Paola Cantù

After recalling some mathematical contributions that are relevant for the structuralist transformation of mathematics, such as abstract algebra, linear algebra, and number theory, this chapter reconstructs Grassmann’s philosophy of mathematics. It is claimed that he contributed to the development of methodological structuralism inasmuch as he clearly separated the study of the most general properties of connections from pure and applied mathematics, basing them on an understanding of generality as conceptual underdetermination, and on the preeminence of the notion of series over that of function. A brief comparison with contemporary philosophical structuralism will clarify Grassmann’s tendency toward a “concept structuralism” rather than an “object structuralism.”

The paper investigates the notions of number, extensive quantity, algebraic quantity, similarity, and probability as introduced in Wolff’s mathematical writings, and evaluates the coherence of these definitions with Wolff’s presentation of the mathematical method. Wolff’s original epistemology is based on the belief that the discussion on the foundation of mathematics and on the history of mathematical ideas is essential not only to pure mathematics but also to its application and teaching. Mathematics is integrated into a larger body of knowledge and subordinated to philosophy, which explains the impurity of definitions and proofs and why certain notions, such as that of similarity and probability, cannot be understood without reference to ontology and logic. Wolff’s investigation of probable propositions allowed for a unique demonstrative methodology applied to philosophical, mathematical, and historical knowledge, because he described probable reasoning as an insufficient knowledge of the requisites for the truth of the premises. This explains the importance assigned to the right order of concepts and the inferential role of definitions. The paper shows the internal coherence between Wolff’s choice of mathematical definitions and his overall epistemology, and underlines the heritage of Descartes, of Barrow’s edition of Euclid’s Elements and of Aristotle’s notion of contiguity rather than the influence exerted by Leibniz’s mathematics or by Newton’s fluxional conception of geometrical quantities.

_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 philosophy of mathematical practice sometimes investigates the social constitution of mathematics but does not always make explicit the philosophical-normative framework that guides the discussion. This chapter investigates some recent proposals in the philosophy of mathematical practice that compare social facts and mathematical objects, discussing similarities and differences. An attempt will be made to identify, through a comparison with three different perspectives in social ontology, the kind of objectivity attributed to mathematical knowledge, the type of representational or non-representational semantics adopted, and the justificatory or coordinative role entrusted to axiomatics. After a brief introduction to key issues in social ontology, Sect. 3 of the chapter offers a survey of contributions by Feferman, Ferreirós, and Cole, highlighting a difference between approaches based on rules, practices, and intentional states, respectively. Section 4.1 also discusses results by Carter, Collin, Giardino, and Pantsar, focusing on differences between the objectivity of knowledge and objectivity of objects and showing that many authors combine a realist, structuralist, and pragmatist perspective, as well as the idea that objectivity comes in degrees. Section 4.2 focuses on the kind of semantics that is adopted in socially oriented approaches. A representational semantics is preferred by authors grounding their views on mental states, whereas a non-representational semantics best fits with views based on practices. Section 5 considers how axiomatics can be understood in a social perspective. Axiomatics generally plays a justificatory role also in theories that aim to explain the social constitution of mathematical knowledge. Yet, if one shifts from approaches based on mental states to approaches anchored on rules or habits, axiomatics can be viewed as an institution based on obligations, functions, coordination problems, and agent’s actions and roles, thus playing an organizational and coordination role.

The paper advocates an epistemological interpretation of the Peano School axiomatics. The construction of axiom systems is presented as a cognitive enterprise unveiling the internal dynamics, evolution, and architecture of axiomatic systems as well as connections to applications. This approach reveals that the study of the relation between axioms and theorems not only serves to reduce a theory to a minimum number of principles and increase the certainty or justification of the latter, but also to study alternative settings of a given mathematical theory. The paper explains why the Peanian conception of axiomatics differs from axiomatic deductivism, if-thenism and scientific deductivism, and analyzes Peano’s conception of rigor. The paper also postulates that Peano had a set-theoretic but also an algorithmic conception of function, and prioritized functions over relations. Two seemingly contradictory aspects of Peano’s axiomatics, global and local reasoning, are illustrated through the examples of rigor and simplicity. Finally, the paper claims that Peano’s axiomatics was a dynamic mathematical enterprise rather than an a priori axiomatization grounded on logical principles, which explains the difference with respect to Frege’s project.

The turn of the last century was a key transitional period for the development of symbolic logic and scientific philosophy. The Peano school, the editorial board of the Revue de Métaphysique et de Morale, and the members of the Vienna Circle are generally mentioned as champions of this transformation of the role of logic in mathematics and in the sciences. The articles contained in this volume aim to contribute to a richer historical and philosophical understanding of these groups and research areas in Italy, France and Austria. Specifically, the contributions focus on the following topics: a detailed investigation of the relation between structuralism and modern mathematics; different notions of definition and interpretation at the turn of last century; a closer understanding of the relation between the Vienna Circle, the Peano School and French philosophy in the first half of the twentieth century.

Recent years have featured the existence of a variety of structuralisms, with an important partition between methodological versus philosophical structuralism. Inside philosophical structuralism, many trends can be identified, corresponding to various ontological stances. We argue here that another main partition has contributed to organize structuralism in the twentieth century, rooted in different technical and theoretical interests. This partition is largely transversal to the ones classically identified. Concretely, the paper will focus on possible differences between an arithmetical and logical notion of structure that can be traced back to the writings of Bertrand Russell and Rudolf Carnap, and a mathematical notion of structure, exemplified in the works by Bourbaki. This coexistence gives rise to a fundamental ambiguity that affects contemporary structuralism. Philosophically, in one case the attention is rather centered on a foundational and reductionist perspective, as featured by the Whitehead-Russell Principia and the Carnapian project of the Aufbau: the scientific construction of the world around the idea of structure. In the other, the focus is on epistemological and dynamical issues, as exemplified by two key issues in Bourbaki’s treatise: understanding the architecture of mathematics, offering a tool-kit to mathematicians. These two distinct meanings still coexist inside contemporary scientific practices and lead to different theoretical interests, as we will show thanks to various recent examples.

This book provides a collection of chapters on the development of scientific philosophy and symbolic logic in the early twentieth century. The turn of the last century was a key transitional period for the development of symbolic logic and scientific philosophy. The Peano school, the editorial board of the Revue de Métaphysique et de Morale, and the members of the Vienna Circle are generally mentioned as champions of this transformation of the role of logic in mathematics and in the sciences. The scholarship contained provides a rich historical and philosophical understanding of these groups and research areas. Specifically, the contributions focus on a detailed investigation of the relation between structuralism and modern mathematics. In addition, this book provides a closer understanding of the relation between symbolic logic and previous traditions such as syllogistics. This volume also informs the reader on the relation between logic, the history and didactics in the Peano School. This edition appeals to students and researchers working in the history of philosophy and of logic, philosophy of science, as well as to researchers on the Vienna Circle and the Peano School.

Two currents of thought play an important role in the contemporary philosophy of mathematics. Although structuralism is not a new idea, it continues to unfold in multiple directions ranging from mathematical practice to its ontological dimension and to be the object of studies, for example as regards the modalities of its genesis. The classical conception of historical epistemology has been broadly enriched recently and is also at the heart of debates that renew the philosophy of science well beyond its classical themes. However, its use in mathematics requires specific analyses because notions such as rigor, experience or ontology have a different meaning and historical weight in mathematics than in the natural sciences. The purpose of this article will be to put some of these debates into perspective by focusing on one of the most significant phenomena of mathematics in the twentieth century—the structuralism of Bourbaki. It will focus in particular on the potentialities of historical epistemology in an attempt to show that its tools enable us to enhance our understanding of some of the issues of structuralism. We will focus on two key notions in the work of Rheinberger, Daston and Hacking, namely epistemic objects and concepts.

In this introductory essay we compare different strategies to study the possibility of applying philosophical theories of social ontology to mathematical practice and vice versa. Analyzing the contributions to the special issue Mathematical practice and social ontology, we distinguish four main strands: (1) to verify whether the very act of producing mathematical knowledge is an intersubjective activity; (2) to explain how the intersubjective nature of mathematics relates to mathematical objectivity; (3) to show how this intersubjectivity-based objectivity is the result of social practice; (4) to understand whether, given the social nature of intersubjectivity-based mathematical objectivity, mathematical objects can be described by analogy with social facts as institutions.

Among the numerous influences and reciprocal interactions between France and Italy at the beginning of the 20th century, it is interesting to investigate the complex case of Louis Rougier’s reception of Italian mathematical logic (including in particular the contributions by some members of the Peano school: Giuseppe Peano, Giovanni Vailati, Alessandro Padoa, and Mario Pieri). This paper aims to investigate the role and the influence of the Peano school on the inversion of this French tendency of philosophers to ignore logic and mathematics. My claim is that Rougier was not only personally interested in Italian mathematics and logic, and in particular in the innovations made by Peano and his school, but used it to support a project of renewal of French logic and scientific philosophy. Rougier recalls the contributions of the school to axiomatics and to mathematical logic, in order to criticize the standard teaching of logic in France, mainly based on syllogistic theory, as well as the view, still defended by his teacher Goblot in 1907, that mathematical proofs are largely based on intuition. For this purpose, any reference to Hilbert’s axiomatics would have sufficed. The reason why Rougier refers also to Peano and his entourage is related to a common interest in linguistics and in epistemology.

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 history of 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. Two introductory essays by the volume’s editors outline the traits of Vailati’s pragmatism and provide insights into the scholar’s life.

The question of the applicability of mathematics is an epistemological issue that was explicitly raised by Kant, and which has played different roles in the works of neo-Kantian philosophers, before becoming an essential issue in early analytic philosophy. This paper will first distinguish three main issues that are related to the application of mathematics: indispensability arguments that are aimed at justifying mathematics itself; philosophical justifications of the successful application of mathematics to scientific theories; and discussions on the application of real numbers to the measurement of physical magnitudes. A refinement of this tripartition is suggested and supported by a historical investigation of the differences between Kant’s position on the problem, several neo-Kantian perspectives, early analytic philosophy, and late 19th century mathematicians. Finally, the debate on the cogency of an application constraint in the definition of real numbers is discussed in relation to a contemporary debate in neo-logicism, in order to suggest a comparison not only with Frege’s original positions, but also with the ideas of several neo-Kantian scholars, including Hölder, Cassirer, and Helmholtz.

At the beginning of the xxth century the high rate of analphabetism and the recent unification of the country, achieved only in 1870, had required a vast program of school and university reforms which were accompanied by a debate on two fundamental questions: whether the university should depend on public funds or become autonomous, and whether the curriculum should be specialized or remain general as in the modern era. The 1859 Casati reform had separated the faculty for literature and philosophy from the faculty for mathematical, physical and natural sciences, thus introducing for the first time specialized curricula.

The article evaluates the Domain Postulate of the Classical Model of Science and the related Aristotelian prohibition rule on kind-crossing as interpretative tools in the history of the development of mathematics into a general science of quantities. Special reference is made to Proclus’ commentary to Euclid’s first book of Elements, to the sixteenth century translations of Euclid’s work into Latin and to the works of Stevin, Wallis, Viète and Descartes. The prohibition rule on kind-crossing formulated by Aristotle in Posterior analytics is used to distinguish between conceptions that share the same name but are substantively different: for example the search for a broader genus including all mathematical objects; the search for a common character of different species of mathematical objects; and the effort to treat magnitudes as numbers.

The paper analyses the role played by the concept of ‘common ground’ in argumentation theories. If a common agreement on all the rules of a discursive exchange is required, either at the beginning or at the end of an argumentative practice, then no violation of the rules is possible. The paper suggests an alternative understanding of ‘common ground’ as something that can change during the development of the argumentative practice, and in particular something that can change without the practice being stopped, just as one sometimes violates the rule of a game without stopping playing. The analysis of some historical examples shows that the violation of rules sometimes leads to a consensual change of a set of beliefs and values. The argumentative rationality thus needs to be reconsidered in the light of the analysis of the conditions under which it would be legitimate to change the rules of an argumentative practice: ‘common ground’ and the right to disagreement, even radical or violent, could then be defended as essential components of argumentative rationality.

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.