Chapter 6. Patterns Across Mathematics
Overview of recurring patterns such as duality, symmetry, local-to-global reasoning, decomposition, recursion, and induction.
Tag
Abstraction
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5.5 Trade-offs in Abstraction
Understanding the benefits and costs of abstraction, and choosing the right level for mathematical work.
5.4 Meta-Mathematical Abstraction
Studying mathematical systems themselves through languages, axioms, models, proofs, and interpretations.
5.3 Categorical Abstraction
Raising abstraction from objects and operations to maps, composition, and universal properties.
5.2 Algebraic Abstraction
Replacing concrete values with symbols and rules to express general patterns.
5.1 Concrete Computation
Working with explicit examples, calculations, and finite procedures as the base level of mathematical reasoning.
Chapter 5. Levels of Abstraction
Overview of how mathematics moves from concrete computation to structural and higher-level reasoning.
4.2 Morphisms and Mappings
Structure-preserving maps, their role in comparison, composition, and transport of mathematical information.
4.1 Structure vs Instance
Distinguishing abstract structures from their concrete instances, and using that distinction to reason across examples.
Chapter 4. Structural Thinking
Overview of structures, mappings, invariants, and classification in mathematics.
3.5 Notation as an Interface
Viewing notation as a designed interface that exposes structure, supports composition, and enables efficient reasoning.
1.2 Emergence of Number Concepts
How numbers move from concrete counting to abstract ideas.
1.1 Abstract Objects and Structures
How mathematics treats objects through the rules they satisfy, the relations they support, and the transformations that preserve them.