Emily Grosholz offers a ‘philosophical account of representation and productive ambiguity.’ She offers a series of demonstrations which purport to exhibit the special kind of cogency she believes is to be found in mathematics as well as scientific demonstrations in which empirical evidence may also intervene. She mentions Galileo, the construction of an antibody mimic, a description of the molecular level of the Maize *Suppressor-Mutator* Transposon, the formal construction of molecular orbitals of benzene mutants, and so on. Her intention in this unusual and well-written book is to do full justice to what she regards as a special rigor of mathematical and scientific reasoning.

Grosholz tells an enthralling story with flair and conspicuously polymathic or interdisciplinary erudition. She aims to insist that such mathematical rationality is more inclusive and multifarious than philosophers of mathematics have admitted during the past century. She points out that demonstration is not only a matter of logic but also the deployment of a variety of modes of representation. Rational persuasion is historically located and context-dependent. A range of semiotic dimensions, the pragmatic, the semiotic and the syntactic must be taken into account.

The introduction is followed by parts on chemistry and geometry, on geometry and seventeenth century mechanics, on geometry and twentieth century typology. A concluding section on logic and typology touches on set theory, arithmetic, logical hierarchy and model theory.

This is a thought-provoking and wide-ranging book. The difficulty, however, is the author's apparent unawareness of the contingent nature of mathematical discourse itself and the failure to describe the varieties of mathematical philosophy which are available. The net result is to concretize a non-experimental philosophy of mathematics and so deprive sociology of mathematics of new employment of the tolerance of the ambiguity which she so rightly hails.

Richard S. Kirby is executive director of the Stuart C. Dodd Institute for Social Innovation in Seattle. He taught business ethics at University of Washington's School of Business Administration and is Visiting Professor of International Finance for the University of Russia's Academy of Education. He was recently selected to become first President of Kepler Space University, a NASA spin-off where he also teaches mathematical and astronomical sciences. He is a government consultant on mathematical sciences and their relevance to profitable government. With Richard J. Spady, his most recent book is *The Leadership of Civilization Building.*