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#abstraction

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Maths › 01. History and Biography ›

1.2 Number Concepts

From objects to ideas Early counting connects directly to objects. People count three stones, five animals, or ten tools. Each number stays tied to a specific collection. Over time, a shift occurs. The idea of “three” starts to exist on its own, independent of what is being counted. This shift matters. Once you understand “three” as a general idea, you can apply it anywhere. Three apples, three days, and three...
mathematics numbers abstraction history May 2, 2026
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....
mathematics patterns structure abstraction May 2, 2026
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...
mathematics abstraction structure method May 2, 2026
Maths › 00. General Mathematics ›

05.4 Meta-Mathematical Abstraction

5.4 Meta-Mathematical Abstraction Meta-mathematical abstraction studies mathematics itself as an object. Instead of working inside one theory, it steps outside and examines the theory's language, axioms, proofs, models, and limits. At the concrete level, we compute. At the algebraic level, we manipulate symbols under rules. At the categorical level, we study objects through maps. At the meta-mathematical level, we study the systems that make those activities possible. A formal theory...
mathematics abstraction metamathematics foundations May 2, 2026
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 :...
mathematics abstraction category-theory structure May 2, 2026
Maths › 00. General Mathematics ›

05.2 Algebraic Abstraction

5.2 Algebraic Abstraction Algebraic abstraction replaces specific values with symbols and replaces repeated calculations with general rules. It keeps operations explicit, but removes dependence on particular instances. A concrete computation shows: $$ (2 + 3)^2 = 25. $$ An algebraic statement expresses the same pattern in general form: $$ (a+b)^2 = a^2 + 2ab + b^2. $$ The second statement applies to all admissible values of $a$ and $b$. A...
mathematics abstraction algebra symbols May 2, 2026
Maths › 00. General Mathematics ›

05.1 Concrete Computation

5.1 Concrete Computation Concrete computation is the entry point of abstraction. It works with explicit objects, finite procedures, and direct evaluation. At this level, every symbol refers to a specific value or object, and every step can be executed without further interpretation. A typical example is a direct calculation: $$ (2 + 3)^2 = 25. $$ Nothing is hidden. The numbers are fixed, the operations are known, and the result...
mathematics abstraction computation examples May 2, 2026
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,...
mathematics abstraction structure foundations May 2, 2026
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....
mathematics structure morphisms maps abstraction May 2, 2026
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...
mathematics structure abstraction instances models May 2, 2026
Maths › 00. General Mathematics ›

03.5 Notation as Interface

3.5 Notation as an Interface Notation is not decoration. It is an interface between ideas and reasoning. A well-designed notation exposes structure, hides irrelevant detail, and supports composition. An interface answers three questions: Question Role of notation What are the objects? Names, symbols, types What operations are allowed? Function application, algebraic operators What properties are visible? Equalities, relations, invariants When notation is effective, you can manipulate expressions without rederiving definitions...
mathematics language notation abstraction interface May 2, 2026
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...
mathematics structure abstraction invariants May 2, 2026
Maths › 00. General Mathematics ›

01.1 Abstract Objects and Structures

1.1 Abstract Objects and Structures Mathematics studies objects by stripping away irrelevant detail. A number, a function, a graph, a space, and a group are not studied as physical things. They are studied through the rules they satisfy, the relations they support, and the transformations that preserve them. An abstract object is an object considered through its mathematical properties rather than its material form. The integer 3 can count apples,...
mathematics foundations abstraction structures May 2, 2026