Methodological Frames: Paul Bernays, Mathematical Structuralism, and Proof Theory

Mathematical structuralism is deeply connected with Hilbert and Bernays’s proof theory and its programmatic aim to ensure the consistency of all of mathematics. That aim was to be reached on the basis of finitist mathematics. Gödel’s second incompleteness theorem forced a step from absolute finitist to relative constructivist proof-theoretic reductions. This mathematical step was accompanied by philosophical arguments for the special nature of the grounding constructivist frameworks. Against that background, this chapter examines Bernays’s reflections on proof-theoretic reductions of mathematical structures to methodological frames via projections. However, these reflections are focused on narrowly arithmetic features of frames. Drawing on broadened meta-mathematical experience, this chapter proposes a more general characterization of frames that has ontological and epistemological significance. The characterization is given in terms of accessibility: domains of objects are accessible if their elements are inductively generated, and principles for such domains are accessible if they are grounded in our understanding of the generating processes.

Carnap’s Structuralist Thesis

This chapter investigates Carnap’s structuralism in his philosophy of mathematics of the 1920s and early 1930s. His approach to mathematics is based on a genuinely structuralist thesis, namely that axiomatic theories describe abstract structures or the structural properties of their objects. The aim in the present article is twofold: first, to show that Carnap, in his contributions to mathematics from the time, proposed three different (but interrelated) ways to characterize the notion of mathematical structure, namely in terms of (i) implicit definitions, (ii) logical constructions, and (iii) definitions by abstraction. The second aim is to re-evaluate Carnap’s early contributions to the philosophy of mathematics in light of contemporary mathematical structuralism. Specifically, the chapter discusses two connections between his structuralist thesis and current philosophical debates on structural abstraction and the on the definition of structural properties.

Saunders Mac Lane: From Principia Mathematica through Göttingen to the Working Theory of Structures

Saunders Mac Lane heard David Hilbert’s weekly lectures on philosophy and utterly believed Hilbert’s declaration that mathematics will know no limits. He studied algebra with Emmy Noether, and both mathematics and philosophy with Hermann Weyl. As a young algebraist he created today’s standard working method for mathematical structure: category theory, with topologist Samuel Eilenberg. As one step, they created the now standard definition of “isomorphism.” They originally saw categories as just a working tool. But in the 1950s, Mac Lane saw Alexander Grothendieck and others radically extend the range of the theory, and in the 1960s, he took up William Lawvere’s idea of categorical foundations. The essay relates all of this to current philosophical structuralism, especially concerning isomorphisms and automorphisms of structures. It concludes by comparing Mac Lane’s motives for structuralist working mathematics with current philosophical motives for structuralist ontology.

Transfer Principles, Klein’s Erlangen Program, and Methodological Structuralism

The present article investigates Felix Klein’s mathematical structuralism underlying his Erlangen program. The aim here is twofold. The first aim is to survey the geometrical background of his 1872 article, in particular, work on the principle of duality and so-called transfer principles in projective geometry. The second aim is more philosophical in character and concerns Klein’s structuralist account of geometrical knowledge. The chapter will argue that his group-theoretic approach is best characterized as a kind of “methodological structuralism” regarding geometry. Moreover, one can identify at least two aspects of the Erlangen program that connect his approach with present philosophical debates, namely (i) the idea to specify structural properties and structural identity conditions in terms of transformation groups and (ii) an account of the structural equivalence of geometries in terms of transfer principles.

The Functional Role of Structures in Bourbaki

In the history of 20th-century mathematical structuralism, the figure of Bourbaki is prominent and sometimes even identified with the philosophical doctrine of structuralism. However, the Bourbaki group consisted of pure mathematicians, among them the greatest of their generation, most of whom seemed little inclined to, and even hesitant about, philosophy. This essay proposes to explore this tension in line with the recent philosophical interest in scientific practice. The working assumption is that the use of the concept of structure in Bourbaki is not mainly conceptual and foundational, but pragmatic and functional. This functional interpretation is governed by the principle of the unity of mathematics. In addition to their deductive “vertical” dimension, taking into account structures can reveal various “horizontal” connections between different theories.

The Ways of Hilbert’s Axiomatics: Structural and Formal

Hilbert’s programmatic papers from the 1920s still shape, almost exclusively, the standard contemporary perspective of his views concerning (the foundations of) mathematics; even his own, quite different work on the foundations of geometry and arithmetic from the late 1890s is often understood from that vantage point. My essay pursues one main goal, namely, to contrast Hilbert’s formal axiomatic method from the early 1920s with his structural axiomatic approach from the 1890s. Such a contrast illuminates the circuitous beginnings of the finitist consistency program and connects the complex emergence of structural axiomatics with transformations in mathematics and philosophy during the 19th century.

Introduction and Overview

The core idea of mathematical structuralism is that mathematical theories, always or at least in many central cases, are meant to characterize abstract structures (as opposed to more concrete, individual objects). As such, structuralism is a general position about the subject matter of mathematics, namely abstract structures; but it also includes, or is intimately connected with, views about its methodology, since studying such structures involves distinctive tools and procedures. The goal of the present collection of essays is to discuss mathematical structuralism with respect to both aspects. This is done by examining contributions by a number of mathematicians and philosophers of mathematics from the second half of the 19th and the early 20th centuries.

Grassmann’s Concept Structuralism

After recalling some mathematical contributions that are relevant for the structuralist transformation of mathematics, such as abstract algebra, linear algebra, and number theory, this chapter reconstructs Grassmann’s philosophy of mathematics. It is claimed that he contributed to the development of methodological structuralism inasmuch as he clearly separated the study of the most general properties of connections from pure and applied mathematics, basing them on an understanding of generality as conceptual underdetermination, and on the preeminence of the notion of series over that of function. A brief comparison with contemporary philosophical structuralism will clarify Grassmann’s tendency toward a “concept structuralism” rather than an “object structuralism.”

Logic of Relations and Diagrammatic Reasoning: Structuralist Elements in the Work of Charles Sanders Peirce

This chapter presents aspects of the work of Charles Sanders Peirce showing that he adhered to a number of pre-structuralist themes. Further, it indicates that Peirce’s position is similar in spirit to the category theoretical structuralist view of Steve Awodey (2004). The first part documents Peirce’s extensive knowledge of, and contribution to, the mathematics of his time, illustrating that relations played a fundamental role. The second part addresses Peirce’s characterization of mathematical reasoning as diagrammatic reasoning, that is, as reasoning done by constructing and observing rational relations in diagrams.

Dedekind’s Mathematical Structuralism: From Galois Theory to Numbers, Sets, and Functions

This essay concerns Dedekind’s “mathematical structuralism,”by which we mean methodological features characteristic for the approach to mathematics in his mature writings. The discussion starts with some background on forerunners, especially Gauss, Dirichlet, and Riemann, whose “conceptual” style of work influenced him strongly. But Dedekind went further than them, by making methodological choices that are more distinctly and fully “structuralist”. This includes his resolute acceptance of actually infinite systems, understood within a “logical” framework, and studied not just axiomatically, but also in terms of isomorphisms and related notions (since 1871). As an illustration, the essay discusses his early adoption and strikingly modern transformation of Galois theory, together with his contributions to algebraic number theory. After that, the essayturns to Dedekind’s more “foundational” contributions, i.e., his writings on the real numbers, the natural numbers, and set theory. It shows that the same methodological choices inform them. Yet his approach kept evolving in subtle ways too, especially in terms of the centrality of functions (Abbildungen, mappings).