Osman Gazi Birgül

This paper consists of two parts and has two aims. The first is to elaborate on the historical-philosophical background of Cantor’s notions of infinity in the context of Spinoza’s influence on him. To achieve this aim, in the first part I compare Spinoza’s and Cantor’s conceptions of actual infinity along with their mathematical implications. Explaining the metaphysical, conceptual, and methodological aspects of Cantor’s expansion of the orthodox finitist conception of number of his time, I discuss how he adopts Spinoza’s motifs to overcome the challenges on his way to proposing his transfinite infinites. To achieve the second aim, i.e., to investigate the philosophical foundations of Cantor’s transfinite infinites, I follow a three-step outline analyzing Cantor’s three principles along with their metaphysical and epistemological-linguistic aspects. I analyze the strengths and weak sides of Lakoff and Núñez’s Basic Metaphor of Infinity and Pantsar’s Process → Object Metaphor, then compare their interpretations with mine. I conclude that the metaphors can be improved when their source domains are limited to the target domains they apply to and when supported by a thorough analysis of the historical-philosophical background of their target domains.

This article discusses the quiddity of the empty set from its epistemological and linguistic aspects. It consists of four parts. The first part compares the concept of _nihil privativum_ and the empty set in terms of representability, arguing the empty set can be treated as a negative and formal concept. It is argued that, unlike Frege’s definition of zero, the quantitative negation with a full scope is what enables us to represent the empty set conceptually without committing to an antinomy. The second part examines the type and scope of the negation in the concept of _nihil privativum_ and the empty set. In the third part the empty set is interpreted as a rigid abstract general term. The uniqueness of the empty set is explained via a widened version of Kripke’s notion of rigidity. The fourth part proposes a construction for the pure singleton, comparing it with Zermelo’s conception of singletons with the Ur-elements. It is argued that the proposed construction does not face the criterion and ontological inflation problems. The first conclusion of the article is that the empty set can be construed as a negative, formal and unique abstract general term, with quantitative negation full in scope. The second conclusion is that the pure singleton constructed out of the empty set construed in this way overcomes the criterion and ontological inflation problems.