Periods in the Use of Euler-type Diagrams

Logicians commonly speak in a relatively undifferentiated way about pre-euler diagrams. The thesis of this paper, however, is that there were three periods in the early modern era in which euler-type diagrams (line diagrams as well as circle diagrams) were expansively used. Expansive periods are characterized by continuity, and regressive periods by discontinuity: While on the one hand an ongoing awareness of the use of euler-type diagrams occurred within an expansive period, after a subsequent phase of regression the entire knowledge about the systematic application and the history of euler-type diagrams was lost. I will argue that the first expansive period lasted from Vives (1531) to Alsted (1614). The second period began around 1660 with Weigel and ended in 1712 with lange. The third period of expansion started around 1760 with the works of Ploucquet, euler and lambert. Finally, it is shown that euler-type diagrams became popular in the debate about intuition which took place in the 1790s between leibnizians and Kantians. The article is thus limited to the historical periodization between 1530 and 1800

Squares of Oppositions, Commutative Diagrams, and Galois Connections for Topological Spaces and Similarity Structures

The aim of this paper is to elucidate the relationship between Aristotelian conceptual oppositions, commutative diagrams of relational structures, and Galois connections.This is done by investigating in detail some examples of Aristotelian conceptual oppositions arising from topological spaces and similarity structures. The main technical device for this endeavor is the notion of Galois connections of order structures

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

Avicenna's Philosophy of Mathematics

I discuss four different aspects of Avicenna’s philosophical views on mathematics, as scattered across his various works. I first explore the negative aspect of his ontology of mathematics, which concerns the question of what mathematical objects (i.e., numbers and geometrical shapes) are not. Avicenna argues that mathematical objects are not independent immaterial substances. They cannot be fully separated from matter. He rejects what is now called mathematical Platonism. However, his understanding of Plato’s view about the nature of mathematical objects differs from both Plato’s actual view and the view that Aristotle attributes to Plato. Second, I explore the positive aspect of Avicenna’s ontology of mathematics, which is developed in response to the question of what mathematical objects are. He considers mathematical objects to be specific properties of material objects actually existing in the extramental world. Mathematical objects can be separated, in mind, from all the specific kinds of matter to which they are actually attached in the extramental word. Nonetheless, inasmuch as they are subject to mathematical study, they cannot be separated from materiality itself. Even in mind they should be considered as properties of material entities. Third, I scrutinize Avicenna’s understanding of mathematical infinity. Like Aristotle, he rejects the infinity of numbers and magnitudes. But he does so by providing arguments that are much more sophisticated than their Aristotelian ancestors. By analyzing the structure of his Mapping Argument against the actuality of infinity, I show that his understanding of the notion of infinity is much more modern than we might expect. Finally, I engage with Avicenna’s views on the epistemology of mathematics. He endorses concept empiricism and judgment rationalism regarding mathematics. He believes that we cannot grasp any mathematical concepts unless we first have had some specific perceptual experiences. It is only through the ineliminable and irreplaceable operation of the faculties of estimation and imagination upon some sensible data that we can grasp mathematical concepts. By contrast, after grasping the required mathematical concepts, independently from all other faculties, the intellect alone can prove mathematical theorems. Other faculties, and in particular the cogitative faculty, can assist the intellect in this regard; but the participation of such faculties is merely facilitative and by no means necessary

The Latin Readers of Algazel, 1150-1600

This dissertation examines how Arabic works found an audience in medieval Europe and became a part of the Latin canon of philosophy. It focuses on a Latin translation of an Arabic philosophical work, Maqasid al-falasifa, by the Muslim theologian al-Ghazali, known as Algazel in Latin. This work became popular because it served as a primer for Arab philosophy and helped Latins understand a tradition that had built upon Greek scholarship for centuries. To find the translation’s audience, this project looks at two sets of evidence. It studies the works of Latin scholars who drew from Algazel’s arguments and illustrates that the translation’s influence was more extensive than historians have previously thought. It also examines copies of the translation in forty manuscripts and broadens the Latin audience of Arab philosophy beyond what historians typically study—the university—to include the anonymous scribes and readers who comprise the often-voiceless majority of medieval literate society. These codices yield details about Algazel’s readers, their interests and concerns, which cannot be gathered from other sources. Scholars spared little expense with these manuscripts since several are quite ornate or contain gold leaf. Many copies possess wide margins where scholars interacted with the text by writing notes, diagrams, pointing hands, warnings, and the occasional doodle. Scribes integrated the work into the established canon by placing Algazel in manuscripts with Christian philosophers from Augustine to Aquinas. The manuscripts also contain marginalia left by generations of readers, which give insight into how scholars read the text and what passages grabbed their attention. The notes indicate that a few readers agreed with ecclesiastical authorities who condemned Algazel’s work since some scholars wrote warnings in the margins alongside passages that they considered dangerous. Thus, Latins paradoxically expended great effort to understand Arab philosophers while simultaneously condemning ideas in the translations as errors. This study expands our understanding of the European interaction with the Arab tradition by examining reading practices with evidence drawn from the readers themselves. It demonstrates that Europeans read translated Arabic works alongside long-standing authorities and treated Arab authors as valuable members of the Latin canon

Premodern Experience of the Natural World in Translation

This innovative collection showcases the importance of the relationship between translation and experience in premodern science, bringing together an interdisciplinary group of scholars to offer a nuanced understanding of knowledge transfer across premodern time and space. The volume considers experience as a tool and object of science in the premodern world, using this idea as a jumping-off point from which to view translation as a process of interaction between diff erent epistemic domains. The book is structured around four dimensions of translation—between terms within and across languages; across sciences and scientific norms; between verbal and visual systems; and through the expertise of practitioners and translators—which raise key questions on what constituted experience of the natural world in the premodern area and the impact of translation processes and agents in shaping experience. Providing a wide-ranging global account of historical studies on the travel and translation of experience in the premodern world, this book will be of interest to scholars in history, the history of translation, and the history and philosophy of science