Lin Hsin Hsin Quantum Security Center
Mathematical Abstraction
DEFINITION
The process of extracting
⭕ underlying structures
⭕ patterns
⭕ properties
from specific instances while removing dependence on real-world objects and concrete contexts
OBJECTIVES
The objective is to allow for the creation of
general theories and self-contained systems where meaning is
derived solely from internal relationships and operations
RULES
The fundamental rules and techniques governing this process include:
Identification of Invariances
Mathematicians isolate properties that remain unchanged across diverse examples, such as the invariant property of successive enumeration when moving from counting physical objects to the abstract notion of natural numbers
Equivalence Relations & Partitioning
Abstraction often involves grouping a domain of entities into classes based on a shared trait or properties (an equivalence relation) and identifying a single abstract object corresponding to each class, such as defining a direction for all parallel lines
Axiomatic Construction
Abstract structures are defined using a minimal set of axioms and primitive terms that capture essential properties, allowing for the deductive derivation of theorems without reference to specific real-world interpretations
Decontextualization & Symbolic Representation
Concrete details eg color or shapes are discarded in favor of symbolic representation (variables and operators), enabling the study of relationships and rules that apply across different systems, such as vectors or matrices
Self-Containment
An abstract mathematical object takes its meaning only from the system within which it is defined, allowing for operations on these objects without the need to reference their original empirical origins
Common examples of this abstraction include
The transition from Euclidean geometry to topology (focusing on properties preserved under continuous deformation rather than precise distances)
From arithmetic to abstractstructure of operations like addition and multiplication in rings and fields rather than specific numerical values