The book explores Peirce's non standard thoughts on a synthetic continuum, topological logics, existential graphs, and relational semiotics, offering full mathematical developments on these areas. More precisely, the following new advances are offered: (1) two extensions of Peirce's existential graphs, to intuitionistic logics (a new symbol for implication), and other non-classical logics (new actions on nonplanar surfaces); (2) a complete formalization of Peirce's continuum, capturing all Peirce's original demands (genericity, supermultitudeness, reflexivity, modality), thanks to an inverse ordinally iterated sheaf of (...) real lines; (3) an array of subformalizations and proofs of Peirce's pragmaticist maxim, through methods in category theory, HoTT techniques, and modal logics. The book will be relevant to Peirce scholars, mathematicians, and philosophers alike, thanks to thorough assessments of Peirce's mathematical heritage, compact surveys of the literature, and new perspectives offered through formal and modern mathematizations of the topics studied. (shrink)

We study the emergence of all references to modern mathematics in Difference and Repetition, and we explain how, using these mathematical approaches, Deleuze constructs an “ideal synthesis of difference” which goes well beyond the usual reductionist perspectives of his commentators.

Peirce’s architectonics, far from rigid, is bended by many plastic transformations, deriving from the cenopythagorean categories, the pragmaticist (modal) maxim, the logic of abduction, the synechistic hypotheses and the triadic classification of sciences, among many other tools capable of molding knowledge. Plasticity, in turn, points to interlacements between mathematics and art, and shapes some associated conceptual forces in the boundary of the disciplines: variation, modulation and invariance; transformability, continuity and discreteness; creative emergence. In this article we focus on this third (...) aspect, through bounded, well defined case studies in the Logic Notebook. The first section describes the manuscript and its interest for a study of creativity, leading to a short speculation on “creative reason” and “plastic imagination” in Peirce. The second section studies five precise cases of creative emergence in the Logic Notebook: differential relatives, existential graphs, sequence diagrams, triadic logic, translatability. Some major surprises occur in those detailed studies. (shrink)

It is possible today to observe in hindsight the epistemological landscape of the twentieth century, and the work of Albert Lautman in mathematical philosophy appears as a profound turning point, opening to a true under- standing of creativity in mathematics and its relation with the real. Little understood in its time or even today, Lautman’s work explores the difficult but exciting intersection where modern mathematics, advanced mathe- matical invention, the structural or unitary relations of mathematical knowledge and, finally, the metaphysical (...) and dialectical tensions underly- ing mathematical activity converge. Well beyond other better-known names in philosophy of mathematics – who are focused above all on ques- tions concerning the logical problem of foundations, important but frag- mentary studies in the vast panorama of modern mathematics – Lautman broaches the emergence of inventiveness in the very broad spectrum of the development of the mathematical real. Group theory, differential geome- try, algebraic topology, differential equations, functional analysis, functions of complex variables and number fields are some of the domains of his preferred examples. He detects in them methods of construction, structu- ration and unification of modern mathematics that he connects to a precise Platonic interpretation in which powerful pairs of ideas serve to organize the edifice of effective mathematics. (shrink)

We show how Peirce's architectonics folds on itself and finds local consequences that correspond to the major global hypotheses of the system. In particular, we study how the pragmaticist maxim can be technically represented in Peirce's existential graphs, well-suited to reveal an underlying continuity in logical operations, and can provide suggestive philosophical analogies. Further, using the existential graphs, we formalize — and prove one direction of — a “local proof of pragmaticism,” trying thus to explain the prominent place that existential (...) graphs can play in the architectonics of pragmaticism, as Peirce persistently advocated. Finally, we present a web of “continuous iterations” of some key Peircean concepts that supports a “lattice of partial proofs” of pragmaticism. (shrink)

The article presents a reading of Badiou’s trilogy, L’Être et l’événement, Logiques des mondes, and L’Immanence des vérités, and points out the mathematical connections with the works of Cohen, Grothendieck, and large cardinal specialists. A synthetic rendering of these connections is first offered, following precise passages in Badiou’s work, then a category-theoretic and Peircean perspective is explored in order to specify the many dialectics in the trilogy, and, finally, some open problems are proposed.

Many natural connections between Peirce and Whitehead have been appearing since Peirce’s manuscripts began to be published in Harvard’s Collected Papers edition. The resemblances are profound, based on speculative philosophies founded on interactions between science and philosophy, in turn erected over a wide understanding of mathematical thought. Some doctoral dissertations have explored the field, but a full monograph was still needed in order to clarify the panorama. Maria Regina Brioschi’s Creativity Between Experience and Cosmos. C.S. Peirce and A.N. Whitehead on (...) Novelty provides the sound, exact, and extremely... (shrink)

Some creativity modes in art and mat he - matics are studied, both looking for specificities and possible transits. Peirce’s “pragmaticist” ar - chitectonics –and, in particular, its attention to reasonableness and creativity– help to weave a counter point between art and literature on a basis ..

En este artículo se muestra una comparación de las lógicas diagramáticas lulianas y peirceanas. En la Sección 1, se introduce una problemática general del tránsito y de sus resoluciones triádicas y espaciales, aprovechando perspectivas de las obras de Aby Warburg y Walter Benjamin. La Sección 2 es una síntesis de los engranajes fundamentales del sistema diagramático de Llull. En la Sección 3, se resumen las características geométrico-topológicas esenciales de los gráficos existenciales de Peirce. Y finalmente, en la Sección 4, se (...) efectúa un rastreo de las apariciones de Llull en los textos de Peirce, y proponemos un reencuentro entre los dos pensadores. (shrink)

Theoria, Volume 87, Issue 4, Page 885-896, August 2021.

This article studies some mathematical concepts which may be ramified in cultural environments, particularly in music and literature. The notions of definability (What may we capture with language, what escapes us?), representation (Which archetypes may be projected onto the world, which types arise from such projections?) and categoricity (Which unitary concepts govern the multiplicity of languages and images?) are narrowly linked with the essential issue of the trans, perhaps the crucial problem of our fleeting times.

Nous étudions le Rapport de Lautman qui ouvre ce volume. Après une description critique des contenus et une discussion des fonds conceptuels du Rapport, nous montrons comment l’émergence postérieure de la théorie des faisceaux répond techniquement aux questionnements et aux préfigurations de Lautman.We study Lautman’s Rapport which opens this volume. After a critical description and a discussion of the contents of the Rapport, we show how the emergence of sheaf theory answers technically many questions raised by Lautman ten years earlier.

We introduce the work of Mariano Picón Salas, who tries to enlarge the Hispanoamerican tradition —historically focused on the web of transits/obstructions between Europe and Latin America— via a larger triangulation that includes the presence of the culture of the United States. His essays from the period 1930-1950 show how pragmatism —understood in Peircean terms as a web of topological transfers of knowledge between the borders of culture— can help to provide a vision that encompasses America as a whole.

As has happened in Latin America, and more generally in the Hispanic World, pragmatism came to our countries mainly through William James and John Dewey (particularly, through his influence in education). Studies in Spanish on Peirce were scarce and superficial until the end of the 20th century. The situation in Colombia follows that pattern. The first valuable Colombian study on Peirce came from Mariluz Restrepo (1993), a fine introduction to Peirce’s semeiotics through the unfolding of the...

No son muchos los textos en filosofía de las matemáticas que intentan dirigirse a la práctica matemática avanzada y, menos aún, aquellos que intentan acercarse a las matemáticas contemporáneas en acción. La lista de tales esfuerzos es bastante reducida—Pólya, Lakatos, Kline, Wilder, Kitcher acercándose a la matemática clásica; De Lorenzo, MacLane, Tymoczko haciéndolo a la matemática moderna; Badiou, Maddy, Patras a la matemática contemporánea—por lo que el trabajo de Corfield, ya sólo por situarse en esa línea minoritaria, merece una cierta (...) atención. El trabajo sin embargo cuenta con amplios méritos propios, más allá de apoyar una saludable línea alternativa, que se contrapone con las tendencias predominantes de la filosofía analítica o de la filosofía del lenguaje aplicadas al universo matemático. Desde el comienzo mismo, gracias a su polémico título, Towards a Philosophy of Real Mathematics, intenta romper con los prejuicios normativos al uso en la filosofía matemática, en particular con las “creencias entre filósofos de que el estudio de las mayores corrientes matemáticas recientes es innecesario”. Las matemáticas “reales” evocan la Apología de Hardy en donde el matemático inglés identifica matemáticas reales con matemáticas avanzadas, ya sea clásicas, del tipo Euler o Gauss, ya sea modernas, del tipo Galois o Riemann. Una amplia introducción presenta una argumentada defensa del valor de una perspectiva filosófica orientada hacia esas matemáticas no elementales, y exhibe algunos de los problemas mayores que emergen en esa aproximación, pero que, en cambio, el “filtro fundacionalista” no deja detectar: el estatuto de los bordes estructurales de las matemáticas, la conectividad de las diferentes teorías matemáticas, la evolución de los conceptos matemáticos, la contingencia del pensamiento matemático, la progresiva riqueza recursiva de las construcciones matemáticas. (shrink)

The article presents two gestures corresponding to two profound new understandings of Peirce's Continuum and Peirce's Existential Graphs. Vargas and Oostra have revolutionized Peirce's mathematical studies, thanks to a first complete model for Peirce's continuum provided by Vargas, and thanks to the emergence of intuitionistic existential graphs provided by Oostra. The article aims at showing how these careful mathematical constructions can be encrypted in very simple gestures.

We study a contemporary need to complement analytic philosophy with pendular, synthetic approaches. We provide new definitions of the dyad analytics/synthetics and complete it with a natural third, horotics. Some historical trends to support a synthetic/horotic paradigm are studied: (i) Peirce’s ideas around his logic of continuity – non Cantorian continuum and existential graphs – emphasizing the importance of mathematical gestures, (ii) Gödel’s understanding of intuitionism as a synthetic counterpart of classical logic, along with a new horotic approach to his (...) work, (iii) Contemporary mathematical achievements (1950-2000), difficult to understand from analytical philosophy perspectives. Finally, we indicate some main features that a systematical synthetic/horotic reasoning should enforce, in order to fulfill its historic sequence. (shrink)

La obra de Charles S. Peirce (1839-1914), considerado por muchos como un filósofo del siglo XIX para el siglo XXI, constituye una verdadera mina de aportaciones originales para entender el complejo y cambiante mundo que nos rodea. Gracias a un permanente empeño de colaboración, los autores de este libro han podido observar el conjunto de una situación dada alrededor de Peirce, concentrándose específicamente en el vaivén comunicativo producido alrededor de la díada Peirce-Mundo Hispánico. En la primera parte se contiene la (...) información disponible sobre el viaje de Peirce a España en 1870, y sus anotaciones sobre España y los españoles dispersas en sus escritos. La segunda parte provee una amplia perspectiva sobre “todo lo que el mundo hispánico ha dicho sobre Peirce” entre los años 1883 y 2000. El trabajo de revisión de la bibliografía consiste en un catálogo razonado de la producción hispánica sobre Peirce y un estudio crítico previo donde se delinean los aportes fundamentales, las carencias y las líneas de tensión de esa producción. -/- . (shrink)

Perhaps due to the period of international crisis, appeals to creativity multiplied in any field. Sure enough, when the status quo cannot grant welfare conditions anymore, something new is needed. And the problem of novelty intertwines that kind of thought that goes by the name of creativity. Philosophically speaking, this request means to question what creativity really is, which are its cognitive processes, whether it is teachable, and where it comes from. A first methodological question is...

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Advances in Peircean Mathematics: The Colombian School ed. by Fernando ZalameaGianluca CaterinaFernando Zalamea (Ed.) Advances in Peircean Mathematics: The Colombian School Berlin, Boston: De Gruyter, 2022. 212 pp. (incl. index).The volume Advances in Peircean Mathematics is an important, very much needed contribution towards a deeper understanding of the impact of Peirce's work especially in the fields of mathematics, logic, and semiotic. It fills a gap in (...) the current literature by recasting some of [End Page 373] Peirce's profound mathematical insights into a formal framework. This is a finely edited editorial project, which summarizes the work of more than two decades done by a group of scholars—the Colombian School—lead by Fernando Zalamea, a pioneer of Peirce studies, and pragmatism in general, in Latin America.The structure of the volume unfolds in three dense chapters, each corresponding to a specific mathematical context connected to various aspects of Peirce's work. Broadly speaking, in the first chapter, Gustavo Arengas proposes the theory of higher category theory as a natural framework to represent the main traits of Peirce's semiotic; in the second chapter, Francisco Vargas develops a thorough set-theoretical model of Peirce's continuum; in the third chapter, Arnold Oostra, in a truly Peircean spirit, formalizes the notion of Intuitionistic Existential Graphs, emphasizing the idea that intuitionistic logic is the most natural context to represent and understand Peirce's diagrammatic logic. The fourth and last chapter, penned by Zalamea, provides the reader with a solid summary of the main results presented in the volume along with a robust road map to navigate through the technical details of the text.Mathematics, as implied by the title of the volume, is at the core of this project, and the authors do not shy away from the complexity, details, and depth of it in order to shed light on some of the most fundamental and groundbreaking nature of Peirce's ideas. That level of rigor and formal soundness is certainly one of the strengths of the volume, although readers who may not be acquainted with at least some basic notions of category theory or with the mathematics of Peirce's work, may find some of the text a bit terse—this is not a casual reading and it does require a certain level of commitment in order to appreciate the depth of the provided insights. On the other hand, all the background material is clearly laid out, making the volume a self-contained work which can be thoroughly enjoyed by a general audience.The real upshot of recasting some of Peirce's intuitions within the fabric of formal mathematics is not only that of providing the instruments to clarify those ideas, but also that of highlighting the role that Peirce's work has been playing in the development of modern mathematics in many relevant aspects, from the very genesis of category theory—which today is widely regarded as a necessary and unifying language in several fields of the mathematical and physical sciences—to the development of non-standard analysis, just to mention a few of them.An elegant instance of such perspective is offered by Gustavo Arengas in the first chapter, where it is argued that ∞-categories can be used to model the natural order of hierarchies implicit in Peirce's conception of the basic triadic relation between sign, object, and interpreter. What, at first glance, may seem just a simple analogy, unfolds onto a methodical and enlightening analysis of Peirce's semiotic, as the author builds [End Page 374] the connections with the relevant categorical constructions in a precise manner. Most of the rest of the chapter focuses on Peirce's notions of the pragmatic and pragmaticist maxim, and the use of basic structures such as monads, limits, presheaves and sheaves as the natural environment to better convey their meaning, both from a mathematical and a philosophical perspective. The reader will find it interesting to realize that, by delving deeper into the understanding of the relational nature of category theory, a great chunk of Peirce's work can be seen as fundamentally intertwined with the apparatus and the language of modern mathematics... (shrink)

This work is a conceptual analysis of certain recent developments in the mathematical foundations of Classical and Quantum Mechanics which have allowed to formulate both theories in a common language. From the algebraic point of view, the set of observables of a physical system, be it classical or quantum, is described by a Jordan-Lie algebra. From the geometric point of view, the space of states of any system is described by a uniform Poisson space with transition probability. Both these structures (...) are here perceived as formal translations of the fundamental twofold role of properties in Mechanics: they are at the same time quantities and transformations. The question becomes then to understand the precise articulation between these two roles. The analysis will show that Quantum Mechanics can be thought as distinguishing itself from Classical Mechanics by a compatibility condition between properties-as-quantities and properties-as-transformations. -/- Moreover, this dissertation shows the existence of a tension between a certain "abstract way" of conceiving mathematical structures, used in the practice of mathematical physics, and the necessary capacity to specify particular states or observables. It then becomes important to understand how, within the formalism, one can construct a labelling scheme. The “Chase for Individuation” is the analysis of different mathematical techniques which attempt to overcome this tension. In particular, we discuss how group theory furnishes a partial solution. (shrink)