Modern Axiomatics Research Articles

Giuseppe Peano and his School: Axiomatics, Symbolism and Rigor

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...

Succès et opportunisme dans les réflexions méthodologiques d’Hilbert et von Neumann

Although there does not seem to be a direct influence of utilitarianism on David Hilbert and his School, there are some pragmatic aspects in the foundational investigations of the father of modern axiomatics and of some of his scholars that can be read in utilitarian terms. In this article, I detail these aspects, focusing on the foundational research of Hilbert throughout his career and the later methodological reflections of John von Neumann on the mathematical sciences. As for research on Hilbert, much attention is paid to the role played by the notion of success as an alternative epistemological criterion for the introduction of powerful conceptual tools within axiomatic theories, as well as for the justification of these theories in the case where consistency proofs were not available. Ironically, after the discovery of Godel’s incompleteness theorems, success became a plain epistemological criterion for the justification of axiomatized sciences, together with all the conceptual tools contained therein. In his later foundational reflections, von Neumann, in particular, a former mathematician of Hilbert’s School, subscribed to an opportunistic methodology of science, i.e. a methodology that tried to coherently develop Hilbert’s thought on success, as well as his axiomatic reform of mathematical sciences. There seems to be, therefore, a clear evolutionary line that can be traced from Hilbert to von Neumann, which also reveals utilitarian aspects (broadly conceived) not only in science, but in axiomatics as well.

What are Implicit Definitions?

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 (1) as specifications of the meaning of the primitive terms of a theory and (2) 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.

The Right Order of Concepts: Graßmann, Peano, Gödel and the Inheritance of Leibniz's Universal Characteristic

This paper tackles the question of whether the order of concepts was still a relevant aspect of scientific rigour in the 19th and 20th centuries, especially in the case of authors who were deeply influenced by the Leibnizian project of a universal characteristic. Three case studies will be taken into account: Hermann Graßmann, Giuseppe Peano and Kurt Gödel. The main claim will be that the choice of primitive concepts was not only a question of convenience in modern hypothetico-deductive investigations, but sometimes also the result of philosophical investigations onto the foundation of scienti­fic disciplines. The question of the "right" order of concepts is an ideal to be followed rather than a task that can be fulfilled, but remains nonetheless an essential part of the axiomatic enterprise. This paper aims to question whether there is in fact such a stark contrast, as there is often claimed to be in the literature, between the debates relating to the right order of concepts and the foundational questions concerning modern axiomatics. The scientific rupture determined by the appearance of hypothetico-deductive systems in mathema­tics and logic should thus not be dissociated from some relevant continuities concerning the ideal of knowledge as the search for a general theory of concepts deriving from some fundamental elements.

Constructive geometrical reasoning and diagrams

Modern formal accounts of the constructive nature of elementary geometry do not aim to capture the intuitive or concrete character of geometrical construction. In line with the general abstract approach of modern axiomatics, nothing is presumed of the objects that a geometric construction produces. This study explores the possibility of a formal account of geometric construction where the basic geometric objects are understood from the outset to possess certain spatial properties. The discussion is centered around Eu, a recently developed formal system of proof (presented in Mumma (Synthese 175:255–287, 2010)) within which Euclid’s diagrammatic proofs can be represented.

Axiomatics Without Foundations. On the Model-theoretical Viewpoint In Modern Axiomatics

Two conflicting interpretations of modern axiomatics will be considered. The logico-analytical interpretation goes back to Pasch, while the model-theoretical approach stems from Hilbert. This perspective takes up the distinction between logic as calculus ratiocinator versus lingua characterica that Heijenoort and Hintikka placed emphasis on. It is argued that the Heijenoort-Hintikka distinction can be carried over from logic to mathematical axiomatics. In particular, the model-theoretical viewpoint is deeply connected to a philosophy of mathematics that is not committed to a foundational perspective, but oriented more at applications and at mathematical practice.

The Road to Modern Logic—An Interpretation

AbstractThis paper aims to outline an analysis and interpretation of the process that led to First-Order Logic and its consolidation as a core system of modern logic. We begin with an historical overview of landmarks along the road to modern logic, and proceed to a philosophical discussion casting doubt on the possibility of a purely rational justification of the actual delimitation of First-Order Logic. On this basis, we advance the thesis that a certain historical tradition was essential to the emergence of modern logic; this traditional context is analyzed as consisting in some guiding principles and, particularly, a set of exemplars (i.e., paradigmatic instances). Then, we proceed to interpret the historical course of development reviewed in section 1, which can broadly be described as a two-phased movement of expansion and then restriction of the scope of logical theory. We shall try to pinpoint ambivalencies in the process, and the main motives for subsequent changes. Among the latter, one may emphasize the spirit of modern axiomatics, the situation of foundational insecurity in the 1920s, the resulting desire to find systems well-behaved from a proof-theoretical point of view, and the metatheoretical results of the 1930s. Not surprisingly, the mathematical and, more specifically, the foundational context in which First-Order Logic matured will be seen to have played a primary role in its shaping.Mathematical logic is what logic, through twenty-five centuries and a few transformations, has become today. (Jean van Heijenoort)

On the Arithmetization of School Geometry in the Setting of Modern Axiomatics.

By arithmetized geometry we mean an axiomatic presentation of geometrywhich presupposes the system of real numbers. In this paper we analyzearithmetized geometry from a semantical point of view and compare it withclassical synthetic exposition of school geometry. We also analyze, from thesame point of view, the didactical use of one particular system ofarithmetized geometry, namely Pogorelov‘s system, in the Greek Lyceum(16–17 year old students).

The Origins of Modern Axiomatics: Pasch to Peano

The Origins of Modern Axiomatics: Pasch to Peano