Abstract

Gottlob Frege's development of his logicist program has had a major impact on the history of logic and on the philosophy of mathematics. The year 1902 marked an important turning point for Frege: it was then that Russell discovered and communicated the inconsistency which could be derived in Frege's system. This problem arose because of the introduction of classes into this system, especially since Frege did not have type distinctions among his classes. ;Yet Frege had very clear type distinctions for his functions. I show how we can easily develop his theory of function levels to get, essentially, a simple theory of types for Fregean functions and concepts. Within this "no-class" development we can prove many of the basic theorems of arithmetic in a way which parallels Frege's own derivations. Numbers appear as second-level concepts in this alternative development; Frege had construed numbers as classes which were considered logical objects in his structure. ;I examine the extent to which this "no-class" construction agrees with Frege's basic principles. This takes on new interest since the recent publication of many of his later papers and correspondence : Frege's writings after 1902 give fresh insights into his original motivations and thought, and they give us a glimpse of his reactions to various proposed solutions of the paradox, many of which are connected with a "no-class" development. ;For Frege the introduction of classes was very much entwined with his goal of construing numbers as logical objects. We are to note the significance of the Fregean thesis that numbers are objects. We are to examine the logical and philosophical repercussions of defining numbers as objects, and of defining them as second-level concepts. I point out the ontological and epistemological aspects of Frege's views here, and I stress their relevance to contemporary discussions in the philosophy of mathematics. We look at the way a Fregean reduction, seen as an explication, can set out to exhibit logical relationships, logical linkages among the truths of arithmetic. The importance of Frege's logicist thesis in the continuing exploration into the nature of number and arithmetic is not to be overlooked