Frege's Intellectual Life As a Logicist Project

I critically discuss Dale Jacquette’s Frege: A Philosophical Biography. First, I provide a short overview of Jacquette’s book. Second, I evaluate Jacquette’s interpretation of Frege’s three major works, Begriffsschrift, Grundlagen der Arithmetik and Grundgesetze der Arithmetik; and conclude that the author does not faithfully represent their content. Finally, I offer some technical and general remarks

Frege, Peano and the Interplay between Logic and Mathematics

In contemporary historical studies, Peano is usually included in the logical tradition pioneered by Frege. In this paper, I shall first demonstrate that Frege and Peano independently developed a similar way of using logic for the rigorous expression and proof of mathematical laws. However, I shall then suggest that Peano also used his mathematical logic in such a way that anticipated a formalisation of mathematical theories which was incompatible with Frege’s conception of logic.Dans les études historiques contemporaines, les contributions de Peano sont généralement envisagées dans le cadre de la tradition logiciste initiée par Frege. Dans cet article, je vais d’abord démontrer que Frege et Peano ont développé de manière indépendante des approches semblables visant à s’appuyer sur la logique pour exprimer rigoureusement des lois mathématiques et les prouver. Ensuite, je soutiendrai cependant que Peano a également utilisé sa logique mathématique d’une manière qui anticipait la formalisation des théories mathématiques, laquelle est incompatible avec la conception de la logique défendue par Frege

Peano's Structuralism and the Birth of Formal Languages

Recent historical studies have investigated the first proponents of methodological structuralism in late nineteenth-century mathematics. In this paper, I shall attempt to answer the question of whether Peano can be counted amongst the early structuralists. I shall focus on Peano's understanding of the primitive notions and axioms of geometry and arithmetic. First, I shall argue that the undefinability of the primitive notions of geometry and arithmetic led Peano to the study of the relational features of the systems of objects that compose these theories. Second, I shall claim that, in the context of independence arguments, Peano developed a schematic understanding of the axioms which, despite diverging in some respects from Dedekind's construction of arithmetic, should be considered structuralist. From this stance I shall argue that this schematic understanding of the axioms anticipates the basic components of a formal language

La Lógica de Gottlob Frege: 1879-1903

In this dissertation I offer a global and detailed reconstruction of the logic developed by Gottlob Frege throughout his career. Even though Frege's logic suffered profound modifications from his initial formulation in Begriffsschrift to its revised version in Grundgesetze, the significant differences between these two works have been rarely taken at face value. I not only argue that these differences exist, but I also explain how they should be understood in the light of the evolution of Frege's thought. First, I suggest a new reconstruction of Begriffsschrift's logic, which amounts to a completely novel reading of its formal system—one that contradicts the core of modern historical studies. In particular, I defend that this logic is not—as it has been repeatedly said—a second-order logic and provide the following reasons. (1) The language is not properly a formal language. (2) In Begriffsschrift there is only one sort of quantification: quantification over arguments. (3) Begriffsschrift's logic does not have a semantics in the modern sense. Second, I offer an explanation of the reasons that drive the evolution of Frege's logic. The transition from Begriffsschrift to Grundgesetze has been seldom addressed and never fully explained. According to my historical analysis, the switch from Frege's position concerning logic in Begriffsschrift to his later conception—finally established in Grundgesetze—can be articulated through the adoption of the distinction between concept and object as the basic element of the formal system. This leads to a formalisation of the notion of concept, which in the end drives to Grundgesetze's notion of function. Finally, I put forward a global analysis of Grundgesetze's logic. In this work, Frege develops a formal system that resembles in many relevant ways a second-order one. I suggest a reconstruction of this formal system that allows us to compare it with Begriffsschrift's. In particular, I formulate precisely every rule of inference proposed by Frege and especially focus on the rules of substitution. Moreover, I reflect on several meta-logical results that can be drawn from this reconstruction

Peano's Structuralism and the Birth of Formal Languages

Recent historical studies have investigated the first proponents of methodological structuralism in late nineteenth-century mathematics. In this paper, I shall attempt to answer the question of whether Peano can be counted amongst the early structuralists. I shall focus on Peano's understanding of the primitive notions and axioms of geometry and arithmetic. First, I shall argue that the undefinability of the primitive notions of geometry and arithmetic led Peano to the study of the relational features of the systems of objects that compose these theories. Second, I shall claim that, in the context of independence arguments, Peano developed a schematic understanding of the axioms which, despite diverging in some respects from Dedekind's construction of arithmetic, should be considered structuralist. From this stance I shall argue that this schematic understanding of the axioms anticipates the basic components of a formal language