Grosholz, Emily R. Representation and Productive Ambiguity in Mathematics and the Sciences.

Roberts, David Lindsay

GROSHOLZ, Emily R. Representation and Productive Ambiguity in Mathematics and the Sciences. New York: Oxford University Press, 2007. xviii + 313 pp. Cloth, $63.00--This book provides a series of case studies supporting the author's claim that ambiguous representation can be central to advances in both mathematics and science.

Chapter 1 briefly treats two exemplary cases and shows then to be imbued with ambiguity: the study of projectile motion by Galileo in the early seventeenth century and the chemical formulas proposed by Jacob Berzelius in the early nineteenth century. Here and throughout Grosholz makes abundant use of C. S. Peirce's distinction between iconic and symbolic representations (that is, those linked to their objects by similarity and those linked by convention), while emphasizing that these two modes will often blend and interchange. Grosholz also here lays out an intellectual genealogy, beginning with the logical positivists, especially Carnap, whose reductionist program she evaluates as "essentially syntactical" (p. 20). The following generation of semantic philosophers (Nancy Cartwright, Ian Hacking, and others) were in turn succeeded by Grosholz's own cohort, who augmented the philosophy of science toolkit with pragmatic considerations, bringing historical context to the forefront. Grosholz claims some credit for including philosophy of mathematics within this pragmatic turn, an extension she sees some otherwise likeminded colleagues (Ursula Klein, Robin Hendry) as having been hesitant to make.

Further groundwork is laid in Chapter 2, "Analysis and Experience." Analysis, for Grosholz, is "the search for conditions of intelligibility" (p. 33). She acknowledges numerous intellectual debts (she is commendably explicit and generous in this regard throughout), from Leibniz, Locke, and Hume to Jean Cavailles, Herbert Breger, and Carlo Cellucci. Leibniz looms especially large; Grosholz is firmly in the camp of those committed to rescuing him from the stunted interpretations of Louis Couturat and Bertrand Russell. Leibniz for Grosholz is not a logicist but rather a pragmatic user of multiple modes of representation, helping to demonstrate "why the philosophical project of fmding the sole correct representation of mathematics is misguided" (p. 48).

Chapters 3, 4, and 5 present close readings of cases involving the interfaces among biology, chemistry, physics, and geometry: constructing and testing an antibody mimic, understanding the transposition of genes, and investigating benzene via molecular orbital theory. Although Grosholz is engaged by the scientific cases in their own right, especially as they illustrate the subtleties of theory reduction, she makes clear that a major motivation for including them is to emphasize commonalities with mathematical practice, and to insist that mathematics should not be considered a unique outlier when compared to the physical and biological sciences.

Descartes, Newton, and Leibniz are treated in Chapters 6, 7, and 8, respectively. Grosholz views Descartes as a genius who persuasively promoted the false hope of ultimate clarity in mathematics and science. His mathematical practice, often inconsistent with his methodological pronouncements, is found to be rich in productive ambiguity. Newton's Principia is likewise rich. A tour of some key propositions from that book reveals the crucial ambiguity of the diagrams found therein, as Newton shifts back and forth between geometry and mechanics, and between the finite and the infinitesimal. Grosholz defends the importance of diagrams against Kantian snobbery. In the Leibniz chapter she emphasizes that "what a diagram means depends on its context of use" (p. 219). Appropriately, the book is abundantly illustrated with figures taken from primary sources being considered.

Chapter 9 begins with a survey of Jules Vuillemin's approach to the philosophy of mathematics and then proceeds to examine De Rham's theorem, which, by employing multiple modes of representation, demonstrates an isomorphism between two sets of algebraic invariants associated with a smoothly triangulated manifold. Nancy Cartwright's views on the abstract and concrete are used to interpret the philosophical significance of this episode.

In the last chapter Grosholz argues that "there cannot be complete speech about mathematical things," (p. 259) and criticizes the foundational proposals of both Bertrand Russell and Penelope Maddy. She looks at mathematical results connecting logic, topology, and algebra, such as the theorem of M. H. Stone, (not W. H., as the book would have it), which asserts that any Boolean algebra can be represented by an algebra of sets associated with an appropriately constructed topological space.

Grosholz's overall position is clear, and each case provides her opportunity for cogent remarks, but probably few readers will find all the case studies equally illuminating or understandable. At times the work resembles a scrapbook of Grosholz's intellectual interests more than a connected argument. Many intriguing questions are left unasked. There is never a hint that investigators at different times or places (all cases are from standard Western science and mathematics) have been better or worse at utilizing ambiguous representation, and no indication that investigators have, or have not, learned to master ambiguity from acquaintance with their predecessors. If scientists and mathematicians are conscious or unconscious of ambiguity does this have any practical consequences?--David Lindsay Roberts, Prince George's Community College.