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Infinity in a rigorous sense is a mathematical concept, but the notion of boundless entities, such as the number series and time, have since antiquity touched a deep philosophical and religious chord in the human heart.

Ancient and medieval conceptions

To the ancient Greek religious sect known as the Pythagoreans, the notion of limit was valued as conferring intelligibility and definition, while the infinite (apeiron) was associated with void and primordial matter, imperfection and instability. Plato (c. 428–327 B.C.E.) captures this negative sensibility in Philebus when he reports that "the men of old" viewed all beings "as consisting in their nature of Limit and Unlimitedness" (16c). Drawing on this background as well as reacting to it, Aristotle (384–322 B.C.E.) adopted the solution of banning anything actually infinite from philosophy. The infinite, he declared, is only "potential," denoting limitless series of successive, finite terms. Time is infinite in this potential sense, without a first beginning or end, but space, which exists all at once, is finite. A similar treatment of infinity is found in Euclidean mathematics, namely in Book 5, definition 4, which allows finite magnitudes as small or as large as desired, but precludes anything actually transfinite.

With the first-century Jewish philosopher Philo and the founder of neoplatonism Plotinus (c. 205–270 C.E.), an actual infinite perfection is attributed in a new positive sense to God to mean that divine perfection transcends every finite case and is immense, eternal, incomprehensible, and unsurpassable. The early Christian leader Augustine of Hippo (354–430 C.E.) in turn stresses in Confessions Book 7 that God is infinite according to a special immaterial measure of perfection, invisible to the bodily eye. The eighth-century theologian John Damascene speaks of God in De Fide Orthodoxa as "a certain sea of infinite substance" (1, 9). Medieval Jewish mystics such as Isaac the Blind and Azriel of Gerona who were active around the thirteenth century enlist the Hebrew en-sof (infinite) to describe the infinite extension of God's thought. Later cabbalists will use the actual infinite as a proper name and refer to "the En-Sof, Blessed be He."

In the mid-thirteenth century, Latin scholastics became concerned with rationalizing divine infinity by framing a coherent philosophical language to discuss various types of infinity and to explore the properties of the actual infinite, such as its noninductive and reflexive character. Two trends are discernible. Thomas Aquinas (c. 1225–1274) built on Aristotle to reach God philosophically as infinite (unrestricted) Being, while his Franciscan counterpart, Bonaventure (1221–1274), drawing more centrally on Augustine, started with a finite degree of ontological perfection and allowed this perfection to be raised to infinity. A new appreciation of the distinction between extension and intensity was thus brought to bear on the infinite, with the notion of intensity serving to mask the paradoxes inherent in the notion of an actual infinite extension. Bonaventure promoted an approach that is introspective rather than cosmological, involving the key premises that the human soul longs for an infinite good (God) and cannot find rest short of reaching it.

Another Franciscan, Peter John Olivi (c. 1248–1298), clarified the difference that exists between a concept taken unrestrictedly (e.g. being) and the determinate infinite case falling under the concept and denoting God (being of infinite intensity). John Duns Scotus (c. 1265–1308), also a Franciscan, formulated on this basis a univocal theocentric metaphysics based on adopting the intensive infinite as the "most perfect concept of God naturally available to us in this lifetime." Finally, by stressing the purely semiotic character of the concept and explaining that denoting God by means of the actual infinite does not imply comprehending God, William of Ockham (1288–1348) helped to secularize the discussion and to give the actual infinite a legitimate place in philosophy. The scientists who introduced ideal elements at infinity in geometry in the seventeenth century, namely Johannes Kepler, René Descartes, and Blaise Pascal, were fully familiar with scholastic mainstreaming of the actual infinite.

Modern conception of infinity

In the seventeenth century, Descartes made infinity a keystone of his metaphysics and philosophy of science. The idea of an actually infinite being is innate in the human mind, he argues, and cannot derive from anything finite, not even by extrapolation. Rather, the human ability to conceptualize the limit of an infinite process proves that the concept of the actual infinite is in us prior to the finite. Descartes also insisted that God alone is actually infinite, so that physical space must be described as merely indefinite rather than infinite. Another seventeenth-century scientist to make creative apologetic use of the actual infinite, based on its mathematical properties, was Blaise Pascal (1623–1662). In his famous "wager" argument, he invoked the disproportion of an infinite reward to urge human beings to bet their lives on God, no matter how small the odds. Pascal also invoked mathematical incommensurability to argue that charity infinitely exceeds a life devoted to science, just as a life of science infinitely exceeds a life spent on material pleasure.

The taste for images of absolute transcendence has waned among theologians in recent times, prompting renewed interest in the potential infinite. Process theology, in particular, inspired by mathematician and philosopher Alfred North Whitehead (1861–1947), has explored metaphors connected with the inner unfolding of time and the evolving universe to depict human beings as partners of God's open-ended creativity. Meanwhile, the actual infinite has found rigorous mathematical expression in transfinite set theory, fathered by mathematician Georg Cantor (1845–1918). Cantor not only extended classical number theory by introducing transfinite numbers but proved that there is a hierarchy of transfinite magnitudes, such that, for instance, the infinite cardinality of the continuum (denoted by c) is larger than the infinite cardinality of the rational numbers (denoted by aleph-zero). The religious dimension of transfinite ideation by no means evaporated on account of this new rigor: Cantor actively sought to enlist Catholic theologians in support of his mathematical discoveries, citing as a personal inspiration Augustine's speculation about God's perfect knowledge of numbers. Cantor's fellow mathematician David Hilbert has perhaps best summarized the dual religious and scientific appeal of infinity in the 1925 address designed to herald Cantor's discovery: "the infinite has always stirred the emotions of mankind more deeply than any other questions; the infinite has stimulated and fertilized reason as few other ideas have; but also the infinite, more than any other notion, is in need of clarification."

See also THOMAS AQUINAS; ARISTOTLE; PLATO; PROCESS THOUGHT; SPACE AND TIME

ANNE A. DAVENPORT