The Nature and Purpose of Numbers

G. Aldo Antonelli
2010 Journal of Philosophy  
Numbers are abstract entities introduced for the purpose of counting. The present paper is dedicated to the explication of this claim, and in particular it addresses the questions of what makes these entities 'abstract,' in what sense they are 'introduced,' and what we mean by 'counting.' Along the way, we investigate the logical status of arithmetic, the function of abstraction principles, and the respective merits of various strategies for reducing arithmetical notions to those of a theory
more » ... t is viewed as more fundamental. The main emphasis is on the conceptual, foundational and philosophical issues, with the technical details fully developed elsewhere. 1 Two main conceptual threads are at the basis of the present approach: a deflationary conception of abstraction and a nonreductionist version of logicism. Each is implemented through a specific device, i.e., respectively, an extra-logical operator representing numerical abstraction and a non-standard (but still first-order) cardinality quantifier. The result is an account of arithmetic characterizing numbers as obtained by abstraction from the equinumerosity relation and emphasizing their cardinal properties (as used in answering "how-many?" questions) over their structural ones. Abstract entities are obtained through the application of an abstraction operator to what Frege would have called a concept, and which we will also refer to as a (possibly complex) predicate -as long as we take care to distinguish predicates from predicate expressions. However, not every mapping of the concepts into the objects represents an instance of abstraction. Abstraction operators are distinguished from other such assignments in that they are assumed to map concepts into objects while respecting a given equivalence relation. Frege, for instance, postulated an abstraction operator assigning objects of a particular kindwhich he called extensions -to concepts in such a way that equi-extensional concepts (i.e., concepts under which the same objects fall) are assigned the same object. This particular postulation, as embodied in an abstraction principle known as Frege's Basic Law V, turned out to be inconsistent. Nonetheless, many other abstraction principles are indeed consistent, and among them, most notably for our
doi:10.5840/jphil2010107415 fatcat:7io7pj4a3vdevhfo6j7e67ltai