Axiomatizing Metatheory: Frege´s New Science

David Hilberts Foundations of Geometry (1899) is considered as a landmark in the history of modern 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 issues concerning his system of axioms in a systematic way. Be- sides using reinterpretations on a large scale to prove various consistency and independence re- sults, he also develops a novel understanding of mathematical theories that was formative for the modern conceptions of logic and mathematics. Interestingly, Gottlob Frege, another leading figure in the development of modern mathematical logic, reacts to Hilberts innovations with almost complete dismissal. In a number of writings, he presents a thorough critique of Hilberts methods and his conception of mathematical theories and the axiomatic method. For the longest time, Hilbert was considered to be the winner of this so-called Frege-Hilbert controversy. Freges criticisms were found to be pedantic and his traditional views on axioms outdated. Largely unnoticed until recently, Frege also presents an interesting proposal of his own as to how independence must be proved in one of his articles on Hilberts Foundations. His proposal is both radical and puzzling: Frege claims that an entirely new science has to be established in order to rigorously prove the independence of axioms in the traditional sense. Although there has been some discussion on various aspects of Freges New Science in the secondary literature, no systematic account of Freges ideas on the matter has been devised so far. The main objec- tive of this project is to fill this gap and to provide such an account. The specific aims are to clarify (1) Freges motivation for introducing a New Science in the first place, (2) what this New Science is supposed to look like and how it relates to concepts and methods in (pre- )modern mathematical logic, and (3) what Freges proposal implies for his philosophy of logic and mathematics in general, and for his views on metalogical investigations in particular. Re- search 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 re- spects contribute to contemporary debates from a broadly Fregean perspective.

David Hilbert's "Foundations of Geometry" is widely considered to be a founding document of modern mathematics because of its novel approach to axiomatics and its in- corporation of metatheoretical issues such as the consistency and mutual independence of axioms. After the publication of the book in 1899, a fierce debate between Hilbert and the father of modern logic, Gottlob Frege, ensued over Hilbert's approach to these issues. In a series of articles dating from 1903 to 1906, Frege not only provides a thorough critique of Hilbert's methodology of proving independence and consistency of axioms and the conception of the axiomatic method on which it is based, but also presents his own ideas on these matters. The main goal of the project was to investigate these ideas both from a historical and systematic perspective. On the historical side, research has focused on advancing our understanding of Frege's views by taking into account the historical context in which he was operating as well as his own background as a nineteenth century geometer. Although some work has been done in this respect in recent times, it turned out that there are some striking lacunae as well. In the course of the project, some of these gaps could be filled by carefully looking at Frege's early work in geometry and putting them into the context of nineteenth century geometry more broadly. Several articles on these topics that were published in top journals not only advanced our understanding of Frege's views in various respects, but our understanding of the development of nineteenth century geometry and its effects on the formation of modern mathematics and logic more generally. On the systematic side, research has focused on providing formal reconstructions of specific aspects relating to Frege's proposal. Two results stand out in this respect. First, based on a procedural understanding of Frege's notoriously elusive notion of "sense", a formal reconstruction of this notion has been developed which is of independent interest for contemporary debates in formal semantics. Secondly, based on a prominent Frege-interpretation proposed by Patricia Blanchette, a formal reconstruction of Frege's conception of provability has been developed that incorporates the possibility of conceptual analysis into the notion of provability and which, again, is of general significance for the epistemology of mathematics. Research on this topic also led to further historical and systematic questions about the relationship between formal and informal provability in mathematics, both in Frege's work and more generally. A project on this subject is currently in preparation.