# Philosophy and History of Mathematics

Logic and mathematics are tools for almost all members of the Department, but they are also objects of investigation. As tools they provide means of rigorously capturing aspects of experience; as objects of investigation they are examined as to their internal coherence, their philosophical justification, and their adequacy for particular purposes. Since ancient times, there has been an intimate connection between philosophical and mathematical thought, a relationship that can be seen in the philosophical reflections of Plato, Descartes, Leibniz, and Kant.

Subtle interactions between philosophy and mathematics can also be seen in the development of mathematics in the 19th century, i.e., in the revolutionary conceptual advances made by Dirichlet, Riemann, Dedekind and others, as well as in the similarly dramatic changes in logic, brought about in large part by Boole, Frege, Peano, Peirce, and Schröder. Complemented by the continuing evolution of the sciences, for example in the work of Hertz, Mach, and Einstein, these developments form the background for the emergence of early analytic philosophy and modern mathematical logic. This historical background reshapes in significant ways the contemporary discussion in the philosophy of mathematics.

The subject is deeply influenced by results of meta-mathematical investigations, but most importantly, for the work in this department, also by mathematical practice. The incompleteness theorems are considered, rightly, to be gems of work in mathematical logic of the last century. They are also viewed as having an enormous philosophical significance; that seems to be right, but only relative to a precise concept of "formal system". The latter is defined using the notion of computability. On the one hand, one can understand then the seemingly conflicting views of Gödel and Turing on mathematical knowledge and the capacities of the human mind. On the other hand, one can go back and see what was supposed to be captured by "formal theories" and uncover the dramatic transformation of mathematics in the 19th century.

The "structuralist" view of mathematics one finds in Dedekind's "foundational" work really emerged out of his concrete work in algebraic number theory. His broad perspective on mathematics impacted that view, which was of course also influenced by Gauss, Dirichlet, and Riemann. Sieg has taken that view as a starting-point and joined it with a quasi-constructive perspective of accessible domains to arrive at an articulation of a "reductive structuralism". This position resolves a number of traditional epistemological and ontological problems. With Awodey and a number of other colleagues, Sieg is editing and translating the marvelous and most significant philosophical essays of Hilbert’s collaborator in the proof theoretic enterprise, Paul Bernays. Sieg's own perspective is developed in his book *Hilbert’s Programs and Beyond* that was published by Oxford University Press.

The philosophy of mathematics has traditionally been concerned with questions of justification and correctness. But as of late a number of researchers have aimed to characterize more general methodological goals and values that influence the decisions that mathematicians make in their everyday practice, for example, when posing questions, formulating definitions, writing proofs, and structuring theories. Using insights from proof theory, formal verification, and the history of mathematics, Avigad has worked to develop more robust accounts of mathematical concepts, methods, and understanding.

Awodey is exploring connections between category theory and structuralism, especially in light of the new Univalent Foundations program. Univalent Foundations includes a new foundational axiom, the Univalence Axiom, according to which isomorphic structures can be identified. This new principle has obvious philosophical consequences, and will doubtless challenge philosophers of mathematics to adjust their views. Philosophers interested in structuralism have begun to recognize the importance of these recent developments, with e.g. several leading philosophers of physics including Ladymann (Bristol) and Halvorson (Princeton) actively engaged in philosophical research on Univalent Foundations. In addition to the comprehensive book *Homotopy Type Theory*, Awodey has written a survey article on the topic of Univalence and structuralism to be published in the journal *Philosophia Mathematica *which will serve to introduce the subject to a wider audience of philosophers of mathematics.

The history of mathematics can be a useful aid in developing a robust and informative philosophy of mathematics. Sieg has explored the logicist and structuralist perspectives that are found in the seminal foundational writings of Dedekind. These perspectives emerged out of the concrete mathematical work Dedekind did in algebraic number theory, and they stand in sharp contrast with those of his contemporary Leopold Kronecker. In work with Rebecca Morris, respectively with Dirk Schlimm, Sieg has analyzed as central features in Dedekind’s work the introduction of *abstract *concepts (structures) and the use of *structure preserving* mappings between different structures (of the same kind). The concept of abstraction involved here is not the classical one, found for example in Kant’s "Logik", but rather one that is exposed in the writings on logic of the contemporaneous and very influential Göttingen philosopher Hermann Lotze.

Avigad has explored methodological aspects of Dedekind's work, especially the development of his theory of ideal divisors, in an effort to understand the way that such methodological considerations interact with philosophical views. With Rebecca Morris, he has studied the history of Dirichlet's seminal theorem on primes in an arithmetic progression, which illuminates the methodological forces that shaped the development of the modern function concept, as well as some of the issues that Frege had to address with his treatment of functions. Avigad has also considered the philosophical views of Kurt Gödel vis-a-vis the meta-mathematical tradition, and has tried to characterize and explain an uneasy tension between Gödel's views and Carnap's.

This concrete work on the dramatic shift in the evolution of mathematics during the 19th century, where some speak of a "transformation of the subject" others of a "revolution," has deep impact on the philosophy of mathematics - a subject that as recently as twenty years ago was pre-occupied with the "Grundlagenstreit" between Brouwer and Hilbert in the 1920s. Both Hilbert and Brouwer are deeply connected to the broad issues alluded to above. Sieg, with colleagues W. Ewald, M. Hallett, and U. Majer, has been working on editing unpublished lecture notes of Hilbert's from the 1890s to the 1930s. They open up a completely novel and fresh perspective on the evolution of Hilbert's foundational thinking and the emergence of proof theory, but that means also on the origins of analytic philosophy.