Reviews

Abstract

Many of us have had the experience of being introduced to algebra via a course on group theory. If your experience was like mine, you were given the axioms and asked to prove a number of consequences. I am willing to believe that many of you fared better than I did in that introductory course. I always feel that I actually started to understand group theory when I took a course from H.S.M. Coxeter at the University of Toronto in which groups were viewed as symmetry groups of polygons. At the time I could not help feeling that I would have done better originally with that sort of introduction and my subsequent teaching was always motivated by the recognition that axioms made more sense when they were presented against a more concrete background. The story of what led to the abstract/axiomatic presentation of mathematics has been told in many places, but one telling is by Leo Corry in his Modern Algebra and the Rise of Mathematical Structures [1]. This approach is typically attributed to Hilbert’s influence, and Corry traces the sequence of texts and approaches that led to Bourbaki and beyond. Bourbaki is often given the credit for providing a definitive formulation of the axiomatic approach, thanks to their presentation from which it often seems that the intuition has been excluded. There is also a literature that looks at ways in which mathematical practice may reflect more general societal and cultural factors. For example, Vladimir Tasic’s Mathematics and the Roots of Postmodernist Thought [6] searches out philosophical and literary connections for recent mathematics. Some of the connections are disputable, but the effort is a reminder of mathematical practice not being isolated. Alma Steingart’s Axiomatics: Mathematical Thought and High Modernism is an attempt to combine the story of abstraction with developments outside of mathematics. In fact, the author claims that mathematics and its drive for abstraction were crucial ingredients in what she identifies as ‘high modernism,’ roughly the period between 1930 and 1970. This runs all the way from applications of mathematics through the social sciences to art and architecture. In telling the story, the author invokes the names of many of the leading mathematicians in the United States through that period and provides a good deal of documentation in the form of quotations and references. Dr. Steingart is a history professor at Columbia who studied mathematics as an undergraduate and researches the interplay between mathematics and politics, and as such she presents this material from a very interesting and well-informed perspective. The introduction assures the reader that this is a history of mathematical thought and not a history of mathematics. In particular, she expresses the belief that no acquaintance with mathematics is required to make sense of her text. However, one might suspect that few without a background in mathematics would find the names