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

256 notes on the concept of the infinitespeculative traditions of the West, pagan and Christian alike, this indeterminacy –when ascribed to the transcendent source of being – came to be understood also as akind of “infinite determinacy”; but even here this determinacy consists in an absolutetranscendence of finite determination. The rule remains indubitable: only that whichis without particularity, definition, limit, location, nature, opposition, or relation is“infinite” in the metaphysical sense. Hence, “matter” – in the sense of u{lh or materiaprima – is understood by classical and mediaeval philosophy as infinite preciselybecause it is utterly devoid of the impress of morfhv or forma, and not because it is inany sense limitlessly extended or divisible. Neither extension nor divisibility, in fact,applies in any way to prime matter; in Aristotelian terms, prime matter belongs entirelyto the realm of toV dunatoVn, the possible, so long as no ejnevrgeia supervenes on it togrant it actual form; its infinity, therefore, is purely privative. Thus, space and time arenot “metaphysically” infinite, despite their interminabilities. Neither, moreover, is anyarithmetic series: the set of even numbers, for instance, while endless, is neverthelessdefinite: it includes and excludes particular members; it is a particular kind of thing; it isbounded by its own nature. In the purely metaphysical sense, in fact, the mathematicalinfinite remains within the realm of the finite.4. This distinction between the mathematical and metaphysical concepts of theinfinite was not obvious to the earliest thinkers of the Greek tradition and was often atbest only a tacit distinction within their reflections. Even in the classical age of Greekthought, the difference between the interminability of “number” and the indeterminacyof “possibility” was not much remarked, although it is obviously implicit in Aristotle’sobservations (in Physics III, 6–7) regarding, on the one hand, spatial extension anddivisibility and, on the other, the purely potential existence of the infinite in the realmof discrete substances.5. Aristotle’s distinction – however undeveloped it may be – leads toward the ratherstriking (but more or less inevitable) conclusion that the infinite, metaphysically conceived,is invariably and necessarily related to the question of being: how is it thatanything – any finite thing, that is – exists? To resort to a somewhat Heideggereanidiom, whereas the mathematical infinite is an essentially “ontic” category, the metaphysicalinfinite is thoroughly “ontological.” It is that over against which the finite is(phenomenologically) set off, or out of which it is extracted, or on which it is impressed,or from which it emanates, or by which it is given. This is true at both extremes of themetaphysical continuum: at the level of prime matter or at the level of the transcendentact or source of being; at the level of absolute privation or at the level of absoluteplenitude.6. From this one can draw a very simple metaphysical distinction between thefinite and the infinite. The realm of the finite is that realm in which the principiumcontradictionis holds true. In an utterly vacuous sense, the more “positive” principiumidentitatis holds true in all worlds and at every level of reality, but the more “negative”principium contradictionis describes that absolute limit by which any finite thing isthe thing it is. In a very real sense, finite existence is noncontradiction. Neither infinitepotentiality nor infinite actuality excludes the coincidence of opposites, inasmuch asneither in and of itself posits anything; only when, on the one hand, potentiality isrealized as a single act (and so becomes “this” thing rather than “that” thing) and, onthe other hand, actuality limits itself by commerce with potentiality (and so becomes