Mereological Axiom of Assotiation of Parts with the Whole Versus Zeno's “Measure” Aporia

Abstract

Introduction. The essence of the cognitive problem is as follows. Each item has different linear dimensions. However, each item is made up of pieces that, therefore, have smaller linear dimensions. Due to the infinite division into parts, the linear dimensions of the smaller and smaller parts must go to zero. They cannot be zero because, due to the combination of parts with zero linear dimensions, it is not possible to obtain an object with linear dimensions larger than zero. At the same time, they cannot have linear sizes larger than zero, because after combining an infinite number of parts with sizes larger than zero, an object with infinitely larger linear dimensions will be obtained. We get a contradiction - the linear dimensions of the parts of the objects can neither be zero nor larger than zero. The main approaches to critique of the Measure aporia are the following: 1) denying the infinite divisibility by assuming the existence of minimal sizes of material objects; 2) acceptance of infinite divisiveness is combined with this or that justification of the ways of overcoming contradiction. The aim and tasks. The aim of the study is to identify the sources of the contradiction of Zeno's “Measure” aporia. To achieve this goal, it has been planned to perform the following tasks: 1) to find out the peculiarities of divisibility on the example of knowledge about the structure of material objects; 2) to compare knowledge about the mentioned features with the Zeno’s material separation model. Research methods. The methodology of the study was to compare a specific example of structuring a material object into parts with Zeno's considerations to identify inconsistencies between them. Research results. The study has found out that the source of the contradiction of the “Measure” aporia is the interpretation of parts formed as a result of the infinite division of the object as separate objects (this accurately describes Occam's statement "Entities should not be multiplied without necessity"). A mereological axiom does not allow such contradictions to arise in the process of reasoning. Discussion. It has been justified that the approaches in which authors accept infinite division and seek the possibility of a consistent explanation of the movement or size of objects are wrong. Conclusions. Conclusions have been obtained: 1) the false notion that an infinite number of parts are formed as separate objects due to the infinitely performed division is the source of contradiction of the “Measure” aporia; 2) the resulting explanation of the source of the contradiction was also used to explain the “Dichotomy” and “Achilles and the tortoise” aporias.Keywords: Zeno's “Measure” aporia, the whole, a part, infinite divisibility, mereology, axiom.