers were not thoroughgoing, since constructivists still countenance
‘constructive reals’. The arguments given here allow not only constructive
reals and classical reals, but also the two above varieties of infinitesimals
their place, but point out in each case that that place is not where it is normally
thought to be. The collective points enable us to settle on a finitary
Analysis, and so a calculus using arbitrarily small, but still measurable units.

Abstract: My aim is to consider one of the most pressing problems for multi-valued logic and second order vagueness, in the light of Peirce's theory of Synechism. I will begin with a presentation of two sets of principles that we may argue form Peirce's work: the first one is directly ...

Final version of the book "Ryszard Kilvington - nieskończoność i geometria" (Richard Kilvington - infinity and geometry) before processing by Łódź University Press (minor, mostly technical changes). Analysis of Kilvington's account on the existence of actual infinities in the created world on the basis of his question "Utrum continuum sit divisibile in infinitum" (from the commentaty on "De generatione"). Kilvington's solutions are presented on the background of later medieval discussions on infinity, continuity and eternity of the world.

En este trabajo analizaremos el tratamiento del problema del continuo y del infinito en el pensamiento de juventud de Leibniz. Mostraremos que, en su abordaje, el filósofo de Leipzig entremezcla problemas físicos, metafísicos y matemáticos. El trabajo se divide en cuatro capítulos. En el primero, examinaremos el contexto en el que Leibniz formuló una de sus tesis más peculiares sobre el problema del continuo, a saber, que hay infinitas partes en acto, así como también algunas consecuencias que se siguen de esta visión. Veremos que en este período, en el que hubo una gran evolución interna, Leibniz planteó algunas nociones muy importantes, como por ejemplo las de lo indivisible y lo infinitamente pequeño. En el segundo capítulo, seguiremos como hilo conductor la distinción planteada por Leibniz al final del período parisino entre ‘tres grados de infinito’, a saber, el infinito en el grado ínfimo, lo máximo en su género y Dios. Se destacan, entre muchas otras cosas, la distinción entre infinito terminado e interminado, y el abordaje de los atributos o formas simples como lo que da razón de las cosas en los diversos géneros y de la unidad de cada género. En el tercer capítulo, examinaremos la concepción del cuerpo de Leibniz del final del período parisino. Veremos, entre otras cosas, que sostuvo un fenomenalismo que no es radical, pues, si diversas mentes son capaces de sentir las mismas cosas, debe haber una causa extramental. Asimismo, examinaremos el problema de explicar que los cuerpos, en los que hay infinitas criaturas, poseen una unidad interna. Finalmente, en el cuarto capítulo, buscaremos establecer continuidades y rupturas entre las concepciones leibnizianas de juventud y las de su primer sistema de madurez.

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.

ABSTRACT: This paper studies the foundations of infinitesimal calculus in the work of José Mariano Vallejo (1779-1846), the most influential mathematician in Spanish during the first half of the 19th century. For this purpose, the analytical foundations of his three main works related to infinitesimal calculus are considered: Memoria sobre la curvatura de las líneas en sus diferentes puntos, sobre el radio de curvatura y sobre las evolutas (1807), and the two editions (1813 and 1832) of the volume of his Tratado Elemental de Matemáticas devoted to calculus. The study shows the evolutionary dynamics of Vallejo’s thought, following the European leading references in the development of mathematical analysis in his time, and how Vallejo combines the successive influence of Lagrange, Lacroix, and Cauchy, according to criteria of educational adequacy and mathematical rigor. In particular, the influences and discrepancies on infinitesimal calculus between Vallejo’s Treatise (1813) and Lacroix’ Traité élémentaire de calcul différentiel et de calcul intégral (1802) are determined, as well as the scope of the introduction of Cauchy’s ideas, that were published for the first time in Spain in the second edition of the volume on infinitesimal calculus of Vallejo’s Treatise (1832).

this article is the third part  from my book: “the invisible world of infinitesimals” Lambert Academic Publishing
“...For physics there is a definite distinction between finiteness and infinity; whereas the mathematician speaks of the infinities and  manipulates them just like finite structures  the physicist with his real measuring roads is bound to finiteness.......Hans Reichenbach (the philosophy of Space and time)”
www.mpantes.gr

L’ipotesi esposta in questo libro è che le radici dell’ontologia dinamica che si delinea negli ultimi dialoghi di Platone, culminante nella definizione dell’essere come dynamis nel Sofista, sono da rintracciare nel dibattito matematico intorno alle grandezze incommensurabili. Esse sono presentate come la soluzione del problema del non-essere, che comporta il passaggio a una forma di razionalità non commisurativa, ma mediativa, che trova la sua massima espressione nel trascendentalismo kantiano e nelle filosofie che da esso derivano, dallo hegelismo all’ermeneutica novecentesca.
La dimostrazione dell’incommensurabilità della diagonale non è però solo il punto di svolta verso un’ontologia della dynamis: è anche il presupposto di un’ontologia intrinsecamente differenziale, che non collassi nella tautologia; di una concezione dell’esistenza in termini posizionali e non quantificazionali, e di una concezione realmente trasformativa ed emancipante della ragione. L’incommensurabile rappresenta il concetto di un’esteriorità o di un eccesso irriducibile a un dominio dato, e quindi il fondamento di quella che qui viene chiamata «una microfisica della libertà».

Na justificação do método de cálculo infinitesimal desenvolvido por Leibniz, o princípio de continuidade desempenha um papel fundamental. Com efeito, seria sob a chancela deste princípio que Leibniz se permitiria tratar a tangente a uma curva como uma espécie de secante, procedimento fundamental ao cálculo infinitesimal. De acordo com Leibniz, essa peculiar espécie de secante se distinguiria das demais pela propriedade de que a distância entre os dois pontos em que ela corta a curva seria "menor do que qualquer distância dada" -o que, para ele, significaria que ela seria "infinitamente pequena" ou mesmo "nula", como se discutirá a seguir. A contraparte deste tratamento da tangente como secante seria considerar a curva como um "polígono inifinitangular", isto é, um polígono possuindo uma "infinidade de lados".

LEIBNIZ: THE DIVINE INFINITUDE AND THE INFINITE IN US
abstract: The true infinite, accord to Leibniz’s  New Essays, is not a quantity, it is prior to any composition and is not formed by adding parts. The infinite, for Leibniz, is actual and it is the property of all things. How the finite creatures know the infinite? In this paper, we investigate what kind of relation is there between the mathematical, quantitative infinite and the knowledge of God’s infinity and actual infinite that exists in the world. Does the ideal order of mathematics instruct us about the real order of the world? And which are the limits of the ideal order in explaining the actual infinity of God and of the world?
keywords: actual infinite, infinitesimal calculus, God, labyrinth, continuous.

In the course of his philosophic career, Charles Peirce made repeated attempts to construct mathematical definitions of the commonsense or experimental notion of'continuity'. In what I will label his Final Definition of Continuity, however, Peirce abandoned the attempt to achieve mathematical definition and assigned the analysis of continuity to an otherwise unnamed extra-mathematical science. In this paper, I identify the Final
Definition, attempt to define its terms, and suggest that it belongs to Peirce's emergent semiotics of vagueness. I argue, further, that it marks the transformation of Peirce's synechism. Before the time of his Final Definition, Peirce adopted a theory of continuity as a foundational principle of metaphysics and assumed this principle might be formalized in a mathematics
of continuity. After the Final Definition, Peirce abandoned his
foundationalism in favor of what he called a critical common-sensism. This is the claim that philosophy (and with it, logic) derives its norms from the observation of actual cognitive practices and that continuity is a distinguishing mark of actual as opposed to merely possible or imagined practices.

Числовая прямая – абстрактное понятие, сформировавшееся в начале XX века, её следует отличать от телесной прямой и геометрической прямой. Телесная прямая, или отрезок – образ, возникший в античности. Геометрическая прямая, или ось, как понятие формируется в период с XVI по XVIII век в математическом анализе. Понятие прямой или кривой как геометрического места точек возникает в XVII веке в первых работах по математическому анализу.
Числовая прямая как концепт сформировалась в работах Кантора и Дедекинда, но сам термин, сначала «числовая шкала», затем «числовая прямая», употребляется с 1912 года.

Many important concepts of the calculus are difficult to grasp, and they may appear epistemologically unjustified. For example, how does a real function appear in “small” neighborhoods of its points? How does it appear at infinity? Diagrams allow us to overcome the difficulty in constructing representations of mathematical critical situations and objects. For example they actually reveal the behavior of a real function not “close to” a point (as in the standard limit theory) but “in” the point. We are interested in our research in the diagrams which play an optical role – microscopes and “microscopes within microscopes”, telescopes, windows, a mirror role (to externalize rough mental models), and an unveiling role (to help create new and interesting mathematical concepts, theories, and structures). In this paper we describe some examples of optical diagrams as a particular kind of epistemic mediator able to perform the explanatory abductive task of providing a better understanding o...

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.

История теоремы Ролля началась в XVII в. с решения алгебраического уравнения методом каскадов и формулирования понятия корневого интервала. В XIX в. Б. Больцано сформулировал на их основе понятие и теорему о непрерывной функции. Эволюция понятия непрерывности в работах О. Коши и К. Вейерштрасса привела к тому, что метод каскадов и метод отделения корней воплотились в две важнейшие теоремы, отражающие свойства непрерывных функций: теорему Ролля и теорему Больцано – Коши. Современная форма этих теорем такова: «Если функция непрерывна на [a, b], дифференцируема в (a, b) и f(a) = f(b), тогда в (a, b) найдется хотя бы одна точка c такая, что f′(c) = 0» и «Если функция непрерывна на [a, b] и имеет разные знаки на краях интервала, то в (a, b) найдется хотя бы одна точка c такая, что f(c) = 0». В историко-математической литературе частично исследована история метода Ролля, но лишь до начала XIX в. и преимущественно с точки зрения истории алгебры.
Провозвестником реформы математического анализа XIX в. был Больцано, который дал первое строго аналитическое доказательство второй из названных теорем. Он полагал, что она выражает основное свойство непрерывной функции, и рассмотрел историю ее возникновения. В реальной же ее истории участвовало немалое число ученых, живших ранее (а потому во многом способствовавших развитию анализа) и их современников, без которых эта история лишается не только важных красок, но и требуемой от исторического свидетельства полноты. Настоящая статья представляет более полную историю вопроса, содержащую анализ работ Б. Больцано, О. Коши, К. Вейерштрасса, Г. Кантора и Н. Н. Лузина, и выделяет роль эволюции понятия среднего значения для развития теории функций.
Rolle’s theorem history began in the XVII century with a solution of an algebraic equation by roots locating method and cascades' method. In XIX century Bolzano formulated a notion and a theorem of continuous function based on them. The evolution of the concept of continuity led the cascades method and the separation of roots method in Cauchy' and Weierstrass' works to be embodied in two important properties of continuous functions: Rolle's theorem and Bolzano-Cauchy’ theorem. The modern form of this theorems as follows. “If a function is continuous at [a, b], differentiable in (a, b), and f(a) = f(b), then at least one such point c will be found in (a, b) that f′(c) = 0”, and the history of the related theorem on the root interval as follows: “If a function is continuous at [a, b] and has different signs at the ends of the interval, then at least one such point c will be found in (a, b) that f(c) = 0”. In the literature on the history of mathematics, they examined the history of Rolle's method, but only until the beginning of the XIX century, and mostly from the perspective of the history of algebra.
Mathematician and theologian Bernard Bolzano was the ingenious forerunner of the ideas of Analysis' reform. He gave a first pure analytical proof to the second mentioned theorem. Bolzano attributed the importance of the key property of a continuous function to this theorem and considered its genesis. In fact, quite a lot of scientists form part of its real history. Some of them lived earlier (and therefore, in many ways contributed to the development of analysis) others were their contemporaries without whom this history would lack not only certain important colors, but the thoroughness a historical evidence needs as well. This article presents a more complete history of the issue containing the analysis of the work of Bolzano, Cauchy, Weierstrass, Cantor, and Luzin, and highlights the role of the notion "mean value" in the evolution of function theory.

"In that semantic tradition of which Frege and Russell are among the most distinguished members, the project of formalizing natural-language sentences is not simply a matter of developing smooth and effective techniques for the representation of reasoning. Over and above the representation of valid inference as valid, and invalid inference as invalid, there is a further objective. Logic in this tradition is what Frege himself famously calls a concept-script, the import of the notion being chiefly that in natural languages, as Frege emphasizes, ‘the connection of words corresponds only partially to the structure of concepts’, thereby compelling the logician to ‘conduct an ongoing struggle against language and grammar, insofar as they fail to give clear expression to the logical’. In the more recent past, a kindred overall approach is forcefully expounded in the work of Quine, who writes, albeit with a positivistic slant, that

the simplification and clarification of logical theory to which a canonical logical notation contributes is not only algorithmic, it is also conceptual ... each elimination of obscure constructions or notions that we manage to achieve, by paraphrase into more lucid elements, is a clarification of the conceptual scheme of science.

The approach is one with which I find myself in general sympathy; indeed the contrast between clear and less-than-clear ‘expressions of the logical’ is vital to the thesis of this work. Though it has not always received the understanding and respect which it deserves, the ideal of a logically transparent language represents, in my estimation, no merely interesting episode in the history of ideas. It embodies, rather, a permanently valid insight, an enduringly valuable ideal for any analytical conception of philosophy."

A finales del siglo XVII existe un intercambio de cartas entre Leibniz y Johann Bernoulli en el que tratan la posible existencia de una referencia para los infinitesimales. Ambos tratan de justificar el cálculo desde posiciones contrapuestas acerca de la realidad de los infinitesimales. Bernoulli defiende vehementemente la existencia del infinito actual mientras que Leibniz lo niega. Algunas de las afirmaciones de Johann Bernoulli podrían leerse hoy en día como predecesoras de las posturas de Cantor.
Artículo publicado en Contrastes 5 (2000) 61-75

A comienzos del siglo XVIII se origina una polémica en la Academia de Ciencias de París a propósito de la fundamentación del cálculo initesimal. Con motivo de la misma Leibniz presentará los infinitesimales como ficciones útiles...
Artículo publicado en Theoria 12:2 (1997) 257-279

In the years following Bertrand Russell's visit in China, fragments from Russell's work on mathematical logic and the foundations of mathematics started to enter the Chinese intellectual world. While up until 1925 Chinese intellectuals like Zhang Shenfu, Zhang Dongsun and others mainly contributed to the dissemination of general notions from Russell's logicism, epistemology as well as mathematical philosophy, two aspiring Chinese mathematicians, Fu Zhongsun and Zhang Bangming, introduced to Chinese readers the first Chinese translation of Russell's Introduction to Mathematical Philosophy. One year after the second edition of the book was published in 1924, Chinese intellectuals started to respond to Russell's "mathematical logic" and philosophy of mathematics, as contained in the work. Consequently, in 1925 the first public debate on this topic broke out in Chinese periodicals. This article aims at providing a summary of these 1925 discussions. It follows the main line of the debate from criticisms of Russell's work to a general debate on "infinitesimals" and "indefinitely cut wooden sticks." Through close examination of the arguments and discursive strategies used by the participants in the debate, the article tries to highlight the state of Chinese discourse on mathematical logic during the time in question as well as the foundations of mathematics, while also shedding some light on the mechanisms and strategies in the undergoing process of the adoption of Western scientific ideas.