Dedekind’s methodology, in his classic booklet on the foundations of arithmetic, has been the topic of some debate. While some authors make it closely analogue to Hilbert’s early axiomatics, others emphasize its idiosyncratic features, most importantly the fact that no axioms are stated and its careful deductive structure apparently rests on definitions alone. In particular, the so-called Dedekind “axioms” of arithmetic are presented by him as “characteristic conditions” in the _definition_ of the complex concept of a _simply infinite_ system. Making (...) sense of Dedekind’s method may be dependent on an analysis of the classical model of deductive science, as presented by authors from the eighteenth and early nineteenth centuries. Studying the modern reconstructions of Euclidean geometry, we show that they did not presuppose deductive independence of the axioms from the definitions. Authors like Wolff elaborated a mathematics _based on definitions_, and the Wolffian model of deductive science shows significant coincidences with Dedekind’s method, despite the great differences in content and approach. Wolff had a conception of definitions as _genetic_, which bears some similarities with Kant’s idea of synthetic definitions: they are understood as positing the content of mathematical concepts and introducing thought objects (_Gedankendinge_) that are the objects of mathematics. The emphasis on the spontaneity of the understanding, which can be found in this philosophical tradition, can also be fruitfully related with Dedekind’s idea of the “free creation” of mathematical objects. (shrink)

Since the application of Postulate I.2 in Euclid’s Elements is not uniform, one could wonder in what way should it be applied in Euclid’s plane geometry. Besides legitimizing questions like this from the perspective of a philosophy of mathematical practice, we sketch a general perspective of conceptual analysis of mathematical texts, which involves an extended notion of mathematical theory as system of authorizations, and an audience-dependent notion of proof.

In 1930, Johann von Neumann, together with Rudolf Carnap and Arend Heyting, participated in a conference held in Königsberg, called “Second Seminar on the Epistemology of Exact Sciences”. The idea behind the reunion of these three researchers was to compose a fairly faithful picture of the three main foundational programs of mathematics at the time: formalism, logicism, and intuitionism. The main objective of this paper is to propose an analysis of the text “The Formalist Foundation of Mathematics” presented by von (...) Neumann in this conference. We show how, from von Neumann's perspective, the problem of the consistency of mathematical theories results from the transformation of a genuinely philosophical question into a genuinely logical-mathematical question and how a proof of consistency could be seen as the solution to the foundational problem. In the final part of the text, we present some considerations about Gerhard Gentzen’s consistency proofs and their importance to contemporary proof theory. (shrink)

In his doctoral dissertation, O. Chateaubriand favored Dedekind’s analysis of the notion of number; whereas in Logical Forms, he favors a fregean approach to the topic. My aim in this paper is to examine the kind of logicism he defends. Three aspects will be considered: the concept of analysis; the universality of arithmetical properties and their definability; the irreducibility of arithmetical objects.

En este artículo proponemos que el examen del concepto de demostración de Oswaldo Chateaubriand en los capítulos 19, 20 y 21 de la Parte II de Logical Forms incluye aspectos retóricos y dialécticos .In this paper we argue that Oswaldo Chateaubriand’s conception of proof, in chapters 19, 20, and 21 of Part II of Logical Forms, incorporates rhetorical aspects , and dialectical aspects.

A concepção padrão de prova é uma concepção lingüística de prova. No entanto,literatura recente reivindica a legitimidade de provas heterogêneas, isto é, que incorporemrecursos visuais ou gráficos. Tal reivindicação implica em um melhor exame da distinçãoentre representação lingüística e representação gráfica ou visual; concomitantemente, elatambém comporta uma análise da dualidade entre discursivo e intuitivo em filosofia dalógica e da matemática. Neste breve artigo examinamos dois exemplos canônicos de provasheterogêneas, salientando, no entanto, seu caráter discursivo. Em uma também brevesecção final introduzimos (...) com caráter conjetural o conceito leibniciano de representaçãoecthetica com vistas a repensar a distinção entre lingüístico e gráfico. (shrink)

A theory of magnitudes involves criteria for their comparison, equivalence and addition. We examine these aspects from an abstract viewpoint, stressing independence and definability. These considerations are triggered by the so-called De Zolt’s principle in the theory of equivalence of plane polygons.

This paper analyzes the theory of area developed by Euclid in the Elements and its modern reinterpretation in Hilbert’s influential monograph Foundations of Geometry. Particular attention is bestowed upon the role that two specific principles play in these theories, namely the famous common notion 5 and the geometrical proposition known as De Zolt’s postulate. On the one hand, we argue that an adequate elucidation of how these two principles are conceptually related in the theories of Euclid and Hilbert is highly (...) relevant for a better understanding of the respective geometrical practices. On the other hand, we claim that these conceptual relations unveil interesting issues between the two main contemporary approaches to the study of area of plane rectilinear figures, i.e., the geometrical approach consisting in the geometrical theory of equivalence and the metrical approach based on the notion of measure of area. Finally, in an appendix logical relations among equivalence, comparison and addition of magnitudes are examined schematically in an abstract setting. (shrink)

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

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 Ontology Summit 2012 explored the current and potential uses of ontology, its methods and paradigms, in big systems and big data: How ontology can be used to design, develop, and operate such systems. The systems addressed were not just software systems, although software systems are typically core and necessary components, but more complex systems that include multiple kinds and levels of human and community interaction with physical-software systems, systems of systems, and the socio-technical environments for those systems which can (...) include cultural, legal, and economic components. The focus themes used for this exploration were Big Systems Engineering, Big Data Challenge, Large Scale Domain Applications, and cross-cutting aspects Ontology Quality, and Federation and Integration of Systems. The Ontology Summit 2012 consisted of over three months of intensive virtual collaborative elaboration of these issues in presentations, panels, and group email. The culmination of these activities was a face-to-face Symposium at the US National Institute of Standards and Technology (NIST), Gaithersburg, MD, USA, 12–13 April 2012. The primary product of this Ontology Summit is the communiqué reported here. But there are other products, some continuing as collaborative, more specifically focused analysis and modeling efforts aligned with various open standards activities. Behind all of these particular products, of course, is the real overriding purpose of the Ontology Summit 2012, which was: the joint collaboration of three distinct communities, the ontology, systems engineering and big systems stakeholder communities, who came together to address common problems, create common understanding and propose common solutions. (shrink)

Resumen Este artículo analiza si en la teoría de los cuatro ethe de la modernidad capitalista, desarrollada por Bolívar Echeverría, no está contenida una debilidad de principio. Se trata de saber si para arribar a sus más altas aportaciones, esta teoría no termina pagando un precio que debiera dar que pensar. ¿Acaso renuncia a la crítica de la ideología, por lo menos en la radicalidad realizada por los autores de la Teoría crítica, inspirados en este punto originalmente por Georg Lukács?This (...) article is analyzing if in the theory of the four ethe of capitalist modernity, developed by Bolívar Echeverría, isn’t included a basic weakness. It is about understanding if for to arrive to its highest contributions, this theory is not paying a too high prize. Does it possibly renounce to the critique of ideology, at least in the radicality fulfilled by the authors of Critical theory, inspired in this point originally by Georg Lukács? (shrink)

Abel Lasalle Casanave’s comments on the rhetorical and dialectical aspects of the discussion of proof in chapters 19 to 21. I center most of my response on two issues raised in his concluding questions; the first being the hermeneutic aspect of the development of mathematics and of mathematical knowledge, and the second “the symbolic conception of mathematics” and the “algebraic mode of thought”.As considerações de Abel Lasalle Casanave dizem respeito aos aspectos retóricos e dialéticos da discussão da noção (...) de prova nos capítulos 19 à 21, e minha réplica está centrada em duas questões levantadas em suas observações finais. A primeira refere-se ao aspecto hermenêutico do desenvolvimento da matemática e do conhecimento matemático; e a segunda à “concepção simbólica da matemática” e ao “modo algébrico de pensar”. (shrink)

In §1 I discuss Dedekind and Frege on the logical and structural analysis of natural numbers and present my view that the logical analysis of the notion of number involves a combination of their analyses. In §2 I answer some of the specific questions that Abel raises in connection with Chapter 9 of Logical Forms.

We review Ferreirós and Lassalle Casanave's recently published book "El árbol de los números". The book is a result of the Brazilian-Spanish conference "Sobre la elucidación del concepto de número: cognición, lógica y práctica matemática" hosted in Sevilla in 2013, and collects new papers on History and Philosophy of Mathematics as well as Mathematical Practice. These papers present results of investigations in Cognitive Sciences, Logic and Epistemology of mathematical certainty.

José Ferreirós y Abel Lasalle Casanave, El árbol de los números: cognición, lógica y práctica matemática, Editorial Universidad de Sevilla, Sevilla, 2015, 256 pp.