Description

What was Bertrand Russell’s role in the development of formal languages? This research project (RUSSELL) aims to clarify the divergences and similarities between the main trends in the development of early modern logic, with a particular focus on Bertrand Russell’s work. Specifically, the plan is to examine Russell’s views on metatheoretical questions and his contribution to the emergence of the notion of formal languages during the early decades of the 20th century.

The RUSSELL project will systematically study the formalisation of logic in the late 19th century and establish a narrative of how the work done by Peano and Schröder laid the foundation for the development of Russell’s logic in the early 20th century. Despite the relevance of this process for a proper understanding of the genesis and development of contemporary logic, the relationship between Russell’s logic, the birth of formal languages, and the contributions of Peano and Schröder has not been thoroughly studied. Generally speaking, this project aims to fill this historiographical gap.

It will engage with current studies on 19th-century logic, particularly those focused on Russell’s contributions, while also exploring the connections between Russell’s philosophy of mathematics and contemporary approaches such as neo-logicism and structuralism.

There are three specific goals that the RUSSELL project seeks to fulfill. First, to determine to what extent Peano’s and Schröder’s logics served as the foundation for Russell’s logic. Second, to characterize Russell’s views on metatheoretical results, both from a historical and contemporary perspective. Third, to assess whether Russell’s work on the theory of types, and more generally his conception of logical systems, can be connected — either partially or fully — to the development of the technical tools required for the formalisation of mathematical and logical theories.

To achieve these goals, we will work with three interconnected hypotheses. First, Russell refined the formal tools developed by Peano and Schröder for his own logicist project. Second, extracting a formal language from the logic of Principia Mathematica (1910–1913) was not among Russell’s programmatic goals or interests. Third, Hilbert’s choice of the theory of types in Principia Mathematica (1910–1913) as the tool for metatheoretical investigations — rather than Schröder’s algebraic logic — can be explained by the fact that it is more straightforward to extract a formal language from the former.

We will first study Russell’s logic, focusing on its notation and syntax. Our attention will be on Russell’s logic of propositions and classes, and we will attempt to reconstruct his logic of relations. From this standpoint, we will then explore possible influences from Peano and Schröder on Russell’s logic. This endeavor is particularly urgent because a systematic comparison of Schröder’s and Peano’s logics with Russell’s is virtually nonexistent in the literature. Finally, we will study the relationship between Russell’s logic and metalogic, examining it from two different perspectives. First, from a historical standpoint, we will provide a rational reconstruction of how Russell’s logic was used to formalize mathematical and logical theories and investigate metatheoretical questions, with a focus on Hilbert’s work and the Göttingen school. Second, from the standpoint of contemporary philosophy of mathematics, we will discuss how logicism (and neo-logicism) fares in the contemporary debate with other approaches in the philosophy of mathematics such as structuralism.

The RUSSELL project will thus adopt a historico-philosophical perspective, combining the methodology of rational reconstruction with philosophical discussion. One of the priorities will be to bring the results of the historical work into the contemporary debate. This will contextualize the development of key concepts and substantiate contemporary motivations by linking them to long-standing strategies that emerged in the late 19th century. As a result, the combination of historical and philosophical perspectives will serve as a methodological advantage.

OBJECTIVES

(1) To determine to what extent Peano’s and Schröder’s logics were taken as a basis for Russell’s logic. This involves two associated goals:

(1.a) To systematically compare Russell’s calculus of relations and Schröder’s algebra of relatives, and to asses of whether Russell maintained significant aspects of Schröder’s algebraic approach to logic beyond his very first works on logic.

(1.b) To study the genesis and development process of The Principles of Mathematics (1903) and Principia Mathematica (1910; 1912; 1913), and to identify specific elements of the logic sketched or developed in these works that can be traced back to Peano’s logic.

(2) To characterize Russell’s view on metatheoretical results. This includes two associated goals:

(2.a) To study Russell’s understanding of the proofs of independence of the axioms of elementary geometry and arithmetic.

(2.b) To evaluate the debate between two major standpoints in the philosophy of mathematics: a logicism point of view — Russell’s position — and a structuralist and model-theoretic point of view, originating in proto-structuralist accounts such as Peano’s, which makes it possible to investigate metatheoretical questions.

(3) To determine whether Russell’s work on the theory of types, and on the conception of logical systems more generally, can be connected — either partially or globally — to the development of the technical tools required for the formalization of mathematical and logical theories.

Research Team

Principal Researcher: Bruno Jacinto (Lisbon).

Team Members: Joan Bertran-San-Millán (Complutense), Paola Cantu (Aix-Marseille), Sébastien Gandon (Clermont Auvergne), José Mestre (Lisbon), Georg Schiemer (Vienna).

Postdoctoral Fellow: New Hire.

Consultant: Bernard Linsky (Alberta).

Publications

Events

Participation in a joint workshop.

One workshop in Lisbon, in 2027.