Deleuze and Mathesis. Deleuze and Fractals. Between Deleuze and Mandelbrot. From Mathesis differentialis to Mathesis fractalis.

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.

Hermeneutic studies of science locate a circle at the heart of scientific practice: scientists only gain knowledge of what they, in some sense, already know. This may seem to threaten the rational validity of science, but one can argue that this circle is a virtuous rather than a vicious one. A virtuous circle is one in which research conclusions are already present in the premises, but only in an indeterminate and underdeveloped way. In order to defend the validity of science, (...) the hermeneuticist must describe a method by which a vague and confused initial knowledge of nature gets transformed into a clear and determinate knowledge of nature. I consider three such methods. The first is regressus demonstrativa, favoured by the physicians of Padua during the fifteenth and sixteenth centuries. The second is mathēsis, introduced by Martin Heidegger in his discussion of seventeenth-century science. The third is Denkstil, a key concept in Ludwik Fleck’s history of syphilology. I conclude by listing three desiderata for a hermeneutic science studies: that it be anti-metaphysical, historical, and sociological. --- Reprinted in: Erich Otto Graf, Martin Schmid & Johannes Fehr (eds.), Fleck and the Hermeneutics of Science (Collegium Helveticum Heft 14) (Zürich, 2016), pp. 85-93. (shrink)

The paper discusses some changes in Bolzano's definition of mathematics attested in several quotations from the Beyträge, Wissenschaftslehre and Grössenlehre: is mathematics a theory of forms or a theory of quantities? Several issues that are maintained throughout Bolzano's works are distinguished from others that were accepted in the Beyträge and abandoned in the Grössenlehre. Changes are interpreted as a consequence of the new logical theory of truth introduced in the Wissenschaftslehre, but also as a consequence of the overcome of Kant's (...) terminology, and of the radicalization of Bolzano's anti‐Kantianism. Bolzano's evolution is understood as a coherent move, once the criticism expressed in the Beyträge on the notion of quantity is compared with a different and larger notion of quantity that Bolzano developed already in 1816. This discussion is enriched by the discovery that two unknown texts mentioned by Bolzano in the Beyträge can be identified with works by von Spaun and Vieth respectively. Bolzano's evolution is interpreted as a radicalization of the criticism of the Kantian definition of mathematics and as an effect of Bolzano's unaltered interest in the Leibnizian notion of mathesis universalis. As a conclusion, the author claims that Bolzano never abandoned his original idea of considering mathematics as a scientia universalis, i.e. as the science of quantities in general, and suggests that the question of ideal elements in mathematics, apart from being a main reason for the development of a new logical theory, can also be considered as a main reason for developing a different definition of quantity. (shrink)

Descartes' methodical doubt is being criticized by naïve realists and others who don't find doubt as a good starting point for metaphysical thought, however, the philosophical achievements of his method have been absorbed in all later philosophies. The objective of this paper is to demonstrate how an inevitable question concerning the foundation of Descartes' mathesis universalis, which led him to investigate this foundation by applying this very method in Metaphysics, has finally enabled him to discover his most important philosophical (...) principle, i.e., the necessary relation between being and reason in the identity of the "I". To demonstrate this, after summarizing the fundamental ideas of mathesis uneversalis, and showing the inner necessity of introducing the methodical doubt to fortify these ideas, the decisive role of metaphysical doubt as the third step of the methodical doubt, and malicious demon as its second phase would be highighted. It would also become clear that the metaphysical doubt has a great importance as a turning point from different aspects: as a knot that ties mathesis universalis with its underlying Metaphysics; as a bridge from within the psychological "I" to the metaphysical "I". (shrink)

The text discusses Linnaeus’ binominal classification and the idea of mathesis unversalis of Leibniz in the ground of Michel Foucault’s conception of ‘epistema’ as well the connexion of XVIII century’s representation of universal order with viewpoints of Descartes (Rules for the Direction of the Mind) and Kant (The Critique of Pure Reason, The Critique of (the Power of) Judgment).

Husserl’s theory of manifolds was developed for the first time in a very short form in the Prolegomena to his Logical Investigations, §§ 69–70 (pp. 248–53), then repeatedly discussed in Ideas I, §§ 71–2 (pp. 148–53), in Formal and Transcendental Logic, §§ 51–4 (pp. 142–54), and finally in the Crisis, § 9 (pp. 20–60). Husserl never lost sight of it: it was his idée fixe. He discussed this theme over forty years, expressing the same, in principle, ideas on it in (...) different terms and versions. His discussions of it, however, were always cut short and inconclusive, so that he never developed his theory of manifolds in detail. Main claim in this essay is that Husserl’s theory of manifolds is twofold. It is (1) a theory of theories, advanced in the good German tradition of Wissenschaftslehre, launched first by Bernard Bolzano. The latter, in turn, is an offspring of the ancient tradition of mathesis universalis, explored in the modernity by Descartes and Leibniz. Further, (2) the theory of manifolds is also a formal theory of everything. Its objective is to provide a systematic account of how the things in the world (its parts) hang together in wholes of quite different kind. It is to be noted that Husserl explicitly speaks about (1) and only implicitly about (2). In this paper we shall also show that the theory of everything is intrinsically connected with the theory of theories. (shrink)

In this article, I deal with the role of hypothesis in the scientific methodology of Isaac Newton, Francis Bacon and René Descartes. The first paragraph is about hypothesis in Newton's lexicon, especially trying to understand the meaning of his famous hypotheses non fingo. The second paragraph deals with Bacon's methodology, arguing especially that his epis-temology was the first to propose an artificial way for inductive inferences, also giving up all hypothesis in science. The third paragraph shows how Descartes, following Bacon's (...) traces and reading his methodology in the light of the idea of a Mathesis Universalis based on mental evidence, structures a full hypothetical-deductive methodology, charging metaphysics with the analytical phase and depriving experiments of any role in it. The fourth paragraph, on Newton again, tries to understand Newton's specific account of hypothesis as an auxiliary phase in the scientific discovery, and what really differentiates Newton from his contemporaries, Boyle and Hooke. (shrink)

In his philosophical structure, Descartes intended to organize, on the basis of mathematics, an efficient method of certainty for research and philosophy. The Mathesis is a body of universal and necessary truths of order and connection between ideas. According to Descartes' epistemology, man has become a researcher and discoverer who must discover himself. For Descartes, certain knowledge is possible, and in his structural cognitive style, action is a function of the form, while Dewey's cognitive style is a function in (...) which form is a function of action and behavior. Examining the consequences of Darwinian views of the world, Dewey rejects Descartes' view: The question of the possibility of human knowledge has been fundamentally wrong. Dewey, by looking again at the absolute, the unconditional, and the certain, in philosophy, sees it only as a means to use if it results in success. Philosophy is not merely a mental organization of eternal and absolute truth, but a technology, method, and technique to be a means for social reform, regulation, and improvement of human and societal behavior. (shrink)

Esta coletânea reúne, organizados na forma de capítulos, sete artigos que exploraram uma ideia central: a de que, subjacente à Lógica de Hegel, há uma mathesis, isto é, uma estruturação matemática que organiza as ideias dessa Lógica, mathesis de que Hegel faz uso, de modo implícito, ao longo do texto da "Ciência da Lógica".

Existe una larga trayectoria en la relación entre matemáticas y narrativa; en este trabajo se muestran los modos específi cos en que se vinculan pese a lo disímiles que, en apariencia, pueden llegar a ser. A partir de un modelo semántico se construye un esquema de análisis narrativo de carácter diagramático y articulación relacional, que considera y se apoya en la noción de ‘Visibilidad’ como dimensión semiótica.

Ce texte est la préface à la réédition d'un certain nombre de textes du philosophe français Jean-Toussaint Desanti, philosophe des mathématiques qui a été marqué par la phénoménologie husserlienne et la pensée de Jean Cavaillès. Nous nous efforçons d'y rassembler les thèmes et thèses centraux de la pensée de Desanti, telle qu’elle s’élabore dans ces textes publiés dans différentes revues : en premier lieu, le thème de l’historicité des idéalités mathématiques et la récusation du sujet comme instance productrice de ces (...) objets. (shrink)

By extending Husserl’s own historico-critical study to include the conceptual mathematics of more contemporary times – specifically category theory and its emphatic development since the second half of the 20th century – this paper claims that the delineation between mathematics and philosophy must be completely revisited. It will be contended that Husserl’s phenomenological work was very much influenced by the discoveries and limitations of the formal mathematics being developed at Göttingen during his tenure there and that, subsequently, the rôle he (...) envisaged for his material a priori science is heavily dependent upon his conception of the definite manifold. Motivating these contentions is the idea of a mathematics which would go beyond the constraints of formal ontology and subsequently achieve coherence with the full sense of transcendental phenomenology. While this final point will be by no means proven within the confines of this paper it is hoped that the very fact of opening up for the possibility of such an idea will act as a supporting argument to the overriding thesis that the relationship between mathematics and phenomenology must be problematised. (shrink)

El análisis de las propiedades geométricas de las configuraciones finitas ha sido uno de los objetivos fundamentales del estudio de las diversas geometrías discretas y de la geometría combinatoria. Este artículo propone plantear la posibilidad de una elucidación de lo matemático desde una perspectiva derivada de las ‘matemáticas en acción’ y no desde una concepción ‘analítico gramatical’ de sus fundamentos, y establecer, al menos, un mínimo umbral de validez, que articule una interpretación semiótica de los signos matemáticos, a través del (...) análisis de un par de ejemplos elementales de geometría configuracional, cuyas construcciones son propuestas por el autor. (shrink)

El presupuesto según el cual el contenido de una manifestación compleja está en función de los contenidos de sus partes componentes, expresa claramente una intuición que solemos tener sobre lo múltiple; implica una reflexión sobre la relación entre el todo y las partes que lo componen; involucra una teoría de las multiplicidades que entraña atributos de naturaleza matemática; presenta el problema de cómo los seres humanos nos relacionamos con los entornos del mundo para generar unidad de sentido. La significación es (...) un proceso de síntesis. Y desentrañar los mecanismos de funcionamiento de dicho proceso es un enigma de carácter eminentemente fenomenológico. La matemática misma, como acto de significación, como conjunto de actos de contenidos intencionales, comparte este carácter y es susceptible de tratamiento fenomenológico. Noesis y semiosis se conjugan para dotar al saber matemático de un contenido, de articulación compleja, que trasciende las perspectivas meramente logicistas, formalistas o intuicionistas. Una primera aproximación a esta problemática es el propósito del presente trabajo. (shrink)

A partir de la conformación de un modelo semántico fundado por la relación de presuposición se construye un entramado algebraico de carácter diagramático y articulación reticular, que considera y se apoya en la noción de orden como dimensión semiótica. Esta articulación relacional se realiza al abarcar diversos niveles de análisis, que van desde un nivel grafémico a un nivel discursivo. Se destacan de modo significativo los procesos complejos de composicionalidad semántica (asociados a modalidades de síntesis) en contraposición con procesos elementales (...) de aditividad (asociados a sumas analíticas). Asimismo, a lo largo del análisis, quedarán establecidos los elementos configuracionales para la articulación reticular de los sucesos pertenecientes a un relato, formando unidades discursivas complejas a través de esquemas narrativos. (shrink)