Paraconsistent Measurement of the Circle

Abstract

A theorem from Archimedes on the area of a circle is proved in a setting where some inconsistency is permissible, by using paraconsistent reasoning. The new proof emphasizes that the famous method of exhaustion gives approximations of areas closer than any consistent quantity. This is equivalent to the classical theorem in a classical context, but not in a context where it is possible that there are inconsistent innitesimals. The area of the circle is taken 'up to inconsistency'. The fact that the core of Archimedes's proof still works in a weaker logic is evidence that the integral calculus and analysis more generally are still practicable even in the event of inconsistency.

Naïve Infinitesimal Analysis: Its Construction and Its Properties

“…They can also specify where the consistency requirement is necessary. These preliminary works in [26] and [52] show how successful a paraconsistent setting 1 If we look historically, the debates of the use of infinitesimals have a long and vivid history. Their early appearance in mathematics was from the Greek atomist philosopher Democritus (around 450 B.C.E.…”