Arithmetic without sets

Extract

During the past 20 years the notion of set has been introduced into school mathematics courses not only at the secondary but also at the primary level. When one considers that neither Newton nor Gauss made any explicit use of the concept, it is rather remarkable that teachers have thought it desirable to teach the concept so early in their school courses. I should like to consider briefly some of the many possible reasons, historical and pedagogic, why “set” has become a school topic. One reason is, of course, the belief which Russell and Whitehead fostered in their Principia mathematica in the first decade of the century, that all mathematical concepts can be reduced to the concept of set. I don’t intend to discuss here the extent to which they failed to justify that belief, although I shall have occasion to mention some of the difficulties which their programme ran into. A second reason is the important part which sets have played in mathematical research this century.