Axiomatics Between Hilbert and the New Math: Diverging Views on Mathematical Research and Their Consequences on Education

David Hilbert is widely acknowledged as the father of the modern axiomatic approach in mathematics. The methodology and point of view put forward in his epoch-making Foundations of Geometry (1899) had lasting influences on research and education throughout the twentieth century. Nevertheless, his own conception of the role of axiomatic thinking in mathematics and in science in general was significantly different from the way in which it came to be understood and practiced by mathematicians of the following generations, including some who believed they were developing Hilbert’s original line of thought. The topologist Robert L. Moore was prominent among those who put at the center of their research an approach derived from Hilbert’s recently introduced axiomatic methodology. Moreover, he actively put forward a view according to which the axiomatic method would serve as a most useful teaching device in both graduate and undergraduate teaching mathematics and as a tool for identifying and developing creative mathematical talent. Some of the basic tenets of the Moore Method for teaching mathematics to prospective research mathematicians were adopted by the promoters of the New Math movement.