We emphasize the role of the choice of vocabulary in formalization of a mathematical area and remark that this is a particular preoccupation of logicians. We use this framework to discuss Kennedy’s notion of ‘formalism freeness’ in the context of various schools in model theory. Then we clarify some of the mathematical issues in recent discussions of purity in the proof of the Desargues proposition. We note that the conclusion of ‘spatial content’ from the Desargues proposition involves arguments which are (...) algebraic and even metamathematical. Hilbert showed that the Desargues proposition implies the coordinatizing ring is associative, which in turn implies the existence of a three-dimensional geometry in which the given plane can be embedded. With W. Howard we give a new proof, removing Hilbert’s ‘detour’ through algebra, of the ‘geometric’ embedding theorem. Finally, our investigation of purity leads to the conclusion that even the introduction of explicit definitions in a proof can violate purity. We argue that although both involve explicit definition, our proof of the embedding theorem is pure while Hilbert’s is not. Thus the determination of whether an argument is pure turns on the content of the particular proof. Moreover, formalizing the situation does not provide a tool for characterizing purity. (shrink)

Purity is known as an ideal of proof that restricts a proof to notions belonging to the ‘content’ of the theorem. In this paper, our main interest is to develop a conception of purity for formal (natural deduction) proofs. We develop two new notions of purity: one based on an ontological notion of the content of a theorem, and one based on the notions of surrogate ontological content and structural content. From there, we characterize which (classical) first-order natural deduction proofs (...) of a mathematical theorem are pure. Formal proofs that refer to the ontological content of a theorem will be called ‘fully ontologically pure’. Formal proofs that refer to a surrogate ontological content of a theorem will be called ‘secondarily ontologically pure’, because they preserve the structural content of a theorem. We will use interpretations between theories to develop a proof-theoretic criterion that guarantees secondary ontological purity for formal proofs. (shrink)

In this paper, we offer a descriptive theory of analogical reasoning in mathematics, stating general conditions under which an analogy may provide genuine inductive support to a mathematical conjecture (over and above fulfilling the merely heuristic role of ‘suggesting’ a conjecture in the psychological sense). The proposed conditions generalize the criteria of Hesse in her influential work on analogical reasoning in the empirical sciences. By reference to several case studies, we argue that the account proposed in this paper does a (...) better job in vindicating the use of analogical inference in mathematics than the prominent alternative defended by Bartha. (shrink)

In this paper I argue for an association between impurity and explanatory power in contemporary mathematics. This proposal is defended against the ancient and influential idea that purity and explanation go hand-in-hand (Aristotle, Bolzano) and recent suggestions that purity/impurity ascriptions and explanatory power are more or less distinct (Section 1). This is done by analyzing a central and deep result of additive number theory, Szemerédi’s theorem, and various of its proofs (Section 2). In particular, I focus upon the radically impure (...) (ergodic) proof due to Furstenberg (Section 3). Furstenberg’s ergodic proof is striking because it utilizes intuitively foreign and infinitary resources to prove a finitary combinatorial result and does so in a perspicuous fashion. I claim that Furstenberg’s proof is explanatory in light of its clear expression of a crucial structural result, which provides the “reason why” Szemerédi’s theorem is true. This is, however, rather surprising: how can such intuitively different conceptual resources “get a grip on” the theorem to be proved? I account for this phenomenon by articulating a new construal of the content of a mathematical statement, which I call structural content (Section 4). I argue that the availability of structural content saves intuitive epistemic distinctions made in mathematical practice and simultaneously explicates the intervention of surprising and explanatorily rich conceptual resources. Structural content also disarms general arguments for thinking that impurity and explanatory power might come apart. Finally, I sketch a proposal that, once structural content is in hand, impure resources lead to explanatory proofs via suitably understood varieties of simplification and unification (Section 5). (shrink)

Peano’s axioms for arithmetic, published in 1889, are ubiquitously cited in writings on modern axiomatics, and his Formulario is often quoted as the precursor of Russell’s Principia Mathematica. Yet, a comprehensive historical and philosophical evaluation of the contributions of the Peano School to mathematics, logic, and the foundation of mathematics remains to be made. In line with increased interest in the philosophy of mathematics for the investigation of mathematical practices, this them...

The paper surveys different notions of implicit definition. In particular, we offer an examination of a kind of definition commonly used in formal axiomatics, which in general terms is understood as providing a definition of the primitive terminology of an axiomatic theory. We argue that such “structural definitions” can be semantically understood in two different ways, namely as specifications of the meaning of the primitive terms of a theory and as definitions of higher-order mathematical concepts or structures. We analyze these (...) two conceptions of structural definition both in the history of modern axiomatics and in contemporary philosophical debates. Based on that, we give a systematic assessment of the underlying semantics of these two ways of understanding the definiens of such definitions, by considering alternative model-theoretic and inferential accounts of meaning. (shrink)

Charlotte Angas Scott (1858–1932) was an internationally renowned geometer, the first British woman to earn a doctorate in mathematics, and the chair of the Bryn Mawr mathematics department for forty years. There she helped shape the burgeoning mathematics community in the United States. Scott often motivated her research as providing a “geometric treatment” of results that had previously been derived algebraically. The adjective “geometric” likely entailed many things for Scott, from her careful illustration of diagrams to her choice of references (...) and citations. This article will focus on Scott’s striking and consistent use of geometric to describe a reality of dynamic points, lines, planes, and spaces that could be manipulated analogously to physical objects. By providing geometric interpretations of algebraic derivations, Scott committed to an early-nineteenth-century aesthetic vision of a “whole” analytical geometry that she adapted to modern research areas. (shrink)

In this paper we isolate a notion that we call “formalism freeness” from Gödel's 1946 Princeton Bicentennial Lecture, which asks for a transfer of the Turing analysis of computability to the cases of definability and provability. We suggest an implementation of Gödel's idea in the case of definability, via versions of the constructible hierarchy based on fragments of second order logic. We also trace the notion of formalism freeness in the very wide context of developments in mathematical logic in the (...) 20th century. (shrink)

There are lots of arguments for, or justifications of, mathematical theorems that make use of principles from physics. Do any of these constitute explanations? On the one hand, physical principles do not seem like they should be explanatorily relevant; on the other, some particular examples of physical justifications do look explanatory. In this article, I defend the idea that physical justifications can and do explain mathematical facts. 1 Physical Arguments for Mathematical Truths2 Preview3 Mathematical Facts4 Purity5 Doubts about Purity: I6 (...) Doubts about Purity: II7 How Physical Arguments Might Explain: I8 How Physical Arguments Might Explain: II9 Conclusion. (shrink)

Recent historical studies have investigated the first proponents of methodological structuralism in late nineteenth-century mathematics. In this paper, I shall attempt to answer the question of whether Peano can be counted amongst the early structuralists. I shall focus on Peano’s understanding of the primitive notions and axioms of geometry and arithmetic. First, I shall argue that the undefinability of the primitive notions of geometry and arithmetic led Peano to the study of the relational features of the systems of objects that (...) compose these theories. Second, I shall claim that, in the context of independence arguments, Peano developed a schematic understanding of the axioms which, despite diverging in some respects from Dedekind’s construction of arithmetic, should be considered structuralist. From this stance I shall argue that this schematic understanding of the axioms anticipates the basic components of a formal language. (shrink)

The aim of this article is to contribute to a better understanding of Frege’s views on semantics and metatheory by looking at his take on several themes in nineteenth century geometry that were significant for the development of modern model-theoretic semantics. I will focus on three issues in which a central semantic idea, the idea of reinterpreting non-logical terms, gradually came to play a substantial role: the introduction of elements at infinity in projective geometry; the study of transfer principles, especially (...) the principle of duality; and the use of counterexamples in independence arguments. Based on a discussion of these issues and how nineteenth century geometers reflected about them, I will then look into Frege’s take on these matters. I conclude with a discussion of Frege’s views and what they entail for the debate about his stance towards semantics and metatheory more generally. (shrink)

A theory of magnitudes involves criteria for their equivalence, comparison and addition. In this article we examine these aspects from an abstract viewpoint, by focusing on the so-called De Zolt’s postulate in the theory of equivalence of plane polygons (“If a polygon is divided into polygonal parts in any given way, then the union of all but one of these parts is not equivalent to the given polygon”). We formulate an abstract version of this postulate and derive it from some (...) selected principles for magnitudes. We also formulate and derive an abstract version of Euclid’s Common Notion 5 (“The whole is greater than the part”), and analyze its logical relation to the former proposition. These results prove to be relevant for the clarification of some key conceptual aspects of Hilbert’s proof of De Zolt’s postulate, in his classicalFoundations of Geometry(1899). Furthermore, our abstract treatment of this central proposition provides interesting insights for the development of a well-behaved theory ofcompatiblemagnitudes. (shrink)

The article investigates one of the key contributions to modern structural mathematics, namely Hilbert’sFoundations of Geometry(1899) and its mathematical roots in nineteenth-century projective geometry. A central innovation of Hilbert’s book was to provide semantically minded independence proofs for various fragments of Euclidean geometry, thereby contributing to the development of the model-theoretic point of view in logical theory. Though it is generally acknowledged that the development of model theory is intimately bound up with innovations in 19th century geometry (in particular, the (...) development of non-Euclidean geometries), so far, little has been said about how exactly model-theoretic concepts grew out of methodological investigations within projective geometry. This article is supposed to fill this lacuna and investigates this geometrical prehistory of modern model theory, eventually leading up to Hilbert’sFoundations. (shrink)

The book contains eight papers previously published in English—three of which also appeared in Mancosu's papers collection —and now translated into French (two of them are co-authored with J....

The paper outlines an interpretation of one of the most important and original contributions of David Hilbert’s monograph Foundations of Geometry , namely his internal arithmetization of geometry. It is claimed that Hilbert’s profound interest in the problem of the introduction of numbers into geometry responded to certain epistemological aims and methodological concerns that were fundamental to his early axiomatic investigations into the foundations of elementary geometry. In particular, it is shown that a central concern that motivated Hilbert’s axiomatic investigations (...) from very early on was the aim of providing an independent basis for geometry. Accordingly, these concerns about an independent grounding for elementary geometry determined very clear methodological constraints in the process of embedding it into a formal axiomatic system. It is argued that Hilbert not only sought to show that geometry could be considered a pure mathematical theory, once it was presented as a formal axiomatic system; he also aimed at showing that in the construction of such an axiomatic system one could proceed purely geometrically, avoiding concept formations borrowed from other mathematical disciplines like arithmetic or analysis. (shrink)

Theoria, Volume 88, Issue 3, Page 629-652, June 2022.

In recent years, several scholars have been investigating Frege’s mathematical background, especially in geometry, in order to put his general views on mathematics and logic into proper perspective. In this article I want to continue this line of research and study Frege’s views on geometry in their own right by focussing on his views on a field which occupied center stage in nineteenth century geometry, namely, projective geometry.

We give a general account of the goals of axiomatization, introducing a variant on Detlefsen’s notion of ‘complete descriptive axiomatization’. We describe how distinctions between the Greek and modern view of number, magnitude, and proportion impact the interpretation of Hilbert’s axiomatization of geometry. We argue, as did Hilbert, that Euclid’s propositions concerning polygons, area, and similar triangles are derivable from Hilbert’s first-order axioms. We argue that Hilbert’s axioms including continuity show much more than the geometrical propositions of Euclid’s theorems and (...) thus are an immodest complete descriptive axiomatization of that subject. (shrink)

The symmetries between points and lines in planar projective geometry and between points and planes in solid projective geometry are striking features of these geometries that were extensively discussed during the nineteenth century under the labels “duality” or “reciprocity.” The aims of this article are, first, to provide a systematic analysis of duality from a modern point of view, and, second, based on this, to give a historical overview of how discussions about duality evolved during the nineteenth century. Specifically, we (...) want to see in which ways geometers’ preoccupation with duality was shaped by developments that lead to modern logic towards the end of the nineteenth century, and how these developments in turn might have been influenced by reflections on duality. (shrink)

The goal of this paper is to connect Rota's discussion of the Husserlian notion of Fundierung with Rota's project of giving combinatorics a foundation in his 1964 paper ‘On the foundations of combinatorial theory I’. Section 2 gives the basic tenets of this seminal paper. Sections 3 and 4 spell out the connections made there between Rota's philosophical writings and his mathematical achievements. Section 5 shows how these two developments fit into Rota's analysis of the place of combinatorics in mathematics.

In this paper, we discuss a proposal for reform in the teaching of Euclidean geometry that reveals the symbiotic relationship between axiomatics and pedagogy. We examine the role of intuition in this kind of reform, as expressed by Mario Pieri, a prominent member of the Schools of Peano and Segre at the University of Turin. We are well aware of the centuries of attention paid to the notion of intuition by mathematicians, mathematics educators, philosophers, psychologists, historians, and others. To set (...) a context for Pieri’s proposal, we only seek to open a small window on views of the pedagogical role of intuition, from primary education to university study that may have informed early 20th century efforts to improve the teaching of geometry at the secondary school level. Pieri addressed the topic of intuition in many of his axiomatizations, including those in projective geometry which was his main area of concentration in foundations. We focus here primarily on his axiom systems for elementary geometry, which embraced the transformational approach of Felix Klein’s vision for the subject. Our goal is to convey Pieri’s thoughts on how to integrate two types of intuition, denoted as sensible and rational, in his endeavors to improve the teaching of the geometry of Euclid. We show how Pieri’s views on geometric intuition and pedagogical reform were either ignored or misrepresented in several notable publications at the turn of the 20th century. In particular, we give Pieri a voice in response to specific comments made in the early 1900s by Federigo Enriques, Ugo Amaldi, and Florian Cajori in widely circulated publications inspired by Klein. (shrink)