Introduction to Special Issue on Potentialism in Mathematics

According to potentialism, the objects with which mathematics is concerned are generated successively, and some of these generative processes cannot be completed. There is room for disagreement about which processes can be completed. A traditional Aristotelian form of potentialism takes a very restrictive view, insisting that at any one stage, there are never more than finitely many such objects, but that we always (i.e., necessarily) have the ability to go on and generate more. Other forms of potentialism take a more relaxed attitude. Potentialism about set theory provides an extreme example. According to this view, it is impossible to complete the process of forming sets from any objects that are available, but any generative process that is indexed by a set-theoretic ordinal can be completed. It is remarkable that throughout most of the history of mathematics, starting (somewhat arbitrarily) with the ancient Greeks, a restrictive form of potentialism dominated mathematical thought. Infinity was seen as only potential, never ‘actual’. That changed radically with Cantor and Dedekind, in the late nineteenth century, partly in the effort to found analysis on a more logically rigorous basis, deploying the concepts of limits of sequences, cuts in the rationals, etc., which seemed to require actually infinite sets. Cantor, of course, went much further with his theory of the transfinite, with never-ending hierarchies of alephs and beth cardinals and transfinite ordinals. Although Brouwerian intuitionism saw a retrenchment of strictly potential infinity, incompatible with any transfinite cardinals or uncountable totalities, classical mathematics embraced the actually infinite, grounded in the axioms of Zermelo–Fraenkel set theory, with the Axiom of Choice, leaving intuitionism to flourish among a small minority of mathematicians (including Bishop and his school, dispensing with choice sequences). Predicativism too rejects Cantor’s higher infinite, e.g., by regarding the collection of sets of natural numbers as a merely potential totality, although the totality of natural numbers is regarded as sufficiently well determined to license the application of classical logic to arithmetical statements.