Architecture of Mathematics describes the logical structure of Mathematics from its foundations to its real-world applications. It describes the many interweaving relationships between different areas of mathematics and its practical applications, and as such provides unique reading for professional mathematicians and nonmathematicians alike. This book can be a very important resource both for the teaching of mathematics and as a means to outline the research links between different subjects within and beyond the subject.
Features
All notions and properties are introduced logically and sequentially, to help the reader gradually build understanding.
Focusses on illustrative examples that explain the meaning of mathematical objects and their properties.
Suitable as a supplementary resource for teaching undergraduate mathematics, and as an aid to interdisciplinary research.
Forming the reader's understanding of Mathematics as a unified science, the book helps to increase his general mathematical culture.
Author(s): Simon Serovajsky
Publisher: CRC Press
Year: 2020
Language: English
Pages: 394
City: Boca Raton
Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Introduction
Floor 1: LANGUAGE
Room 1.1 Alphabet
Room 1.2 Syntax
Room 1.3 Semantics
Floor 2: SETS
Room 2.1 Sets
Room 2.2 Subsets
Room 2.3 Set product
Room 2.4 Correspondences
Room 2.5 Relations
Room 2.6 Operators
Room 2.7 Equinumerosity
Floor 3: NUMBERS
Section I CARDINALITIES
Room 3.1 Zero number
Room 3.2 Natural numbers
Section II SOLUTIONS
Room 3.3 Integer numbers
Room 3.4 Rational numbers
Room 3.5 Algebraic numbers
Section III CUTS
Room 3.6 Real numbers
Section IV TUPLES
Room 3.7 Complex numbers
Room 3.8 Quaternions
Floor 4: OBJECTS
Block A ORDERED OBJECTS
Room 4A.1 Preordered sets
Room 4A.2 Partially ordered sets
Room 4A.3 Special ordered sets
Block B ALGEBRAIC OBJECTS
Section I OPERATIONS
Room 4B.1 Operations
Section II SETS WITH INTERIOR COMPOSITION LAWS
Subsection 1 Groupoids
Room 4B.2 Groupoids
Room 4B.3 Monoids
Room 4B.4 Groups
Subsection 2 Rings
Room 4B.5 Rings
Room 4B.6 Bodies and fields
Subsection 3 Lattices
Room 4B.7 Lattices
Room 4B.8 Boolean algebras
Section III SETS WITH EXTERIOR COMPOSITION LAWS
Subsection 1 Groups with operators
Room 4B.9 Modules
Room 4B.10 Vector spaces
Subsection 2 Rings with operators
Room 4B.11 Algebras
Section IV UNIVERSAL ALGEBRAS
Room 4B.12 Universal algebras
Block C TOPOLOGICAL OBJECTS
Section I TOPOLOGICAL SPACES
Room 4C.1 General topological spaces
Room 4C.2 Determination of topological spaces
Room 4C.3 Special topological spaces
Section II METRIC SPACES
Room 4C.4 General metric spaces
Room 4C.5 Special metric spaces
Block D MEASURABLE OBJECTS
Section I MEASURABLE SPACES
Room 4D.1 Measurable spaces
Section II MEASURE SPACES
Room 4D.2 Measures
Room 4D.3 Integrals
Floor 5: COMPOSITES
Section I MIXED STRUCTURES
Room 5.1 Consistence of structures
Section II TOPOLOGICAL ALGEBRAIC OBJECTS
Room 5.2 Topological groupoids
Room 5.3 Topological groups
Room 5.4 Topological vector spaces
Room 5.5 Normed vector spaces
Room 5.6 Banach spaces
Room 5.7 Hilbert spaces
Section III APPLICATIONS
Room 5.8 Derivatives
Floor 6: SYNTHESIS
Section I STRUCTURES
Room 6.1 Scale of sets
Room 6.2 Structures
Section II CATEGORIES
Subsection 1 Categories
Room 6.3 General categories
Room 6.4 Non-concrete categories
Room 6.5 Functors
Subsection 2 Concepts
Room 6.6 Special morphisms
Room 6.7 Subobjects and quotient objects
Room 6.8 Product of objects
Room 6.9 Initial and terminal objects
Subsection 3 Beginning
Room 6.10 Ordered objects
Room 6.11 Algebraic objects
Room 6.12 Short addition
Index
