LOGIC & PUZZLES / MATHEMATICAL THINKING
What makes a
proof work?
A proof is not a collection of persuasive examples; it is a reasoned path from assumptions to a conclusion.
THE MAIN IDEA
Every important step needs a warrant.
A mathematical proof establishes that a statement follows from definitions, assumptions, and previously justified results. Trying several examples is useful for discovering a pattern, but finitely many examples cannot usually prove a claim about infinitely many possible values. One counterexample, however, is enough to disprove a universal statement: if someone claims that every integer with a property satisfies a conclusion, a single valid integer that fails it ends the claim. Direct proofs build the result from the assumptions. A proof by contradiction temporarily assumes the opposite and derives an impossible outcome. Mathematical induction proves a base case and then shows that if a statement holds at one natural number, it holds at the next; both steps are required to establish the claim for all numbers in that sequence. Each proof technique relies on precise definitions and an explicitly stated domain. A claim about positive integers may not hold for zero, negative numbers, real values, or objects outside that set. Notation compresses a chain of reasoning, but it should not hide an unsupported jump such as dividing by a quantity that might be zero. When learning a proof, expand every implication in ordinary language: what is known at this line, which definition or theorem is used, and what conclusion follows? Draw a diagram when it clarifies geometry, but remember that a sketch can look true by scale while failing for a narrow boundary case. Check whether equality is permitted, whether quantities are positive, and whether a construction works for all allowed inputs. A computer can test millions of examples and help find counterexamples, yet finite computation by itself may not prove a theorem over an infinite domain. Formal proof assistants can verify proofs when the definitions and proof object are carefully supplied, but even then the intended statement and assumptions need human scrutiny. Not every valuable mathematical exploration ends with proof: a conjecture is an informed proposal, an empirical pattern is evidence, and a theorem is a claim supported by a valid argument. Label the status clearly. The same habits improve non-mathematical arguments too, although factual claims in the world need empirical evidence rather than deductive proof from definitions. A clear proof is not about sounding authoritative. It is about letting another person inspect the complete chain, test its premises, and understand why no permitted case was left out.
TRY THIS
Find the hidden assumption in a familiar rule.
Take the statement “the sum of two odd integers is even.” Write each odd integer in the form 2k+1 and 2m+1, add them, and factor out two. State why the result is even and which values k and m may take.