This paper is a discussion of the treatment of Leibniz's conception of substance in Heidegger's The Metaphysical Foundations of Logic. I explain Heidegger's account, consider its relation to recent interpretations of Leibniz in the Anglophone secondary literature, and reflect on the ways in which Heidegger's methodology may illuminate what it is to read Leibniz and other figures in the history of philosophy.

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. (shrink)

We seek to elucidate the philosophical context in which one of the most important conceptual transformations of modern mathematics took place, namely the so-called revolution in rigor in infinitesimal calculus and mathematical analysis. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind,and Weierstrass. The dominant current of philosophy in Germany at the time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our (...) main thesis is that Marburg neo-Kantian philosophy formulated a sophisticated position towards the problems raised by the concepts of limits and infinitesimals. The Marburg school neither clung to the traditional approach of logically and metaphysically dubious infinitesimals, nor whiggishly subscribed to the new orthodoxy of the “great triumvirate” of Cantor, Dedekind, and Weierstrass that declared infinitesimals conceptus nongrati in mathematical discourse. Rather, following Cohen’s lead, the Marburg philosophers sought to clarify Leibniz’s principle of continuity, and to exploit it in making sense of infinitesimals and related concepts. (shrink)

Gottfried Wilhelm Leibniz (1646–1716) was one of the great thinkers of the seventeenth and eighteenth centuries and is known as the last “universal genius”. He made deep and important contributions to the fields of metaphysics, epistemology, logic, philosophy of religion, as well as mathematics, physics, geology, jurisprudence, and history. Even the eighteenth century French atheist and materialist Denis Diderot, whose views could not have stood in greater opposition to those of Leibniz, could not help being awed by his achievement, writing (...) in his Encyclopedia, “Perhaps never has a man read as much, studied as much, meditated more, and written more than Leibniz… What he has composed on the world, God, nature, and the soul is of the most sublime eloquence. If his ideas had been expressed with the flair of Plato, the philosopher of Leipzig would cede nothing to the philosopher of Athens.” (Vol. 9, p. 379) Indeed, Diderot's mood was almost despairing in a remark from another piece, which also has a great deal of truth in it: “When one compares the talents one has with those of a Leibniz, one is tempted to throw away one's books and go die quietly in the dark of some forgotten corner.” More than a century later, Gottlob Frege, who fortunately did not cast his books away in despair, expressed similar admiration, declaring that “in his writings, Leibniz threw out such a profusion of seeds of ideas that in this respect he is virtually in a class of his own.” (“Boole's logical Calculus and the Concept script” in Posthumous Writings , p. 9) The aim of this entry is primarily to introduce Leibniz's life and summarize and explicate his views in the realms of metaphysics, epistemology, philosophical theology, and natural philosophy. (shrink)

: Few texts summarize and at the same time compound the challenges of their author's philosophy so sharply as Hermann Cohen's Das Prinzip der Infinitesimalmethode und seine Geschichte (1883). The book's meaning and style are greatly illuminated by placing it in the scientific, political, and academic context of late-nineteenth century Germany. As this context changed, so did both the reception of the philosophy of the infinitesimal and of the Marburg school more generally. A study of this transformation casts significant light (...) on the political relevance of the philosophy of science in theWilhelmine era. As a means of following this development across time, Cohen's text is read through its changing reception in the philosophy of his closest disciple, Ernst Cassirer. (shrink)

In the present paper, the author aims at laying the foundations of a symbolics of technical gesture, according to the thesis that symbolic faculty is another face of the technological one, and that they are both in truth two sides of the same coin. Accordingly, the author suggests to rename the whole dimen-sion as “meta-environmentality”. The analysis is carried out on the basis of a specific comparison between Cassirer’s definition of “animal symbolicum” and its scientific consistence in the light of (...) modern palaeontology. “Animal symbol-icum” is here compared with Leroi-Gourhan’s homo technologicus, and Cassi-rer’s ideas on human identity tested starting from paleoanthropological data. The result of the inquiry lead us to recognize the urgency of integrating Cassi-rer’s argument with the primacy of the technological capacity, but a deep anal-ysis of the characterizing attributes of the latter compels us to uphold the symbolic attitude of the technological dimension. The author then sketches a basic description of the guidelines of a symbolic theory of technology, and tries to show how the basic elements of such an approach were familiar both to Cassirer and Leroi-Gourhan. As a consequence of the whole theory, the author elaborates a chronological analysis of human identity, whose basic result is the determination of the future as main temporal dimen-sion of human acting. (shrink)

This paper explores some of the constructive dimensions and specifics of human theoretic cognition, combining perspectives from (Husserlian) genetic phenomenology and distributed cognition approaches. I further consult recent psychological research concerning spatial and numerical cognition. The focus is on the nexus between the theoretic development of abstract, idealized geometrical and mathematical notions of space and the development and effective use of environmental cognitive support systems. In my discussion, I show that the evolution of the theoretic cognition of space apparently follows (...) two opposing, but in truth, intrinsically aligned trajectories. On the epistemic plane, which is the main focus of Husserl’s genetic phenomenological investigations, theoretic conceptions of space are progressively constituted by way of an idealizing emancipation of spatial cognition from the concrete, embodied intentionality underlying the human organism’s perception of space. As a result of this emancipation, it ultimately becomes possible for the human mind to theoretically conceive of and posit space as an ideal entity that is universally geometrical and mathematical. At the same time, by synthesizing a range of literature on spatial and mathematical cognition, I illustrate that for the theoretic mind to undertake precisely this emancipating process successfully, and further, for an ideal and objective notion of geometrical and mathematical space to first of all become fully scientifically operative, the cognitive support provided by a range of specific symbolic technologies is central. These include lettered diagrams, notation systems, and more generally, the technique of formalization and require for their functioning various cognitively efficacious types of embodiment. Ultimately, this paper endeavors to understand the specific symbolic-technological dimensions that have been instrumental to major shifts in the development of idealized, scientific conceptions of space. The epistemic characteristics of these shifts have been previously discussed in genetic phenomenology, but without devoting sufficient attention to the constructive role of symbolic technologies. At the same time, this paper identifies some of the irreducible phenomenological and epistemic dimensions that characterize the functioning of the historically situated, embodied and distributed theoretic mind. (shrink)

During the first phase of Greek mathematics a proof consisted in showing or making visible the truth of a statement. This was the epagogic method. This first phase was followed by an apagogic or deductive phase. During this phase visual evidence was rejected and Greek mathematics became a deductive system. Now epagoge and apagoge, apart from being distinguished, roughly according to the modern distinction between inductive and deductive procedures, were also identified on account of the conception of generality as continuity. (...) Epistemology of mathematics today only remembers the distinction, forgetting where they agreed, in this manner not only destroying the unity of the perceptual and conceptual but also forgetting what could be gained from Aristotelian demonstrative science. (shrink)

Resumen En el presente trabajo analizo el concepto de cuerpo en los escritos maduros de Leibniz. En el marco del debate contemporáneo acerca del estatus de la sustancia corpórea examino el entramado de las tres nociones sobre las que se construye la metafísica leibniziana de los cuerpos, a saber, la de materia prima, la de cuerpo orgánico y la de extensión. En cada caso evalúo el estatus ontológico que Leibniz les reconoce así como el papel que cumplen en su metafísica.In (...) this paper I examine the concept of body in Leibniz’s mature writings. In the context of the contemporary debate on the status of the corporeal substance, I analyze the relation between the three notions from which he builds his metaphysics of bodies, namely, the primary matter, the organic body, and the extension. For each case I evaluate their ontological status and the role they play in his metaphysics. (shrink)

the domain of Art critique and becoming a philosophical argument. How can we think of Perspective as symbolic Form? Is Perspective really a symbolic form? Why is Perspective so important? Because at the beginning of the Modern Age, Perspective as spiritual figure grounds many symbolic or even many scientific constructions. We could we say that perspective open the foundation of modern science as such. The “Geometrization” of Vision, beginning with perspective, will be for us the interpretative key in order to (...) understand the Modern Age as a whole. This understanding will allow us to understand the anthropologic dimension arising from the Modern Age, called „Perspectivism“. Assuming that perspective was neither only an invention of painting nor of geometry nor of philosophy, taken as singular fields of human inquiry, we will try to sketch the genesis of “perspective” from an interdisciplinary point of view. By doing so, we will also try to fix its deep significance for the anthropology of the Modern Age. Living and feeling in a perspectival world is the real invention of the Modern Age, one that overcame the closed Cosmos of the Middle Ages in order to reveal to mankind its own potential. Our interdisciplinary approach will proceed from many points of view and converge on the idea of a new kind of human experience. Such an interdisciplinary approach will open new questions about our present time. Are we justified in thinking of our experience today as perspectival? What does it mean today to think from perspectives in the manifold dimensions of our living and to face to the complexity of our times? (shrink)

The article attempts to reconsider the relationship between Leibniz’s and Kant’s philosophy of geometry on the one hand and the nineteenth century debate on the foundation of geometry on the other. The author argues that the examples used by Leibniz and Kant to explain the peculiarity of the geometrical way of thinking are actually special cases of what the Jewish-German mathematician Felix Hausdorff called “transformation principle”, the very same principle that thinkers such as Helmholtz or Poincaré applied in a more (...) general form in their celebrated philosophical writings about geometry. The first two parts of the article try to show that Leibniz’s and Kant’s philosophies of geometry, despite their differences, appear to be preoccupied with the common problem of the impossibility to grasp conceptually the intuitive difference between two figures (such as a figure and its scaled, displaced or mirrored copy). In the third part, it is argued that from the perspective of Hausdorff’s philosophical-geometrical reflections, this very same problem seems to find a more radical application in Helmholtz’s or Poincaré’s thought experiments on the impossibility of distinguishing distorted copies of our universe from the original one. I draw the conclusion that in Hausdorff’s philosophical work, which has received scholarly attention only recently, one can find not only an original attempt to frame these classical arguments from a set-theoretical point of view, but also the possibility of considering the history of philosophy of geometry from an uncommon perspective, where especially the significance of Kant’s infamous appeal to “intuition” can be judged by more appropriate standards. (shrink)

Marburg Neo-Kantianism has attracted substantial interest among contemporary philosophers drawn by its founding idea that the success of advanced theoretical science is a given fact and it is the task of philosophical inquiry to ground the objectivity of scientific achievement in its a priori sources (Cohen and Natorp 1906, p. i). The Marburg thinkers realized that recent advances and developments in the mathematical sciences had changed the character of Kant’s transcendental project, demanding new methods and approaches to establish the objectivity (...) of the latest insights into physical reality. Hermann Cohen, one of the founders of the Marburg School, attempted to base the objectivity of science on the concepts of .. (shrink)

Ernst Cassirer’s epistemological trilogy – Substanzbegriff und Funktionsbegriff (1910), Zur Einsteinschen Relativitätstheorie (1921) and Determinismus und Indeterminismus (1937) – is well known to Western scholars, some of whom recently devoted a number of in-depth and interesting studies to Cassirer’s epistemology. Nonetheless, they overlooked aspects of Cassirer’s concept of idealisation and his model of science as found in his last epistemological work: Determinismus und Indeterminismus. In this essay I will consider these two almost disregarded aspects of Cassirer’s epistemology in order to (...) provide a better understanding of his idea of science. (shrink)

Leibniz propôs mais de um conceito para descrever filosoficamente a substância. Ironia instrutiva, esta pluralidade que tem por fim uma explicação unificada da realidade culminou em uma definição dos componentes fundamentais do mundo – as mônadas – como unidades que abrigam a multiplicidade. Estas substâncias, bem como a sua função essencial de se exprimirem mutuamente, apenas se tornam plenamente inteligíveis quando observamos os conceitos anteriores, de que a filosofia madura de Leibniz herdou algumas intuições. Guiando-nos pelas obras-chave do filósofo e (...) físico alemão, procuraremos retraçar esse percurso intelectual, a fim não de esboçar um hipotético sistema seu, mas de _unificar_ o tratamento que ele dispensou à _unidade_. Tal hipótese de leitura concorda fundamentalmente com aquela sustentada por Michel Fichant, mas traz pontuais acréscimos ou desacordos, que serão indicados neste artigo. (shrink)

The founders of the Marburger Schule of Neo-Kantianism, Hermann Cohen and Paul Natorp, laid an emphasis upon a Platonic understanding of mathematics and logic as the paradigmatic epistemological basis of philosophy. Their successors, namely Ernst Cassirer and Nicolai Hartmann, made obvious, however, that new biological thinking can have a strong influence on ontology as well as on the theory of knowledge. They could show that biology was no longer to be treated as a metaphysical system in that pejorative meaning of (...) metaphysics which Kant had so severely been opposing. Against the anti-realistic approach of the older neo-Kantians who wanted to eliminate Kant's thing-in-itself (Ding an sich), both Cassirer and Hartmann returned to a form of realism by way of Hegel's philosophical results and reflections on his method. This makes clear that their realism is still to be taken as a more of less monistic idealism. However, considering that modern biology as an antimetaphysical force became influential in their system for the first time and that it did so in two different but - on second thoughts - complementary ways, it becomes clear that it was necessary to change the paradigms of any neo-Kantian philosophy. Cassirer proved this by his development as a philosopher with a strong historical impetus while Hartmann as a more systematical philosopher only pointed in that direction. (shrink)

A summary is always incomplete, but in this instance it is essentially so; as is explained in the final chapter, any essential question whatever one could ask about the subjectmatter of this thesis can only be answered by the complete exposition this thesis presents. This summary can, therefore, only indicate the general outline of the thesis. The object of the thesis is to make clear a historical and systematic development of logical and ontological structures concerning force in Leibniz, Kant, and (...) Hegel. Viewed as a whole, this development presents a dynamic conceptual structure which applies to the universe as a totality which determines itself as a self-mediating process of continual change. In Division 1, it is demonstrated that the metaphysical system of the so-called mature period of Leibniz presents this self-mediating process in its implicit form. In this system the universe determines itself by way of a circular structure; the universe as a totality is represented in the complete substances (which constitute the universe) and expressed by them; representation and expression are united in the concept of force (which is the internal activity of the complete substances) - force constitutes the totality the universe is. It is pointed out that this circular structure is in fact a unification of two levels, viz. universality (that is, the universe as one whole) and singularity (that is, the individual parts of the universe), in which each level contains the other; this mutual containment is called hyperception. Typical for Leibniz’s system is the containment of the universal in the singular. The actual mediation which is requested to establish the circularity is not expounded in Leibniz’s system, but remains implicit. In Division 2, it is demonstrated that in the metaphysical system of the so-called pre-critical period of Kant Leibniz’s system is analyzed and that, as a result of this, Kant’s system is the analytical form of the implicit self-mediation which is presupposed in Leibniz’s system. It is pointed out how this analysis leads to the emergence of a third level, which Leibniz’s system lacks, viz. particularity (that is, the actual relatedness of the individual parts, which establishes the universe as a totality). On this level, Kant’s concept of force plays its crucial role of mediating between universality and singularity; however, Kant does not succeed in fitting the three levels together and as a consequence the mediating force stands for itself instead of being actually mediating activity. In Division 3, it is demonstrated how in Hegel’s system the final unification of the three levels is established, how the mediation is conceived in the structure of the self-actualizing essence, and how this system sublates the other two systems (it understands them as forms of reflection; reflection is an essential category in Hegel’s system). It is pointed out that, whereas in Leibniz’s form of hyperception a contradiction is implied (as concerns the containment of the universal in the singular), this is not the case in Hegel’s system (because of the actual mediation between the universal and the singular). (shrink)