Reasoning and Proof

Inductive reasoning generalises from examples. It suggests what might be true and never guarantees it.

Deductive reasoning

Deductive reasoning starts from accepted facts and derives a conclusion that must follow. This is what a proof is.

Counterexamples

One counterexample destroys a general claim. No number of confirming examples proves one.

The asymmetry

Disproving takes one case; proving takes an argument covering every case. That imbalance is why proof exists.

If-then statements

Most geometric facts are conditionals. Swapping the two halves gives the converse, which may well be false.

Both are useful

Inductive reasoning finds the conjecture and deductive reasoning confirms it. Mathematics needs both.

What counts as an argument

A proof is a chain where every claim follows from a definition, a postulate, or a previously proved result. Measuring an accurate drawing establishes one case and proves nothing general.

One counterexample disproves

Proving a general claim requires an argument covering every case; disproving it needs one example. That asymmetry is worth internalising, and it makes disproof much easier than proof.

A converse needs its own proof

A statement being true does not make its converse true. Some converses hold and many do not, so each direction is a separate claim requiring separate justification.

Some terms are left undefined

Point, line and plane are taken as primitive, because defining everything would make the definitions circular. Starting from a few undefined terms is what stops the regress.