Leibniz: Philosophy of Mathematics and Logic

In this chapter, I consider the largely overlooked influence of E. W. von Tschirnhaus' treatise on method, the Medicina mentis, on Wolff's early philosophical project (in both its conception and execution). As I argue, part of Tschirnhaus' importance for Wolff lies in the use he makes of principles gained from experience as a foundation for the scientific enterprise in the context of his broader philosophical rationalism. I will show that this lesson from Tschirnhaus runs through Wolff's earliest philosophical discussions, and (...) indeed continues to inform his major texts in logic and mathematics just before the publication of the German Metaphysics. In the end, my discussion has the effect of revealing Tschirnhaus to be an exceptionally important influence on Wolff's development, perhaps even as important (or so I suggest) as Leibniz. (shrink)

Consider an infinite series whose items are each explained by their immediate successor. Does such an infinite explanation explain the whole series or does it leave something to be explained? Hume arguably claimed that it does fully explain the whole series. Leibniz, however, designed a very telling objection against this claim, an objection involving an infinite series of book copies. In this paper, I argue that the Humean claim can, in certain cases, be saved from the Leibnizian “infinite book copies” (...) objection, and that this provides an interesting way to defuse some cosmological arguments for the existence of God and to give a non-theistic but complete explanation of the Universe. In the course of my argumentation, I also show that circular explanations can be “self-explanatory” as well: explaining two items by each other can explain the couple of items tout court. (shrink)

This paper reconsiders Leibniz’s conception of the nature of possible things and offers a novel interpretation of the actualization of possible substances. This requires analyzing a largely neglected notion, the reality of individual essences. Thus far scholars have tended to construe essences as representational items in God’s intellect. We acknowledge that finite essences have being in the divine intellect but insist that they are also grounded in the infinite essence of God, as limitations of it. Indeed, we show that it (...) is critical to understand that this dependence on God’s essence is prior to the dependence on God through divine ideas. Here it is crucial to distinguish questions concerning the ontological status of essences from questions concerning their reality. This yields a fresh view of Leibniz’s theory of creation, which takes seriously his claim that the same thing is first a mere possibility but after creation an actually existent substance. (shrink)

In 1682, Leibniz published an essay containing his solution to the classic problem of squaring the circle: the alternating converg-ing series that now bears his name. Yet his attempts to disseminate his quadrature results began seven years earlier and included four distinct approaches: the conventional (journal article), the grand (treatise), the impostrous (pseudepigraphia), and the extravagant (medals). This paper examines Leibniz’s various attempts to disseminate his series formula. By examining oft-ignored writings, as well as unpublished manuscripts, this paper answers the (...) question of how one of the greatest mathematicians sought to introduce his first great geometrical discovery to the world. (shrink)

The essential idea of Leibniz’s Theodicy was little understood in his time but has become one of the organizing themes of modern mathematics. There are many phenomena that are possible locally but for purely mathematical reasons impossible globally. For example, it is possible to build a spiral staircase that is rising at any given point, but it is impossible to build one that is rising at all points and comes back to where it started. The necessity is mathematically provable, so (...) not subject to exception by divine power. Leibniz’s Theodicy argues that God could improve the universe locally in many ways, but not globally. This paper defends Leibniz, giving positive reasons for believing that there are so many necessary interconnections between goods and evils that God is faced with a choice like the classic Trolley case, where all of the scenarios that could be chosen upfront contain evils, but some more than others. Local changes for the better seem easy to imagine, but a proper understanding of global constraints undermines the initial impression that they can be done without global cost. The paper concludes by explaining how the context of the Leibnizian argument makes it reasonable to pursue the issue of whether there are no worlds better than this one. (shrink)

The first collection of Leibniz's key writings on the binary system, newly translated, with many previously unpublished in any language. -/- The polymath Gottfried Wilhelm Leibniz (1646–1716) is known for his independent invention of the calculus in 1675. Another major—although less studied—mathematical contribution by Leibniz is his invention of binary arithmetic, the representational basis for today's digital computing. This book offers the first collection of Leibniz's most important writings on the binary system, all newly translated by the authors with many (...) previously unpublished in any language. Taken together, these thirty-two texts tell the story of binary as Leibniz conceived it, from his first youthful writings on the subject to the mature development and publication of the binary system. -/- As befits a scholarly edition, Strickland and Lewis have not only returned to Leibniz's original manuscripts in preparing their translations, but also provided full critical apparatus. In addition to extensive annotations, each text is accompanied by a detailed introductory “headnote” that explains the context and content. Additional mathematical commentaries offer readers deep dives into Leibniz's mathematical thinking. The texts are prefaced by a lengthy and detailed introductory essay, in which Strickland and Lewis trace Leibniz's development of binary, place it in its historical context, and chart its posthumous influence, most notably on shaping our own computer age. (shrink)

Para favorecer la interacción disciplinar y recuperar la dimensión práctica del conocimiento matemático en la escuela secundaria, Yves Chevallard plantea la necesidad de introducir en los programas de estudio la matemática mixta. La matemática mixta, cuyo apogeo tuvo lugar en Europa entre los siglos XVI y XVIII, se propone el abordaje de problemas surgidos por fuera de la propia matemática valiéndose de nociones mecánicas -como la de centro de gravedad y fuerza centrífuga- y del empleo de variados instrumentos para realizar (...) las construcciones requeridas. En este trabajo consideramos a la matemática mixta en la cultura matemática de la segunda mitad del siglo XVII y, en dicho contexto, presentamos dos casos de la práctica matemática de G. W. Leibniz en los que la investigación matemática involucra el diseño de máquinas. La consideración de tales casos permite situar a la matemática mixta, siguiendo la terminología de Ian Hacking, en un terreno medio entre representar e intervenir. El terreno medio entre representar e intervenir es un valle fértil donde la matemática aporta a, y se nutre de, una multiplicidad de otros dominios. También es una vía de acceso para historizar el conocimiento matemático, destacando las condiciones materiales para su desarrollo y el importante rol de los problemas. Palabras clave: Práctica matemática, Resolución de problemas, Siglo XVII, Máquinas, Cultura material. (shrink)

The concept of ‘ideas’ plays central role in philosophy. The genesis of the idea of continuity and its essential role in intellectual history have been analyzed in this research. The main question of this research is how the idea of continuity came to the human cognitive system. In this context, we analyzed the epistemological function of this idea. In intellectual history, the idea of continuity was first introduced by Leibniz. After him, this idea, as a paradigm, formed the base of (...) several fundamental scientific conceptions. This idea also allowed mathematicians to justify the nature of real numbers, which was one of central questions and intellectual discussions in the history of mathematics. For this reason, we analyzed how Dedekind’s continuity idea was used to this justification. As a result, it can be said that several fundamental conceptions in intellectual history, philosophy and mathematics cannot arise without existence of the idea of continuity. However, this idea is neither a purely philosophical nor a mathematical idea. This is an interdisciplinary concept. For this reason, we call and classify it as mathematical and philosophical invariance. (shrink)

This volume tells the story of the legacy and impact of the great German polymath Gottfried Wilhelm Leibniz (1646-1716). Leibniz made significant contributions to many areas, including philosophy, mathematics, political and social theory, theology, and various sciences. The essays in this volume explores the effects of Leibniz’s profound insights on subsequent generations of thinkers by tracing the ways in which his ideas have been defended and developed in the three centuries since his death. Each of the 11 essays is concerned (...) with Leibniz’s legacy and impact in a particular area, and between them they show not just the depth of Leibniz’s talents but also the extent to which he shaped the various domains to which he contributed, and in some cases continues to shape them today. With essays written by experts such as Nicholas Jolley, Pauline Phemister, and Philip Beeley, this volume is essential reading not just for students of Leibniz but also for those who wish to understand the game-changing impact made by one of history’s true universal geniuses. (shrink)

El cálculo infinitesimal elaborado por Leibniz en la segunda mitad del siglo XVII tuvo, como era de esperarse, muchos adeptos pero también importantes críticos. Uno pensaría que cuatro siglos después de haber sido presentado éste, en las revistas, academias y sociedades de la época, habría ya poco qué decir sobre el mismo; sin embargo, cuando uno se acerca al cálculo de Leibniz –tal y como me sucedió hace tiempo– fácilmente puede percatarse de que el debate en torno al cálculo leibniziano (...) ha trascendido las fronteras temporales del siglo XVII y XVIII y aún sigue vigente, al menos en parte y sobre ciertos puntos específicos. Lo anterior resulta un tanto inquietante, entre otras cosas porque implica que hay que revisar el cálculo leibniziano para tratar de entender el motivo por el que se sigue debatiendo actualmente. El propósito de este artículo no es presentar las tesis principales del cálculo, tampoco hacer una defensa del mismo y mucho menos desarrollar una crítica de sus fundamentos matemáticos, epistémicos u ontológicos. Mi propósito es menos ambicioso, pretendo mostrar, en primer lugar, dos de las primeras objeciones que se formularon contra el cálculo infinitesimal y que fueron hechas por dos contemporáneos de Leibniz, a saber, Rolle y Nieuwentijit; por otro lado, y sirviéndome de lo anterior, quiero presentar dos comentaristas actuales que abordan el tema del cálculo. Lo anterior con el objetivo de ilustrar cuál es el estado de la cuestión, es decir, en qué momento está el debate sobre el cálculo infinitesimal y en qué se enfocan actualmente los comentaristas al hablar sobre él, pues al hacerlo –e incluso a veces sin pretenderlo– mantienen abierto el debate sobre los aciertos y desaciertos del método infinitesimal. (shrink)

To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on pro- cedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian infinitesimal (...) calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones. (shrink)

In this paper, an interpretation of "necessity", inspired by a Leibnizian idea and based on the method of dialogical logic, is introduced. The semantic rules corresponding to such an account of necessity are developed, and then some peculiarities, and some potential advantages, of the introduced dialogical explanation, in comparison with the customary explanation offered by the possible worlds semantics, are briefly discussed.

This book presents new research into key areas of the work of German philosopher and mathematician Gottfried Wilhelm Leibniz (1646-1716). Reflecting various aspects of Leibniz's thought, this book offers a collection of original research arranged into four separate themes: Science, Metaphysics, Epistemology, and Religion and Theology. With in-depth articles by experts such as Maria Rosa Antognazza, Nicholas Jolley, Agustín Echavarría, Richard Arthur and Paul Lodge, this book is an invaluable resource not only for readers just beginning to discover Leibniz, but (...) also for scholars long familiar with his philosophy and eager to gain new perspectives on his work. (shrink)

Our computational metaphysics group describes its use of automated reasoning tools to study Leibniz’s theory of concepts. We start with a reconstruction of Leibniz’s theory within the theory of abstract objects (henceforth ‘object theory’). Leibniz’s theory of concepts, under this reconstruction, has a non-modal algebra of concepts, a concept-containment theory of truth, and a modal metaphysics of complete individual concepts. We show how the object-theoretic reconstruction of these components of Leibniz’s theory can be represented for investigation by means of automated (...) theorem provers and finite model builders. The fundamental theorem of Leibniz’s theory is derived using these tools. (shrink)

In Part 1 of this short book, Rescher provides an overview of the nature and source of Leibniz’s interest in the theory and practice of cryptanalysis, including his unsuccessful bid to secure an apprentice for John Wallis (1616-1703) with a view to perpetuating the Englishman’s remarkable deciphering abilities. In Part 2, perhaps the most interesting part of the book, Rescher offers his account of the inner workings of Leibniz’s cipher machine. Part 3 provides a brief pictorial history of such machines (...) and related technologies. Finally, in Part 4 Rescher analyzes some of Leibniz’s own relatively scant attempts at decryption. Though there is little of direct philosophical interest in this work, its account of Leibniz’s forays into the field of encryption and in particular its reconstruction of his remarkable machine are fascinating and should hold considerable appeal for those interested in Leibniz’s pursuits more broadly and in the histories of cryptography and machine design. (shrink)

Aunque no es posible ofrecer aquí un análisis detallado de las diferentes cues tionesimplicadas en la teoría de las representaciones en la Modernidad, sí pode mos atender auno de sus aspectos más relevantes, presente en la exposición de Leibniz, Wolff,Baumgarten y Kant, a saber: el reconocimiento progresivo y la vin dicación de unconocimiento específicamente sensible como un tipo específico de conocimiento en lafilosofía escolar alemana del siglo XVIII. A pesar de que Leibniz atribuye al conocimiento claro y confuso el grado (...) más bajo en su esquema de los niveles del conocimiento, tambiénes cierto que su exposición apunta ya a la irre ductibilidad de este ámbito en unconocimiento finito como el humano. Por lo demás, lo confuso aquí adquiere sentido en elsistema del saber en la medida en que Leibniz considera que todos los conocimientos obre cuestiones de hecho, si bien falibles, se encuentran en conexión con la verdad en lamedida en que parti cipan de la misma, al expresar el fundamento último de lo real y sersusceptibles de perfectibilidad en el seno de una subjetividad dinámica que comienza aser entendida como unidad vital. En Wolff y Baumgarten se profundiza en la vía que esabierta ya por Leibniz: la exposición de los niveles de conocimiento en el marco de unaimbricación de la lógica de la representación con una psicología vitalista, en la que sereconocerá la experiencia del sentimiento como expresión subjetiva de la perfectibilidaddel conocimiento humano. Aunque la filosofía de Kant supondrá una ruptura con laexposición escolar de esta teoría de los niveles de conocimiento o los tipos derepresentación, lo cierto es que el nacimiento de su teoría de la sensi bilidad tiene lugar enel marco histórico de esta concepción tradicional. (shrink)

Logique et existenceDeleuze à propos des « conditions du réel »Pour Deleuze, l’un des problèmes fondamentaux d’une théorie de la pensée est de savoir comment la pensée peut quitter la sphère du possible pour penser le réel, c’est-àdire pour penser l’existence elle-même. La position du réel semble être hors du concept. Des pré-kantiens comme Leibniz approchaient ce problème par le biais de la distinction entre vérités d’essence et vérités d’existence, alors que des post-kantiens comme Maimon l’approchaient par la distinction entre (...) les conditions de l’expérience possible et celles de l’expérience réelle. La logique classique définit la sphère du possible par trois principes logiques – l’identité, la non-contradiction et le tiers-exclu – et la présente étude examine les trois grandes trajectoires qui, dans cette histoire de la philosophie, ont tenté d’utiliser l’un de ces trois principes classiques pour pénétrer l’existence ellemême : 1) Leibniz cherchait à étendre le principe de d’identité àl’existence entière ; 2) Hegel cherchait à étendre le principe de non-contradiction à la totalité de l’expérience ; et 3) le groupe des penseurs appelés de manière assez large « existentialistes » cherchait à étendre le principe du tiers-exclu à la totalité de l’existence. La conclusion examine les raisons pour lesquelles Deleuze a été fasciné par chacune de ces tentatives philosophiques pour « penser l’existence », tout en pensant néanmoins qu’elles ont toutes échoué ; et pourquoi aussi il a fini par développer sa propre réponse au problème en faisant appel à un principe de différence.Logica e EsistenzaLe ‘Condizioni del reale’ in DeleuzePer Deleuze, uno dei problemi fondamentali per una teoria del pensiero è: come può il pensiero abbandonare la sfera del possibile per pensare il reale, ossia, pensare l’esistenza stessa? La posizione del reale sembra essere fuori dal concetto. Prekantiani come Leibniz affrontano questo problema in termini di distinzione fra verità dell’essenza e verità dell’esistenza, mentre post-kantiani come Maimon affrontano il problema in termini di distinzione fra condizioni dell’esperienza possibile e condizioni dell’esperienza reale. La logica classica ha definito la sfera del possibile secondo tre principi logici – identità, non-contraddizione, terzo escluso – e questo saggio analizza tre grandi ‘parabole’ della storia della filosofia che hanno tentato di usare uno di questi tre principi della logica per penetrare l’esistenza stessa: Leibniz hanno tentato di estendere il principio di identità a tutta l’esistenza; Hegel hanno tentato di estendere il principio di non-contraddizione a tutta l’esistenza; il gruppo di pensatori chiamati “esistenzialisti” ha tentato di estendere il principio del terzo escluso all’esistenza. La conclusione analizza sia le ragioni per le quali Deleuze era affascinato da ciascuno di questi tentativi filosofici di “pensare l’esistenza” nonostante fosse convinto che essi avessero fallito, sia i motivi per cui egli in conclusione traccia la propria risposta al problema facendo appello al principio della differenza. (shrink)

El punto de partida del calculus universalis leibniziano es la teoría aristotélica del silogismo, pero Leibniz se independiza de las ideas de Aristóteles para desarrollar su propio sistema lógico, mucho más general, aplicando el instrumento combinatorio a la silogística. Lo que propone es una importante modificación del modelo demostrativo axiomático, mediante la creación de cálculos lógico-simbólicos que no se limitan a los ámbitos tradicionales de la deducción, sino que admiten procedimientos discursivos más complejos que los de la lógica clásica, ampliando (...) con ello la noción de racionalidad. (shrink)

Explications of the reconstruction of Leibniz’s metaphysics that Deleuze undertakes in 'The Fold: Leibniz and the Baroque' focus predominantly on the role of the infinitesimal calculus developed by Leibniz.1 While not underestimat- ing the importance of the infinitesimal calculus and the law of continuity as reflected in the calculus of infinite series to any understanding of Leibniz’s metaphysics and to Deleuze’s reconstruction of it in The Fold, what I propose to examine in this paper is the role played by other (...) developments in mathematics that Deleuze draws upon, including those made by a number of Leibniz’s near contemporaries – the projective geometry that has its roots in the work of Desargues (1591–1661) and the ‘proto-topology’ that appears in the work of Du ̈rer (1471–1528) – and a number of the subsequent developments in these fields of mathematics. Deleuze brings this elaborate conjunction of material together in order to set up a mathematical idealization of the system that he considers to be implicit in Leibniz’s work. The result is a thoroughly mathematical explication of the structure of Leibniz’s metaphysics. What is provided in this paper is an exposition of the very mathematical underpinnings of this Deleuzian account of the structure of Leibniz’s metaphysics, which, I maintain, subtends the entire text of The Fold. (shrink)

The reconstruction of Leibniz’s metaphysics that Deleuze undertakes in The Fold provides a systematic account of the structure of Leibniz’s metaphysics in terms of its mathematical foundations. However, in doing so, Deleuze draws not only upon the mathematics developed by Leibniz—including the law of continuity as reflected in the calculus of infinite series and the infinitesimal calculus—but also upon developments in mathematics made by a number of Leibniz’s contemporaries—including Newton’s method of fluxions. He also draws upon a number of subsequent (...) developments in mathematics, the rudiments of which can be more or less located in Leibniz’s own work—including the theory of functions and singularities, the Weierstrassian theory of analytic continuity, and Poincaré’s theory of automorphic functions. Deleuze then retrospectively maps these developments back onto the structure of Leibniz’s metaphysics. While the Weierstrassian theory of analytic continuity serves to clarify Leibniz’s work, Poincaré’s theory of automorphic functions offers a solution to overcome and extend the limits that Deleuze identifies in Leibniz’s metaphysics. Deleuze brings this elaborate conjunction of material together in order to set up a mathematical idealization of the system that he considers to be implicit in Leibniz’s work. The result is a thoroughly mathematical explication of the structure of Leibniz’s metaphysics. This essay is an exposition of the very mathematical underpinnings of this Deleuzian account of the structure of Leibniz’s metaphysics, which, I maintain, subtends the entire text of The Fold. (shrink)

Leibniz is well known for his formulation of the infinitesimal calculus. Nevertheless, the nature and logic of his discovery are seldom questioned: does it belong more to mathematics or metaphysics, and how is it connected to his physics? This book, composed of fourteen essays, investigates the nature and foundation of the calculus, its relationship to the physics of force and principle of continuity, and its overall method and metaphysics. The Leibnizian calculus is presented in its origin and context together with (...) its main contributors: Archimedes, Cavalieri, Wallis, Hobbes, Pascal, Huygens, Bernoulli, and Nieuwentijt. Many of us know and probably have used the Leibnizian formula: to .. (shrink)

This paper is intended to examine Leibniz's Principle of Continuity, as well as the conditions of its application in leibnizian infinitesimal calculus, in the light of some aspects of formal language developed by Leibniz. It aims at evaluating whether, and to what extent, that principle can be justified on the basis of the logic of Leibniz.

In Hegel ou Spinoza,1 Pierre Macherey challenges the influence of Hegel’s reading of Spinoza by stressing the degree to which Spinoza eludes the grasp of the Hegelian dialectical progression of the history of philosophy. He argues that Hegel provides a defensive misreading of Spinoza, and that he had to “misread him” in order to maintain his subjective idealism. The suggestion being that Spinoza’s philosophy represents, not a moment that can simply be sublated and subsumed within the dialectical progression of the (...) history of philosophy, but rather an alternative point of view for the development of a philosophy that overcomes Hegelian idealism. Gilles Deleuze also considers Spinoza’s philosophy to resist the totalising effects of the dialectic. Indeed, Deleuze demonstrates, by means of Spinoza, that a more complex philosophy antedates Hegel’s, which cannot be supplanted by it. Spinoza therefore becomes a significant figure in Deleuze’s project of tracing an alternative lineage in the history of philosophy, which, by distancing itself from Hegelian idealism, culminates in the construction of a philosophy of difference. It is Spinoza’s role in this project that will be demonstrated in this paper by differentiating Deleuze’s interpretation of the geometrical example of Spinoza’s Letter XII (on the problem of the infinite) in Expressionism in Philosophy, Spinoza,2 from that which Hegel presents in the Science of Logic.3. (shrink)

This paper seeks to indicate some connections between a major philosophi- cal project of the seventeenth century, the conception of a mathesis universalis, and the practice of baroque poetry. I shall argue that these connections consist in a peculiar view of language and systems of notation which was particularly common in European baroque culture and which provided the necessary conceptual background for both poetry and the mathesis universalis.

Llull and Leibniz both subscribed to conceptual atomism: the belief that the majority of concepts are compounds constructed from a relatively small number of primitive concepts. Llull worked out techniques for finding the logically possible combinations of his primitives, but Leibniz criticized Llull’s execution of these techniques. This article argues that Leibniz was right about things being more complicated than Llull thought but that he was wrong about the details. The paper attempts to correct these details.

The aim of this paper is to show that topology has a bearing on Leibniz’s Principle of the Identity of Indiscernibles (PII). According to (PII), if, for all properties F, an object a has property F iff object b has property F, then a and b are identical. If any property F whatsoever is permitted in PII, then Leibniz’s principle is trivial, as is shown by “identity properties”. The aim of this paper is to show that topology can make a (...) contribution to the problem of giving criteria of how to restrict the domain of properties to render (PII) non-trivial. In topology a wealth of different Leibnizian principles of identity can be defined - PII turns out to be just the weakest topological separation axiom T0 in disguise, stronger principles of can be defined with the aid of higher separation axioms Ti, i > 0. Topologically defined properties have a variety of nice features, in particular they are stable in a natural sense. Topologically defined properties do not have a monopoly on defining “good” properties. In the final section of the paper it is show that the topological approach is closely related to Gärdenfors’s approach of conceptual spaces based on the concept of convexity. (shrink)

A standard strategy for defending a claim of non-identity is one which invokes Leibniz’s Law. (1) Fa (2) ~Fb (3) (∀x)(∀y)(x=y ⊃ (∀P)(Px ⊃ Py)) (4) a=b ⊃ (Fa ⊃ Fb) (5) a≠b In Kalderon’s view, this basic strategy underlies both Moore’s Open Question Argument (OQA) as well as (a variant formulation of) Frege’s puzzle (FP). In the former case, the argument runs from the fact that some natural property—call it “F-ness”—has, but goodness lacks, the (2nd order) property of its (...) being an open question whether everything that instantiates it is good to the conclusion that goodness and F-ness are distinct. And in the latter case, the argument runs from the fact that that Hesperus has, but Phosphorus lacks, the property of being believed by the ancient astronomers to be visible in the evening sky to the conclusion that Hesperus and Phosphorus are distinct. Kalderon argues that both the OQA and FP fail because in neither case is there good reason to believe that both (1) and (2) are true. The reason we are tempted to believe that they are true is because we mistake de dicto claims for de re claims. In order for FP to go through, the truth of the following de re claims needs to be established: FP1) Hesperus was believed by the ancient astronomers to be visible in the evening sky. (shrink)