On Euclidean diagrams and geometrical knowledge

We argue against the claim that the employment of diagrams in Euclidean geometry gives rise to gaps in the proofs. First, we argue that it is a mistake to evaluate its merits through the lenses of Hilbert’s formal reconstruction. Second, we elucidate the abilities employed in diagram-based inferences in the Elements and show that diagrams are mathematically reputable tools. Finally, we complement our analysis with a review of recent experimental results purporting to show that, not only is the Euclidean diagram-based practice strictly regimented, it is rooted in cognitive abilities that are universally shared.; Argumentamos en contra de la afirmación de que el uso de diagramas en la geometría euclidiana da lugar a vacíos o lagunas en las pruebas. En primer lugar, mostramos que es un error evaluar sus méritos a través de las lentes de la reconstrucción formal de Hilbert. En segundo lugar, esclarecemos las habilidades empleadas en las inferencias basadas en los diagramas en los Elementos, y mostramos que los diagramas son herramientas matemáticas respetables. Finalmente, complementamos nuestro análisis con una revisión de resultados experimentales recientes que pretenden mostrar que la práctica diagramática euclidiana no solo está estrictamente regimentada, sino que también está enraizada en ciertas habilidades cognitivas universalmente compartidas

Introduction

There has been little overt discussion of the experimental philosophy of logic or mathematics. So it may be tempting to assume that application of the methods of experimental philosophy to these areas is impractical or unavailing. This assumption is undercut by three trends in recent research: a renewed interest in historical antecedents of experimental philosophy in philosophical logic; a “practice turn” in the philosophies of mathematics and logic; and philosophical interest in a substantial body of work in adjacent disciplines, such as the psychology of reasoning and mathematics education. This introduction offers a snapshot of each trend and addresses how they intersect with some of the standard criticisms of experimental philosophy. It also briefly summarizes the specific contribution of the other chapters of this book

On an assumption of geometric foundation of numbers

In line with the latest positions of Gottlob Frege, this article puts forward the hypoth- esis that the cognitive bases of mathematics are geometric in nature. Starting from the geometry axioms of the Elements of Euclid, we introduce a geometric theory of propor- tions along the lines of the one introduced by Grassmann in Ausdehnungslehre in 1844. Assuming as axioms, the cognitive contents of the theorems of Pappus and Desargues, through their configurations, in an Euclidean plane a natural field structure can be iden- tified that reveals the purely geometric nature of complex numbers. Reasoning based on figures is becoming a growing interdisciplinary field in logic, philosophy and cognitive sciences, and is also of considerable interest in the field of education, moreover, recently, it has been emphasized that the mutual assistance that geometry and complex numbers give is poorly pointed out in teaching and that a unitary vision of geometrical aspects and calculation can be clarifying

Reliability of mathematical inference

Of all the demands that mathematics imposes on its practitioners, one of the most fundamental is that proofs ought to be correct. This is also a demand that is especially hard to fulfill, given the fragility and complexity of mathematical proof. This essay considers some of ways that mathematics supports reliable assessment, which is necessary to maintain the coherence and stability of the practice

On Euclidean diagrams and geometrical knowledge

We argue against the claim that the employment of diagrams in Euclidean geometry gives rise to gaps in the proofs. First, we argue that it is a mistake to evaluate its merits through the lenses of Hilbert’s formal reconstruction. Second, we elucidate the abilities employed in diagram-based inferences in the Elements and show that diagrams are mathematically reputable tools. Finally, we complement our analysis with a review of recent experimental results purporting to show that, not only is the Euclidean diagram-based practice strictly regimented, it is rooted in cognitive abilities that are universally shared

Reliability of mathematical inference

Of all the demands that mathematics imposes on its practitioners, one of the most fundamental is that proofs ought to be correct. It has been common since the turn of the twentieth century to take correctness to be underwritten by the existence of formal derivations in a suitable axiomatic foundation, but then it is hard to see how this normative standard can be met, given the differences between informal proofs and formal derivations, and given the inherent fragility and complexity of the latter. This essay describes some of the ways that mathematical practice makes it possible to reliably and robustly meet the formal standard, preserving the standard normative account while doing justice to epistemically important features of informal mathematical justification

Universal intuitions of spatial relations in elementary geometry

FSW - Self-regulation models for health behavior and psychopathology - ou

An Open Logic Approach to EPM

open2noEPM is a high operative and didactic versatile tool and new application areas are envisaged continuously. In turn, this new awareness has allowed to enlarge our panorama for neurocognitive system EPM is a high operative and didactic versatile tool and new application areas are envisaged continuosly. In turn, this new awareness has allowed to enlarge our panorama for neurocognitive system behavior understanding, and to develop information conservation and regeneration systems in a numeric self-reflexive/reflective evolutive reference framework. Unfortunately, a logically closed model cannot cope with ontological uncertainty by itself; it needs a complementary logical aperture operational support extension. To achieve this goal, it is possible to use two coupled irreducible information management subsystems, based on the following ideal coupled irreducible asymptotic dichotomy: "Information Reliable Predictability" and "Information Reliable Unpredictability" subsystems. To behave realistically, overall system must guarantee both Logical Closure and Logical Aperture, both fed by environmental "noise" (better… from what human beings call "noise"). So, a natural operating point can emerge as a new Trans-disciplinary Reality Level, out of the Interaction of Two Complementary Irreducible Information Management Subsystems within their environment. In this way, it is possible to extend the traditional EPM approach in order to profit by both classic EPM intrinsic Self-Reflexive Functional Logical Closure and new numeric CICT Self-Reflective Functional Logical Aperture. EPM can be thought as a reliable starting subsystem to initialize a process of continuous self-organizing and self-logic learning refinement. understanding, and to develop information conservation and regeneration systems in a numeric self-reflexive/reflective evolutive reference framework. Unfortunately, a logically closed model cannot cope with ontological uncertainty by itself; it needs a complementary logical aperture operational support extension. To achieve this goal, it is possible to use two coupled irreducible information management subsystems, based on the following ideal coupled irreducible asymptotic dichotomy: "Information Reliable Predictability" and "Information Reliable Unpredictability" subsystems. To behave realistically, overall system must guarantee both Logical Closure and Logical Aperture, both fed by environmental "noise" (better… from what human beings call "noise"). So, a natural operating point can emerge as a new Trans-disciplinary Reality Level, out of the Interaction of Two Complementary Irreducible Information Management Subsystems within their environment. In this way, it is possible to extend the traditional EPM approach in order to profit by both classic EPM intrinsic Self-Reflexive Functional Logical Closure and new numeric CICT Self-Reflective Functional Logical Aperture. EPM can be thought as a reliable starting subsystem to initialize a process of continuous self-organizing and self-logic learning refinement.Fiorini, Rodolfo; Degiacomo, PieroFiorini, Rodolfo; Degiacomo, Pier

Recent studies on signs: Commentary and perspectives

In this commentary, I reply to the fourteen papers published in the Sign Systems Studies special issue on Peirce’s Theory of Signs, with a view on connecting some of their central themes and theses and in putting some of the key points in those papers into a wider perspective of Peirce’s logic and philosophy