Abstract

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...