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

from potential infinity to actual infinity 33God’s essence. In this sense, Thomas rejects an interpretation of infinity as a quantity,as it was understood by ancient philosophers (Aquinas 1934, I, 7, 1). He does not referto the theological heritage of Gregory and the Areopagite, but to John of Damascus(Johannes Damascenus 1955, I,2,4). 72Turning to the question as to whether or not infinity is possible in the world or inmathematics, Thomas changes from a qualitative notion of infinity to a quantitativenotion of infinity. In this context, his distinction between essence and existence isoperative. In God, essence and existence coincide; therefore, infinity can be attributedto God. 73 However, this coincidence of essence and existence is not applicable to hiscreatures. Therefore, in the final analysis infinities cannot be infinities in creation. 74 Hethus rejects not only the existence of infinite magnitudes in the finite creation 75 but alsotheir existence as infinite magnitudes in geometry (Aquinas 1934, I,7,3) 76 and actualinfinity in mathematics. 77 Thus, he did not contribute to science by his understandingof infinity.Basically, it was the simultaneous application of the distinction of esse and essentiaboth to the infinite God and to the creatures that led St. Thomas Aquinas to theconviction that neither in the world nor in mathematics can an actual infinity bepossible. Then, of course, the whole picture changes as soon as the God-world relationis no longer seen in the categories of essence and esse. In fact, this is the case withNicholas of Cusa.Besides the huge Aristotelian influence on Western rational theology, there has alsobeen a tremendous influence by the Areopagite on Western theology. The next decisivestep toward an intellectual clarification of actual infinity was made by a Germanadherent and admirer of the Areopagite. This step is found in the contribution byNicholas of Cusa, whose thought will now be examined.72 “Sed contra est quod Damascenus dicit (De fide orthodoxa, lib. 1, cap. 4): ‘Deus est infinitus et aeternus etincircumscriptibilis’” (Aquinas 1934,I,7,1).73 “Ad primum sic proceditur. Videtur quod aliquid aliud quam Deus posit esse infinitum per essentiam. Virtusenim rei proportionatur essentiae ejus. Si igitur essential Dei est infinita, oportet quod ejus virtus sit infinita.Ergo potest producere effectum infinitum, cum quantitas virtutis per effectum cognoscatur” (Aquinas 1934,I, 7, 2).74 “Ad primum ergo dicendum quod hoc est contra rationem facti, quod essential rei sit ipsum esse ejus,quia esse subsistens non est esse creatum: unde contra rationem facti est, quod sit simpliciter infinitum.Sicut igitur Deus, licet habeat potentiam infinitam, non tamen potest facere aliquid non factum, hoc enimesset contradictoria esse simul; ita non potest facere aliquid infinitum simpliciter” (Aquinas 1934, I,7,2 ad1).75 Again he relates infinity to the hylomorphism, arguing that matter in no way can be conceived of as infinite,whereas only form might have an infinite magnitude. Real infinity thus can be attributed only to God:“dicendum quod aliquid praeter Deum potest esse infinitum secundum quid, sed non simpliciter” (Aquinas1934,I,7,2resp.).76 St. Thomas Aquinas refers to geometrical objects when he claims that in mathematics there are no infinitemagnitudes: “Unde, cum forma quanti, inquantum hujusmodi, sit figura, oportebit, quod habeat aliquamfiguram. Et sic erit finitum: est enim figura, quae termino vel terminis comprehenditur” (Aquinas 1934,I, 7, 3).77 “Ad tertium dicendum quod, licet quibusdam positis, alia poni non sit eis oppositum: tamen infinita poniopponitur cuilibet speciei multitudinis. Unde non est possible esse aliquam multitudinem actu infinitam”(Aquinas 1934, I, 7, 4 ad tertium).