One is a science, the other an art; one useful, the other seemingly decorative, but mathematics and music share common origins in cult and mystery and have been linked throughout history. Emblems of Mind is Edward Rothstein’s classic exploration of their profound similarities, a journey into their “inner life.” Along the way, Rothstein explains how mathematics makes sense of space, how music tells a story, how theories are constructed, how melody is shaped. He invokes the poetry (...) of Wordsworth, the anthropology of Le;vi-Strauss, the imagery of Plato, and the philosophy of Kant. Math and music, Rothstein shows, apply comparable methods as they create their abstractions, display similar concerns with ratio and proportion, and depend on metaphors and analogies to create their meanings. Ultimately, Rothstein argues, they reveal the ways in which we come to understand the world. They are images of the mind at work and play; indeed, they are emblems of Mind itself. Jacques Barzun called this book “splendid.” Martin Gardner said it was “beautifully written, marvelous and entertaining.” It will provoke all serious readers to think in new ways about the grand patterns in art and life. “Lovely, wistful. . . . Rothstein is a wonderful guide to the architecture of musical space, its tensions and relations, its resonances and proportions. . . . His account of what is going on in the music is unfailingly felicitous.”— New Yorker “Provocative and exciting. . . . Rothstein writes this book as a foreign correspondent, sending dispatches from a remote and mysterious locale as a guide for the intellectually adventurous. The remarkable fact about his work is not that it is profound, as much of the writing is, but that it is so accessible.”— Christian Science Monitor. (shrink)

The Topos of Music is the upgraded and vastly deepened English extension of the seminal German Geometrie der Töne. It reflects the dramatic progress of mathematical music theory and its operationalization by information technology since the publication of Geometrie der Töne in 1990. The conceptual basis has been vastly generalized to topos-theoretic foundations, including a corresponding thoroughly geometric musical logic. The theoretical models and results now include topologies for rhythm, melody, and harmony, as well as a classification (...) theory of musical objects that comprises the topos-theoretic concept framework. Classification also implies techniques of algebraic moduli theory. The classical models of modulation and counterpoint have been extended to exotic scales and counterpoint interval dichotomies. The probably most exciting new field of research deals with musical performance and its implementation on advanced object-oriented software environments. This subject not only uses extensively the existing mathematical music theory, it also opens the language to differential equations and tools of differential geometry, such as Lie derivatives. Mathematical performance theory is the key to inverse performance theory, an advanced new research field which deals with the calculation of varieties of parameters which give rise to a determined performance. This field uses techniques of algebraic geometry and statistics, approaches which have already produced significant results in the understanding of highest-ranked human performances. The book's formal language and models are currently being used by leading researchers in Europe and Northern America and have become a foundation of music software design. This is also testified by the book's nineteen collaborators and the included CD-ROM containing software and music examples. (shrink)

In recent years, several remarkable results have shown that certain theorems of finite combinatorics are unprovable in certain logical systems. These developments have been instrumental in stimulating research in both areas, with the interface between logic and combinatorics being especially important because of its relation to crucial issues in the foundations of mathematics which were raised by the work of Kurt Godel. Because of the diversity of the lines of research that have begun to shed light on these issues, (...) there was a need for a comprehensive overview which would tie the lines together. This volume fills that need by presenting a balanced mixture of high quality expository and research articles that were presented at the August 1985 AMS-IMS-SIAM Joint Summer Research Conference, held at Humboldt State University in Arcata, California.With an introductory survey to put the works into an appropriate context, the collection consists of papers dealing with various aspects of 'unprovable theorems and fast-growing functions'. Among the topics addressed are: ordinal notations, the dynamical systems approach to Ramsey theory, Hindman's finite sums theorem and related ultrafilters, well quasiordering theory, uncountable combinatorics, nonstandard models of set theory, and a length-of-proof analysis of Godel's incompleteness theorem. Many of the articles bring the reader to the frontiers of research in this area, and most assume familiarity with combinatorics and/or mathematical logic only at the senior undergraduate or first-year graduate level. (shrink)

Kennedy shows that Plato gave his dialogues a similar musical structure, dividing each dialogue into twelve parts and inserting symbols at each twelfth to mark a musical note. These passages are either harmonious or dissonant and traverse the ups and downs of a known musical scale. Many of Plato's early followers insisted that Plato used symbols to conceal his own views within the dialogues, but modern scholars have denied this. Kennedy, an expert in Pythagorean mathematics and music (...) class='Hi'>theory, is able to show that Plato's dialogues contain a system of symbols that are undetectable by those without knowledge of obsolete Greek mathematics. The book begins with a concise and accessible introduction to Plato's symbolic schemes and the role of allegory in ancient times. The author then annotates the musical symbols in two of Plato's most popular dialogues, the Symposium and Euthyphro, and shows that Plato used the musical scale as an outline for structuring his narratives. (shrink)

Foreword: Mathesis as a philosophy of the beauty of music -- The constructive relationships between music and mathematics : the Pythagorean conception of universal music and its spread in the worldview of later periods -- Semantic interpretation of the interaction between music and mathematics : mystic middle ages and the sacral baroque -- Constructive aspects of the interaction between music and other arts -- Musica mathematica in practice : aspects of analysis -- Constructive (...) aspects of music composition -- Semantic aspects of music composition -- The mathematized musical graph -- Implications of modern mathematical theories. (shrink)

Known today mostly as an author of Romantic short stories and fairy tales for children, Prince Vladimir Odoyevskiy was a distinguished thinker of his time, philosopher and bibliophile. The scope of his interests includes also history of magic arts and alchemy, German Romanticism, Church music. An attempt to understand the peculiarity of eight specific modes used in chants of Russian Orthodox Church led him to his own musical theory based upon well-known writings by Zarlino, Leibniz, Euler, Prony. He (...) realized his theoretical ideas in an enharmonical piano with nineteen notes for every octave. His call for a different musical scale remained largely ignored in the nineteenth century, until the topic was raised anew by twentieth-century composers and musicians. (shrink)

The Consolations of Philosophy by Boethius, whose English translators include King Alfred, Geoffrey Chaucer, and Queen Elizabeth I, ranks among the most remarkable books to be written by a prisoner awaiting the execution of a tyrannical death sentence. Its interpretation is bound up with his other writings on mathematics and music, on Aristotelian and propositional logic, and on central themes of Christian dogma. -/- Chadwick begins by tracing the career of Boethius, a Roman rising to high office under (...) the Gothic King Theoderic the Great, and suggests that his death may be seen as a cruel by-product of Byzantine ambitions to restore Roman imperial rule after its elimination in the West in AD 476. Subsequent chapters examine in detail his educational programme in the liberal arts designed to avert a threatened collapse of culture and his ambition to translate into Latin everything he could find on Plato and Aristotle. -/- Boethius has been called `last of the Romans, first of the scholastics'. This book is the first major study in English of a writer who was of critical importance in the history of thought. (shrink)

There is no topic that attracts more attention, more energy and time and devotion, than love. As long as there's been recorded history, love has taken center seat as the inspiration for countless paintings, instigator of wars, muse of untold poets and musicians. And just as poetry, art and music have the ability to communicate something about love that is difficult to articulate with words, the same is true of mathematics. Of course, mathematics can't easily help us (...) translate the emotional side of love, emotions rarely behave in a neatly ordered, rational and easily predictable way. It is difficult to quantify the rollercoaster of romance or to define how lovers might feel via a set of simple equations. But that doesn't mean that mathematics isn't crucial to understanding love. Love, like most things in life, is full of patterns. And mathematics is ultimately the study of patterns, from predicting the weather to the fluctuations of the stock market, the movement of planets or the growth of cities. These patterns twist and turn and warp and evolve just as the rituals of love do. In [this book, the author] takes the listener on a fascinating journey through the patterns that define our love lives, tackling some of the most common yet complex questions pertaining to love: What's the chance of us finding love? What's the chance that it will last? How does online dating work, exactly? When should you settle down? How can you avoid divorce? When is it right to compromise? Can game theory help us decide whether or not to call? From evaluating the best strategies for online dating to defining the nebulous concept of beauty, Dr. Fry proves, with great insight, wit and fun, that math is a surprisingly useful tool to negotiate the complicated, often baffling, sometimes infuriating, always interesting, patterns of love. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy of Music Education Review 11.2 (2003) 141-156 [Access article in PDF] Music Education as Critical PracticeA Naturalist View Lauri Väkevä University Of Oulu, Finland I This essay defends naturalism as a framework for philosophy of music education. I have three general reasons for supporting naturalism. First, by taking naturalism seriously we can keep our philosophies up-to-date with scientific inquiry. Second, naturalism can emancipate us from (...) transcendental pseudo-questions and instead help us to deal with the empirical problems of everyday life. Third, with naturalism, we can free ourselves from a priori theories of morals and aesthetics and thus view the active part that normative and aesthetic concerns play in human praxis. Even if I endorse naturalism, I do not subscribe to scientism. 1 Rather, I submit to a pragmatist view that stresses the role of philosophy (and science) in dealing with the questions of our life practices. Of the recent perspectives in the philosophy of music education my approach comes closest to the critical praxialism discussed in the MayDay group. 2All in all, I find a lot in common with the praxialist discussion (within or outside the MayDay group) and the kind of pragmatism that follows John Dewey's mature philosophy. 3 However, it seems to me that the naturalist bent of Deweyan pragmatism has not been thoroughly analyzed in our field. 4 One reason for this [End Page 141] might be the bad report that 'hard' or reductive naturalism has been given in connection with the experimental paradigm of music education research. 5 I want to argue that the critical potential of Deweyan pragmatism should not (and cannot) be separated from a naturalist approach. Taken together, critical and naturalist views can contribute to any philosophy, especially in fields that have a long tradition of leaning on otherworldly explanations.In what follows, I begin by outlining (in broad brushstrokes) the historical position of pragmatism and its status in present day philosophy. Next, I focus on the implications of Deweyan naturalist pragmatism and the possibility of maintaining a critical pragmatist attitude with naturalist presumptions in a pedagogical context. In the final part of the essay, I outline an interpretative framework for a critical pragmatism of music education that takes advantage of the naturalizing tendencies in contemporary philosophy without losing its grasp on the things that matter musically. II For a while, pragmatism had a bad name in academic philosophy. This was especially true after the linguistic turn in the first half of the last century. 6 By making the linguistic turn, many philosophers hoped to disassociate themselves from the speculative tradition(s) once and for all. Logical analysts also made it more difficult for philosophers to ignore the findings of scientists. Thus they paved the way for contemporary reductive naturalism that takes science as the measure of what can be known. 7However, analytic philosophers were not alone in their attempt to provide philosophy with scientific rigor. Classical pragmatists, too, had integrated a naturalist penchant in their output. Embracing naturalism with an open-ended experimental spirit, they wanted to focus on "the problems of men" as the chief subject matter of philosophy. 8 To many analytic philosophers, this seemed like a return to armchair speculation with an ingredient of sentimental ethics thrown in for good measure. Worse still, it looked as though the classical pragmatists were giving in to the spirit of the corporate West: speaking of the truth as "cash-value" surely seemed like a blasphemy to intellectuals who sought from formal logic and mathematics a purity of mind to clear up the messiness of everyday experience. 9Logical analysis also had its shortcomings. While many mainstream philosophers turned to logical analysis for clarity and certainty, at the same time they submitted to Cartesian-Kantian mind-body dualism with its problematic epistemology. Thus, they kept supporting the tensions between the rationalist and empiricist or analytic and synthetic domains of knowledge that haunted modernist neo-Kantian arguments.These distinctions turned out to be easy targets for postanalytic critics. 10 [End Page 142]Classical pragmatists anticipated this... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy of Music Education Review 11.2 (2003) 141-156 [Access article in PDF] Music Education as Critical PracticeA Naturalist View Lauri Väkevä University Of Oulu, Finland I This essay defends naturalism as a framework for philosophy of music education. I have three general reasons for supporting naturalism. First, by taking naturalism seriously we can keep our philosophies up-to-date with scientific inquiry. Second, naturalism can emancipate us from (...) transcendental pseudo-questions and instead help us to deal with the empirical problems of everyday life. Third, with naturalism, we can free ourselves from a priori theories of morals and aesthetics and thus view the active part that normative and aesthetic concerns play in human praxis. Even if I endorse naturalism, I do not subscribe to scientism. 1 Rather, I submit to a pragmatist view that stresses the role of philosophy (and science) in dealing with the questions of our life practices. Of the recent perspectives in the philosophy of music education my approach comes closest to the critical praxialism discussed in the MayDay group. 2All in all, I find a lot in common with the praxialist discussion (within or outside the MayDay group) and the kind of pragmatism that follows John Dewey's mature philosophy. 3 However, it seems to me that the naturalist bent of Deweyan pragmatism has not been thoroughly analyzed in our field. 4 One reason for this [End Page 141] might be the bad report that 'hard' or reductive naturalism has been given in connection with the experimental paradigm of music education research. 5 I want to argue that the critical potential of Deweyan pragmatism should not (and cannot) be separated from a naturalist approach. Taken together, critical and naturalist views can contribute to any philosophy, especially in fields that have a long tradition of leaning on otherworldly explanations.In what follows, I begin by outlining (in broad brushstrokes) the historical position of pragmatism and its status in present day philosophy. Next, I focus on the implications of Deweyan naturalist pragmatism and the possibility of maintaining a critical pragmatist attitude with naturalist presumptions in a pedagogical context. In the final part of the essay, I outline an interpretative framework for a critical pragmatism of music education that takes advantage of the naturalizing tendencies in contemporary philosophy without losing its grasp on the things that matter musically. II For a while, pragmatism had a bad name in academic philosophy. This was especially true after the linguistic turn in the first half of the last century. 6 By making the linguistic turn, many philosophers hoped to disassociate themselves from the speculative tradition(s) once and for all. Logical analysts also made it more difficult for philosophers to ignore the findings of scientists. Thus they paved the way for contemporary reductive naturalism that takes science as the measure of what can be known. 7However, analytic philosophers were not alone in their attempt to provide philosophy with scientific rigor. Classical pragmatists, too, had integrated a naturalist penchant in their output. Embracing naturalism with an open-ended experimental spirit, they wanted to focus on "the problems of men" as the chief subject matter of philosophy. 8 To many analytic philosophers, this seemed like a return to armchair speculation with an ingredient of sentimental ethics thrown in for good measure. Worse still, it looked as though the classical pragmatists were giving in to the spirit of the corporate West: speaking of the truth as "cash-value" surely seemed like a blasphemy to intellectuals who sought from formal logic and mathematics a purity of mind to clear up the messiness of everyday experience. 9Logical analysis also had its shortcomings. While many mainstream philosophers turned to logical analysis for clarity and certainty, at the same time they submitted to Cartesian-Kantian mind-body dualism with its problematic epistemology. Thus, they kept supporting the tensions between the rationalist and empiricist or analytic and synthetic domains of knowledge that haunted modernist neo-Kantian arguments.These distinctions turned out to be easy targets for postanalytic critics. 10 [End Page 142]Classical pragmatists anticipated this... (shrink)

The processes taking place in modern science tend to integrate various scientific disciplines in studying any scientific phenomenon. This explains the search of methodology based on the techniques of different sciences involved in formation of the subject of research, which in this article is represented by the digital music art. Interpreted as a branch of music art with digital specificity, digital music art incorporates electronic music presented by specific, algorithmic, and electronic music. A signature characteristic (...) of digital music art consists in its interdisciplinarity, which suggests the combined effect of separate parts of the system and leads to self-organization, which corresponds with the scientific tasks of synergetics. This substantiates the relevance for elaboration of interdisciplinary methodology for studying digital music art in the context of research mechanisms developed within synergetics. The scientific lies in the analysis of digital art through the prism of the qualities inherent to synergetics. The disciplines united by digital art (mathematics, set theory, probability theory, information theory, combinatorics, game theory, cybernetics, computer science, acoustics, sociology, communication, and biology) are interdisciplinary, which means that in ensemble with other disciplines they form a complex interdisciplinary knowledge of structure of digital music art. At the same time, the disciplines that comprise scientific framework of digital music art are attributed to different sciences: humanities, natural sciences, mathematics, social sciences, and technical sciences. In this context, the research of digital music art is simultaneously of interdisciplinary and multiscientific nature. The author established a peculiar impact of interdisciplinarity upon digital art, which generates the qualities inherent to synergetics – nonlinearity, disequilibrium, openness, chaotic nature, entropy, indeterminacy, and dissipation. (shrink)

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and updated (...) by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Singing Democracy:Music and Politics in Jean-Jacques Rousseau's ThoughtJulia SimonComment? Tous les intervalles de mon Clavecin sont altérés?... Fi, le vilain instrument; ne m'en parlez plus.... Je veux chanter.—Anton Bemetzrieder, Leçons de ClavecinDemocratic theory of the eighteenth century, and particularly Rousseau's, is suffused with the idealism and lack of pragmatism that make it both immensely compelling and extraordinarily frustrating. Conceived under the decaying edifice of the absolute monarchy, (...) it strives toward perfection, offering theoretical formulations that often defy practical application. Yet this theory continues to inspire democratic practice and political debates even more than 200 years after its writing. In one passage of Du contrat social Rousseau asserts, "If there were a people of Gods, it would govern itself democratically. Such a perfect government is not suited to men."1 And yet the thrust of the work and the critical force of Rousseau's corpus encourage and challenge us to strive toward democracy, as impossible a goal as it may be for mere mortals.Alongside Rousseau's works in social and political theory lies another vast body of theoretical work in music. Rarely, if ever, are they read in tandem. Music in the eighteenth century suffers from many of the same difficulties that political theory does, especially the tendency toward a level of abstraction that defies practical application. Like the problem of democracy in the eighteenth century, music also presents the temptation of retreat into a world of abstract perfection based on mathematical certainty with little relation to practical reality. [End Page 433] Perhaps nowhere is the tension so acute between the ideal and the real in music as in the domain of tuning specifically, in keyboard tuning, and the question of tempering. So as in the political domain, there is a potentially wide gulf between music theory and music practice.If there is any hope of realizing democracy in the way that Rousseau understood it, an exploration of his music theory may provide the crucial bridge between theory and practice. Rousseau's theoretical work in music, and singing in particular, covers a number of distinct subject areas. First, his proposal for a new system of musical notation links directly to democratic impulses in his political and social theory. Second, the kinds of emotions that music stirs are related in Rousseau's thought to forms of expression that remain closer to their natural origins. Finally, Rousseau's preference for melody, as opposed to harmony, relates directly to his conception of the political sphere and specifically to the relationship between the general will and the workings of the body politic in a democracy. Unexpectedly, reading Rousseau's musical theory alongside his democratic theory produces a nuanced and moderate view of the social contract that deepens our understanding of the relationship between relative and absolute values in politics as well as in music and offers a model for practice.Musical Notation and DemocracyA distinctly democratic impulse motivates the Projet concernant de nouveaux signes pour la musique for Rousseau's revision of the musical notation system is designed to increase accessibility to music. With characteristic rhetoric Rousseau introduces his project for "simplifying" musical notation with a short, to-the-point paragraph: "This project tends to make music easier to write down, easier to learn and much less diffuse."2 Following this statement of purpose, Rousseau relates, in a contrasting rhetorical style—replete with specialized vocabulary—the current state of affairs in musical notation:This quantity of lines, keys, transpositions, of sharps, flats, naturals, of simple and complex measures, of whole-notes, half-notes, quarter-notes, eighth-notes, sixteenth-notes, thirty-second notes, of rests, half-rests, quarter-rests, eighth-rests, sixteenth-rests, etc. gives a quantity of signs and combinations from which result two principle inconveniences: the first, to occupy too great a volume, and the second to overload the memory of schoolchildren in such a way that the ear being formed and the organs having acquired all the necessary facility long before one is [End Page 434] capable of singing from a book, it follows that the difficulty is more in the observation of rules than in the execution of... (shrink)

This book endeavours to pinpoint the relations between musical, and especially instrumental, practice and the evolving conceptions of pitch systems. It traces the development of ancient melodic notation from reconstructed origins, through various adaptations necessitated by changing musical styles and newly invented instruments, to its final canonical form. It thus emerges how closely ancient harmonic theory depended on the culturally dominant instruments, the lyre and the aulos. These threads are followed down to late antiquity, when details recorded by Ptolemy (...) permit an exceptionally clear view. Dr Hagel discusses the textual and pictorial evidence, introducing mathematical approaches wherever feasible, but also contributes to the interpretation of instruments in the archaeological record and occasionally is able to outline the general features of instruments not directly attested. The book will be indispensable to all those interested in Greek music, technology and performance culture and the general history of musicology. (shrink)

Perfect Harmony and Melting Strains assembles interdisciplinary essays investigating concepts of harmony during a transitional period, in which the Pythagorean notion of a harmoniously ordered cosmos competed with and was transformed by new theories about sound - and new ways of conceptualizing the world. From the perspectives of philosophy, literary scholarship, and musicology, the contributions consider music's ambivalent position between mathematical abstraction and sensibility, between the metaphysics of harmony and the physics of sound. Essays examine the late medieval and (...) early modern history of ideas concerning the nature of music and cosmic harmony, and trace their transformations in early modern musico-literary discourses. Within this framework, essays further offer original readings of important philosophical, literary, and musicological works. This interdisciplinary volume brings into focus the transformation of a predominant Renaissance worldview and of music's scientific, theological, literary, as well as cultural conceptions and functions in the early modern period, and will be of interest to scholars of the classics, philosophy, musicology, as well as literary and cultural studies. (shrink)

We provide a new perspective on the relation between the space of description of an object and the appearance of novelties. One of the aims of this perspective is to facilitate the interaction between mathematics and historical sciences. The definition of novelties is paradoxical: if one can define in advance the possibles, then they are not genuinely new. By analyzing the situation in set theory, we show that defining generic (i.e., shared) and specific (i.e., individual) properties of elements (...) of a set are radically different notions. As a result, generic and specific definitions of possibilities cannot be conflated. We argue that genuinely stating possibilities requires that their meaning has to be made explicit. For example, in physics, properties playing theoretical roles are generic; then, generic reasoning is sufficient to define possibilities. By contrast, in music, we argue that specific properties matter, and generic definitions become insufficient. Then, the notion of new possibilities becomes relevant and irreducible. In biology, among other examples, the generic definition of the space of DNA sequences is insufficient to state phenotypic possibilities even if we assume complete genetic determinism. The generic properties of this space are relevant for sequencing or DNA duplication, but they are inadequate to understand phenotypes. We develop a strong concept of biological novelties which justifies the notion of new possibilities and is more robust than the notion of changing description spaces. These biological novelties are not generic outcomes from an initial situation. They are specific and this specificity is associated with biological functions, that is to say, with a specific causal structure. Thus, we think that in contrast with physics, the concept of new possibilities is necessary for biology. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:A Humanist History of Mathematics?Regiomontanus's Padua Oration in ContextJames Steven ByrneIn the spring of 1464, the German astronomer, astrologer, and mathematician Johannes Müller (1436–76), known as Regiomontanus (a Latinization of the name of his hometown, Königsberg in Franconia), offered a course of lectures on the Arabic astronomer al-Farghani at the University of Padua. The only one of these to survive is his inaugural oration on the history and (...) utility of the mathematical arts.1 Regiomontanus tells his audience that the purpose of the oration is torelate first the origin of our arts, and among which nations they first began to be cultivated, in what way they were at last translated from various foreign tongues into Latin, which of our ancestors were famed in these disciplines, and to whom in our lifetimes recognition should be granted.2To this end, he offers a history of the quadrivial arts (arithmetic, geometry, music, and astronomy) and other important mathematical disciplines from [End Page 41] antiquity to his own time, praises their utility, and exhorts his audience to revive the languishing study of mathematics at Padua. Astrology, a discipline with which Regiomontanus himself was closely associated, is singled out for particular praise.Traditionally, Regiomontanus's Padua oration has been seen through the lens of its rather obvious humanism. In particular, the Padua oration has come to be understood as the rhetorical embodiment of the fifteenth-and-sixteenth-century revival of ancient (Greek) mathematics. Just as this revival is inextricable from the rise of humanism, so Regiomontanus has come to be seen as an exemplar of humanist mathematics.3 It is the aim of this paper to examine the Padua oration in the context both of contemporary humanist rhetoric and of Regiomontanus's own intellectual background in order to argue that, while the oration is stylistically consistent with humanist norms, the vision of mathematics presented in it is also deeply grounded in the university mathematical curriculum and in Regiomontanus's own reading of mathematical texts.Regiomontanus was educated primarily at the University of Vienna (he was also briefly at the University of Leipzig), where he enrolled in 1450, completed his baccalaureate in 1452 and became a master in 1457.4 He remained at Vienna until 1461, when the death of his friend and teacher Georg Peurbach prompted him to travel to Italy with his patron, Cardinal Bessarion.5 In Regiomontanus's day Vienna was probably the most important of the German universities, rivaled only by Prague, whose prestige had declined after it was stripped of its theology faculty in the aftermath of the Hussite Wars.6 The curriculum at the University of Vienna was modeled [End Page 42] after that of Paris, and Parisian scholars played a major role both in its founding in 1365 and in its re-establishment (this time with a theology faculty) in 1384.7Most importantly for the purposes of this paper, the Viennese mathematical curriculum of Regiomontanus's day included all of the traditional authorities taught at Paris in the fourteenth century. For arithmetic and algebra, various "algorisms" (prose or poetry instructions for carrying out arithmetical operations that often also included a small amount of number theory) were the basic texts, supplemented in the fourteenth century by the Quadripartitum numerorum of Jean de Murs. Jordanus de Nemorare's De numeris datis and al-Khwarizmi's Algebra were common sources for those engaged in more advanced studies (i.e., they were not normally the subject of ordinary lectures, but were readily available to interested students, and would perhaps have been the subject of occasional extraordinary lectures). Euclid's Elements, supplemented by the commentaries of Pappus and Campanus, was the central text for geometry, and a number of medieval texts on practical and speculative geometry were in circulation as well. For astronomy, the Sphere of Sacrobosco and the Theorica planetarum were the most commonly used teaching texts, sometimes supplemented by al-Farghani's Elements of Astronomy (the subject of Regiomontanus's Padua lectures). Advanced students could read numerous more specific treatises by Arabic and Latin authors. For optics, the Perspectiva communis of John Peckham was the most common basic text, with texts... (shrink)

Leonhard Euler (1707–1783) treated in his study „De motu vibratorio tympanorum [6], published in 1766, the theory of the vibrating rectangular and circular membrane mathematically in such a comprehensive way that little more had to be added, considering today's standards. However, he omitted to interprete his results physically. Therefore his uncomparable work found little recognition, especially in the field of musical accoustics.

Flora Levin explores how and why music was so important to the ancient Greeks. She examines the distinctions that they drew between the theory of music as an art ruled by number and the theory wherein number is held to be ruled by the art of music. These perspectives generated more expansive theories, particularly the idea that the cosmos is a mirror-image of music's structural elements and, conversely, that music by virtue of its (...) cosmic elements - time, motion, and the continuum - is itself a mirror-image of the cosmos. These opposing perspectives gave rise to two opposing schools of thought, the Pythagorean and the Aristoxenian. Levin argues that the clash between these two schools could never be reconciled. Her book shows how the Greeks' appreciation of the profundity of music's interconnections with philosophy, mathematics, and logic led to groundbreaking intellectual achievements that no civilisation has ever matched. (shrink)

Professor Michael Edgeworth McIntyre is an eminent scientist who has also had a part-time career as a musician. From a lifetime's thinking, he offers this extraordinary synthesis exposing the deepest connections between science, music, and mathematics, while avoiding equations and technical jargon. He begins with perception psychology and the dichotomization instinct and then takes us through biological evolution, human language, and acausality illusions all the way to the climate crisis and the weaponization of the social media, and beyond (...) that into the deepest parts of theoretical physics - demonstrating our unconscious mathematical abilities. He also has an important message of hope for the future. Contrary to popular belief, biological evolution has given us not only the nastiest, but also the most compassionate and cooperative parts of human nature. This insight comes from recognizing that biological evolution is more than a simple competition between selfish genes. Rather, he suggests, in some ways it is more like turbulent fluid flow, a complex process spanning a vast range of timescales. Professor McIntyre is a Fellow of the Royal Society of London (FRS) and has worked on problems as diverse as the Sun's magnetic interior, the Antarctic ozone hole, jet streams in the atmosphere, and the psychophysics of violin sound. He has long been interested in how different branches of science can better communicate with each other and with the public, harnessing aspects of neuroscience and psychology that point toward the deep 'lucidity principles' that underlie skilful communication. (shrink)

From Ancient Greek times, music has been seen as a mathematical art, and the relationship between mathematics and music has fascinated generations. This collection of wide ranging, comprehensive and fully-illustrated papers, authored by leading scholars, presents the link between these two subjects in a lucid manner that is suitable for students of both subjects, as well as the general reader with an interest in music. Physical, theoretical, physiological, acoustic, compositional, and analytical relationships between mathematics and (...) music are unfolded and explored with focus on tuning and temperament, the mathematics of sound, bell-ringing and modern compositional techniques. (shrink)

Music theorists of almost all ages employ a concept of "Nature" to justify observations or statements about music. The understanding of what "Nature" is, however, is subject to cultural and historical differences. In tracing these explanatory strategies and their changes in music theories between c. 1600 and 1900, these essays explore (for the first time in a book-length study) how the multifarious conceptions of nature, located variously between scientific reason and divine power, are brought to bear on (...) music theory and how they affect our understanding of music. (shrink)

This work aims to demonstrate that a universal model of man can be constructed using the precise mathematical formulation of a few philosophical theses about the nature of human beings. The author uses it to find theoretical solutions for several unexplained phenomena in psychophysics.

Little is known about the harmonic theorist Archestratus (probably early 3rd century BC). Our only substantial information comes from Porphyry, who quotes a brief comment by a certain Didymus on his epistemological stance, and seeks to justify it through reflection on a rather startling technical doctrine which Archestratus propounded; and from Philodemus, who comments scathingly on his view of the relation between harmonic theory and philosophy. Neither passage is easy to interpret; this paper tries to make sense of them, (...) and to set them in an intelligible relation to one another. It argues that the doctrine recorded by Porphyry becomes comprehensible when it is placed against the background of Aristoxenus' work in harmonics, and it discusses Porphyry's inferences about the way in which his epistemological position diverged from that of Aristoxenus. It argues that Philodemus' report gives evidence of Archestratus' interest in issues of central concern to philosophy and in particular of an engagement with Aristotelian thought; it tries to identify some specific questions which attracted his attention, and to explain how he seems to have answered them, and why. It suggests that the two reports can be brought together as elements in a single, though fragmentary picture, and finally that Archestratus can be assigned an interesting though minor role in the history of Peripatetic philosophy and science. (shrink)

The effect of music in Greek and Latin literature -- The impact and value of music according to ancient theorists -- The value of music in systematic analysis : philosophical and psychological considerations.

Much controversy surrounds Schenker's mature theory and its attempt to explain musical pitch motion. Becoming Heinrich Schenker brings a new perspective to Schenker's theoretical work, showing that ideas characteristic of his mature theory, although in many respects fundamentally different, developed logically out of his earlier ideas. Robert P. Morgan provides an introduction to Schenker's mature theory and traces its development through all of his major publications, considering each in detail and with numerous music examples. Morgan also (...) explores the relationship between Schenker's theory and his troubled ideology, which crucially influenced the evolution of his ideas and was heavily dependent upon both the empirical and idealist strains of contemporary German philosophical thought. Relying where possible on quotations from Schenker's own words, this book offers a balanced approach to his theory and a unique overview of this central music figure, generally considered to be the most prominent music theorist of the twentieth century. (shrink)

The aim of this paper is to examine how Adorno's aesthetic and musicological thinking was received in Czech and Slovak musicology in the decades between the 60s and the 80s. The focus is on the Czech and Slovak translation of some of Adorno’s musicological treatises and lectures – especially those concerning his views on the Second Vienna School and the musical poetics of its immediate successors – which were published in former Czechoslovakia. The study offers an interesting perspective on Adorno’s (...) relatively unknown lecture Form der neuen Musik and its related, although not identical, Czech version Formové princípy súčasnej hudby [Formal Principles of Contemporary Music] as well as on his discussion with some Slovak composers and musicologists published as Dnes je možné iba radikálne kritické myslenie [Today, Only Radical Critical Thinking is Possible]. The study also considers other scientific texts by Adorno in relation to the above-mentioned translations of his works. The analysis, reflection, and interpretation of Adorno’s works in former Czechoslovakia, as well as their contemporary reception, turn out to be sporadic in the examined period. The purpose of this research is to revive awareness of their significance and to give a new impulse to their reassessment within the current musicological and philosophical reflection. (shrink)

Ever since the early decades of this century, there have emerged a number of competing schools of ecology that have attempted to weave the concepts underlying natural resource management and natural-historical traditions into a formal theoretical framework. It was widely believed that the discovery of the fundamental mechanisms underlying ecological phenomena would allow ecologists to articulate mathematically rigorous statements whose validity was not predicated on contingent factors. The formulation of such statements would elevate ecology to the standing of a rigorous (...) scientific discipline on a par with physics. However, there was no agreement as to the fundamental units of ecology. Systems ecologists sought to identify the fundamental organization that tied the physical and biological components of ecosystems into an irreducible unit: the ecosystem was their fundamental unit. Population ecologists sought, instead, to identify the biological mechanisms regulating the abundance and distribution of plant and animal species: to these ecologists, the individual organism was the fundamental unit of ecology, and the physical environment was nothing more than a stage upon which the play of individuals in perennial competition took place. As Joel Hagen has pointed out, the two schools were thus dividied by fundamentally different and irreconcilable assumptions about the nature of ecosystems.Notwithstanding these divisive efforts to elevate the image of ecology, the discipline remained in the shadows of American academia until the mid-1960s, when systems ecologists succeeded in projecting ecology onto the national scene. They did so by seeking closer involvement with practical problems: they argued before Congress that their approach to the theoretical problems of ecology was uniquely suited to the solution of the impending “environmental crisis.” With the establishment of the International Biological Program, they succeeded in attracting unprecedented levels of funding for systems ecology research. Theoretical population ecologists, on the other hand, found themselves consigned to the outer regions of this new institutional landscape. The systems ecologists' successful capture of the limelight and the purse brought the divisions between them and population ecologists into sharper relief — hence the hardening of the division of ecology observed by Hagen.45I have argued that the population biologist Richard Levins, prompted by these institutional developments, sought to challenge the social position of systems ecology, and to assert the intellectual priority of theoretical population ecology. He attempted to do so by articulating a nontrivial and rather carefully thought out classification of ecological models that led to the disqualification of systems analysis as a legitimate approach to the study of ecological phenomena. I have suggested that — ultimately —Levins's case against systems analysis in ecology rested on the view that an aspiration to realism and prediction was incompatible with an interest in theoretical issues, a concern that he equated with the search for generality. He sought to reinforce this argument by exploiting the fact that systems ecologists had staked their future on the provision of technical solutions to the problems of the “environmental crisis”: he associated systems ecologists' aspiration to realism and precision with a concern for practical issues, trading on the widely accepted view that practical imperatives are incompatible with the aims of scientific inquiry.46 These are plausible, but nonetheless questionable, claims which have now become an integral part of ecological knowledge. And finally, I hope to have shown how even the most abstract levels of scientific argument are shaped by political considerations, and how discussions of the conceptual development of modern ecology might benefit from a greater consideration of its historical and social dimensions.47. (shrink)

In my dissertation I analyze the structure of fundamental physical theories. I start with an analysis of what an adequate primitive ontology is, discussing the measurement problem in quantum mechanics and theirs solutions. It is commonly said that these theories have little in common. I argue instead that the moral of the measurement problem is that the wave function cannot represent physical objects and a common structure between these solutions can be recognized: each of them is about a clear three-dimensional (...) primitive ontology that evolves according to a law determined by the wave function. The primitive ontology is what matter is made of while the wave function tells the matter how to move. One might think that what is important in the notion of primitive ontology is their three-dimensionality. If so, in a theory like classical electrodynamics electromagnetic fields would be part of the primitive ontology. I argue that, reflecting on what the purpose of a fundamental physical theory is, namely to explain the behavior of objects in three--dimensional space, one can recognize that a fundamental physical theory has a particular architecture. If so, electromagnetic fields play a different role in the theory than the particles and therefore should be considered, like the wave function, as part of the law. Therefore, we can characterize the general structure of a fundamental physical theory as a mathematical structure grounded on a primitive ontology. I explore this idea to better understand theories like classical mechanics and relativity, emphasizing that primitive ontology is crucial in the process of building new theories, being fundamental in identifying the symmetries. Finally, I analyze what it means to explain the word around us in terms of the notion of primitive ontology in the case of regularities of statistical character. Here is where the notion of typicality comes into play: we have explained a phenomenon if the typical histories of the primitive ontology give rise to the statistical regularities we observe. (shrink)