Symbolic Notation Research Articles

The history of the emergence of symbolic notation in mathematics in the 17th century and transformation of mathematical practice

The article analyzes the intellectual and cultural preconditions that made it possible to create symbolic notation in mathematics of the 17th century. It is shown that the reprints of Euclid’s Elements in the XVI century formed an intellectual and cultural background that allowed Viet and Stevens to transform the classical concept of arithmos into a symbolic concept of number, which made the creation of symbolic notation possible. It is argued that the development of symbolic notation is not a simple change in the way of notation, but significantly changed the mathematical practice, making it possible to introduce new objects and through symbolic transformations.

Design Rules, Volume 2: Chapter 6—The Value Structure of Technologies, Part 1: Mapping Functional Components

The purpose of this chapter is to build a robust and versatile language that is capable of representing the value structure of large technical systems. The language is based on elements I have labeled functional components. The language is more abstract than the language of technical recipes and task networks, thus it is capable of hiding details. However, the language also makes it possible to “trace back” from each named functional component to a technical recipe and a corresponding task network. Finally, although the language itself is non-mathematical, the value structure it reveals can be used to specify equations and prove mathematical propositions about the value of technical systems. The plan of the chapter as follows. I first explain why it is difficult to value technologies using standard economic methods based prices, quanities and probabilities. I then describe a methodology that shows how functional components are combined through technology to create a particular artifact or technical system. The methodology uses symbolic notation to clarify relationships between and among functional components. I illustrate these relationships using an ancient technology—the technology for making a garment from pieces of cloth. I go on to describe commonly observed patterns within technical systems, including optional features; composite functions; and platforms.

Integer number sense and notation: A case study of a student with a mathematics learning disability

Although approximately 6% of students have a mathematics learning disability (MLD) also known as dyscalculia, little is known about how MLD impacts students beyond basic arithmetic. In this study we focused on one mathematical topic foundational to algebra – integer operations – and conducted a videotaped design experiment with one student with MLD. Through 14 teaching episodes we explored the ways in which standard mathematical tools (e.g., symbols, representations) were inaccessible and evaluated the design of alternative tools. Our detailed retrospective analysis revealed that the student had an unconventional understanding of integer quantities and symbolic notation, which resulted in issues of accessibility and persistent difficulties. Deliberate attempts to address inaccessibility revealed nuances in the student’s understanding, and suggests that both number sense and notational issues needed to be addressed in tandem. Implications for instruction are discussed.

Myths and mythconceptions: what does it mean to be a programming language, anyhow?

Modern software does not stand alone; it is embedded in complex physical and sociotechnical systems. It relies on computational support from interdependent subsystems as well as non-code resources such as data, communications, sensors, and interactions with humans. Both general-purpose programming languages and mainstream programming language research focus on symbolic notations with well-defined abstractions that are intended for use by professionals to write programs that solve precisely specified problems. There is a strong emphasis on correctness of the resulting programs, preferably by formal reasoning. However, these languages, despite their careful design and formal foundations, address only a modest portion of modern software and only a minority of software developers. Several persistent myths reinforce this focus. These myths express an idealized model of software and software development. They provide a lens for examining modern software and software development practice: highly trained professionals are outnumbered by vernacular developers. Writing new code is dominated by composition of ill-specified software and non-software components. General-purpose languages may be less appropriate for a task than domain-specific languages, and functional correctness is often a less appropriate goal than overall fitness for task. Support for programming to meet a specification is of little help to people who are programming in order to understand their problems. Reasoning about software is challenged by uncertainty and nondeterminism in the execution environment and by the increasingly dominant role of data, especially with the advent of systems that rely on machine learning. The lens of our persistent myths illuminates the dissonance between our idealized view of software development and common practice, which enables us to identify emerging opportunities and challenges for programming language research.

Creating DALI, a Large Dataset of Synchronized Audio, Lyrics, and Notes

The DALI dataset is a large dataset of time-aligned symbolic vocal melody notations (notes) and lyrics at four levels of granularity. DALI contains 5358 songs in its first version and 7756 for the second one. In this article, we present the dataset, explain the developed tools to work the data and detail the approach used to build it. Our method is motivated by active learning and the teacher-student paradigm. We establish a loop whereby dataset creation and model learning interact, benefiting each other. We progressively improve our model using the collected data. At the same time, we correct and enhance the collected data every time we update the model. This process creates an improved DALI dataset after each iteration. Finally, we outline the errors still present in the dataset and propose solutions to global issues. We believe that DALI can encourage other researchers to explore the interaction between model learning and dataset creation, rather than regarding them as independent tasks.

Symbolic fractions elicit an analog magnitude representation in school-age children

A fundamental question about fractions is whether they are grounded in an abstract nonsymbolic magnitude code similar to that postulated for whole numbers. Mounting evidence suggests that symbolic fractions could be grounded in mechanisms for perceiving nonsymbolic ratio magnitudes. However, systematic examination of such mechanisms in children has been lacking. We asked second- and fifth-grade children (prior to and after formal instructions with fractions, respectively) to compare pairs of symbolic fractions, nonsymbolic ratios, and mixed symbolic–nonsymbolic pairs. This paradigm allowed us to test three key questions: (a) whether children show an analog magnitude code for rational numbers, (b) whether that code is compatible with mental representations of symbolic fractions, and (c) how formal education with fractions affects the symbolic–nonsymbolic relation. We examined distance effects as a marker of analog ratio magnitude processing and notation effects as a marker of converting across numerical codes. Second and fifth graders’ reaction times and error rates showed classic distance and notation effects. Nonsymbolic ratios were processed most efficiently, with mixed and symbolic notations being relatively slower. Children with more formal instruction in symbolic fractions had a significant advantage in comparing symbolic fractions but had a smaller advantage for nonsymbolic ratio stimuli. Supplemental analyses showed that second graders relied on numerator distance more than holistic distance and that fifth graders relied on holistic fraction magnitude distance more than numerator distance. These results suggest that children have a nonsymbolic ratio magnitude code and that symbolic fractions can be translated into that magnitude code.

The Role of Notations in Mathematics

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.

Governance of the water-energy-food nexus: insights from four infrastructure projects in the Lower Mekong Basin

The social relations and biophysical flows that link water, food, and energy systems are said to form a ‘nexus’. Efforts to steer or otherwise exert influence on decisions that impact upon these nexus links, including to ignore them, take place at multiple levels, vary in complexity, and have implications for who benefits and who is burdened by those relations and flows. This paper examines how nexus links have been governed, using four medium- to large-scale water infrastructure projects in Laos and Thailand as probes into problematic issues of coordination, anticipation, inclusion, and attribution. Project documents, media reports, and published analyses were coded to extract information about nexus links, narratives, and decisions. Nexus interactions were summarized using a novel symbolic notation and then classified along a scale of increasing structural complexity as pairs, chains, and loops. The key finding from the analysis of the four projects was that nexus governance was fragmented, reactive, exclusive, and opaque. Coordination among ministries was limited with inter-ministerial bodies, and integrated development plans ineffective at guiding project design or operation decisions in the presence of bureaucratic competition. Anticipation of cross-sectoral concerns was rare, despite scope to identify them early in feasibility studies, and assessment activities; instead they were only acknowledged after public pressure. Inclusion of the needs of vulnerable and affected groups was limited, although poverty alleviation, and other social benefits were a significant element in project justification narratives. Attribution of responsibility was difficult as many key decisions took place behind closed doors, while project information was withheld, raising further governance issues of transparency and accountability. Structural complexity in the nexus links made addressing governance problems even more challenging.

Inversion effects on mental arithmetic in English- and Polish-speaking adults

In some languages the order of tens and units in number words is inverted compared with the symbolic digital notation (e.g., German 23 → "dreiundzwanzig," literally: "three-and-twenty"). In other languages only teen-numbers are inverted (e.g., English 17 → "seventeen"; Polish 17 → "siedemnaście" literally "seventeen"). Previous studies have focused on between group comparisons of inverted and non-inverted languages and showed that number word inversion impairs performance on basic numerical tasks and arithmetic. In two independent experiments, we investigated whether number word inversion affects addition performance within otherwise non-inverted languages (Exp. 1: English, Exp. 2: Polish). In particular, we focused on the influence of inverted (I; English: teen-numbers ⩾ 13, Polish: numbers 11-19) and non-inverted (N) summands with sums between 13 and 39. Accordingly, three categories of addition problems were created: N + N, N + I, and I + I with problem size matched across categories. Across both language groups, we observed that problems with results in the 20 and 30 number range were responded to faster when only non-inverted summands were part of the problems as opposed to problems with one or two inverted summands. In line with this, the cost of a carry procedure was the largest for two inverted summands. The results support the notion that both language-specific and language-invariant aspects contribute to addition problem-solving. In particular though, regarding language-specific aspects, the results indicate that inverted number word formation of teens influences place-value processing of Arabic digits even in otherwise non-inverted languages.

Emblematic Mechanisms and Psychoanalysis

In the paper the parallels between the emblematic “mechanisms” of signification and the psychoanalytic theory of Sigmund Freud as well as Carl Gustav Jung have been studied. The Austrian psychiatrist has discovered template schemes that become a visual delineation, the blueprint for developing his scientific vocabulary, methodology, classification of psycho-emotional behavioral types in mythological plots. The Eros and Thanatos images handling, the exploitation of mythical tales about Oedipus and Electra, Prometheus, Narcissus, and many other ones to specify the behavioral complexes denote the presence of “emblematic methodology” in the formation of psychoanalytic conceptions and categories. His interpretations of famous mythological plots are boiled down to emblematic reduction.
 Carl Gustav Jung frequently selected symbolic notations as his research targets, which were a denotative space for expressing internal mental receptions and historic constellations of cultural axiology. In his writings we see the intention to assemble the concepts of image (iconic) and socio-cultural idea (conventional) into a sole compound that syncretically denote unity of meaning. Such an arrangement of iconic-conventional interdetermination is often significative elbowroom in Jung the decoding of which may allow to discern complex mental reflections. Notwithstanding the fact that he considers a symbol to be the standard unit of cognitive-cultural experience “conservation”, its functional semantics definition is fulfilled in emblematic patterns. This emblematic-cognitive form is not only a method of determining the initial images-ideas of the unconscious, “the mythological figures” of inner conflicts, typical experience of generations, but also the principle of justification and expression of his theory conceptual foundation. To a certain extent, it is an element of the Swiss psychologist’s scientific thinking style and language.

Forgotten Histories of the Audiobook

This article investigates the different affordances of magnetic tape and print as they are entextualized in various co(n)texts by writers, ethnographers, and musicians throughout the Americas in the late 1960s. I analyze printed books made from tape recordings—Cuban anthropologist Miguel Barnet and his interview subject Esteban Montejo’s Biografía de un cimarrón (Biography of a Runaway Slave, 1966), Rodolfo Walsh’s true-crime denunciation ¿Quién mató a Rosendo? (Who killed Rosendo?, 1968), and Andy Warhol’s experimental a: a novel (1968)—to ask why these writers transduced their recordings into print rather than release them as audiobooks, how or if listening to those tapes would alter the meaning of their printed entextualizations, and what musical interactions with the same media in the same contexts can tell us about the limits both of print and of symbolic musical notation. Tracing the intersection of musical and literary works, the article argues that a writerly ethics of distortion, rather than fidelity, arises from this mutual encounter with sound on tape, and ponders how dialogic audiobooks might contest older issues of power and representation for those writers, North and South, who worked in support of marginalized (Afro-Cuban, working class, and queer) subjects.

Symbolic Tensor Neural Networks for Digital Media – from Tensor Processing via BNF Graph Rules to CREAMS Applications

This tutorial material on Convolutional Neural Networks (CNN) and its applications in digital media research is based on the concept of Symbolic Tensor Neural Networks. The set of STNN expressions is specified in Backus-Naur Form (BNF) which is annotated by constraints typical for labeled acyclic directed graphs (DAG). The BNF induction begins from a collection of neural unit symbols with extra (up to five) decoration fields (including tensor depth and sharing fields). The inductive rules provide not only the general graph structure but also the specific shortcuts for residual blocks of units. A syntactic mechanism for network fragments modularization is introduced via user defined units and their instances. Moreover, the dual BNF rules are specified in order to generate the Dual Symbolic Tensor Neural Network (DSTNN). The joined interpretation of STNN and DSTNN provides the correct flow of gradient tensors, back propagated at the training stage. The proposed symbolic representation of CNNs is illustrated for six generic digital media applications (CREAMS): Compression, Recognition, Embedding, Annotation, 3D Modeling for human-computer interfacing, and data Security based on digital media objects. In order to make the CNN description and its gradient flow complete, for all presented applications, the symbolic representations of mathematically defined loss/gain functions and gradient flow equations for all used core units, are given. The tutorial is to convince the reader that STNN is not only a convenient symbolic notation for public presentations of CNN based solutions for CREAMS problems but also that it is a design blueprint with a potential for automatic generation of application source code.

Drawing for Assessing Learning Outcomes in Chemistry

Extant research on the efficacy of drawing for assessing learning in chemistry suggests it may provide richer information on the quality of students’ mental models. Here, we examine the relative utility of formative assessments that involve drawing modalities as well as verbal and symbolic modalities. We analyzed the drawings, written explanations, and symbolic notations made by 36 students learning about chemical reactivity from a dynamic visualization. Our analysis demonstrates that drawing affords the emphasis of different conceptual features across different representational modalities. Students represented spatiotemporal structures (i.e., particle composition and location) more frequently in their drawings but represented spatiotemporal transformations (i.e., particle motion and interaction) more frequently in their written explanations. Students represented more conceptual features in both their drawings and written explanations than they did in the symbolic notation commonly used in the chemical sciences. Our findings suggest that assessment items that involve drawing can help evaluators construct richer models of student knowledge, but they should not be used in isolation from verbal or symbolic modalities.

'MusiMath' and 'Academic Music' - Two music-based intervention programs for fractions learning in fourth grade students.

Music and mathematics require abstract thinking and using symbolic notations. Controversy exists regarding transfer from musical training to math achievements. The current study examined the effect of two integrated intervention programs representing holistic versus acoustic approaches, on fraction knowledge. Three classes of fourth graders attended 12 lessons on fractions: One class attended the ‘MusiMath’ holistic program (n = 30) focusing on rhythm within the melody. Another class attended the ‘Academic Music’ acoustic program (Courey et al., Educ Stud Math 81:251, 2012) (n = 25) which uses rhythm only. The third class received regular fraction lessons (comparison group, n = 22). Students in both music programs learned to write musical notes and perform rhythmic patterns through clapping and drumming as part of their fraction lessons. They worked toward adding musical notes to produce a number (fraction), and created addition/subtraction problems with musical notes. The music programs used a 4/4 time signature with whole, half, quarter and eighth notes. In the math lessons, the students learned the analogy between musical durations and 12,14,18 fractions, but also practiced fractions other than 12,14,18. Music and math were assessed before, immediately following, and 3‐ and 6‐months post‐intervention. Pre‐ to post‐intervention analyses indicated that only the ‘MusiMath’ group showed greater transfer to intervention‐trained and untrained fractions than the comparison group. The ‘Academic Music’ group showed a trend on trained fractions. Although both music groups outperformed the comparison group 3‐ and 6‐months post‐intervention on trained fractions, only the ‘MusiMath’ group demonstrated greater gains in untrained fractions. Gains were more evident in trained than in untrained fractions. A video abstract of this article can be viewed at https://youtu.be/uJ_KWWDO624

Embedding and Sustaining Inclusive Practice to Support Disabled Students in Online and Blended Learning

UK higher education data has shown persistent differences in degree outcomes for specific student groups. Consequently, the Office for Students (the UK government’s higher education regulator) are funding 17 projects to address these inequalities. Building on its expertise, our institution is leading the IncSTEM project alongside colleagues from two other universities, to evaluate, scale up and promote inclusive teaching and learning practice within Science, Technology, Engineering and Mathematics (STEM) disciplines in higher education.There are challenges with inclusive distance learning, many of which are emphasised in STEM through the prevalence of practical and field activities, the widespread use of groupwork, and the use of text that is rich in symbolic notation. Online and blended learning approaches, including access to digital learning resources, bring opportunities for more inclusive practice, but can also lead to unforeseen and unquantified barriers for students. Integrating an inclusive approach to teaching and learning requires universities to embed and sustain practices that consider the diverse needs of students throughout curriculum design and delivery, bringing benefits to all students.In this paper, we present data on staff perceptions and practices regarding accessibility and inclusion for disabled students, explore examples of inclusive practice, and discuss how these can be applied by practitioners in order to create a higher education environment in which students of all backgrounds and characteristics are able to succeed.

The concept of force in population dynamics

The area of population dynamics has a rich history of the development and analysis of models of biological and social phenomena using ordinary differential equations. This paper describes a method for understanding the influence one variable exerts on another in such models as a force, with the relative effects of these forces providing a narrative explanation of the curvature in variable behaviour. Using the stock/flow form of a model, a symbolic notation is developed that identifies the forces with the causal pathways of the model’s feedback loops. A force is measured by its impact, defined as the ratio of acceleration to rate of change, computed by differentiation along its associated pathway between variables. Different phases of force dominance are determined to enhance the standard stability analysis of the models, providing an explanation of model behaviour in Newtonian mechanical terms. The concepts developed are applied to well-known models from mathematical biology: the Spruce Budworm model, where force dominance identifies scenarios that give clarity to intervention points; and the Lotka–Volterra predator–prey model where the analysis highlights the importance of dissipative forces in achieving stability. Conclusions are drawn on the explanatory power of this approach, with suggestions made for future work.

Technical Notation as a Tool for Basic Research in Relational Frame Theory

A core overarching aim of Relational Frame Theory (RFT) research on language and cognition is the prediction and influence of human behavior with precision, scope, and depth. However, the conceptualization and delineation of empirical investigations of higher-order language and cognition from a relational framing theoretical standpoint is a challenging task that requires a high degree of abstract reasoning and creativity. To that end, we propose using symbolic notation as seen in early RFT experimental literature as a possible functional-analytical tool to aid in the articulation of hypotheses and design of such experiments. In this article, we provide examples of aspects of cognition previously identified in RFT literature and how they can be articulated rather more concisely using technical notation than in-text illustration. We then provide a brief demonstration of the utility of notation by offering examples of several novel experiments and hypotheses in notation format. In two tables, we provide a “key” for understanding the technical notation written herein, which other basic-science researchers may decide to draw on in future. To conclude, this article is intended to be a useful resource to those who wish to carry out basic RFT research on complex language and cognition with greater technical clarity, precision, and broad scope.

The Way They Did Theory Then

ABSTRACTIn this article, I take sixteenth-century mathematics as a test case for considering at its most general level what doing theory entails. Ideally, doing theory aims to change rather than merely describe reality, and in order to do so, it represents that reality differently. Such transformations, I argue, need special techniques of representation. In early modern mathematics, two techniques in particular enable new theoretical content: symbolic notation, and the real continuum. Together they create a language that can redescribe mathematical objects, for example, so that a single general formula can solve for all specific instances, and simple counting numbers can be reidentified in the real continuum as limits. I argue for the formative contribution of the imagination in such developments, that is, for an ability to reassemble givens of a problem in such a way as to solve it; and to concoct new objects or possibilities that do not exist in reality. In the sixteenth century, imaginary numbers (roots of negative values) most explicitly demonstrate the importance of speculating beyond what is real. As we consider what doing theory in the twenty-first century entails, it is worth remembering, in light of the article’s analysis, how form or technique itself has agency, and the ability to transform theoretical content.

Massively Parallel Simulations of Binary Black Hole Intermediate-Mass-Ratio Inspirals

We present a highly-scalable framework that targets problems of interest to the numerical relativity and broader astrophysics communities. This framework combines a parallel octree-refined adaptive mesh with a wavelet adaptive multiresolution and a physics module to solve the Einstein equations of general relativity in the BSSN formulation. The goal of this work is to perform advanced, massively parallel numerical simulations of Intermediate Mass Ratio Inspirals (IMRIs) of binary black holes with mass ratios on the order of 100:1. These studies will be used to generate waveforms as used in LIGO data analysis and to calibrate semi-analytical approximate methods. Our framework consists of a distributed memory octree-based adaptive meshing framework in conjunction with a node-local code generator. The code generator makes our code portable across different architectures. The equations corresponding to the target application are written in symbolic notation and generators for different architectures can be added independent of the application. Additionally, this symbolic interface also makes our code extensible, and as such has been designed to easily accommodate many existing algorithms in astrophysics for plasma dynamics and radiation hydrodynamics. Our adaptive meshing algorithms and data-structures have been optimized for modern architectures with deep memory hierarchies. This enables our framework to have achieve excellent performance and scalability on modern leadership architectures. We demonstrate excellent weak scalability up to 131K cores on ORNL's Titan for binary mergers for mass ratios up to 100.

Diagrams and alien ways of thinking

The recent wave of data on exoplanets lends support to METI ventures (Messaging to Extra-Terrestrial Intelligence), insofar as the more exoplanets we find, the more likely it is that “exominds” await our messages. Yet, despite these astronomical advances, there are presently no well-confirmed tests against which to check the design of interstellar messages. In the meantime, the best we can do is distance ourselves from terracentric assumptions. There is no reason, for example, to assume that all inferential abilities are language-like. With that in mind, I argue that logical reasoning does not have to be couched in symbolic notation. In diagrammatic reasoning, inferences are underwritten, not by rules, but by transformations of self-same qualitative signs. I use the Existential Graphs of C. S. Peirce to show this. Since diagrams are less dependent on convention and might even be generalized to cover non-visual senses, I argue that METI researchers should add some form of diagrammatic representations to their repertoire. Doing so can shed light, not just on alien minds, but on the deepest structures of reasoning itself.