Methods & Domains

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Catalogue presentation and source-binding data

Domain registry (classification is not complete coverage)

Foundations and logic (foundations_logic) — The foundational structures of mathematical language, axioms, sets, types, provability and formal reasoning.
Arithmetic and number systems (arithmetic_number_systems) — Numbers, operations, extensions of number systems and basic quantitative structures.
Algebra (algebra) — Operational structures, equations, symmetry and algebraic systems.
Geometry (geometry) — Shape, space, distance, angle and geometric structures.
Topology (topology) — Continuity, neighborhoods, convergence and properties preserved under continuous deformation.
Analysis (analysis) — Limits, continuity, differentiation, integration and infinite processes.
Number theory (number_theory) — Integers and their arithmetic structures.
Combinatorics and discrete mathematics (combinatorics_discrete) — Finite or countable structures, counting, graphs and discrete relations.
Probability and statistics (probability_statistics) — Randomness, distributions, inference and uncertainty in data.
Dynamical systems (dynamical_systems) — Systems whose states evolve through time or iteration.
Differential equations (differential_equations) — Relations and evolution described by derivatives.
Numerical and computational mathematics (numerical_computational) — Approximation algorithms, error, stability and computational implementations.
Optimization and operations research (optimization_operations_research) — Finding the best decisions under constraints.
Mathematical physics (mathematical_physics) — Mathematical structures and methods in physical theories.
Mathematical biology (mathematical_biology) — Mathematical models of living systems.
Mathematics of economics and finance (economics_finance) — Mathematical models of economic, risk and financial systems.
Mathematics of computer science (computer_science_mathematics) — Mathematical foundations of algorithms, computation, information and formal languages.
Mathematics education (mathematics_education) — Mathematical learning, teaching, assessment and curricula.
History, philosophy and methodology (history_philosophy_methodology) — The history of mathematical ideas, standards of validity and problem-solving methods.
Formalized mathematics (formalized_mathematics) — Expressing, checking and replaying mathematics with proof assistants and formal systems.

Methods & Domains

Direct proof

Derive the conclusion directly from known conditions through valid reasoning.

When to use it

The conclusion follows step by step from definitions or known theorems.

Basic steps

Expand the conditions.
Select relevant definitions or theorems.
Derive the conclusion step by step.

Common failures

Silently using an unproved proposition.

Proof by contradiction

Assume the conclusion fails and derive a contradiction.

When to use it

Negating the conclusion makes a conflict with known conditions easier to obtain.

Basic steps

Negate the goal.
Derive consequences.
Obtain a contradiction.

Common failures

The contradiction is unrelated to the original assumption.

Mathematical induction

Establish an infinite family of propositions through base cases and an inductive step.

When to use it

Propositions are organized by natural numbers or a recursive structure.

Basic steps

Prove the base case.
State the inductive hypothesis.
Prove the next step.

Common failures

Using the inductive hypothesis outside its permitted scope.

Construction

Explicitly build a mathematical object satisfying the conditions.

When to use it

An existence problem asks for a concrete object.

Basic steps

Analyze the constraints.
Propose a candidate construction.
Verify every condition.

Common failures

Showing an example without covering the general case.

Invariant

Find a quantity preserved by an operation or evolution.

When to use it

States change repeatedly while some properties may be preserved.

Basic steps

Define the states.
Propose a candidate quantity.
Prove preservation.
Compare the initial and final states.

Common failures

The proposed quantity is not actually preserved.

Symmetry

Use equivalence under transformations to reduce the cases of a problem.

When to use it

Several cases are equivalent under a transformation.

Basic steps

Identify the transformation.
Prove the problem is preserved.
Reduce to representative cases.

Common failures

Mistaking approximate symmetry for exact symmetry.

Extremal principle

Choose a largest, smallest or earliest object to force a structural conclusion.

When to use it

An extreme element exists in a finite set or a well-ordered structure.

Basic steps

Choose an extreme object.
Analyze its constraints.
Derive the conclusion.

Common failures

The extremum does not exist or its attainment has not been established.

Pigeonhole principle

Infer a collision when there are more objects than containers.

When to use it

The problem has finitely many categories and an imbalance in counts.

Basic steps

Define the objects and containers.
Compare their counts.
Infer a collision.

Common failures

The container categories are not mutually exclusive.

Diagonalization

Construct an object differing from every listed item in at least one position.

When to use it

The goal is non-enumerability or escaping a given family.

Basic steps

Assume an enumeration.
Choose a difference for each item.
Construct the new object.

Common failures

The new object is outside the permitted space.

Compactness

Reduce an infinite or global problem to finite or convergent data.

When to use it

Local information is needed to obtain a global conclusion.

Basic steps

Choose a compact structure.
Extract finite or convergent data.
Transfer the property.

Common failures

The compactness hypotheses are not satisfied.

Localization

Study local data first, then examine how it can be glued into a global result.

When to use it

Local behavior can control the global problem.

Basic steps

Choose a local scale.
Prove the local property.
Check the gluing.

Common failures

Local solutions cannot be glued globally.

Linearization

Replace a nonlinear problem by a linear one at a local or approximation level.

When to use it

First-order local behavior dominates.

Basic steps

Choose an expansion point.
Take the linear principal part.
Estimate the remainder.

Common failures

Ignoring the range in which the remainder estimate applies.

Coordinates

Translate geometric or abstract relations into coordinates and algebraic expressions.

When to use it

Convenient coordinates can be chosen for the geometric relations.

Basic steps

Choose a coordinate system.
Translate the objects.
Work algebraically.
Translate the meaning back.

Common failures

The coordinate choice introduces a degeneracy.

Generating function

Encode sequence or combinatorial data as a formal power series.

When to use it

A recurrence, counting problem or convolution structure is present.

Basic steps

Construct the generating function.
Transform the relations.
Solve the relations.
Extract coefficients.

Common failures

Confusing convergence with formal operations.

Fourier methods

Decompose functions into frequency components to analyze operations and structure.

When to use it

Translation, periodicity or convolution structure is present.

Basic steps

Choose a transform.
Move to the frequency domain.
Handle frequency components.
Apply the inverse transform.

Common failures

Ignoring function-space and convergence hypotheses.

Probabilistic method

Use random selection to prove existence of a structure or estimate its properties.

When to use it

Direct construction is difficult but average behavior can be controlled.

Basic steps

Define a random model.
Estimate probabilities or expectations.
Infer existence.

Common failures

The random model does not cover the target objects.

Numerical approximation

Use finite-precision algorithms to approximate a continuous or infinite problem.

When to use it

An analytic solution is unavailable or too costly.

Basic steps

Choose a discretization.
Analyze the error.
Perform the computation.
Check stability.

Common failures

Treating an approximation as an exact proof.

Computational experiment

Use programs to generate cases and explore patterns or candidate counterexamples.

When to use it

Many cases are needed or manual exploration is difficult.

Basic steps

Formalize the computable problem.
Fix the environment.
Execute and record.
Analyze the limitations.

Common failures

Presenting finite testing as a universal proof.

Counterexample search

Systematically search an explicit domain for an input that falsifies a proposition.

When to use it

The proposition can be checked on a finite or sampled domain.

Basic steps

Define the declared domain.
Build the decision procedure.
Execute the search.
Preserve the limitations.

Common failures

Treating not finding a counterexample as a proof.

Formal verification

Use a kernel checker in an explicit formal system to verify an exact statement.

When to use it

A machine-replayable universally quantified statement is needed.

Basic steps

Define the formal model.
Build the semantic bridge.
Compile the proof.
Reject placeholders.

Common failures

The formal statement and encyclopedia semantics are not fully aligned.