Linguistic aspects of musical and mathematical notation

1992 ◽  
pp. 169 ◽  
Author(s):  
James D. McCawley
Keyword(s):  
Mathematical Notation

Enlightening Symbols

2016 ◽  
Author(s):  
Joseph Mazur
Keyword(s):  
Sixteenth Century ◽  
Number System ◽  
Psychological Effects ◽  
Mathematical Notation ◽  
Rhetorical Style ◽  
The Past ◽  
Mathematical Symbols ◽  
Before And After ◽  
New Ideas ◽  
Different Cultures

While all of us regularly use basic mathematical symbols such as those for plus, minus, and equals, few of us know that many of these symbols weren't available before the sixteenth century. What did mathematicians rely on for their work before then? And how did mathematical notations evolve into what we know today? This book explains the fascinating history behind the development of our mathematical notation system. It shows how symbols were used initially, how one symbol replaced another over time, and how written math was conveyed before and after symbols became widely adopted. Traversing mathematical history and the foundations of numerals in different cultures, the book looks at how historians have disagreed over the origins of the number system for the past two centuries. It follows the transfigurations of algebra from a rhetorical style to a symbolic one, demonstrating that most algebra before the sixteenth century was written in prose or in verse employing the written names of numerals. It also investigates the subconscious and psychological effects that mathematical symbols have had on mathematical thought, moods, meaning, communication, and comprehension. It considers how these symbols influence us (through similarity, association, identity, resemblance, and repeated imagery), how they lead to new ideas by subconscious associations, how they make connections between experience and the unknown, and how they contribute to the communication of basic mathematics. From words to abbreviations to symbols, this book shows how math evolved to the familiar forms we use today.

Curious Beginnings

2016 ◽  
Author(s):  
Joseph Mazur
Keyword(s):  
Historical Event ◽  
Mathematical Notation ◽  
Human Language ◽  
Mathematical Writing ◽  
Finite Set ◽  
And Algebra ◽  
Simple Equations ◽  
Infinite Set

This chapter traces the beginnings of mathematical notation. For tens of thousands of years, humans had been leaving signification marks in their surroundings, gouges on trees, footprints in hard mud, scratches in skin, and even pigments on rocks. A simple mark can represent a thought, indicate a plan, or record a historical event. Yet the most significant thing about human language and writing is that speakers and writers can produce a virtually infinite set of sounds, declarations, notions, and ideas from a finite set of marks and characters. The chapter discusses the emergence of the alphabet, counting, and mathematical writing. It also considers the discovery of traces of Sumerian number writing on clay tablets in caves from Europe to Asia, the use of Egyptian hieroglyphics, and algebra problems in the Rhind (or Ahmes) papyrus that presented simple equations without any symbols other than those used to indicate numbers.

Fractal Geometry as a Bridge between Realms

2013 ◽  
pp. 25-43 ◽  
Author(s):  
Terry Marks-Tarlow
Keyword(s):  
Fractal Geometry ◽  
Open Systems ◽  
Autonomous Systems ◽  
Mathematical Notation ◽  
Deep Space ◽  
The Unconscious ◽  
Self And Other ◽  
Branching Structures ◽  
Self Similar ◽  
Multidimensional Objects

This chapter describes fractal geometry as a bridge between the imaginary and the real, mind and matter, conscious and the unconscious. Fractals are multidimensional objects with self-similar detail across size and/or time scales. Jung conceived of number as the most primitive archetype of order, serving to link observers with the observed. Whereas Jung focused upon natural numbers as the foundation for order that is already conscious in the observer, I offer up the fractal geometry as the underpinnings for a dynamic unconscious destined never to become fully conscious. Throughout nature, fractals model the complex, recursively branching structures of self-organizing systems. When they serve at the edges of open systems, fractal boundaries articulate a paradoxical zone that simultaneously separates as it connects. When modeled by Spencer-Brown’s mathematical notation, full interpenetration between inside and outside edges translates to a distinction that leads to no distinction. By occupying the infinitely deep “space between” dimensions and levels of existence, fractal boundaries contribute to the notion of intersubjectivity, where self and other become most entwined. They also exemplify reentry dynamics of Varela’s autonomous systems, plus Hofstadter’s ever-elusive “tangled hierarchy” between brain and mind.

Developing Covariational Reasoning Among Students Using Contexts of Formulas

2020 ◽  
Vol 02 (04) ◽  
pp. 2050016
Author(s):  
Andrzej Sokolowski
Keyword(s):  
Mathematical Knowledge ◽  
First Year ◽  
Mathematical Notation ◽  
College Level ◽  
Physics Students ◽  
First Year College ◽  
Covariational Reasoning ◽  
Standard Formula ◽  
Physically Identical

Multiple studies have been conducted to assess students’ ability to apply covariational reasoning to sketching graphs in physics. This study is supported by research on developing students’ skills in sketching functions in mathematics. It attempts to evaluate physics students’ ability to apply these skills to identify critical algebraic attributes of physics formulas for their potential to be sketched. Rather than seeking formulas’ physical interpretation, this study is posited to challenge students’ skills to merge their mathematical knowledge within physics structures. A group of thirty ([Formula: see text]) first-year college-level physics students were provided with two physically identical equations that described the object’s position. However, one equation was expressed in functional mathematical notation, whereas the other in a standard formula notation. The students were asked to classify the symbols in each formula as variables or parameters and determine these formulas’ potential to be graphed in respective coordinates. The analysis revealed that 93% of these students considered function notation as possessing sketchable potential against 13% who envisioned such potential in the standard formula notation. Further investigations demystified students’ confusion about the classification of the symbols used in the formula notation. These results opened up a gate for discussing the effects of algebraic notations in physics on activating students’ covariational skills gained in mathematics courses. Suggestions for improving physics instructions stemming from this study are discussed.

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