Paola Cantù

Argumentation theory underwent a significant development in the Fifties and Sixties: its revival is usually connected to Perelman's criticism of formal logic and the development of informal logic. Interestingly enough it was during this period that Artificial Intelligence was developed, which defended the following thesis (from now on referred to as the AI-thesis): human reasoning can be emulated by machines. The paper suggests a reconstruction of the opposition between formal and informal logic as a move against a premise of an argument for the AI-thesis, and suggests making a distinction between a broad and a narrow notion of algorithm that might be used to reformulate the question as a foundational problem for argumentation theory.

The aim of the paper is to analyze Hölder’s understanding of geometry and measurement presented in Intuition and Reasoning [Hölder 1900], “The Axioms of Quantity and the Theory of Measurement” [Hölder 1901], and The Mathematical Method [Hölder 1924]. The paper explores the relations between a) Hölder’s demarcation of geometry from arithmetic based on the notion of given concepts, b) his philosophical stance towards Kantian apriorism and empiricism, and c) the choice of Dedekind’s continuity in the axiomatic formulation of the theory of magnitudes. The paper shows that the choices made at the axiomatic level reflect Hölder’s epistemological framework, and individuates the originality of his approach in the case by case analysis of deductive procedures.

The philosophy of mathematics has been accused of paying insufficient attention to mathematical practice: one way to cope with the problem, the one we will follow in this paper on extensive magnitudes, is to combine the `history of ideas' and the `philosophy of models' in a logical and epistemological perspective. The history of ideas allows the reconstruction of the theory of extensive magnitudes as a theory of ordered algebraic structures; the philosophy of models allows an investigation into the way epistemology might affect relevant mathematical notions. The article takes two historical examples as a starting point for the investigation of the role of numerical models in the construction of a system of non-Archimedean magnitudes. A brief exposition of the theories developed by Giuseppe Veronese and by Rodolfo Bettazzi at the end of the 19th century will throw new light on the role played by magnitudes and numbers in the development of the concept of a non-Archimedean order. Different ways of introducing non-Archimedean models will be compared and the influence of epistemological models will be evaluated. Particular attention will be devoted to the comparison between the models that oriented Veronese's and Bettazzi's works and the mathematical theories they developed, but also to the analysis of the way epistemological beliefs affected the concepts of continuity and measurement.

Giuseppe Veronese is known for his studies on spaces with more dimensions; less known are his “philosophical” writings, that concern the foundations of geometry and mathematics and explain the reasons for constructing a non-Archimedean geometry (several years before David Hilbert’s Grundlagen) and the formulation of a concept of continuity that admits infinitely big and small quantities. After sketching some relevant aspects of Veronese’s epis-temology, the article will analyse the relation between geometry and spatial intuition by a comparison of Veronese’s, Helmholtz’s and Poincaré’s conceptions of the geometrical space. According to Veronese, the representational space, and the geometrical space as well, are not Euclidean nor have a definite number of dimensions: that is why one can represent himself a space with four dimensions and even a non-Archimedean continuum.

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.

TABLE OF CONTENTS I. La rinascita novecentesca 1. Chaïm Perelman: la nuova retorica 2. Stephen Toulmin: la pratica logica e l’uso degli argomenti 3. Ragionamento e linguaggio: la logica naturale di Jean-Blaise Grize II. La logica informale 1. Informale vs. formale? 2. Il concetto di argomento 3. La ripresa della teoria di Paul Grice 4. La ricostruzione degli argomenti 5. La valutazione degli argomenti: le fallacie 6. Il network problem III. Dialogo e dialettica 1. La logica dialogica di Paul Lorenzen 2. Charles L. Hamblin e i sistemi dialettici 3. Else Barth e la dialettica formale 4. Jaakko Hintikka: logica interrogativa e teoria dei giochi IV. Pragmatica e dialettica 1. La Pragma-Dialectics di van Eemeren e Grootendorst: un ideale filosofico della razionalità 2. Ricostruzione e valutazione degli argomenti 3. L’analisi delle fallacie 4. Fallacie e biases: tra argomentazione e psicologia V. Intersoggettività e impegni dialogici 1. La New Dialectic e il ragionamento interpersonale: Douglas Walton ed Erik Krabbe 2. Contesti dialogici e commitment store 3. Teoria delle fallacie e presumptive reasoning 4. La fondazione della Informal Logic 5. Argomentazione, informatica e scienza cognitiva VI. Razionalità e fondazione 1. Razionalità discorsiva: la pragmatica universale di Jürgen Habermas 2. Razionalità inferenziale e semantica: la pragmatica normativa di Robert B. Brandom 3. La pragmatica trascendentale di Karl O. Apel 4. Modelli di razionalità 5. Fondazione o giustificazione VII. Argomentazione e pratiche sociali 1. Ragionamento pratico 2. Etiche del discorso 3. Diritto e politiche dell’argomentazione.

_Logic and Pragmatism_ features a number of the key writings of Giovanni Vailati (1863–1909), the Italian mathematician and philosopher renowned for his work in mechanics, geometry, logic, and epistemology. The selections in this book—many of which are available here for the first time in English—focus on Vailati’s significant contributions to the field of pragmatism. Accompanying these pieces are introductory essays by the volume’s editors that outline the traits of Vailati’s pragmatism and provide insights into the scholar’s life.

The paper aims to establish if Grassmann’s notion of an extensive form involved an epistemological change in the understanding of geometry and of mathematical knowledge. Firstly, it will examine if an ontological shift in geometry is determined by the vectorial representation of extended magnitudes. Giving up homogeneity, and considering geometry as an application of extension theory, Grassmann developed a different notion of a geometrical object, based on abstract constraints concerning the construction of forms rather than on the homogeneity conditions required by the modern version of the theory of proportions. Secondly, Grassmann’s conception of mathematical knowledge will be investigated. Parting from the traditional definition of mathematics as a science of magnitudes, Grassmann considered mathematical forms as particulars rather than universals: the classification of the branches of mathematics was thus based on different operational rules, rather than on empirical criteria of abstraction or on the distinction of different species belonging to a common genus. It will be argued that a different notion of generalization is thus involved, and that the knowledge of mathematical forms relies on the understanding of the rules of generation of the forms themselves. Finally, the paper will analyse if Grassmann’s approach in the first edition of the Ausdehnungslehre should be explained in terms of the notion of purity of method, and if it clashes with Grassmann’s later conventionalism. Although in the second edition the features of the operations are chosen by convention, as it is the case for the anti-commutative property of the multiplication, the choice is oriented by our understanding of the resulting forms: a simplification in the algebraic calculus need not correspond to a simplification in the ‘dimensional’ interpretation of the result of the multiplicative operation.