Project Description

In 1899, David Hilbert published his famous Foundations of Geometry in an effort to place geometry on a solid footing. Hilbert's Foundations are considered a landmark in the history of logic and mathematics by philosophers, mathematicians and logicians alike. In his "Festschrift", Hilbert presents a modern axiomatization of Euclidean geometry and for the first time addresses metatheoretical questions concerning his axiom system in a systematic way. Besides using reinterpretations on a large scale to prove various consistency and independence results, he also develops a novel understanding of mathematical theories that was formative for the modern conceptions of logic and mathematics.

Interestingly, Frege reacts to Hilbert's innovations with almost complete dismissal. In a series of articles and several letters to Hilbert, he presents a thorough critique of Hilbert's methods and his conception of mathematical theories and the axiomatic method. For the longest time, Hilbert was considered to be the winner of this scholarly debate. Frege's criticisms were found to be pedantic and his traditional views on axioms outdated. Largely ignored until recently, in one of his articles on Hilbert's Foundations, Frege also presents a proposal of his own as to how independence must be proved. His proposal is both radical and puzzling: Frege claims that a "new science" with "its own, specific basic truths" needs to be established in order to rigorously prove the independence of axioms. Although there has been some discussion regarding various aspects of Frege's New Science in the literature, no systematic account of Frege's ideas on the matter has been devised so far. The main objective of this project is to fill this gap and to provide such an account.

Specifically, I aim

  • to make clear what Frege's New Science is supposed to look like exactly and how it relates to concepts and methods in modern and pre-modern mathematical logic.
  • to clarify Frege's motivation for introducing a New Science in the first place, and what his proposal implies for his philosophy of logic and mathematics.
  • to determine whether, and, if so, to what extent, Frege's conception of metatheoretical investigations might be reflected in contemporary discussions in the philosophy of logic.

The basic contention concerning the first question is that Frege's New Science is best understood in terms of an axiomatized theory that includes theories of truth and provability within a sufficiently rich object language. Accordingly, the project's main objective is to provide a rigorous reconstruction of Frege's inchoate ideas by drawing on recent developments in the study of axiomatic theories of truth, informal provability, and related concepts, and to answer the other questions in the light of the provided reconstruction.

The hope is that research on the project will not only have an impact on our understanding of the philosophy of logic and mathematics of one of the central figures in the development of modern logic and thereby foster our understanding of the history of logic and mathematics, but also in various respects contribute to contemporary debates from a broadly Fregean perspective.


Back to main site