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The Thomist 64 (2000): 101-25 CANTOR'S TRANSFINITE NUMBERS AND TRADITIONAL OBJECTIONS TO ACTUAL INFINITY JEANW. RIOUX Benedictine College Atchison, Kansas Georg Cantor was one of the most prominent mathematicians of the late nineteenth and early twentieth centuries. His development of a theory of transfinite numbers resurrected philosophical questions about infinity and led to a division of mathematics into schools of thought such as formalism and intuitionism. Cantor's published attempts to justify his mathematical theories were directed not only toward the mathematicians of his day but also toward philosophers, both ancient and contemporary. His efforts on the latter front were rooted in his desire to deal with objections to the very idea of the actual infinite in quantity, and he attached great importance to those objections that came from traditional philosophy.1 I intend to review the basic philosophical issues: Cantor's claims to a workable mathematics of real and actually infinite quantities, his response to Aristotelian objections to those claims, and my reflections on whether Cantor finally settled the matter, as he had hoped. 1 I shall focus upon Cantor's philosophical defense of transfinite numbers against this fundamental Aristotelian position as it is found in part 5 of his "Ueber unendliche, lineare Punktmannigfaltigkeiten," Mathematische Annalen 21 :545-86 (published separately as Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ei11 mathematisch-philosophischer Versuch in der Lehre des Unend.lichen [Leipzig: B. G. Teubner, 1883], hereafter Grumilage11). Cantor saw Aristotle as the source of the Scholastic position on infinity, and in the Gnmdlagen he addressed the basic error involved in all 'finitist' reasoning, as he saw it. A few years after writing the Grundlagen, Cantor published papers dealing with specific Scholastic arguments against the actual infinite in detail. For example, see Robin Small, "Cantor and the Scholastics," American Catholic Philosophical Quarterly 66 (1992): 407-28, where the question concerns the eternity of the world. 101 102 JEANW. RIOUX Cantor was well aware of the prevailing mathematical and philosophical climates of his day.2 As he drew nearer to a completed theory of transfinite numbers, he became increasingly interested in justifying it, not only as a consistent and practical3 exercise of mathematical thought, but also as one having a basis in the real world,4 as somehow providing as real a view of the natural world as did the relatively unproblematic theory of finite integers. While other mathematicians seemed unconcerned with such metaphysical questions, Cantor devoted much of his time and effort to addressing them, especially later in his life.5 Ironically, Cantor'sfascination with these metaphysical aspects of the theory, so foreign to his contemporaries in mathematics, turned out to be somewhat prophetic, since the dubious ontological character of transfinite numbers was to figure prominently in later developments in mathematics.6 I will begin by giving an overview of Cantor's transfinite number theory, focusing in particular upon his claim that transfinite numbers possess an objective, or real, infinity which is actual, not merely potential.7 I. SUMMARY OF CANTOR'S TRANSFINITE NUMBER THEORY Cantor discovered thatmerely 'potential' infinity, the sortwith which mathematicians were comfortable to that day, was not the 2 See Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Princeton, N.J.: Princeton University Press, 1979), 120-22. 3 As he notes in Grundlagen, § 8, the success in application of a mathematical concept is a major factor in whether it is accepted as legitimate or abandoned. 4 Cantor meant atleastthree thingsby 'real number': 'real' as distinguished from complex, rational, or irrational numbers; 'real' as existing in the understanding, as consistent and definite ideas in the mind; and 'real' as existing in the extramental world. Cantor held that all numbers that are real in the second sense are images of those that are real in the third sense. For more on the distinction between the last two types of reality, see Grundlagen, § 8. See also Dauben, Georg Cantor, 125-26. 5 Though many of the mathematicians of his day were able to look past Cantor's concern with the philosophical and theological implications of his transfinite theory (and in this way many came to defend the theory), to Cantor they were an inseparable part of his...


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