Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

a note on infinity in mathematics, philosophy, and theology 277complex and endlessly fascinating, but even a short treatment is beyond the possibilitiesof this limited chapter. Fortunately, this volume contains several chapters thattogether offer a wealth of insight and references to this history; I would call particularattention to the chapters by Wolfgang Achtner (Chapter 1) and David Bentley Hart(Chapter 12). 3 Here Hart begins by stating that “there is not – nor has there ever been –any single correct or univocal concept of the infinite.” Granted Hart’s point, what I wantexmine here is a very basic distinction between that aspect of the concept of the infinitethat by and large dominated this history, namely, the infinite as the negation of thefinite, and the new element in the concept of the infinite found in modern mathematics,specifically that of Georg Cantor. To portray this distinction in its simplest form, whilerecognizing that this might, in fact, be an oversimplification, I touch briefly on a fewkey examples in this rich history. 4As is well known, the ancient Greeks defined the concept of infinity as apeiron,the unbounded. In so doing, they unequivocally distinguished between the infinite andthe finite: something is infinite if it has no boundary or end, or if it is chaotic andlacks structure or order. Thus, the infinite is defined by contrast with the finite: theinfinite is totally different from, even opposed to, those things that make up ordinary,finite experience. In this view, underlying the world of finite entities is a formless andindeterminate infinity, and finite entities are good precisely because they are formed,bounded, and determinate in contrast with the infinite. Evidence of this contrast surfacedin the form of paradoxes, such as the famous race between Achilles and the tortoise, thatdate back at least to Zeno of Elea (490–430 BCE). These paradoxes were understoodto arise precisely because of the sharp contrast between the finite and the infinite.There are, in fact, roots in Greek thought on the infinite far earlier than Zeno’swritings. Anaximander (610–547 BCE) conceived of the infinite as a spherical, limitlesssubstance that is eternal, inexhaustible, and lacks boundaries or distinctions. Theworld of finite entities was seen as arising out of that which is infinite. Pythagoras(570–500 BCE) rejected the infinite as having anything to do with the real world. Hecontended that all actual things are finite and representable by the natural, or whole,numbers. Pythagoras taught that the geometrical forms in one, two, and three dimensionsarise when a mathematical limit is imposed on the underlying infinite structure.Plato (428–348 BCE) believed that the Good must be definite rather than indefinite, andtherefore must be finite rather than infinite. According to Plato, God as the demiurgeimposes limitations (i.e., intelligible forms) on preexistent, inchoate matter, giving riseto the structured world around us as an ordered whole instead of a formless, unintelligible,infinite chaos. In all these cases, the infinite took on a negative quality comparedwith the finite.With the philosopher Aristotle (384–322 BCE), however, we find a rather differentconception of infinity, and one that continued in Western thought without serious3 See “Notes on the Concept of the Infinite in the History of Western Metaphysics,” p. 1.4 Hart also makes a crucial distinction, at the outset, between the physical/mathematical and the metaphysical/ontological meanings of “infinity.” While accepting this distinction, the purpose of my paper is to begin withthe ways in which the mathematical concept of infinity has changed with the work of Cantor and explore theimplications of this change for the ways we implicitly use the mathematical sense of infinity in discussingmetaphysical issues in the context of theology focused on revelation and on the divine attributes (followingPannenberg).