Infinite Divisibility: Aristotle and Zeno The discussion below, then, is not meant to be a comprehensive survey of all aspects of infinity (or even all aspects of this notion as it appears within mathematics, logic, and metaphysics), but is instead intended to provide a basic understanding of two important themes underlying hundreds of years of thought on the topic. Much of the history of mathematics and philosophy can be seen as an (often indirect) inquiry into the coherence of such collections and how they differ from finite collections.Īs a result, this entry will for the most part ignore other interesting, but less central, issues within the literature on infinity, including infinitesimals, mereological theories containing gunk, unrestrictedly general quantification, nonstandard set theories, and nonmathematical uses of the term "infinite" (e.g., theological understandings of the infinite).
The second aspect to be addressed is already implicit in the first -the idea that there can be infinitely large collections at all. The puzzles that arise from such division are central both to philosophical thinking about notions such as part and whole and to the mathematical analysis of lines, surfaces, and other continuous objects. The first is infinite divisibility, the idea that an object can, in some sense (and perhaps only ideally), be divided into an infinite collection of smaller and smaller parts. This entry focuses on two aspects of the infinite. As a result, it is unsurprising that philosophers throughout history have thought deeply about what infinity is, whether the notion is coherent, whether there are infinite entities (or infinitely many entities), and how we can know about such entities if they exist. Any serious thought about the nature of space, time, God (or gods), mathematics, and motion quickly leads to more general concerns regarding the notion, or notions, of infinity intimately tied up with such issues. The notion of infinity, and the problems, both philosophical and mathematical, that arise from it have been a central concern for over two millennia.
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