Less then Something, but More then Nothing - Zeno, Infinitesimals and the Continuum Paradox

• Main Author: Dolenc, Sašo
• Format: Article in Journal/Newspaper
• Language: Slovenian
• Published: ZRC SAZU, Založba ZRC 2016
• Subjects: philosophy of science; philosophy of nature; continuum; infinitesimals; history of mathematics; infinity; Zeno’s aporias; filozofija znanosti; filozofija narave; kontinuum; infinitezimali; zgodovina matematika; neskončnost; Zenonove aporije; phil; relig; sami
• Online Access: https://ojs.zrc-sazu.si/filozofski-vestnik/article/view/3565

The hypothesis is that a line can be divided into an infinite number of parts. The question is whether extension is a property of these infinitely small parts of the line. If the infinitely small parts are extensional then they can be further divided. On the other hand, if they have no extension how can they compose the line? The sum of non-extensional units can not be extensional. None of the answers explains the problem which is thus further considered to be the continuum paradox. Although the infinitesimal calculus does not solve the continuum paradox, it finds a way to deal with it by introducing infinitesimals as infinitely small parts that are neither extensional nor non-extensional. Even if infinitesimals have no extension, they can compose a line. Če predpostavimo, da je črta neskončno deljiva, jo lahko razdelimo na neskončno majhne dele, za katere ni jasno ali so razsežni ali ne. Če imajo razsežnost, jih lahko še naprej delimo, torej še niso neskončno razdeljeni. Če pa nimajo razsežnosti, potem ni jasno, kako lahko iz njih sestavimo razsežno črto, saj vsota nerazsežnih enot ne more biti razsežna. Nobeden od obeh možnih odgovorov problema ne razreši, zato problem obravnavamo kot paradoks kontinuuma. Čeprav infinitezimalni račun paradoksa kontinuuma ne razreši, vpelje infinitezimale kot neskončno majhne dele, ki niso ne točke ne daljice. Iz infinitezimalov lahko sestavimo črto, čeprav sami nimajo razsežnosti.