The oldest trick in logic: assume the opposite and watch it collapse
To prove something is true, you can pretend it is false and follow the consequences until they fall apart. This move, reductio ad absurdum, runs from a satirical Greek poem about gods drawn as horses to Euclid's geometry. Yet some schools of mathematics still refuse to accept proofs built this way.
The Latin name means reduction to absurdity, and the technique goes by many others: argument to absurdity, apagogical argument, indirect proof and, in mathematics, proof by contradiction. It establishes a claim by showing that its opposite leads to something ridiculous or self-contradictory. Formal logic encodes it as an inference rule. Although mathematicians use it freely, it is nonconstructive, and not every school of mathematical thought accepts it.
The absurdity can take different forms. One kind clashes with experience: the Earth cannot be flat, because if it were finite and flat people would tumble off the edge. Another produces a strict logical contradiction, as in the proof that no smallest positive rational number exists, since halving any candidate gives a smaller one. The standard mathematical recipe is to assume a statement is false, derive two assertions that contradict each other, invoke the law of noncontradiction, and conclude the statement must be true. A special variety proves that something exists by showing that assuming nothing has the property leads to contradiction.
The earliest known example comes from Xenophanes of Colophon, who lived from about 570 to 475 BCE. Mocking Homer for giving the gods human flaws, he noted that people imagine gods with human bodies, yet horses and oxen able to draw would picture them as horses and oxen. The gods cannot be both, so the whole practice of projecting human traits onto them collapses.
Greek mathematicians relied on the method for basic results, Euclid and Archimedes among the earliest. Plato's early dialogues turned it into a formal method of questioning, the elenchus or Socratic method. Socrates would accept an opponent's harmless-sounding claim, then lead him step by step, adding background assumptions, until the claim produced an absurd result and the speaker was left in aporia, a state of puzzlement.
Source: Reductio ad absurdum
