Reductio ad absurdum in the context of Nonconstructive proof

The method of exhaustion (Latin: methodus exhaustionis) is a method of finding the area of a shape by inscribing inside it a sequence of polygons (one at a time) whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the nth polygon and the containing shape will become arbitrarily small as n becomes large. As this difference becomes arbitrarily small, the possible values for the area of the shape are systematically "exhausted" by the lower bound areas successively established by the sequence members.

The method of exhaustion typically required a form of proof by contradiction, known as reductio ad absurdum. This amounts to finding an area of a region by first comparing it to the area of a second region, which can be "exhausted" so that its area becomes arbitrarily close to the true area. The proof involves assuming that the true area is greater than the second area, proving that assertion false, assuming it is less than the second area, then proving that assertion false, too.

Quadrature of the Parabola (Greek: Τετραγωνισμὸς παραβολῆς) is a treatise on geometry, written by Archimedes in the 3rd century BC and addressed to his Alexandrian acquaintance Dositheus. It contains 24 propositions regarding parabolas, culminating in two proofs showing that the area of a parabolic segment (the region enclosed by a parabola and a line) is ${\displaystyle {\tfrac {4}{3}}}$ that of a certain inscribed triangle.

It is one of the best-known works of Archimedes, in particular for its ingenious use of the method of exhaustion and in the second part of a geometric series. Archimedes dissects the area into infinitely many triangles whose areas form a geometric progression. He then computes the sum of the resulting geometric series, and proves that this is the area of the parabolic segment. This represents the most sophisticated use of a reductio ad absurdum argument in ancient Greek mathematics, and Archimedes' solution remained unsurpassed until the development of integral calculus in the 17th century, being succeeded by Cavalieri's quadrature formula.

To be ridiculous is to be something highly incongruous or inferior, sometimes deliberately so to make people laugh or get their attention, and sometimes unintendedly so as to be considered laughable and earn or provoke ridicule and derision. It comes from the 1540s Latin "ridiculosus" meaning "laughable", from "ridiculus" meaning "that which excites laughter", and from "ridere" meaning "to laugh". "Ridiculous" is an adjective describing "the ridiculous".

In common usage, "ridiculousness" is used as a synonym for absurdity or nonsense. From a historical and technical viewpoint, "absurdity" is associated with argumentation and reasoning, "nonsense" with semantics and meaning, while "ridiculous" is most associated with laughter, superiority, deformity, and incongruity. Reductio ad absurdum is a valid method of argument, while reductio ad ridiculum is invalid. Argument by invective declaration of ridiculous is invalid, while arguments involving declarations of nonsense may summarize a cogent semantic problem with lack or meaning or ambiguity.