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

226 infinities in cosmologyTo this remark concerning the existence of the probability measure on an infinitecollection of universes we should add the following caveat: It cannot be excluded thaton higher levels of complexity, such as intelligent life, an “individualization principle”could be operating. Philosophers, for a long time, postulated its existence (for instance,Leibniz’s indiscernibility principle), and we can find its traces even in elementarymathematics. For example, in the set of real numbers, each number is an individualentity that is never repeated in the entire uncountable infinite set of reals. The “individualizationprinciple,” in this case, consists in both peculiar properties of a givenreal number and in the ordering properties of the whole set of reals. If such a principleworks with respect to such apparently simple entities as real numbers, should we notexpect that something analogous could be at work at much higher levels of complexity?The general conclusion of the present section is that, in contrast with “infinitely distant”and “infinitely divergent” in the standard cosmology, the concept of infinity, as itis involved in speculations around the multiverse idea, has not yet been mathematicallytamed; 18 consequently, it must be looked on with extreme caution. In these matters, aslong as good mathematics is absent, we do not know what we are talking about.10.7 Theological LessonWhen the First Vatican Council, at its third session, declared that God is “almighty,eternal, immeasurable, incomprehensible, infinite in will, understanding and everyperfection,” 19 it only articulated a long tradition existing in Christian theology andphilosophy. In this tradition infinity was usually understood as that which has no limit,boundary, or end and which surpasses any measure or magnitude. It is obvious thatsuch an understanding of Divine Infinity cannot be “mathematically tamed,” which iswhy, when we apply the concept of infinity to God, we are relegated to its analogous ormetaphorical use. The awareness of this fact in Christian theology often assumed theform of an apophatic or negative theology (via negativa) – a conviction that everythingwe assert about God is, in fact, about what God is not, rather than about what God is.Such an approach played an important role in the early Christian theology (Pseudo-Dionysius, the Cappadocian Fathers, etc.), but also in the later theological thinking(e.g., Meister Eckhart, St. John of the Cross). 20It is natural that apophatic tendencies can be found in the writings of those mathematicallyinspired authors who speculated about theological implications of the conceptof infinity. It was Descartes, inspired by Nicholas of Cusa, who said that “there ismore reality in the infinite substance than in the finite” (Descartes, 1641); 21 consequently,he regarded infinity as a primary mathematical concept and all other concepts18 There are attempts to put some order into the multiverse concept; see Ellis (2007) and McCabe (2005).19 Session 3, April 24, 1870: Dogmatic Constitution on the Catholic Faith, chap. 1. See: The Scripture Documents.An Anthology of Official Catholic Teachings, ed. by D. B. Bechard, The Order of Saint Benedict, Collegeville,Minnesota, 2002, pp. 14–18.20 An interesting study of apophatic theology can be found in Hadot (1987).21 This does not prevent Descartes from treating infinity in an apophatic way: “And I must not imagine that Ido not apprehend the infinite by a true idea, but only by the negation of the finite, in the same way that Icomprehend repose and darkness by the negation of motion and light” (Descartes 1641).