13.3 Normal Forms
Normalization of proofs, elimination of detours, and structural simplification of derivations.
Tag
Structure
18 notes
10.1 Structure of a Paper
How a mathematical paper or article is organized so that readers can follow the main ideas.
6.5 Recursion and Induction
Defining objects step by step and proving properties by following the same construction.
6.4 Decomposition and Composition
Breaking complex objects into simpler parts and building larger structures from controlled combinations.
6.3 Local-to-Global Principles
How mathematics studies small pieces first and then assembles them into statements about the whole.
6.1 Duality
Understanding how reversing structure reveals parallel theories and results.
Chapter 6. Patterns Across Mathematics
Overview of recurring patterns such as duality, symmetry, local-to-global reasoning, decomposition, recursion, and induction.
5.5 Trade-offs in Abstraction
Understanding the benefits and costs of abstraction, and choosing the right level for mathematical work.
5.3 Categorical Abstraction
Raising abstraction from objects and operations to maps, composition, and universal properties.
Chapter 5. Levels of Abstraction
Overview of how mathematics moves from concrete computation to structural and higher-level reasoning.
4.5 Examples: Groups, Spaces, Graphs
Concrete examples showing structural thinking across algebra, topology, and graph theory.
4.4 Invariants and Classification
How preserved quantities and properties support comparison, classification, and structural reasoning.
4.3 Isomorphism as Sameness
How isomorphism formalizes structural sameness and separates equality from equivalence.
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.
1.3 Equality, Identity, and Equivalence
Different notions of sameness in mathematics: strict equality, structural identity, and equivalence relations.
Chapter 1. Nature of Mathematical Objects
Overview of abstract objects, structures, equality, finiteness, and viewpoints in mathematics.