pure mathematics
Theory without applications
Pure mathematics is mathematics pursued for its own sake, without concern for practical application.
Where applied mathematics solves problems in physics, engineering, or economics, pure mathematics explores abstract structures, proofs, and logical relationships: number theory, topology, abstract algebra.
The distinction is somewhat artificial—discoveries in pure mathematics often find unexpected applications decades later.
Demonstrated logical conclusion
- lemmaBase form, or helper theorem
- corollaryLogical natural consequence
- conjectureGuess from limited facts
- postulateAssumed starting point
deduction · induction · rigor · proposition · contradiction · counterexample · reasoning · implication
Pure math studying integers
- topologyStudy of spatial arrangement
- algebraMath using symbols
- geometryScience of spatial relationships
- analysisMethodical study of parts
combinatorics · set theory · graph theory · field theory · differential equations · probability theory
Self-evident starting principle
- predicateSentence's non-subject portion
- quantifierWord expressing amount
- syllogismFormal deductive argument structure
- inferenceConclusion reached through reasoning
tautology · validity · soundness · completeness · consistency · formalism · incompleteness · decidability
Refined, graceful quality
- beautyAesthetic excellence
- symmetryInvariance under transformation
- simplicityLack of complexity
- harmonyAgreement or musical blend
minimalism · proportion · balance · clarity · precision · coherence · economy · abstraction
Many and various
- groupAssembled collection of things
- ringClosed curve shape
- fieldArea of specialty
- moduleIndependent functional component
lattice · algebra · isomorphism · tensor · subspace · basis · dimension · kernel
Size of a set
- continuum hypothesisClaim about cardinal sizes
- uncountableNot able to enumerate
- countableAble to be tallied
- ordinalNumber denoting position
Cantor · boundless · infinite series · convergence · divergence · limit · Zeno · paradox
theorem
Demonstrated logical conclusion
- lemmaBase form, or helper theorem
- corollaryLogical natural consequence
- conjectureGuess from limited facts
postulate · deduction · induction · rigor · proposition · contradiction · counterexample · reasoning · implication · demonstration
number theory
Pure math studying integers
- topologyStudy of spatial arrangement
- algebraMath using symbols
- geometryScience of spatial relationships
analysis · combinatorics · set theory · graph theory · field theory · differential equations · probability theory
axiom
Self-evident starting principle
- predicateSentence's non-subject portion
- quantifierWord expressing amount
- syllogismFormal deductive argument structure
inference · tautology · validity · soundness · completeness · consistency · formalism · incompleteness · decidability · recursion
elegance
Refined, graceful quality
- beautyAesthetic excellence
- symmetryInvariance under transformation
- simplicityLack of complexity
harmony · minimalism · proportion · balance · clarity · precision · coherence · economy · abstraction · generalization
manifold
Many and various
module · lattice · algebra · isomorphism · tensor · subspace · basis · dimension · kernel · image
cardinality
Size of a set
- continuum hypothesisClaim about cardinal sizes
- uncountableNot able to enumerate
- countableAble to be tallied
ordinal · Cantor · boundless · infinite series · convergence · divergence · limit · Zeno · paradox · well-ordering
- exact sciencesphysics, chemistry, math, algebra, geometry
- pure researchbasic science, theoretical physics, philosophy, quantum mechanics, cosmology
pure mathematicsapplied mathematics
Problem-solving through quantitative methods
applied science · engineering · physics · applied physics · statistics
pure mathematicsexperimental physics
Testing theories through observation
natural science · physical science · applied science · experimental research · field research
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