Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of _which number_ is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of _notation_. The purpose of this article is to explore the relationship between (...) the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects. (shrink)

Notations are cognitive systems involving distinctive psychological functions, behaviors, and material forms. Seen through this lens, two main types – semasiography and visible language – are fundamentally differentiated by their material prehistories, emphasis on iconography, and the centrality of language's combinatorial faculty. These fundamental differences suggest that key qualities (iconicity, expressiveness, concision) are difficult to conjoin in a single system.

Die Theorie der Notation wurde von Nelson Goodman ([Goodman 1968]: Kap. 4) im Zusam­menhang mit der Frage nach den Iden­titäts­krite­rien für Kunstwer­ke ent­wickelt. Eine Nota­tion ist ein Zeichen­system, das ein syntak­tisches oder seman­tisches Krite­rium dafür ermög­licht, welche Gegen­stände oder Ereig­nisse Einzel­fälle eines bestimm­ten Werks sind. Ein solches Krite­rium ist dann notwen­dig, wenn Werke mehre­re Einzel­fälle zulas­sen, deren Iden­tität nicht durch ihre Entste­hungsge­schichte bestimmt ist. Da dies in para­digma­tischer Weise in der Musik der Fall ist, führe ich den Begriff der (...) Nota­tion für die Musik ein, ehe ich mich kurz der Lite­ratur und ausführ­licher den bilden­den Künsten zuwen­de. Die resul­tieren­den Iden­titäts­krite­rien sollen auch für Musik, Texte und Bilder ohne Kunstwerk­status gelten, auch wenn wir hier in der Regel nicht von Werken sprechen. (shrink)

Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship (...) between the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects. (shrink)

Mathematical notation includes a vast array of signs. Most mathematical signs appear to be symbolic, in the sense that their meaning is arbitrarily related to their visual appearance. We explored the hypothesis that mathematical signs with iconic aspects—those which visually resemble in some way the concepts they represent—offer a cognitive advantage over those which are purely symbolic. An early formulation of this hypothesis was made by Christine Ladd in 1883 who suggested that symmetrical signs should be used to convey (...) commutative relations, because they visually resemble the mathematical concept they represent. Two controlled experiments provide the first empirical test of, and evidence for, Ladd's hypothesis. In Experiment 1 we find that participants are more likely to attribute commutativity to operations denoted by symmetric signs. In Experiment 2 we further show that using symmetric signs as notation for commutative operations can increase mathematical performance. (shrink)

The main goal of this essay is to propose and make plausible a framework for developing a philosophical account of musical notation. The proposed framework countenances four elements of notation: symbols (abstract objects that collectively constitute the backbone of a ‘system’ of notation), their characteristic ‘forms’ (for example, shapes, understood abstractly), the concrete instances, or ‘engravings’, of those forms, and the meanings of the symbols. It is argued that these elements are distinct. Along the way, several preliminary (...) arguments are given for how one ought to understand them—for example, it is suggested that engravings represent symbols rather than instantiate forms, although they are characteristically seen to represent a symbol by being seen to instantiate an associated form. Having proposed this framework, the essay explores the nature of musical instructions, as the meanings of symbols, and offers an argument in favor of the commonly held (but recently challenged) view that those meanings are imperative. Specifically, composites of musical notation (paradigmatically, musical scores) primarily express instructional meaning, and denote something like ‘sonic structures’ only secondarily, in virtue of their primary, imperative, meaning. (shrink)

What does it take for two logics to be mere notational variants? The present paper proposes a variety of different ways of cashing out notational variance, in particular isolating a constraint on any reasonable account of notational variance which makes plausible that the only kinds of translations which can witness notational variance are what are sometimes called definitional translations.

Expressively equivalent logical languages can enunciate logical notions in notationally diversified ways. Frege’s Begriffsschrift, Peirce’s Existential Graphs, and the notations presented by Wittgenstein in the Tractatus all express the sentential fragment of classical logic, each in its own way. In what sense do expressively equivalent notations differ? According to recent interpretations, Begriffsschrift and Existential Graphs differ from other logical notations because they are capable of “multiple readings.” We refute this interpretation by showing that there are at least three different kinds (...) of such multiple readings. While readings of the first kind do not capture any essential difference among notations but only among vocabularies, corresponding to readings of the second and the third kind two general parameters according to which notations may differ are defined: linearity vs. non-linearity, and tabularity vs. non-tabularity. This answers the question of how there can be substantially different but expressively equivalent logical notations. (shrink)

One of the central debates among British chemists during the 1830s concerned the use of symbols to represent elements and compounds. Chemists such as Edward Turner, who desired to use symbolic notation mainly for practical reasons, eventually succeeded in fending off metaphysical objections to their approach. These objections were voiced both by the philosopher William Whewell, who wished to subordinate the chemists' practical aims to the rigid standard of algebra, and by John Dalton, whose hidebound opposition to abbreviated (...) class='Hi'>notation symbolized the suspicion with which older British chemists perceived continental innovations. It is argued that the success of chemists like Turner in this debate reflects their larger success in the 1830s in achieving disciplinary autonomy and in beginning to align themselves more closely with prevailing chemical practice across the Channel. (shrink)

In recent years, dependency grammars have established themselves as valuable tools in theoretical and computational linguistics. To many linguists, dependency grammars and the more standard constituency-based formalisms are notational variants. We argue that, beyond considerations of formal equivalence, cognition may also serve as a background for a genuine comparison between these different views of syntax. In this paper, we review and evaluate some of the most common arguments and evidence employed to advocate for the cognitive or neural reality of dependency (...) grammars in linguistics, psycholinguistics, or neurolinguistics. We then raise the possibility that the abilities to represent and track, alternatively or in parallel, constituency and dependency structures co-exist in human cognition and are constitutive of syntactic competence. (shrink)

Multicultural environments require learning multiple number notations wherein some are encountered more frequently than others. This leads to differences in exposure and consequently differences in usage between notations. We find that differential notational usage imposes a significant neurocognitive load on number processing. Despite simultaneous acquisition, forty-two adult binumerate populations, familiar with two positional writing systems namely Hindu Nagari digits and Hindu Arabic digits, reported significantly lower preference and usage for Nagari as compared to Arabic. Twenty-four participants showed significantly increased reaction (...) times and reduced accuracy while performing magnitude comparison tasks in Nagari with respect to Arabic. Functional magnetic resonance imaging revealed that processing Nagari elicited significantly greater activity in number processing and attention networks. A direct subtraction of networks for Nagari and Arabic notations revealed a neural circuit comprising of bilateral intra-parietal sulcus, inferior and mid frontal gyri, fusiform gyrus and the anterior cingulate cortex (FDR p<0.005). Additionally, whole brain correlation analysis showed that activity in the left inferior parietal region was modulated by task performance in Nagari. We attribute the increased activation in Ng to increased task difficulty due to infrequent exposure and usage. Our results reiterate the role of the left intra-parietal sulcus in modulating performance in numeric tasks and highlight that of the attention network for monitoring symbolic notation mode in binumerates. (shrink)

This article establishes itself first in a kind of slough, a lack of inspiration, and transvalues this via Fred Wah’s poem “Ikebana” and Roland Barthes’s celebration of haiku as a form that “lacks inspiration.” Following Barthes on “the minimal act of writing that is Notation,” this article explores and theorizes the status of the notational in and for poetics. The article registers and sustains the ambiguity in notatio, notationis and suggests that the notational points to a conceptual dialectic between (...) condensation and dilation. Poets Tonya Foster and W. S. Graham offer key cases, as do the author’s own poems. The article moves from haikus and other short forms (including the imagist poem) to haibuns (a mixed form of verse and prose) to questions of accent as a mode of differentiation from the surround. Drawing on Claudia Rankine and Jacques Rancière (as well as on classical rhetoric), the article also shows how the notational as censure might register antagonisms and not only accents or gestures (as notational forms are typically glossed). Alert to the enmeshment of notational forms with other media—such as the novel and photography—the article suggests in its final turn that a notational poetics can also vivify the (sometimes latent) diegetic aspect of short, minimal, “uninspired,” supposedly “immediate” notational forms and practices, as we see in the dance of notation and annotation, or in the unfolding of haiku within mixed forms such as the haibun. (shrink)

This paper presents a diagrammatic notation for visualizing epistemic entities and relations. The notation was created during the Visualizing Worldviews project funded by the University of Toronto’s Jackman Humanities Institute and has been further developed by the scholars participating in the university’s Research Opportunity Program. Since any systematic diagrammatic notation should be based on a solid ontology of the respective domain, we first outline the current state of the scientonomic ontology. We then proceed to providing diagrammatic tools (...) for visualizing the epistemic entities and relations of this ontology. These basic diagramming techniques allow us to construct diagrams of various types for both synchronic and diachronic visualizations. The paper concludes by highlighting some future research directions. As the notation presented here is de facto accepted and used in scientonomy, the paper suggests no modifications. (shrink)

In this article we investigate whether the following conjecture is true or not: does the addition-free theta functions form a canonical notation system for the linear versions of Friedman’s well-partial-orders with the so-called gap-condition over a finite set of n labels. Rather surprisingly, we can show this is the case for two labels, but not for more than two labels. To this end, we determine the order type of the notation systems for addition-free theta functions in terms of (...) ordinals less than ε0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _0$$\end{document}. We further show that the maximal order type of the Friedman ordering can be obtained by a certain ordinal notation system which is based on specific binary theta functions. (shrink)

The Turing degrees of infinite levels of the Ershov hierarchy were studied by Liu and Peng [8]. In this paper, we continue the study of Turing degrees of infinite levels and lift the study of density property to the levels beyond ω2. In doing so, we rely on notations with some nice properties. We introduce the concept of normalizing notations and generate normalizing notations for higher levels. The generalizations of the weak density theorem and the nondensity theorem are proved for (...) higher levels in the Ershov hierarchy. Furthermore, we also investigate the minimal degrees in the infinite levels of the Ershov hierarchy. (shrink)

Ontological pluralism is the view that there are different ways to exist. It is a position with deep roots in the history of philosophy, and in which there has been a recent resurgence of interest. In contemporary presentations, it is stated in terms of fundamental languages: as the view that such languages contain more than one quantifier. For example, one ranging over abstract objects, and another over concrete ones. A natural worry, however, is that the languages proposed by the pluralist (...) are mere notational variants of those proposed by the monist, in which case the debate between the two positions would not seem to be substantive. Jason Turner has given an ingenious response to this worry, in terms of a principle that he calls ‘logical realism’. This paper offers a counter-response on behalf of the ‘notationalist’. I argue that, properly applied, the principle of logical realism is no threat to the claim that the languages in question are notational variants. Indeed, there seems to be every reason to think that they are. (shrink)

It is argued that standard music notation pictures Aristotle’s time (time, as Aristotle conceived of it) in a number of important respects, which concern its micro-structure. It is then argued that this allows us to see some features of Aristotle’s time more clearly. Most importantly, Aristotelian instants can be pictured by bar-lines. This allows us to see as how radically devoid of any content Aristotelian instants should be interpreted. Thus, attention to music notation may show why Aristotle was (...) not a presentist. (shrink)

This volume contains English translations of Frege's early writings in logic and philosophy and of relevant reviews by other leading logicians. Professor Bynum has contributed a biographical essay, introduction, and extensive bibliography.

In his Languages of Art, Nelson Goodman proposes a theory of artistic notation that includes foundational requirements for any system of symbols we might use to specify and communicate the features of an artwork, in architecture or any other art form. Goodmans' theory usefully explains how notation can reveal linguistic-like phenomena of various art forms. But not all art forms can enjoy benefits of a full-blown notational system, in Goodman's view, and he suggests that architecture's symbol systems fall (...) short in this regard. It is a shortcoming of architecture, he believes, that its notation cannot communicate the sum of a given work's essential features. Against this view it may be argued that artworks are generally and inimically historical in character, such that an inability to capture this dimension may result in a failure to pick out the identity of a work. As a consequence, the suggestion of that inability looks like a shortcoming of the general notational theory rather than of the art form or its notation per se. I defend Goodman's background formalism against the criticism that an ahistoricist notation cannot possibly be adequate. But I reject his view that no actual architectural notation can satisfy the formal criteria of his theory. In particular, I propose that the foundational theory for Computer Aided Design (CAD) described by the architect William Mitchell satisfies Goodman's criteria and so yields a notation that enables communication of an architectural work's essential features. If, as I suggest, such a notation is feasible, then we have an additional result that Goodman foresees: a means of abstracting future architectural works from their historical contexts. This in turn yields a consequence at which Goodman only hints-that such architectural works can be intellectually grasped and physically constructed along purely formalist lines. One practical result here is that we introduce economical solutions to architectural design. On a theoretical level, we also expose the inessential character of historical properties to creating or understanding architecture, at least with respect to future possible works. (shrink)

Spekkens has introduced a toy theory (Spekkens in Phys. Rev. A 75(3):032110, 2007) in order to argue for an epistemic view of quantum states. I describe a notation for the theory (excluding certain joint measurements) which makes its similarities and differences with the quantum mechanics of stabilizer states clear. Given an application of the qubit stabilizer formalism, it is often entirely straightforward to construct an analogous application of the notation to the toy theory. This assists calculations within the (...) toy theory, for example of the number of possible states and transformations, and enables superpositions to be defined for composite systems. (shrink)

A linear notation for Charles S. Peirce's alpha and beta diagrammatic systems of existential graphs is presented. These two systems are equivalent to propositional and first-order logic. Some differences between the linear and graphical notation are analyzed, revealing some of the strengths and weaknesses of Peirce's system.

This paper argues that many so-called digital technologies can be construed as notational technologies, explored through the example of Monegraph, an art and digital asset management platform built on top of the blockchain system originally developed for the cryptocurrency bitcoin. As the paper characterizes it, a notational technology is the performance of syntactic notation within a field of reference, a technologized version of what Nelson Goodman called a “notational system.” Notational technologies produce abstracted entities through positive and reliable, or (...) constitutive, tests of socially acceptable meaning. Accordingly, this account deviates from typical narratives of blockchains, instead demonstrating that blockchain technologies are effective at managing digital assets because they produce abstracted identities through the performance of notation. Since notational technologies rely on configurations of socially acceptable meaning, this paper also provides a philosophical account of how blockchain technologies are socially embedded. (shrink)

The paper builds upon a growing body of critical research on the proliferating use of bibliometrics as a means to evaluate academic research, but brings to it a specifically Foucauldian, genealogical approach. The paper has three parts. Part 1 situates bibliometrics as a new technology of neoliberal, biopolitical governmentality, alongside the host of other ‘metrics’ that have emerged in the last two decades. Part 2 analyses bibliometrics’ antecedents in prior notational practices in the Western heritage, highlighting how forms of noting (...) have almost always had political valences tied to projects of control or subversion. Part 3 then delineates the specific features of bibliometrics as a new form of notation, highlighting the latest forms of academic subjectivity bibliometrics suppose and increasingly are summoning into being. (shrink)

G. Jäger gave in Arch. Math. Logik Grundlagenforsch. 24 , 49-62, a recursive notation system on a basis of a hierarchy Iαß of α-inaccessible regular ordinals using collapsing functions following W. Buchholz in Ann. Pure Appl. Logic 32 , 195-207. Jäger's system stops, when ordinals α with Iα0 = α enter. This border is now overcome by introducing additional a hierarchy Jαß of weakly inaccessible Mahlo numbers, which is defined similarly to the Jäger hierarchy. An ordinal μ is called (...) Mahlo, if every normal-function f : μ → μ has regular fixpoints. Collapsing is defined for both Mahlo and simply regular ordinals such that for every Mahlo ordinal μ out of the J-hierarchy Ψμα is a regular σ such that Iσ0 = σ. For these regular σ again collapsing functions Ψσ are defined. To get a proper systematical order into the collapsing procedure, a pair of ordinals is associated to σ and α, and the definition of Ψσα is given by recursion on a suitable well-ordering of these pairs. Thus a fairly large system of ordinal notations can be established. It seems rather straightforward, how to extend this setting further. (shrink)

The terms of a mathematical problem become precise and concise if they are expressed in an appropriate notation, therefore notations are useful to mathematics. But are notations only useful, or also essential? According to prevailing view, they are not essential. Contrary to this view, this paper argues that notations are essential to mathematics, because they may play a crucial role in mathematical discovery. Specifically, since notations may consist of symbolic notations, diagrammatic notations, or a mix of symbolic and diagrammatic (...) notations, notations may play a crucial role in mathematical discovery in different ways. Some examples are given to illustrate these ways. (shrink)

Buchholz presented a method to build notation systems for infinite sequent-style derivations, analogous to well-known systems of notation for ordinals. The essential feature is that from a notation one can read off by a primitive recursive function its n th predecessor and, e.g. the last rule applied. Here we extend the method to the more general setting of infinite terms, in order to make it applicable in other proof-theoretic contexts as well as in recursion theory. As examples, (...) we use the method to 1. give a new proof of a well-known trade-off theorem , which says that detours through higher types can be eliminated by the use of transfinite recursion along higher ordinals, and 2. construct a continuous normalization operator with an explicit modulus of continuity. (shrink)

Well over a century after its introduction, Frege's two-dimensional Begriffsschrift notation is still considered mainly a curiosity that stands out more for its clumsiness than anything else. This paper focuses mainly on the propositional fragment of the Begriffsschrift, because it embodies the characteristic features that distinguish it from other expressively equivalent notations. In the first part, I argue for the perspicuity and readability of the Begriffsschrift by discussing several idiosyncrasies of the notation, which allow an easy conversion of (...) logically equivalent formulas, and presenting the notation's close connection to syntax trees. In the second part, Frege's considerations regarding the design principles underlying the Begriffsschrift are presented. Frege was quite explicit about these in his replies to early criticisms and unfavorable comparisons with Boole's notation for propositional logic. This discussion reveals that the Begriffsschrift is in fact a well thought-out and carefully crafted notation that intentionally exploits the possibilities afforded by the two-dimensional medium of writing like none other. (shrink)

The pragmatic notion of assertion has an important inferential role in logic. There are also many notational forms to express assertions in logical systems. This paper reviews, compares and analyses languages with signs for assertions, including explicit signs such as Frege’s and Dalla Pozza’s logical systems and implicit signs with no specific sign for assertion, such as Peirce’s algebraic and graphical logics and the recent modification of the latter termed Assertive Graphs. We identify and discuss the main ‘points’ of these (...) notations on the logical representation of assertions, and evaluate their systems from the perspective of the philosophy of logical notations. Pragmatic assertions turn out to be useful in providing intended interpretations of a variety of logical systems. (shrink)

This volume contains English translations of Frege's early writings in logic and philosophy and of relevant reviews by other leading logicians. Professor Bynum has contributed a biographical essay, introduction, and extensive bibliography.

This chapter contains sections titled: Introduction A Methodology for Ontology The Need for a Criterion of Ontological Commitment The Role of a Canonical Notation The Ontology of Mathematics The Notion of Existence The Ontology of Natural Sciences Do Intensions Belong to the Furniture of the World? How to Treat Intensional Contexts without Positing Intensions Fiction, Intentional Objects and Existence Lesniewski's Ontology Acknowledgments.

This paper investigates provability and non-provability of well-foundedness of ordinal notations in weak theories of bounded arithmetic. We define a notion of well-foundedness on bounded domains. We show that T21 and S22 can prove the well-foundedness on bounded domains of the ordinal notations below 0 and Γ0. As a corollary, the class of polynomial local search problems, PLS, can be augmented with cost functions that take ordinal values below 0 and Γ0 without increasing the class PLS.

This paper investigates provability and non-provability of well-foundedness of ordinal notations in weak theories of bounded arithmetic. We define a notion of well-foundedness on bounded domains. We show that T21 and S22 can prove the well-foundedness on bounded domains of the ordinal notations below 0 and Γ0. As a corollary, the class of polynomial local search problems, PLS, can be augmented with cost functions that take ordinal values below 0 and Γ0 without increasing the class PLS.

The emotional expression of laughter in opera scores and performance was evaluated by converting notation to temporal data and contrasting it with the conversational laughter it emulates. The potency of scored and sung laughter was assayed by its ability to trigger contagion in audiences.

In the study of modal and nonclassical logics, translations have frequently been employed as a way of measuring the inferential capabilities of a logic. It is sometimes claimed that two logics are “notational variants” if they are translationally equivalent. However, we will show that this cannot be quite right, since first-order logic and propositional logic are translationally equivalent. Others have claimed that for two logics to be notational variants, they must at least be compositionally intertranslatable. The definition of compositionality these (...) accounts use, however, is too strong, as the standard translation from modal logic to first-order logic is not compositional in this sense. In light of this, we will explore a weaker version of this notion that we will call schematicity and show that there is no schematic translation either from first-order logic to propositional logic or from intuitionistic logic to classical logic. (shrink)

Peano was one of the driving forces behind the development of the current mathematical formalism. In this paper, we study his particular approach to notational design and present some original features of his notations. To explain the motivations underlying Peano's approach, we first present his view of logic as a method of analysis and his desire for a rigorous and concise symbolism to represent mathematical ideas. On the basis of both his practice and his explicit reflections on notations, we discuss (...) the principles that guided Peano's introduction of new symbols, the choice of characters, and the layout of formulas. Finally, we take a closer look, from a systematic and historical perspective, at one of Peano's most striking innovations, his use of dots for the grouping of subformulas. (shrink)

As is well known, it was only in 1926 that a comprehensive mathematical theory of braids was published—that of Emil Artin. That said, braids had been researched mathematically before Artin’s treatment: Alexandre Theophile Vandermonde, Carl Friedrich Gauß and Peter Guthrie Tait had all attempted to introduce notations for braids. Nevertheless, it was only Artin’s approach that proved to be successful. Though the historical reasons for the success of Artin’s approach are known, a question arises as to whether other approaches to (...) deal with braids existed, approaches that were developed after Artin’s article and were essentially different from his approach. The answer, as will be shown, is positive: Modesto Dedò developed in 1950 another notation for braids, though one, which was afterward forgotten or ignored. This raises a more general question: what was the role of Artin’s notation, or, respectively, Dedò’s, that enabled either the acceptance or the neglect of their theories? More philosophically, can notation be an epistemic technique, prompting new discoveries, or rather, can it also operate an as obstacle? The paper will analyze the method introduced by Dedò to notate braids, and also its history and implications. It aims to show that Dedò, in contrast to Artin, focused on factorizations of braids and the algebraic relations between the operations done on these factorizations. Dedò’s research was done against the background of Oscar Chisini’s research of algebraic curves on the one hand and of Artin’s successful notation of braids on the other hand. Taking this into account, the paper will in addition look into the epistemic role of notation, comparing Dedò’s work with Artin’s, as both presented different notations of braids and their deformations. (shrink)

The author of the Moretum, who is quite unlikely to be identified with Virgil himself, designed a clear winter setting as the background to his narration. An indepth scrutiny of winter images and topics dispersed throughout the poem (astronomical opening, repeated allusions to the winter fire, description of the farmer’s occupations) shows that the poet was a careful reader of the Georgics and borrowed from Virgil’s passages related to the winter season. The crafting and tone of several verses confirm the (...) witty and parodic intentions of the author. (shrink)

The philosophical underpinnings of this article are the Peircian notion of the triadic nature of the sign as iconic, linguistic and indexical, and the use of the sign as a ‘Zeug’ or thing as a means of pointing to or deixis in the context of creative activity in the classroom. This involves Lyotard's conception of desire as the generation of a space where the pupil can be affected by what the world donates. Both deixis and desire take on added value (...) in relation to the psychoanalytic theories of Winnicott in terms of the role of transitional object use in the generation of ‘potential space’, and Nussbaum's notion of tyche or being on the lookout for what the world has to offer. In terms of education, the central section of the article looks at the way these notions can be applied to teaching and learning in the secondary art classroom. This involves the use of what I have termed transitional notation, combining all three Peircian sign modes on a surface of inscription that is ‘ready-to-hand and that operates as what the anthropologist Alfred Gell termed an ‘index of agency’. (shrink)

Music is said to consist of melody, rhythm, and harmony. Sound is assumed to be something that automatically follows once musical structure is determined. Sound, which is what actually impinges on our eardrums, has been so long forgotten in the history of musical theory. It is ironic that we do not talk about the music which we hear every day but rather are exclusively concerned about the abstracted structure behind it. This is a legacy of ancient Greek ideas about music, (...) which most contemporary musical discourses are based on. Under this tradition, imperceptible music has priority over perceptible music in reality. On the contrary, music was explained in the framework of sound, tone, and music in ancient China. Although different thinkers tried to define music in a variety of ways, sound was never completely expelled from their musical theorizations. On one hand, music was regarded as a continuum of sound and tone, namely, perceptible music and intelligible music in its structure. On the other hand, music was considered as something that could not be reduced to its structure or notation but could be associated with much broader contexts. This article explores two different traditions in ancient Greece and China in terms of their musical ontology. By answering the question, “What is music as such?” differently, they developed different views on issues such as the relationship of music and emotion, the role of music in society, the symbolization of music, and so on. (shrink)