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-calledGrundlagenstreitwas in full swing. An emotional man like Brouwer, who (...) easily suffered under stress, was already on edgewhen Fraenkel's bookZehn Vorlesungen Über die Grundlegung der Mengenlehre, [Fraenkel 1927] was about to appear.With theGrundlagenstreitreaching (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)

Servant leadership is increasingly gaining interest inside and outside academia. This article builds and extends current theorizing by describing the process that introduces compassionate love as a practical translation for the need to serve, which was positioned by Greenleaf as the core of servant leadership. This article takes a virtues perspective and shows how servant leadership may encourage a more meaningful and optimal human functioning with a strong sense of community to current-day organizations. In essence, we propose that a leader’s (...) propensity for compassionate love will encourage a virtuous attitude in terms of humility, gratitude, forgiveness and altruism. This virtuous attitude will give rise to servant leadership behavior in terms of empowerment, authenticity, stewardship and providing direction. (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)

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)

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)

Accounts and analyses of the formation of scholarly and scientific communities in the early modern period by means of memory and collective identity.

There are two principles that lend Brouwer's mathematics the extra power beyond arithmetic. Both are presented in Brouwer's writings with little or no argument. One, the principle of bar induction, will not concern us here. The other, the continuity principle for numbers, occurs for the first time in print in [4]. It is formulated and immediately applied to show that the set of numerical choice sequences is not enumerable. In fact, the idea of the continuity property can be dated fairly (...) precisely, it is to be found in the margin of Brouwer's notes for his course on Pointset Theory of 1915/16. The course was repeated in 1916/17 and he must have inserted his first formulation of the continuity principle in the fall of 1916 as new material right at the beginning of the course.In modern language, the principle readswhere α and β range over choice sequences of natural numbers, m and x over natural numbers, and stands for ⟨α, α, …, α⟩, the initial segment of α of length m.An immediate consequence of WC-N is that all full functions are continuous, and, as a corollary, that the continuum is unsplittable [28]. Note that WC-N is incompatible with Church's thesis, [22], section 4.6.After Brouwer asserted WC-N, Troelstra was the first to ask in print for a conceptual motivation, but he remained an exception; most authors followed Brouwer by simply asserting it, cf. [18].Let us note first that in one particular case the principle is obvious indeed, namely in the case of the lawless sequences. The notion of lawless sequence surfaced fairly late in the history of intuitionism. Kreisel introduced it in [17] for metamathematical purposes. There is a letter from Brouwer to Heyting in which the phenomenon also occurs [7]. This is an important and interesting fact since it is the only time that Brouwer made use of a possibility expressly stipulated in, e.g., [5], see below. (shrink)

Servant leadership has been theorized as a model where the moral virtue of humility co-exists with action-driven behavior. This article provides an empirical study that tests how these two apparently paradoxical aspects of servant leadership interact in generating follower engagement, while considering the hierarchical power of the leader as a contingency variable. Through a three-way moderation model, a study was conducted based on a sample of 232 people working in a diverse range of companies. The first finding is that humble (...) leaders showed the highest impact on follower engagement regardless of their hierarchical position. Less humble leaders in lower hierarchical positions seem to be able to compensate for that through a strong action-oriented leadership style. Most notably for leaders in high hierarchical positions, the moral virtue of humility seems to strengthen the impact of their action-oriented leadership the most. These findings provide empirical support and a better understanding of the interplay between the moral virtue of humility and the action-oriented behaviors of servant leadership. (shrink)

Summary In this paper we examine the reaction of the Leiden low-temperature laboratory of Heike Kamerlingh Onnes to new ideas in quantum theory. Especially the contributions of Albert Einstein (1906) and Peter Debye (1912) to the theory of specific heat, and the concept of zero-point energy formulated by Max Planck in 1911, gave a boost to solid state research to test these theories. In the case of specific heat measurements, Kamerlingh Onnes's laboratory faced stiff competition from Walter Nernst's Institute of (...) Physical Chemistry in Berlin. In fact, Berlin got the better of it because Leiden lacked focus. After the liquefaction of helium in 1908, Kamerlingh Onnes transformed his laboratory into an international facility for low temperature research, and for this reason it was impossible to make headway with the specific heat measurements. In the case of zero-point energy, Leiden developed a magnetic research programme to test the concept. Initially the balance of evidence seemed to be tipping in favour of zero-point energy. After 1914, however, Leiden would desert the theory in fovour, of a concept from calssical physics. A curious move that illustrates Kamerlingh Onnes's discomfort with the new quantum theory. (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)

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.

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)

Summary In this paper we examine the reaction of the Leiden low-temperature laboratory of Heike Kamerlingh Onnes to new ideas in quantum theory. Especially the contributions of Albert Einstein (1906) and Peter Debye (1912) to the theory of specific heat, and the concept of zero-point energy formulated by Max Planck in 1911, gave a boost to solid state research to test these theories. In the case of specific heat measurements, Kamerlingh Onnes's laboratory faced stiff competition from Walter Nernst's Institute of (...) Physical Chemistry in Berlin. In fact, Berlin got the better of it because Leiden lacked focus. After the liquefaction of helium in 1908, Kamerlingh Onnes transformed his laboratory into an international facility for low temperature research, and for this reason it was impossible to make headway with the specific heat measurements. In the case of zero-point energy, Leiden developed a magnetic research programme to test the concept. Initially the balance of evidence seemed to be tipping in favour of zero-point energy. After 1914, however, Leiden would desert the theory in fovour, of a concept from calssical physics. A curious move that illustrates Kamerlingh Onnes's discomfort with the new quantum theory. (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)

Dirk van Dalen’s biography studies the fascinating life of the famous Dutch mathematician and philosopher Luitzen Egbertus Jan Brouwer. Brouwer belonged to a special class of genius; complex and often controversial and gifted with a deep intuition, he had an unparalleled access to the secrets and intricacies of mathematics. Most mathematicians remember L.E.J. Brouwer from his scientific breakthroughs in the young subject of topology and for the famous Brouwer fixed point theorem. Brouwer’s main interest, however, was in the foundation (...) of mathematics which led him to introduce, and then consolidate, constructive methods under the name ‘intuitionism’. This made him one of the main protagonists in the ‘foundation crisis’ of mathematics. As a confirmed internationalist, he also got entangled in the interbellum struggle for the ending of the boycott of German and Austrian scientists. This time during the twentieth century was turbulent; nationalist resentment and friction between formalism and intuitionism led to the Mathematische Annalen conflict. It was here that Brouwer played a pivotal role. The present biography is an updated revision of the earlier two volume biography in one single book. It appeals to mathematicians and anybody interested in the history of mathematics in the first half of the twentieth century. (shrink)

Aan de hand van recente ontdekkingen op het gebied van neurowetenschappen, biologische evolutie, paleontologie, evolutionaire psychologie en evolutionaire dynamicamodellen, leggen Dirk Van Duppen en Johan Hoebeke uit hoe pro-sociale gedragingen en altruïsme het succes van de ‘homo sapiens’ evolutionair hebben bepaald. Vermits de mens als meest kwetsbaar en meest prematuur dier op de wereld is gekomen, was zijn enige manier om te overleven een biologisch-culturele co-evolutie waarbij zijn intelligente vermogens zich enkel konden ontwikkelen door zijn ingeslepen drang om zich (...) sociaal te gedragen. Dit wetenschappelijk gegrond beeld van de ‘homo sapiens’ staat volledig haaks op dit van de neoliberale ‘homo economicus’ die enkel op individuele winstmaximalisatie gericht is en op de competitiedrang, die als enige drijfkracht voor de vooruitgang wordt beschouwd. De discrepantie tussen de biologisch-culturele fundamenten van onze soort en de actuele overheersende ideologie waaraan wij continu onderworpen worden, zou het stijgend ongenoegen en het ‘mal vivre’ in onze maatschappijen kunnen verklaren. Zoals José Saramago het uitte: ‘Als de omstandigheden zo bepalend zijn voor de mens, laat ons dan die omstandigheden meer menselijk maken.’. (shrink)

The related fields of fractal image encoding and fractal image analysis have blossomed in recent years. This book, originating from a NATO Advanced Study Institute held in 1995, presents work by leading researchers. It is developing the subjects at an introductory level, but it also has some recent and exciting results in both fields. The book contains a thorough discussion of fractal image compression and decompression, including both continuous and discrete formulations, vector space and hierarchical methods, and algorithmic optimizations. The (...) book also discusses multifractal approaches to image analysis, segmentation, and recognition, including medical applications. (shrink)

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

With the proliferation of networked electronic communication came daunting capabilities to collect, process, combine and store data, resulting in hitherto unseen transformational pressure on the concepts of trust, security and privacy as we know them. The Future Internet will bring about a world where real life will integrate physical and digital life. Technology development for data linking and mining, together with unseen data collection, will lead to unwarranted access to personal data, and hence, privacy intrusion. Trust and identity lie at (...) the basis of many human interactions and transactions, and societies have developed legitimate concern for privacy being essential for freedom and creativity. The burgeoning development of the Information Society, particularly during the past fifteen years, transcended the societal readiness to respond to the transformational change evoked by ICT. We have reached the eleventh hour for the preservation of trust and privacy as elements that can be transposed into our digital future. Europe has been at the forefront in recognizing the importance of privacy protection in relation to digital data, witness the advanced European legislation in this domain. The European Commission recognizes that appropriate measures need to combine technology development with legal means, user awareness and tools supporting data controllers to comply with law in an accountable and transparent way, and that empower users with a controlling stake in managing their personal data. Activities are underway at many levels. European RTD programmes play their role in supporting research in trustworthy ICT, privacy enhancing technologies, privacy-by-design in service layers as well as in networks, enabling technologies such as cryptography, and in generalized frameworks for trust and privacy-protective identity management. (shrink)

There are two principles that lend Brouwer's mathematics the extra power beyond arithmetic. Both are presented in Brouwer's writings with little or no argument. One, the principle of bar induction, will not concern us here. The other, the continuity principle for numbers, occurs for the first time in print in [4]. It is formulated and immediately applied to show that the set of numerical choice sequences is not enumerable. In fact, the idea of the continuity property can be dated fairly (...) precisely, it is to be found in the margin of Brouwer's notes for his course on Pointset Theory of 1915/16. The course was repeated in 1916/17 and he must have inserted his first formulation of the continuity principle in the fall of 1916 as new material right at the beginning of the course.In modern language, the principle readswhere α and β range over choice sequences of natural numbers, m and x over natural numbers, and stands for ⟨α, α, …, α⟩, the initial segment of α of length m.An immediate consequence of WC-N is that all full functions are continuous, and, as a corollary, that the continuum is unsplittable [28]. Note that WC-N is incompatible with Church's thesis, [22], section 4.6.After Brouwer asserted WC-N, Troelstra was the first to ask in print for a conceptual motivation, but he remained an exception; most authors followed Brouwer by simply asserting it, cf. [18].Let us note first that in one particular case the principle is obvious indeed, namely in the case of the lawless sequences. The notion of lawless sequence surfaced fairly late in the history of intuitionism. Kreisel introduced it in [17] for metamathematical purposes. There is a letter from Brouwer to Heyting in which the phenomenon also occurs [7]. This is an important and interesting fact since it is the only time that Brouwer made use of a possibility expressly stipulated in, e.g., [5], see below. (shrink)

Servant leadership and other ethical and moral approaches to leadership have been criticized for focusing on followers to the potential detriment of other stakeholders, specifically shareholders. With individual data collected from 485 respondents nested in 55 similar stores in a single company, within a large metropolitan area in France, we tested a multilevel model whereby servant leadership relates positively to business-unit performance measured by profit growth—a key indicator for shareholders—through the mediation of employee flourishing and revenue growth. With financial performance (...) data collected approximately 15 months after servant leadership was measured, results supported the hypothesized relationships. At the individual level, this study also showed the association of servant leadership and employee flourishing to be negatively moderated by individual perceptions of power distance orientation: the weaker the power distance orientation, the stronger the influence of servant leadership on employee flourishing. Improving on prior studies that relied on data aggregated at the group level and proximal indicators of performance, this multilevel study sheds light on how servant leadership, a prominent form of ethically conscious leadership, may resolve the fundamental tension leaders face vis-à-vis the potentially diverging interests of their stakeholders. (shrink)