mathematical proofs
Rigorous logical arguments
A mathematical proof is a rigorous logical argument demonstrating that a statement must be true, proceeding from accepted axioms and previously established results.
Proofs are the foundation of mathematical certainty—unlike empirical sciences, mathematics proves things absolutely.
The proof of the Pythagorean theorem appears in most geometry textbooks.
To prove something mathematically is to establish it beyond doubt, not merely to verify it with examples.
- corollaryLogical natural consequence
- Euclid's ElementsFoundational geometry and number theory text
- theoremDemonstrated logical conclusion
- lemmaBase form, or helper theorem
- axiomSelf-evident starting principle
first principles · Hilbert's problems · reductio ad absurdum · conjecture · number theory · calculus · if and only if · necessary condition · sufficient condition · proposition · logical inference · truth tables · geometry · tautology · Venn diagrams · algebra · therefore · steps
- mathaddition, subtraction, multiplication, division, algebra
- mathematicsalgebra, arithmetic, calculus, geometry, number
- ancient mathalgebra, arithmetic, geometry, fractions, addition
- liberal artshumanities, trivium, philosophy, history, classical education
- classical physicsNewton, Newton's laws, Newtonian physics, laws of motion, force
- math historyEuclid, Pythagoras, Newton, Leibniz, Euler
mathematical proofsconjectures
Unverified assumptions or theories
hunches · guesses · speculation · wild guesses · gut feelings
mathematical proofsempirical evidence
Proof from measurable observation
inductive reasoning · trial and error · heuristics · approximations · intuition
mathematical proofsfallacies
Invalid logical arguments
open problems · faulty reasoning · circular reasoning · non sequiturs · refutations