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2022 | Buch

Motivation and Background Kapitel lesen Erstes Kapitel lesen

Functorial Semiotics for Creativity in Music and Mathematics

verfasst von: Guerino Mazzola, Sangeeta Dey, Zilu Chen, Yan Pang

Verlag: Springer International Publishing

Buchreihe : Computational Music Science

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book presents a new semiotic theory based upon category theory and applying to a classification of creativity in music and mathematics. It is the first functorial approach to mathematical semiotics that can be applied to AI implementations for creativity by using topos theory and its applications to music theory.

Of particular interest is the generalized Yoneda embedding in the bidual of the category of categories (Lawvere) - parametrizing semiotic units - enabling a Čech cohomology of manifolds of semiotic entities. It opens up a conceptual mathematics as initiated by Grothendieck and Galois and allows a precise description of musical and mathematical creativity, including a classification thereof in three types. This approach is new, as it connects topos theory, semiotics, creativity theory, and AI objectives for a missing link to HI (Human Intelligence).

The reader can apply creativity research using our classification, cohomology theory, generalized Yoneda embedding, and Java implementation of the presented functorial display of semiotics, especially generalizing the Hjelmslev architecture. The intended audience are academic, industrial, and artistic researchers in creativity.

Inhaltsverzeichnis

Frontmatter

Orientation

Frontmatter
1. Motivation and Background
Summary
We present the motivation for the development of this functorial semiotics as a bridge between Human Intelligence and Artificial Intelligence.
Guerino Mazzola, Sangeeta Dey, Zilu Chen, Yan Pang

General Concepts

Frontmatter
2. Semiotics
Summary
This chapter gives an overview of semiotics as developed by Charles Sanders Peirce, Ferdinand de Saussure, Louis Hjelmslev, and Roland Barthes.
Guerino Mazzola, Sangeeta Dey, Zilu Chen, Yan Pang
3. Functorial Semantics Category
Summary
This chapter presents the category of functorial semantics, which formalizes the semiotic objects together with the connecting morphisms. These objects are called H-jets, “H” standing for Hjelmslev, who introduced the vertical dimension in a comprehensive semiotic.
Guerino Mazzola, Sangeeta Dey, Zilu Chen, Yan Pang
4. Examples
Summary
This chapter presents a number of essential examples of H-jets: pointers, sets,
Guerino Mazzola, Sangeeta Dey, Zilu Chen, Yan Pang
5. Semantic and Expressive Topology
Summary
This chapter presents a (classical) topology on H-jet collections that relates to semantic aspects. Dually, we shall introduce an expressive topology.
Guerino Mazzola, Sangeeta Dey, Zilu Chen, Yan Pang

Semantic Math

Frontmatter
6. Concept Mathematics
Summary
Part III deals with the very definition of conceptual mathematics and then a discussion of first attempts to generate a systematic approach to conceptual mathematics.
Guerino Mazzola, Sangeeta Dey, Zilu Chen, Yan Pang
7. Yoneda
Summary
We discuss some global consequences of Yoneda’s Lemma and its philosophy.
Guerino Mazzola, Sangeeta Dey, Zilu Chen, Yan Pang
8. Semantic Representations
Summary
This chapter opens the question about the semantic “loading” of a mathematical concept. This representation relates to the categories of H-jets, where concept are conceived as “sources” of a semantic extension or, dually, as “sinks” of an expressive extension.
Guerino Mazzola, Sangeeta Dey, Zilu Chen, Yan Pang
9. Cech Cohomology
Summary
We discuss two approaches to Cech cohomology: function spaces for global filters, and functorial cohomology associated with the semantic topology.
Guerino Mazzola, Sangeeta Dey, Zilu Chen, Yan Pang
10. Semiotic Classification of Creative Strategies
Summary
The following approach is a semiotic one, it presents creativity as an extension of a given semiotic system .We shall use the functorial semiotic theory developed in the previous chapters to unfold this idea in a mathematically explicit way.
Guerino Mazzola, Sangeeta Dey, Zilu Chen, Yan Pang

Applications and Consequences

Frontmatter
11. Applications and Consequences
Summary
Music (and more generally the arts) creates semiotic structures independently of ‘external’ reference contents. We discuss this qualification as a conceptual challenge.
Guerino Mazzola, Sangeeta Dey, Zilu Chen, Yan Pang

References, Index

Frontmatter
12. Conclusions and Perspectives
Summary
This chapter reviews how the mathematics of new concepts of music that we discuss in the book can be used to better understand creativity and its logic. The reading of this book will support your ability to become a more versatile music creator. By tying together the concepts of functorial semiotics, you will expand your ability to compose methodically and strategically, rather than simply waiting for inspiration to strike.
Guerino Mazzola, Sangeeta Dey, Zilu Chen, Yan Pang
Backmatter
Metadaten
Titel
Functorial Semiotics for Creativity in Music and Mathematics
verfasst von
Guerino Mazzola
Sangeeta Dey
Zilu Chen
Yan Pang
Copyright-Jahr
2022
Electronic ISBN
978-3-030-85190-3
Print ISBN
978-3-030-85189-7
DOI
https://doi.org/10.1007/978-3-030-85190-3

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