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)

Alfred North Whtiehead stated that mathematics and music compete with each other for the honor of being the most novel achievement of the human imagination. Actually, there is no rivalry or competition, for the two enterprises interpenetrate once the investigation is pursued far enough. Though on opposite ends of the science-humanities spectrum, each discipline points toward the other and elicits the same sort of reverential puzzlement. This essay considers the seminal interconnection between music and mathematics and concludes that (...) this unexpected conjunction of opposites bespeaks a higher, unseen reality. Music cum mathematics is an optics that allows us to discover, celebrate, and participate in God's ongoing Creation. (shrink)

A green philosopher's peripeteia.--Physics and metaphysics of music.--The roots of arithmetic.--Critique of new geometrical abstractions.--The philosophical value of science.

Jean Dieudonné, the spokesman of the group of French mathematicians named Bourbaki, called mathematics the music of reason. This metaphor invites a phenomenological account of the affective, in contrast to the epistemic and discursive, nature of mathematics: What constitutes its charm? Mathematical reasoning is described as a perceptual experience, which in Husserl’s late philosophy would be a case of passive synthesis. Like a melody, a mathematical proof is manifest in an affective identity of a temporal object. Rather than an (...) exercise for its own sake, this account sheds a different light on both the epistemic limitation of mathematical science, and the discursive problem of social responsibility in mathematics – two issues at the heart of Husserl’s critique of science as well as of mid-20th century mathematics, for which Nicolas Bourbaki stands as a monument of rigor. (shrink)

Creativity is an essential factor in ensuring the sustainable development of a society. Improving students’ creativity has gained much attention in education, especially in Science, Technology, Engineering, Arts, and Mathematics education. In a quasi-experimental design, this study examines the effectiveness of a project-based STEAM program on the development of creativity in Chinese elementary school science education. We selected two fourth-graders classes. One received a project-based STEAM program, and the other received a conventional science teaching over 6 weeks. Students’ creativity was (...) assessed before and after the intervention using a multi-method approach, including a test of divergent thinking, a story completion through the Consensus Assessment Technique, a creative self-efficacy measure, and a group-based creative project. Moreover, all students received a test of their science knowledge after the intervention. The results showed that compared with the control group, the creativity of the experimental group students improved significantly for 6 weeks at both individual and group level, even though their knowledge in science were comparable. This result confirmed the effectiveness of a project-based STEAM educational program improving elementary school students’ creativity. Implications are discussed. (shrink)

Mathematics and music in practice and performance, and in learning and teaching, share many characteristics, such as beauty and harmony, memory and intuition and mind or intellect. These raise the principles of processing information in mathematics and music and, by implication, the role of an acquaintance with the essentials of perception, abstraction and affective connaturality in teacher education. This paper compares mathematics and music and considers the acquisition of knowledge and skills through the external and internal senses (...) and emotions, utilizing the role of knowledge through multiple intelligences. In doing so it does not canvas the utilities of mathematics and music as fields of human endeavour so much as their role in the cultivation of serenity and knowledge in the cultured mind. This is a theoretical paper but it is based on nearly a century of teaching from the combined work of the two authors in the teaching of music and mathematics. The paper highlights the importance of inspiration in teaching, inspiration built on a thorough basis of the foundations of anthropology to include the emotions as well as the intellect. While teacher education programs rightly concern themselves with knowledge of the field of study, knowledge of pedagogy, they do not always consider the ability to inspire which is at the heart of managing and mentoring people. (shrink)

Summary In 1670, the Bolognese mathematician Pietro Mengoli published his Speculationi di musica, a highly original work attempting to found the mathematical study of music on the anatomy of the ear. His anatomy was idiosyncratic and his mathematics extraordinarily complex, and he proposed a unique double mechanism of hearing. He analysed in detail the supposed behaviour of the subtle part of the air inside the ear, and the patterns of strokes made on the eardrum by simultaneous sounds. Most strikingly, (...) he divided the musical octave into a continuous set of regions which he colour-coded to show their effects on a listener. His work did not find its way into the mainstream of seventeenth-century mathematical studies of music, but when examined in its context it has the potential to shed light on that discipline, as well as being of considerable interest in its own right. Here, I focus on the anatomical and mathematical basis of Mengoli's work. (shrink)

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 chief of these five essays is the effort of a composer and conductor, deeply attached to Kant and widely read in mathematics and popularized physics, to disclose in "music's innate design" a key to the nature of "subliminal" reality, the "Ever Present." Result: both the order of music and the order of subliminal reality are elliptical. The accompanying essays, of chiefly biographical interest, present a philosophical critique of the new mathematics and geometry by a doctrinaire young Kantian. (...) -- L. K. B. (shrink)

For 4E cognitive science, minds are embodied, embedded, enacted, and extended. Proponents observe that we regularly ‘offload’ our thinking onto body and world: we use gestures and calculators to augment mathematical reasoning, and smartphones and search engines as memory aids. I argue that music is a beyond-the-head resource that affords offloading. Via this offloading, music scaffolds access to new forms of thought, experience, and behaviour. I focus on music’s capacity to scaffold emotional consciousness, including the self-regulative processes (...) constitutive of emotional consciousness. In developing this idea, I consider the ‘material’ and ‘worldmaking’ character music, and I apply these considerations to two case studies: music as a tool for religious worship, and music as a weapon for torture. (shrink)

In this book, the author argues that what constitutes music in various societies is culturally based, not the result of some universal aspect of human physical and psychological make-up. This is true not only in non-Western music cultures, but in the West as well. Contrary to popular belief among musicians and the general public, the basis of Western music and acoustics is not scientific, but superstitious. Pythagorean mathematics as it relates to harmonics does not work, a fact (...) that mathematicians and scientists have known for years and this has important implications, for it means that Western musical aesthetics and educational programmes in music are not rooted in the laws of physics, as is often maintained, but in a belief system that is an arbitrary product of human imagination. (shrink)

This article analyses the conditions under which mathematics could enter the field of fourteenth-century music. It distinguishes between descriptive and argumentative uses of mathematics. Jean de Murs’ uses of arithmetic to study musical time is an example of the former, Jean de Boen’s study of the division of the whole tone an example of the latter. It is furthermore explained how the mathematical descriptions appear to bring into agreement two types of constraint, namely the physical characteristics of sound and (...) the aesthetic principles of the medieval discourse about music. Within these constraints, mathematics manages to fulfill different argumentative roles: it has an ontological function when music is seen as a part of the quadrivium; but an explicative function in the framework of the scientia media and, in an more innovative spirit for Jean de Boens, it provides a definition of the possible in the argumentation about the division of the whole tone. (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)

This essay explores some aspects of the relation between philosophy and music. First, how music can inspire philosophy; second, how philosophy can inspire music. Mathematics as a middle term between music and philosophy, the idea of wholeness in a musical composition or a philosophical text, music as a mode of thought displaying traits such as logic, coherence, and sense—these are some ways in which music and philosophy may be seen to be connected. Also, composers (...) sometimes have explicit recourse to philosophical ideas in advancing their music, there being prominent examples of this in the twentieth century. Lastly, given there is such a thing as the philosophy of music, might there also be the music of philosophy? (shrink)

This is the first critical edition of a fascinating medieval work on music, written in Iraq in the tenth century. It is accompanied by an English translation and full annotation. The Epistle examines not just the technical, scientific, and mathematical aspects of music, but its cosmic, psychological, and spiritual dimensions.

Studies by Gardiner and colleagues connecting musical pitch and arithmetic learning support Rips et al.'s proposal that natural number concepts are constructed on a base of innate abilities. Our evidence suggests that innate ability concerning sequence ( or BSC) is fundamental. Mathematical engagement relating number to BSC does not develop automatically, but, rather, should be encouraged through teaching.

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 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)

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)

Mathematics is a powerful tool for describing and developing our knowledge of the physical world. It informs our understanding of subjects as diverse as music, games, science, economics, communications protocols, and visual arts. Mathematical thinking has its roots in the adaptive behavior of living creatures: animals must employ judgments about quantities and magnitudes in the assessment of both threats (how many foes) and opportunities (how much food) in order to make effective decisions, and use geometric information in the environment (...) for recognizing landmarks and navigating environments. Correspondingly, cognitive systems that are dedicated to the processing of distinctly mathematical information have developed. In particular, there is evidence that certain core systems for understanding different aspects of arithmetic as well as geometry are employed by humans and many other animals. They become active early in life and, particularly in the case of humans, develop through maturation. Although these core systems individually appear to be quite limited in application, in combination they allow for the recognition of mathematical properties and the formation of appropriate inferences based upon those properties. In this overview, the core systems, their roles, their limitations, and their interaction with external representations are discussed, as well as possibilities for how they can be employed together to allow us to reason about more complex mathematical domains. (shrink)

Mathematicians sometimes judge a mathematical proof to be beautiful and in doing so seem to be making a judgement of the same kind as aesthetic judgements of works of visual art, music or literature. Mathematical proofs are also appraised for explanatoriness: some proofs merely establish their conclusions as true, while others also show why their conclusions are true. This paper will focus on the prima facie plausible assumption that, for mathematical proofs, beauty and explanatoriness tend to go together. To (...) make headway we need to have some grip on what it is for a proof to be beautiful, and for that we need some account of judgements of beauty in general. That is the concern of the first section. The second section faces the problem that it is far from obvious how abstract entities, such as mathematical proofs, can be beautiful, strictly and literally speaking. Reasons are given for the view that they can be. The third section introduces the distinction between proofs which explain their conclusions and proofs which do not. Finally, the question whether, for mathematical proofs, the beautiful and the explanatory tend to coincide is addressed. It is argued that we have reason to doubt that explanatory proofs tend to be beautiful, and insufficient reason to believe or disbelieve that beautiful proofs tend to be explanatory. (shrink)

Understanding everyday behavior relies heavily upon understanding our ability to improvise, how we are able to continuously anticipate and adapt in order to coordinate with our environment and others. Here we consider the ability of musicians to improvise, where they must spontaneously coordinate their actions with co-performers in order to produce novel musical expressions. Investigations of this behavior have traditionally focused on describing the organization of cognitive structures. The focus, here, however, is on the ability of the time-evolving patterns of (...) inter-musician movement coordination as revealed by the mathematical tools of complex dynamical systems to provide a new understanding of what potentiates the novelty of spontaneous musical action. We demonstrate this approach through the application of cross wavelet spectral analysis, which isolates the strength and patterning of the behavioral coordination that occurs between improvising musicians across a range of nested time-scales. Revealing the sophistication of the previously unexplored dynamics of movement coordination between improvising musicians is an important step towards understanding how creative musical expressions emerge from the spontaneous coordination of multiple musical bodies. (shrink)

The ties between music and philosophy are strong and venerable, as they date back to the very beginnings of the latter. According to the ancient tale, Pythagoras, when passing by a smithy one day, noticed that the hammers make sounds of different pitch and, more importantly, that some of the pitch combinations feel pleasant on the ear while the others sound rather harsh. Intrigued by this phenomenon, the ancient sage began to further investigate it with the so called monochord (...) (being just a plank with a string attached to it which could be shortened at any chosen position to alter the pitch). Thus he discovered that the successions of two sounds that sound pleasant (or, to use a more technical term, consonant) could be achieved if the string is being shortened by 1:2, 2:3 or 3:4. Pythagoras could not be more satisfied with this finding, as it proved his claim that the universe is, on its hidden and true level, based upon an intelligible mathematical structure, and, moreover, that a key to understanding this structure is the Holy Tetractys, a figure representing the first four natural numbers in the form of a regular triangle. (shrink)

This article studies some mathematical concepts which may be ramified in cultural environments, particularly in music and literature. The notions of definability (What may we capture with language, what escapes us?), representation (Which archetypes may be projected onto the world, which types arise from such projections?) and categoricity (Which unitary concepts govern the multiplicity of languages and images?) are narrowly linked with the essential issue of the trans, perhaps the crucial problem of our fleeting times.

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)

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)