Analisi critica d'opera, incentrata sulla recezione dell'opera di Tolomeo durante il Medioevo e sul pensiero scientifico-filosofico dell' Età Medievale e Rinascimentale.

Beth models of analysis are used in model theoretic proofs of the disjunction and existence property. By glueing strings of models one obtains a model that combines the properties of the given models. The method asks for a common generalization of Kripke and Beth models. The proof is carried out in intuitionistic analysis plus Markov's Principle. The main new feature is the external use of intuitionistic principles to prove their own preservation under glueing.

This article analyzes the pediatric material in the Arabic commentaries (written tenth–fifteenth centuries) on the Hippocratic Aphorisms by exploring the traces of its late-antique origins and highlighting the influences of contemporary Islamic sources. This study demonstrates, first, how the commentaries assimilate Galenic pediatric theory through intricate elaborations and innovations; and second, that the commentators on the Aphorisms exhibit a strict theoretical interest in the causes and nature of childood diseases as opposed to their remedies. Consequently, it shows that therapeutic pediatric (...) material was restricted to other nonexegetical genres, such as the encyclopedias of Ibn Sīnā and Ibn Rabban al-Ṭabarī. The commentary material is thus a unique example of medieval Islamic theoria or theoretical medicine, which transmitted and transformed Galenic and early medieval theoretical explanations of childhood illnesses through the centuries. (shrink)

In the present paper the story is told of the brief and far from tranquil encounter of L.E.J. Brouwer and A. Fraenkel. The relationship which started in perfect harmony, ended in irritation and reproaches.The mutual appreciation at the outset is beyond question. All the more deplorable is the sudden outbreak of an emotional disagreement in 1927. Looking at the Brouwer–Fraenkel episode, one should keep in mind that at that time the so-called Grundlagenstreit was in full swing. An emotional man like (...) Brouwer, who easily suffered under stress, was already on edgewhen Fraenkel's book Zehn Vorlesungen Über die Grundlegung der Mengenlehre, [Fraenkel 1927] was about to appear.With the Grundlagenstreit reaching (in print!) a level of personal abuse unusual in the quiet circles of pure mathematics, Brouwer was rather sensitive, where the expositions of his ideas were concerned. So when he thought that he detected instances of misconception and misrepresentation in the case of his intuitionism, he felt slighted. We will mainly look at Brouwer's reactions. since the Fraenkel letters have not been preserved.The late Mrs. Fraenkel kindly put the Brouwer letters that were in her possession at my disposal. I am grateful to the Fraenkel family for the permission to use the material.I am indebted to Andreas Blass for his valuable suggestions and corrections.Abraham Fraenkel (then still called Adolf) was one of the first non-partisan mathematicians, if not the first, who developed a genuine interest in intuitionism. (shrink)

The present volume is intended as an all-round introduction to constructivism. Here constructivism is to be understood in the wide sense, and covers in particular Brouwer's intuitionism, Bishop's constructivism and A.A. Markov's constructive recursive mathematics. The ending "-ism" has ideological overtones: "constructive mathematics is the (only) right mathematics"; we hasten, however, to declare that we do not subscribe to this ideology, and that we do not intend to present our material on such a basis.

On October 4, 1937, Zermelo composed a small note entitled “Der Relativismus in der Mengenlehre und der sogenannte Skolemsche Satz”(“Relativism in Set Theory and the So-Called Theorem of Skolem”) in which he gives a refutation of “Skolem's paradox”, i.e., the fact that Zermelo-Fraenkel set theory—guaranteeing the existence of uncountably many sets—has a countable model. Compared with what he wished to disprove, the argument fails. However, at a second glance, it strongly documents his view of mathematics as based on a world (...) of objects that could only be grasped adequately by infinitary means. So the refutation might serve as a final clue to his epistemological credo.Whereas the Skolem paradox was to raise a lot of concern in the twenties and the early thirties, it seemed to have been settled by the time Zermelo wrote his paper, namely in favour of Skolem's approach, thus also accepting the noncategoricity and incompleteness of the first-order axiom systems. So the paper might be considered a late-comer in a community of logicians and set theorists who mainly followed finitary conceptions, in particular emphasizing the role of first-order logic (cf. [8]). However, Zermelo never shared this viewpoint: In his first letter to Gödel of September 21, 1931, (cf. [1]) he had written that the Skolem paradox rested on the erroneous assumption that every mathematically definable notion should be expressible by a finite combination of signs, whereas a reasonable metamathematics would only be possible after this “finitistic prejudice” would have been overcome, “a task I have made my particular duty”. (shrink)

Empirical analyses of the ethics of corporations with the aim to improve the state of corporate ethics are rare. This paper develops an integrated, normative model of corporate ethics by conceptualizing the ethical quality of organizations and by relating this contextual quality to various expressions of immoral behavior. This so-called Ethics Qualities Model for organizations, which contains 21 ethical qualities, allows one to assess the ethical content of institutional groups of individuals. A proper conceptualization is highly relevant both for the (...) empirical corroboration of business ethics theories and for managerial purposes, such as judging individual and group performance or informing external stakeholders. The empirical applicability of the model is illustrated by an explorative case study of a large, globally operating financial institution. This case-study demonstrates that the corporate ethical qualities differ with respect to their perceived optimality as well as to their estimated impact on (un)ethical conduct. The various results provide managers with many clues to understand their organization and to take effective measures to improve the ethical content of their organization. (shrink)

On October 4, 1937, Zermelo composed a small note entitled “Der Relativismus in der Mengenlehre und der sogenannte Skolemsche Satz”(“Relativism in Set Theory and the So-Called Theorem of Skolem”) in which he gives a refutation of “Skolem's paradox”, i.e., the fact that Zermelo-Fraenkel set theory—guaranteeing the existence of uncountably many sets—has a countable model. Compared with what he wished to disprove, the argument fails. However, at a second glance, it strongly documents his view of mathematics as based on a world (...) of objects that could only be grasped adequately by infinitary means. So the refutation might serve as a final clue to his epistemological credo.Whereas the Skolem paradox was to raise a lot of concern in the twenties and the early thirties, it seemed to have been settled by the time Zermelo wrote his paper, namely in favour of Skolem's approach, thus also accepting the noncategoricity and incompleteness of the first-order axiom systems. So the paper might be considered a late-comer in a community of logicians and set theorists who mainly followed finitary conceptions, in particular emphasizing the role of first-order logic (cf. [8]). However, Zermelo never shared this viewpoint: In his first letter to Gödel of September 21, 1931, (cf. [1]) he had written that the Skolem paradox rested on the erroneous assumption that every mathematically definable notion should be expressible by a finite combination of signs, whereas a reasonable metamathematics would only be possible after this “finitistic prejudice” would have been overcome, “a task I have made my particular duty”. (shrink)

Luitzen Egburtus Jan Brouwer founded a school of thought whose aim was to include mathematics within the framework of intuitionistic philosophy; mathematics was to be regarded as an essentially free development of the human mind. What emerged diverged considerably at some points from tradition, but intuitionism has survived well the struggle between contending schools in the foundations of mathematics and exact philosophy. Originally published in 1981, this monograph contains a series of lectures dealing with most of the fundamental topics such (...) as choice sequences, the continuum, the fan theorem, order and well-order. Brouwer's own powerful style is evident throughout the work. (shrink)

From the reviews: "A good textbook can improve a lecture course enormously, especially when the material of the lecture includes many technical details. Van Dalen's book, the success and popularity of which may be suspected from this steady interest in it, contains a thorough introduction to elementary classical logic in a relaxed way, suitable for mathematics students who just want to get to know logic. The presentation always points out the connections of logic to other parts of mathematics. The (...) reader immediately see the logic is "just another branch of mathematics" and not something more sacred." Acta Scientiarum Mathematicarum, Hungary. (shrink)

Although the main responsibility for informed consent of medical procedures rests with doctors, nurses’ roles are also important, especially as patient advocates. Nurses’ preparation for this role in settings with a hierarchical and communal culture has received little attention. We explored the views of hospital managers and nurses regarding the roles of nurses in informed consent and factors influencing these roles. We conducted a qualitative study in a private, multispecialty hospital in Indonesia. Semi-structured interviews were conducted with seven managers. Two (...) rounds of focus group discussions with nurses (n = 27) were conducted. Constant comparative approach was used in the analysis. Nurses can act as manager, witness, information giver, and advocate in the informed consent process. These roles are influenced by nurses’ preparedness, hospital culture and policy, patients’ understanding, family involvement, and cost-related issues. In preparation for these tasks, nurses should acquire communication skills, clinical knowledge, and legal and ethical knowledge. (shrink)

Dedicated to Dana Scott on his sixtieth birthday.It is common knowledge that for a short while Hermann Weyl joined Brouwer in his pursuit of a revision of mathematics according to intuitionistic principles. There is, however, little in the literature that sheds light on Weyl's role and in particular on Brouwer's reaction to Weyl's allegiance to the cause of intuitionism. This short episode certainly raises a number of questions: what made Weyl give up his own program, spelled out in “Das Kontinuum”, (...) how did Weyl come to be so well-informed about Brouwer's new intuitionism, in what respect did Weyl's intuitionism differ from Brouwer's intuitionism, what did Brouwer think of Weyl's views,…? To some of these questions at least partial answers can be put forward on the basis of some of the available correspondence and notes. The present paper will concentrate mostly on the historical issues of the intuitionistic episode in Weyl's career.Weyl entered the foundational controversy with a bang in 1920 with his sensational paper “On the new foundational crisis in mathematics”. He had already made a name for himself in the foundations of mathematics in 1918 with his monograph “The Continuum” [18]; this contained in addition to a technical logical-mathematical construction of the continuum, a fairly extensive discussion of the shortcomings of the traditional construction of the continuum on the basis of arbitrary—and hence also impredicative—Dedekind cuts. (shrink)

Brouwer and Weyl recognized that the intuitive continuum requires a mathematical analysis of a kind that set theory is not able to provide. As an alternative, Brouwer introduced choice sequences. We first describe the features of the intuitive continuum that prompted this development, focusing in particular on the flow of internal time as described in Husserl's phenomenology. Then we look at choice sequences and their logic. Finally, we investigate the differences between Brouwer and Weyl, and argue that Weyl's conception of (...) choice sequences is defective on several counts. (shrink)

There are basically two ways to view intuitionistic logic: as a philosophical‐foundational issue in mathematics; or as a technical discipline within mathematical logic. Considering first the philosophical aspects, for they will provide the motivation for the subject, this chapter follows L. E. J. Brouwer, the founding father of intuitionism. Although Brouwer himself contributed little to intuitionistic logic as seen from textbooks and papers, he did point the way for his successors.

In the present paper the story is told of the brief and far from tranquil encounter of L.E.J. Brouwer and A. Fraenkel. The relationship which started in perfect harmony, ended in irritation and reproaches.The mutual appreciation at the outset is beyond question. All the more deplorable is the sudden outbreak of an emotional disagreement in 1927. Looking at the Brouwer–Fraenkel episode, one should keep in mind that at that time the so-called Grundlagenstreit was in full swing. An emotional man like (...) Brouwer, who easily suffered under stress, was already on edgewhen Fraenkel's book Zehn Vorlesungen Über die Grundlegung der Mengenlehre, [Fraenkel 1927] was about to appear.With the Grundlagenstreit reaching a level of personal abuse unusual in the quiet circles of pure mathematics, Brouwer was rather sensitive, where the expositions of his ideas were concerned. So when he thought that he detected instances of misconception and misrepresentation in the case of his intuitionism, he felt slighted. We will mainly look at Brouwer's reactions. since the Fraenkel letters have not been preserved.The late Mrs. Fraenkel kindly put the Brouwer letters that were in her possession at my disposal. I am grateful to the Fraenkel family for the permission to use the material.I am indebted to Andreas Blass for his valuable suggestions and corrections.Abraham Fraenkel was one of the first non-partisan mathematicians, if not the first, who developed a genuine interest in intuitionism. (shrink)

John Foxal was an English Franciscan active primarily in Italy in the latter half of the fifteenth century. His philosophical and theological positions were Scotist, and in his works he comments upon various texts of John Duns Scotus with reference to the Scotist school of thought active in the early fourteenth century. In this article we supplement the list of extant manuscripts, provide a revised list of Foxal’s treatises and offer an edition of one of these, the Tractatus de propositione (...) per se nota. (shrink)

This chapter contains sections titled: Logic: The Proof Interpretation Analysis: Choice Sequences Further Semantics.

On October 4, 1937, Zermelo composed a small note entitled “Der Relativismus in der Mengenlehre und der sogenannte Skolemsche Satz” in which he gives a refutation of “Skolem's paradox”, i.e., the fact that Zermelo-Fraenkel set theory—guaranteeing the existence of uncountably many sets—has a countable model. Compared with what he wished to disprove, the argument fails. However, at a second glance, it strongly documents his view of mathematics as based on a world of objects that could only be grasped adequately by (...) infinitary means. So the refutation might serve as a final clue to his epistemological credo.Whereas the Skolem paradox was to raise a lot of concern in the twenties and the early thirties, it seemed to have been settled by the time Zermelo wrote his paper, namely in favour of Skolem's approach, thus also accepting the noncategoricity and incompleteness of the first-order axiom systems. So the paper might be considered a late-comer in a community of logicians and set theorists who mainly followed finitary conceptions, in particular emphasizing the role of first-order logic. However, Zermelo never shared this viewpoint: In his first letter to Gödel of September 21, 1931, he had written that the Skolem paradox rested on the erroneous assumption that every mathematically definable notion should be expressible by a finite combination of signs, whereas a reasonable metamathematics would only be possible after this “finitistic prejudice” would have been overcome, “a task I have made my particular duty”. (shrink)

The original Brouwerian counter examples were algorithmic in nature; after the introduction of choice sequences, Brouwer devised a version which did not depend on algorithms. This is the origin of the creating subject technique. The method allowed stronger refutations of classical principles. Here it is used to show that negative dense subsets of the continuum are indecomposable.