This ambitious work puts forward a new account of mathematics-as-language that challenges the coherence of the accepted idea of infinity and suggests a startlingly new conception of counting. The author questions the familiar, classical, interpretation of whole numbers held by mathematicians and scientists, and replaces it with an original and radical alternative-what the author calls non-Euclidean arithmetic. The author's entry point is an attack on the notion of the mathematical infinite in both its potential and actual forms, an attack organized around his claim that any interpretation of...

Two features of mathematics stand out: its menagerie of seemingly eternal objects (numbers, spaces, patterns, functions, categories, morphisms, graphs, and so on), and the hieroglyphics of special notations, signs, symbols, and diagrams associated with them. The author challenges the widespread belief in the extra-human origins of these objects and the understanding of mathematics as either a purely mental activity about them or a formal game of manipulating symbols. Instead, he argues that mathematics is a vast and unique man-made imagination machine controlled by writing. Mathematics as...