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Maths › 03. Logic and Foundations ›

13.3 Normal Forms

13.3 Normal Forms Normal forms for proofs are simplified shapes of derivations. The aim is to remove unnecessary detours and expose the essential structure of a proof. In propositional logic, formulas have normal forms such as CNF and DNF. In proof theory, proofs can also have normal forms. A proof may be correct but still contain redundant steps. Normalization rewrites the proof into a cleaner form. Detours in Proofs A...

Maths › 00. General Mathematics ›

10.1 Structure of a Paper

10.1 Structure of a Paper A mathematical paper is a written path through an idea. It does not only list results. It explains what problem is being studied, why the problem matters, what tools are used, and what has been proved. For a beginner, it is useful to think of a paper as a carefully organized explanation. Each part has a job. The title tells the reader what the paper...

Maths › 00. General Mathematics ›

06.5 Recursion and Induction

6.5 Recursion and Induction Recursion and induction are paired patterns. Recursion defines objects step by step. Induction proves statements about objects built step by step. The natural numbers give the basic model. Start with $0$. Then repeatedly apply the successor operation: $$ 0,\quad 1,\quad 2,\quad 3,\quad \dots $$ Each number is either the starting value or obtained from an earlier number. This recursive construction supports the principle of induction. To...

Maths › 00. General Mathematics ›

06.4 Decomposition and Composition

6.4 Decomposition and Composition Decomposition and composition are inverse patterns. Decomposition breaks a complex object into simpler parts. Composition builds a complex object from simpler parts. Much of mathematics moves between these two directions. Decomposition is useful because large objects are often hard to understand directly. A problem becomes easier when its pieces can be studied separately. Composition is useful because it explains how local or partial information forms a...

Maths › 00. General Mathematics ›

06.3 Local-to-Global

6.3 Local-to-Global Principles A local-to-global principle studies an object by examining small parts and then asking whether those local facts determine the whole. This pattern appears whenever a large structure is too complex to understand directly. Local information is information available near a point, on a small region, or on a restricted part of an object. Global information concerns the entire object. For example, a curve may look like a...

Maths › 00. General Mathematics ›

06.1 Duality

6.1 Duality Duality is a recurring pattern in mathematics where one concept is paired with another by reversing roles, directions, or operations. A statement about one object often has a corresponding statement about its dual, obtained by systematically exchanging parts of the structure. At a simple level, duality appears in operations. Intersection and union of sets behave as opposites under complement: $$ (A \cup B)^c = A^c \cap B^c, \quad...

Maths › 00. General Mathematics ›

06. Patterns Across Mathematics

Chapter 6. Patterns Across Mathematics Mathematics is organized by recurring patterns. The same forms of reasoning appear in algebra, geometry, topology, analysis, combinatorics, computation, and logic. Once these patterns are recognized, new subjects become easier to learn because their methods are no longer isolated. This chapter studies several of the most common patterns. Duality appears when a concept has a counterpart obtained by reversing direction, exchanging roles, or changing perspective....

Maths › 00. General Mathematics ›

05.5 Trade-offs in Abstraction

5.5 Trade-offs in Abstraction Abstraction makes mathematics powerful, but it is never free. Each level of abstraction preserves some information and suppresses other information. The benefit is generality. The cost is distance from concrete examples. At a concrete level, everything is visible. We can compute, test, and inspect. The object is close to intuition. But concrete reasoning often fails to scale. A calculation may explain one case without explaining the...

Maths › 00. General Mathematics ›

05.3 Categorical Abstraction

5.3 Categorical Abstraction Categorical abstraction studies objects through their relationships. Instead of asking only what an object contains, it asks how the object maps to and from other objects. This shift is important. At the algebraic level, we focus on operations inside a structure. At the categorical level, we focus on morphisms between structures and on how those morphisms compose. A category has objects and morphisms. If $$ f :...

Maths › 00. General Mathematics ›

04.5 Examples

4.5 Examples: Groups, Spaces, Graphs Structural thinking becomes clearer through examples. This section shows how the same principles appear in different areas. We focus on three domains: groups, topological spaces, and graphs. Each has its own objects, morphisms, invariants, and classification problems. 1. Groups A group is a set $G$ with a binary operation satisfying associativity, identity, and inverses. Objects and maps Objects: groups $(G, \ast)$ Morphisms: group homomorphisms $$...

Maths › 00. General Mathematics ›

05. Levels of Abstraction

Chapter 5. Levels of Abstraction Mathematics is written and understood across several levels of abstraction. The same idea may appear as a concrete computation, an algebraic manipulation, a structural statement, or a general principle expressed through mappings and relationships. This chapter gives an overview of these levels and explains how they interact. At the most concrete level, mathematics deals with explicit objects and calculations. Numbers are evaluated, expressions are simplified,...

Maths › 00. General Mathematics ›

04.4 Invariants

4.4 Invariants and Classification An invariant is a property, quantity, or structure that remains unchanged under a chosen class of transformations. Invariants are central to structural thinking. They allow us to compare objects without depending on representation. If two objects are isomorphic, their structural invariants agree. Therefore, if an invariant differs, the objects cannot be isomorphic. For example, two finite sets with different cardinalities cannot be bijective. Two vector spaces...

Maths › 00. General Mathematics ›

04.2 Morphisms and Mappings

4.2 Morphisms and Mappings A morphism is a map that preserves structure. It is the correct notion of transformation for a given mathematical setting. A plain function sends elements of one set to elements of another set. $$ f : X \to Y $$ A morphism does more. It respects the operations, relations, or laws that define the structures on $X$ and $Y$. For groups, the morphisms are group homomorphisms....

Maths › 00. General Mathematics ›

04.1 Structure vs Instance

4.1 Structure vs Instance A structure is a pattern. An instance is a concrete realization of that pattern. This distinction is central to mathematical reasoning. It lets us prove results once at the structural level and then apply them to many examples. A group is a structure. The integers under addition are an instance of that structure. The nonzero rational numbers under multiplication are another instance. The symmetries of a...

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04.3 Isomorphism

4.3 Isomorphism as Sameness Isomorphism is the mathematical notion of structural sameness. Two objects are isomorphic when there is a reversible map between them that preserves the relevant structure. Equality is usually too strict. Two objects can have different representations and still behave identically for the purposes of a theory. For example, the additive group of integers and the additive group of even integers are not equal as sets: $$...

Maths › 00. General Mathematics ›

04. Structural Thinking

Chapter 4. Structural Thinking Mathematics becomes powerful when it shifts from individual objects to structures. Instead of studying each example in isolation, structural thinking identifies patterns that apply across many instances. This chapter gives an overview of that approach. A structure consists of objects together with operations, relations, and laws. Groups, vector spaces, graphs, and topological spaces are all examples. Once a structure is defined, attention moves from the objects...

Maths › 00. General Mathematics ›

01. Nature of Objects

Chapter 1. Nature of Mathematical Objects Mathematics begins with objects, but it does not treat them as physical things. It treats them through structure. A number, a function, a space, or a graph is understood by how it behaves, not by how it is represented. This chapter gives an overview of that viewpoint. The first step is abstraction. Instead of focusing on individual examples, mathematics identifies patterns. These patterns are...

Maths › 00. General Mathematics ›

01.3 Equality, Identity, and Equivalence

1.3 Equality, Identity, and Equivalence Mathematics depends on precise notions of sameness. The symbol = appears simple, but it carries different meanings depending on context. This section separates three levels: equality, identity, and equivalence. Equality is the strictest notion. Two objects are equal if they are the same object in the underlying system. a = b In set theory, equality is extensional. Two sets are equal when they have exactly...

#structure

13.3 Normal Forms

10.1 Structure of a Paper

06.5 Recursion and Induction

06.4 Decomposition and Composition

06.3 Local-to-Global

06.1 Duality

06. Patterns Across Mathematics

05.5 Trade-offs in Abstraction

05.3 Categorical Abstraction

04.5 Examples

05. Levels of Abstraction

04.4 Invariants

04.2 Morphisms and Mappings

04.1 Structure vs Instance

04.3 Isomorphism

04. Structural Thinking

01. Nature of Objects

01.3 Equality, Identity, and Equivalence